TPTP Problem File: SLH0492^1.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_02915_121271__14431406_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1408 ( 465 unt; 115 typ; 0 def)
% Number of atoms : 4974 (1520 equ; 0 cnn)
% Maximal formula atoms : 38 ( 3 avg)
% Number of connectives : 18442 ( 515 ~; 19 |;1041 &;15362 @)
% ( 0 <=>;1505 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 794 ( 794 >; 0 *; 0 +; 0 <<)
% Number of symbols : 113 ( 110 usr; 11 con; 0-4 aty)
% Number of variables : 3764 ( 781 ^;2877 !; 106 ?;3764 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:47.629
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
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simula1175017789036123529lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula9164532646813739645lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula3069223088535412739lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
simula3219528783544830217lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_lambda_lambda > list_l5168835686021156086lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
simula1230958359699143427lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_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_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_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_member_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
member2667661554319814297lambda: list_lambda_lambda > set_li7878742357367608272lambda > $o ).
thf(sy_v_Ua____,type,
ua: list_lambda_lambda ).
thf(sy_v_u____,type,
u: lambda_lambda ).
thf(sy_v_x41,type,
x41: lambda_lambda ).
thf(sy_v_x42,type,
x42: 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_U,axiom,
ua != nil_lambda_lambda ).
% U
thf(fact_3_cube,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).
% cube
thf(fact_4_ex__un__null,axiom,
? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X @ T2 )
= X )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ Y @ T3 )
= Y )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_5_Trgs_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ).
% Trgs.cases
thf(fact_6_Cube_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_7_Cube_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_8_Resid_Osimps_I1_J,axiom,
! [Uu: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ nil_lambda_lambda @ Uu )
= nil_lambda_lambda ) ).
% Resid.simps(1)
thf(fact_9_Con__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 )
!= nil_lambda_lambda ) ) ).
% Con_sym
thf(fact_10_cong__symmetric,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ).
% cong_symmetric
thf(fact_11_cong__transitive,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) ) ) ) ) ).
% cong_transitive
thf(fact_12_ide__backward__stable,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ A ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% ide_backward_stable
thf(fact_13_prfx__transitive,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_14_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_15_uU,axiom,
paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ u @ ua ) ).
% uU
thf(fact_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_36_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_37_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T5 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U3 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_38_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_39_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_40_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_41_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_42_mem__Collect__eq,axiom,
! [A: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A @ ( collec621444948832755541lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_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_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_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_49_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_50_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va2 ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_51_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_52_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= ( lambda_App @ ( lambda_resid @ T @ T5 ) @ ( lambda_resid @ U @ U3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_53_lambda__calculus_Olambda_Odistinct_I5_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( lambda_Nil
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(5)
thf(fact_54_lambda__calculus_OCube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% lambda_calculus.Cube
thf(fact_55_lambda__calculus_OArr__Trg,axiom,
! [X4: lambda_lambda] :
( ( lambda_Arr @ X4 )
=> ( lambda_Arr @ ( lambda_Trg @ X4 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_56_lambda__calculus_OCon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
= ( ( lambda_resid @ U @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_sym
thf(fact_57_lambda__calculus_OArr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ T @ U ) ) ) ).
% lambda_calculus.Arr_resid
thf(fact_58_lambda__calculus_OArr__not__Nil,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( T != lambda_Nil ) ) ).
% lambda_calculus.Arr_not_Nil
thf(fact_59_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_60_lambda__calculus_OCon__implies__Arr1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Con_implies_Arr1
thf(fact_61_lambda__calculus_OCon__implies__Arr2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% lambda_calculus.Con_implies_Arr2
thf(fact_62_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_63_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_64_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_65__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_66_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_67_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_68__C2_C,axiom,
paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ).
% "2"
thf(fact_69_coinitial__ide__are__cong,axiom,
! [A: list_lambda_lambda,A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_sources__cong__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_78_sources__are__cong,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_79_source__is__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% source_is_ide
thf(fact_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_94_append_Oright__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ A @ nil_lambda_lambda )
= A ) ).
% append.right_neutral
thf(fact_95_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_96_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_97_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_98_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs )
= ( append_lambda_lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_99_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_100_append_Oassoc,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ A @ B ) @ C )
= ( append_lambda_lambda @ A @ ( append_lambda_lambda @ B @ C ) ) ) ).
% append.assoc
thf(fact_101_not__Cons__self2,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_102_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs @ Us ) )
& ( ( append_lambda_lambda @ Us @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us )
= Zs )
& ( Ys
= ( append_lambda_lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_103_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us2 ) )
=> ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_104_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_105_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_106_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_107_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_108_remdups__adj_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ~ ! [X: lambda_lambda,Y3: lambda_lambda,Xs2: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_109_neq__Nil__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
= ( ? [Y4: lambda_lambda,Ys2: list_lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_110_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys3: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y3 @ Ys3 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys3: list_lambda_lambda] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_111_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_112_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_113_Cons__eq__appendI,axiom,
! [X2: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_114_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_115_append_Oleft__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_116_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_117_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_118_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys3: list_lambda_lambda,Y3: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ Y3 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_119_Cons__eq__append__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_120_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( cons_lambda_lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X2 @ Xs ) ) )
| ? [Ys4: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X2 @ Ys4 ) )
& ( ( append_lambda_lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_121_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_122_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_123_joinable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X5: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X5 ) ) ) ).
% joinable_def
thf(fact_124_Resid__Arr__Src,axiom,
! [T4: list_lambda_lambda,A: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_Arr_Src
thf(fact_125_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_126_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_127_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_128_coinitial__iff,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T5 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ) ).
% coinitial_iff
thf(fact_129_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_130_in__sourcesE,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A ) ) ) ).
% in_sourcesE
thf(fact_131_sources__con__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_132_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_133__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_134__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_135__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_136__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_137__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_138__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_139__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_140__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_141_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_142_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_143_Srcs__are__con,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A3 ) ) ) ).
% Srcs_are_con
thf(fact_144_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_145_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_146_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_147_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_148_resid__ide__arr,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_149_resid__arr__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ A )
= T ) ) ) ).
% resid_arr_ide
thf(fact_150_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_151_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_152_ide__def,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
= ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
= A ) ) ) ).
% ide_def
thf(fact_153_ideE,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ~ ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
!= A ) ) ) ).
% ideE
thf(fact_154_con__transitive__on__ide,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ B )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ C )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_155_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_156_con__imp__coinitial__ax,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ? [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 @ T )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_157_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_158_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_159_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_160_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_161_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_162_ide__implies__arr,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% ide_implies_arr
thf(fact_163_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_164_arr__char,axiom,
! [T4: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% arr_char
thf(fact_165_sources__are__con,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 ) ) ) ).
% sources_are_con
thf(fact_166_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_167_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_168_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_169_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_join__of__arr__src_I1_J,axiom,
! [T: list_lambda_lambda,A: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_189_join__of__arr__src_I2_J,axiom,
! [T: list_lambda_lambda,A: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_190_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_191_lambda__calculus_Ocon__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% lambda_calculus.con_char
thf(fact_192__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_193_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_194_ideI,axiom,
! [A: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
= A )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ) ).
% ideI
thf(fact_195_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_196_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_197_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_198_in__sourcesI,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A )
=> ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_sourcesI
thf(fact_199_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_200_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_201_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_202_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_203_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_204_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_205_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_206_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_207_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_208_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_209_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T3 @ U4 )
=> ( con_li79537494059641180lambda @ Resid @ T3 @ U4 ) )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_210_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U4 )
=> ( con_lambda_lambda @ Resid @ T3 @ U4 ) )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_211_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_212_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_213_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_214_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_215_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_216_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_217_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_218_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_219_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_220_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_221_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_222_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_223_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_224_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_225_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_226_partial__magma_Ointro,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X6: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP @ X6 @ T3 )
= X6 )
& ( ( OP @ T3 @ X6 )
= X6 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ Y3 @ T2 )
= Y3 )
& ( ( OP @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X6 ) ) )
=> ( partia748546660539056525lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_227_partial__magma_Ointro,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X6: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ X6 @ T3 )
= X6 )
& ( ( OP @ T3 @ X6 )
= X6 ) )
& ! [Y3: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ Y3 @ T2 )
= Y3 )
& ( ( OP @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X6 ) ) )
=> ( partia3924075026459072775lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_228_partial__magma_Oex__un__null,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ X @ T2 )
= X )
& ( ( OP @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP @ Y @ T3 )
= Y )
& ( ( OP @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_229_partial__magma_Oex__un__null,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ X @ T2 )
= X )
& ( ( OP @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ Y @ T3 )
= Y )
& ( ( OP @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_230_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X3: list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ( ( OP2 @ X3 @ T6 )
= X3 )
& ( ( OP2 @ T6 @ X3 )
= X3 ) )
& ! [Y4: list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ( ( OP2 @ Y4 @ T6 )
= Y4 )
& ( ( OP2 @ T6 @ Y4 )
= Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_231_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X3: lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( ( OP2 @ X3 @ T6 )
= X3 )
& ( ( OP2 @ T6 @ X3 )
= X3 ) )
& ! [Y4: lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( ( OP2 @ Y4 @ T6 )
= Y4 )
& ( ( OP2 @ T6 @ Y4 )
= Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_232_Con__single__ide__ind,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_241__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_242__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_243__092_060Lambda_062_Ojoinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
= ( ? [X5: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X5 ) ) ) ).
% \<Lambda>.joinable_def
thf(fact_244__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_245__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_246__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_247__092_060Lambda_062_Osource__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.source_is_ide
thf(fact_248__092_060Lambda_062_Osources__are__cong,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) ) ) ) ).
% \<Lambda>.sources_are_cong
thf(fact_249__092_060Lambda_062_Osources__cong__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.sources_cong_closed
thf(fact_250__092_060Lambda_062_Oresid__ide_I1_J,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ T )
=> ( ( lambda_resid @ T @ A )
= T ) ) ) ).
% \<Lambda>.resid_ide(1)
thf(fact_251__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_cong
thf(fact_252__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A3 )
=> ( A = A3 ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_eq
thf(fact_253__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_254__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_255__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_256__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_257__092_060Lambda_062_Oide__backward__stable,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ A ) )
=> ( ide_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.ide_backward_stable
thf(fact_258__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_259__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_260__092_060Lambda_062_Ojoin__of__arr__src_I2_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ A @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(2)
thf(fact_261__092_060Lambda_062_Ojoin__of__arr__src_I1_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ A @ T @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(1)
thf(fact_262__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_263__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_264__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_265__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_266__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_267__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_268__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_269__092_060Lambda_062_Osources__are__con,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A3 ) ) ) ).
% \<Lambda>.sources_are_con
thf(fact_270__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_271__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_272__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_273__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_274__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_275__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_276__092_060Lambda_062_Oresid__source__in__targets,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( member_lambda_lambda @ ( lambda_resid @ A @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.resid_source_in_targets
thf(fact_277__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_278__092_060Lambda_062_Oide__implies__arr,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( arr_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.ide_implies_arr
thf(fact_279__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_280__092_060Lambda_062_Oin__sourcesE,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( con_lambda_lambda @ lambda_resid @ T @ A ) ) ) ).
% \<Lambda>.in_sourcesE
thf(fact_281__092_060Lambda_062_Osources__con__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.sources_con_closed
thf(fact_282__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T5 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T5 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_283__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T5 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_284__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_285__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_286__092_060Lambda_062_Ocon__ide__are__eq,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( A = A3 ) ) ) ) ).
% \<Lambda>.con_ide_are_eq
thf(fact_287__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A4 )
& ( con_lambda_lambda @ lambda_resid @ A4 @ T )
& ( con_lambda_lambda @ lambda_resid @ A4 @ U ) ) ) ).
% \<Lambda>.con_imp_coinitial_ax
thf(fact_288__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_289__092_060Lambda_062_Ocon__transitive__on__ide,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( ide_lambda_lambda @ lambda_resid @ C )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ C )
=> ( con_lambda_lambda @ lambda_resid @ A @ C ) ) ) ) ) ) ).
% \<Lambda>.con_transitive_on_ide
thf(fact_290__092_060Lambda_062_OideE,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( lambda_resid @ A @ A )
!= A ) ) ) ).
% \<Lambda>.ideE
thf(fact_291__092_060Lambda_062_Oide__def,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( con_lambda_lambda @ lambda_resid @ A @ A )
& ( ( lambda_resid @ A @ A )
= A ) ) ) ).
% \<Lambda>.ide_def
thf(fact_292__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_293__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_294__092_060Lambda_062_Oresid__arr__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A )
=> ( ( lambda_resid @ T @ A )
= T ) ) ) ).
% \<Lambda>.resid_arr_ide
thf(fact_295__092_060Lambda_062_Oresid__ide__arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ T ) ) ) ) ).
% \<Lambda>.resid_ide_arr
thf(fact_296__092_060Lambda_062_Otarget__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.target_is_ide
thf(fact_297__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_298__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_299__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_300__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_301__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_302__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_303__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_304_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_305__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_306_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_307_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_308_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_309_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_310__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_311_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_312_Srcs__con__closed,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_313__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_314_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).
% cong_respects_seq
thf(fact_315_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T7: list_lambda_lambda,U5: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U5 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_316_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_317_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_318_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_319_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_320_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_321_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_322_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_323_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_324_confluence__single,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T3 @ U4 )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U4 ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% confluence_single
thf(fact_325__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_326__092_060Lambda_062_Oin__sourcesI,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A )
=> ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.in_sourcesI
thf(fact_327__092_060Lambda_062_OideI,axiom,
! [A: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( ( lambda_resid @ A @ A )
= A )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ) ).
% \<Lambda>.ideI
thf(fact_328__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_329__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_330_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_331_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_332_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_333_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_334_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_335_Resid__single__ide_I2_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_single_ide(2)
thf(fact_336_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_337_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_338_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_339_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_340_cong__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T7: list_lambda_lambda,U5: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T7 @ U5 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T7 @ U5 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_341_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_342_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_343_Con__single__ideI_I2_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_344_Con__single__ideI_I1_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_345_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_346_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_347_cong__cons,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,T5: lambda_lambda,U5: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ ( cons_lambda_lambda @ T5 @ U5 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ U5 ) @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_348_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_349_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_350_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_351_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_352_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_353_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_354_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_355_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_356_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_357_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_358_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_359_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_360_partial__magma_Onull__eqI,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ! [T3: list_lambda_lambda] :
( ( ( OP @ N @ T3 )
= N )
& ( ( OP @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_361_partial__magma_Onull__eqI,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ! [T3: lambda_lambda] :
( ( ( OP @ N @ T3 )
= N )
& ( ( OP @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_362_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ( OP @ ( partia6873046715171720854lambda @ OP ) @ T )
= ( partia6873046715171720854lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_363_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ ( partia1461432607296028304lambda @ OP ) @ T )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_364_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ( OP @ T @ ( partia6873046715171720854lambda @ OP ) )
= ( partia6873046715171720854lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_365_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ T @ ( partia1461432607296028304lambda @ OP ) )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_366_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V4: list_lambda_lambda,W2: list_lambda_lambda,W3: list_lambda_lambda,T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( member2667661554319814297lambda @ V @ NN )
=> ( ( member2667661554319814297lambda @ V4 @ NN )
=> ( ( member2667661554319814297lambda @ W2 @ NN )
=> ( ( member2667661554319814297lambda @ W3 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ V )
= ( source6119337877621565212lambda @ Resid @ W2 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ V4 )
= ( source6119337877621565212lambda @ Resid @ W3 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ W2 )
= ( target4026644918049360146lambda @ Resid @ W3 ) )
=> ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T5 @ V4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T5 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T5 @ W3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T5 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_367_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V4: lambda_lambda,W2: lambda_lambda,W3: lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V4 @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( member_lambda_lambda @ W3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V4 )
= ( source5627170679743844886lambda @ Resid @ W3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W2 )
= ( target5490931411331770124lambda @ Resid @ W3 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T5 @ V4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T5 @ W3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376__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_377__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_378__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_379__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_380__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_381__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_382__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T5 @ U3 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_383__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_384__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_385__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_386__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_387__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_388_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_389_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_390_target__is__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% target_is_ide
thf(fact_391_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_392_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_393_resid__source__in__targets,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% resid_source_in_targets
thf(fact_394__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_395_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_396_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_397_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_398__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_399__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_400_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_401__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_402__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_403_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_404_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_405_coterminal__iff,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T5 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ) ).
% coterminal_iff
thf(fact_406_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_407_targets__eqI,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).
% targets_eqI
thf(fact_408_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_409_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_410_sources__eqI,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).
% sources_eqI
thf(fact_411_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_412_lambda__calculus_Onull__char,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= lambda_Nil ) ).
% lambda_calculus.null_char
thf(fact_413__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_414__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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_Int__subset__iff,axiom,
! [C2: set_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
= ( ( ord_le1926335974945931754lambda @ C2 @ A2 )
& ( ord_le1926335974945931754lambda @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_423_Int__subset__iff,axiom,
! [C2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
= ( ( ord_le3064231726521830256lambda @ C2 @ A2 )
& ( ord_le3064231726521830256lambda @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_424_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_425_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_426_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_427_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_428_inf__bot__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_right
thf(fact_429_inf__bot__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_right
thf(fact_430_inf__bot__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_left
thf(fact_431_inf__bot__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_left
thf(fact_432__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_433__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_434_Srcs__eqI,axiom,
! [T4: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) ) ) ).
% Srcs_eqI
thf(fact_435__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_436__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_437__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_438_empty__Collect__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ P ) )
= ( ! [X3: list_lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_439_empty__Collect__eq,axiom,
! [P: lambda_lambda > $o] :
( ( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ P ) )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_440_Collect__empty__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( ( collec8830145823237851355lambda @ P )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_441_Collect__empty__eq,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_442_all__not__in__conv,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ! [X3: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ X3 @ A2 ) )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% all_not_in_conv
thf(fact_443_all__not__in__conv,axiom,
! [A2: set_lambda_lambda] :
( ( ! [X3: lambda_lambda] :
~ ( member_lambda_lambda @ X3 @ A2 ) )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% all_not_in_conv
thf(fact_444_empty__iff,axiom,
! [C: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ C @ bot_bo1397037886704484924lambda ) ).
% empty_iff
thf(fact_445_empty__iff,axiom,
! [C: lambda_lambda] :
~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).
% empty_iff
thf(fact_446_subset__antisym,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_447_subset__antisym,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_448_subsetI,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A2 )
=> ( member_lambda_lambda @ X @ B3 ) )
=> ( ord_le1926335974945931754lambda @ A2 @ B3 ) ) ).
% subsetI
thf(fact_449_subsetI,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A2 )
=> ( member2667661554319814297lambda @ X @ B3 ) )
=> ( ord_le3064231726521830256lambda @ A2 @ B3 ) ) ).
% subsetI
thf(fact_450_inf__right__idem,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_451_inf__right__idem,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_452_inf_Oright__idem,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ B )
= ( inf_in3354698378728726462lambda @ A @ B ) ) ).
% inf.right_idem
thf(fact_453_inf_Oright__idem,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ).
% inf.right_idem
thf(fact_454_inf__left__idem,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_455_inf__left__idem,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_456_inf_Oleft__idem,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( inf_in3354698378728726462lambda @ A @ B ) ) ).
% inf.left_idem
thf(fact_457_inf_Oleft__idem,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ).
% inf.left_idem
thf(fact_458_inf__idem,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_459_inf__idem,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_460_inf_Oidem,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ A )
= A ) ).
% inf.idem
thf(fact_461_inf_Oidem,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ A )
= A ) ).
% inf.idem
thf(fact_462_Int__iff,axiom,
! [C: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
= ( ( member2667661554319814297lambda @ C @ A2 )
& ( member2667661554319814297lambda @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_463_Int__iff,axiom,
! [C: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
= ( ( member_lambda_lambda @ C @ A2 )
& ( member_lambda_lambda @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_464_IntI,axiom,
! [C: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ A2 )
=> ( ( member2667661554319814297lambda @ C @ B3 )
=> ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_465_IntI,axiom,
! [C: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ A2 )
=> ( ( member_lambda_lambda @ C @ B3 )
=> ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_466_inf_Obounded__iff,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) )
= ( ( ord_le1926335974945931754lambda @ A @ B )
& ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_467_inf_Obounded__iff,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) )
= ( ( ord_le3064231726521830256lambda @ A @ B )
& ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_468_le__inf__iff,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) )
= ( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
& ( ord_le1926335974945931754lambda @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_469_le__inf__iff,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) )
= ( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
& ( ord_le3064231726521830256lambda @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_470_subset__empty,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% subset_empty
thf(fact_471_subset__empty,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% subset_empty
thf(fact_472_empty__subsetI,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A2 ) ).
% empty_subsetI
thf(fact_473_empty__subsetI,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A2 ) ).
% empty_subsetI
thf(fact_474_bot__set__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ bot_bo430095131446760353mbda_o ) ) ).
% bot_set_def
thf(fact_475_bot__set__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).
% bot_set_def
thf(fact_476_ex__in__conv,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ? [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A2 ) )
= ( A2 != bot_bo1397037886704484924lambda ) ) ).
% ex_in_conv
thf(fact_477_ex__in__conv,axiom,
! [A2: set_lambda_lambda] :
( ( ? [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A2 ) )
= ( A2 != bot_bo1271416176687173302lambda ) ) ).
% ex_in_conv
thf(fact_478_equals0I,axiom,
! [A2: set_li7878742357367608272lambda] :
( ! [Y3: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ Y3 @ A2 )
=> ( A2 = bot_bo1397037886704484924lambda ) ) ).
% equals0I
thf(fact_479_equals0I,axiom,
! [A2: set_lambda_lambda] :
( ! [Y3: lambda_lambda] :
~ ( member_lambda_lambda @ Y3 @ A2 )
=> ( A2 = bot_bo1271416176687173302lambda ) ) ).
% equals0I
thf(fact_480_equals0D,axiom,
! [A2: set_li7878742357367608272lambda,A: list_lambda_lambda] :
( ( A2 = bot_bo1397037886704484924lambda )
=> ~ ( member2667661554319814297lambda @ A @ A2 ) ) ).
% equals0D
thf(fact_481_equals0D,axiom,
! [A2: set_lambda_lambda,A: lambda_lambda] :
( ( A2 = bot_bo1271416176687173302lambda )
=> ~ ( member_lambda_lambda @ A @ A2 ) ) ).
% equals0D
thf(fact_482_emptyE,axiom,
! [A: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ A @ bot_bo1397037886704484924lambda ) ).
% emptyE
thf(fact_483_emptyE,axiom,
! [A: lambda_lambda] :
~ ( member_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ).
% emptyE
thf(fact_484_Collect__mono__iff,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) )
= ( ! [X3: lambda_lambda] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_485_Collect__mono__iff,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) )
= ( ! [X3: list_lambda_lambda] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_486_set__eq__subset,axiom,
( ( ^ [Y5: set_lambda_lambda,Z2: set_lambda_lambda] : ( Y5 = Z2 ) )
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A5 @ B4 )
& ( ord_le1926335974945931754lambda @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_487_set__eq__subset,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] : ( Y5 = Z2 ) )
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A5 @ B4 )
& ( ord_le3064231726521830256lambda @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_488_subset__trans,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C2 )
=> ( ord_le1926335974945931754lambda @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_489_subset__trans,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C2 )
=> ( ord_le3064231726521830256lambda @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_490_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_491_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_492_subset__refl,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A2 @ A2 ) ).
% subset_refl
thf(fact_493_subset__refl,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A2 @ A2 ) ).
% subset_refl
thf(fact_494_subset__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
! [T6: lambda_lambda] :
( ( member_lambda_lambda @ T6 @ A5 )
=> ( member_lambda_lambda @ T6 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_495_subset__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
! [T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T6 @ A5 )
=> ( member2667661554319814297lambda @ T6 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_496_equalityD2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( A2 = B3 )
=> ( ord_le1926335974945931754lambda @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_497_equalityD2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( A2 = B3 )
=> ( ord_le3064231726521830256lambda @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_498_equalityD1,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( A2 = B3 )
=> ( ord_le1926335974945931754lambda @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_499_equalityD1,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( A2 = B3 )
=> ( ord_le3064231726521830256lambda @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_500_subset__eq,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A5 )
=> ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_501_subset__eq,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A5 )
=> ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_502_equalityE,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( A2 = B3 )
=> ~ ( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ~ ( ord_le1926335974945931754lambda @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_503_equalityE,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( A2 = B3 )
=> ~ ( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ~ ( ord_le3064231726521830256lambda @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_504_subsetD,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( member_lambda_lambda @ C @ A2 )
=> ( member_lambda_lambda @ C @ B3 ) ) ) ).
% subsetD
thf(fact_505_subsetD,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( member2667661554319814297lambda @ C @ A2 )
=> ( member2667661554319814297lambda @ C @ B3 ) ) ) ).
% subsetD
thf(fact_506_in__mono,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,X2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( member_lambda_lambda @ X2 @ A2 )
=> ( member_lambda_lambda @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_507_in__mono,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,X2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( member2667661554319814297lambda @ X2 @ A2 )
=> ( member2667661554319814297lambda @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_508_inf__left__commute,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) )
= ( inf_in3354698378728726462lambda @ Y2 @ ( inf_in3354698378728726462lambda @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_509_inf__left__commute,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) )
= ( inf_in1766959559283430712lambda @ Y2 @ ( inf_in1766959559283430712lambda @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_510_inf_Oleft__commute,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ B @ ( inf_in3354698378728726462lambda @ A @ C ) )
= ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_511_inf_Oleft__commute,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ B @ ( inf_in1766959559283430712lambda @ A @ C ) )
= ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_512_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( B3
= ( inf_in3354698378728726462lambda @ K @ B ) )
=> ( ( inf_in3354698378728726462lambda @ A @ B3 )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_513_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_lambda_lambda,K: set_lambda_lambda,B: set_lambda_lambda,A: set_lambda_lambda] :
( ( B3
= ( inf_in1766959559283430712lambda @ K @ B ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B3 )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_514_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A2
= ( inf_in3354698378728726462lambda @ K @ A ) )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_515_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_lambda_lambda,K: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A2
= ( inf_in1766959559283430712lambda @ K @ A ) )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_516_inf__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_517_inf__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_518_inf_Ocommute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_519_inf_Ocommute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_520_inf__assoc,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Z )
= ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_521_inf__assoc,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Z )
= ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_522_inf_Oassoc,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C )
= ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) ) ) ).
% inf.assoc
thf(fact_523_inf_Oassoc,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C )
= ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ).
% inf.assoc
thf(fact_524_inf__sup__aci_I1_J,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_525_inf__sup__aci_I1_J,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_526_inf__sup__aci_I2_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Z )
= ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_527_inf__sup__aci_I2_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Z )
= ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_528_inf__sup__aci_I3_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) )
= ( inf_in3354698378728726462lambda @ Y2 @ ( inf_in3354698378728726462lambda @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_529_inf__sup__aci_I3_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) )
= ( inf_in1766959559283430712lambda @ Y2 @ ( inf_in1766959559283430712lambda @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_530_inf__sup__aci_I4_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_531_inf__sup__aci_I4_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_532_Int__left__commute,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C2 ) )
= ( inf_in3354698378728726462lambda @ B3 @ ( inf_in3354698378728726462lambda @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_533_Int__left__commute,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) )
= ( inf_in1766959559283430712lambda @ B3 @ ( inf_in1766959559283430712lambda @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_534_Int__left__absorb,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_535_Int__left__absorb,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_536_Int__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_537_Int__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B4 @ A5 ) ) ) ).
% Int_commute
thf(fact_538_Int__absorb,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_539_Int__absorb,axiom,
! [A2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_540_Int__assoc,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ C2 )
= ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_541_Int__assoc,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_542_IntD2,axiom,
! [C: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
=> ( member2667661554319814297lambda @ C @ B3 ) ) ).
% IntD2
thf(fact_543_IntD2,axiom,
! [C: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
=> ( member_lambda_lambda @ C @ B3 ) ) ).
% IntD2
thf(fact_544_IntD1,axiom,
! [C: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
=> ( member2667661554319814297lambda @ C @ A2 ) ) ).
% IntD1
thf(fact_545_IntD1,axiom,
! [C: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
=> ( member_lambda_lambda @ C @ A2 ) ) ).
% IntD1
thf(fact_546_IntE,axiom,
! [C: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
=> ~ ( ( member2667661554319814297lambda @ C @ A2 )
=> ~ ( member2667661554319814297lambda @ C @ B3 ) ) ) ).
% IntE
thf(fact_547_IntE,axiom,
! [C: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
=> ~ ( ( member_lambda_lambda @ C @ A2 )
=> ~ ( member_lambda_lambda @ C @ B3 ) ) ) ).
% IntE
thf(fact_548_inf_OcoboundedI2,axiom,
! [B: set_lambda_lambda,C: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_549_inf_OcoboundedI2,axiom,
! [B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_550_inf_OcoboundedI1,axiom,
! [A: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_551_inf_OcoboundedI1,axiom,
! [A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_552_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B5: set_lambda_lambda,A6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_553_inf_Oabsorb__iff2,axiom,
( ord_le3064231726521830256lambda
= ( ^ [B5: set_li7878742357367608272lambda,A6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_554_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_555_inf_Oabsorb__iff1,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_556_inf_Ocobounded2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_557_inf_Ocobounded2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_558_inf_Ocobounded1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_559_inf_Ocobounded1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_560_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( A6
= ( inf_in1766959559283430712lambda @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_561_inf_Oorder__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( A6
= ( inf_in3354698378728726462lambda @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_562_inf__greatest,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ Z )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_563_inf__greatest,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ Z )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_564_inf_OboundedI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_565_inf_OboundedI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_566_inf_OboundedE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) )
=> ~ ( ( ord_le1926335974945931754lambda @ A @ B )
=> ~ ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_567_inf_OboundedE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) )
=> ~ ( ( ord_le3064231726521830256lambda @ A @ B )
=> ~ ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_568_inf__absorb2,axiom,
! [Y2: set_lambda_lambda,X2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_569_inf__absorb2,axiom,
! [Y2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_570_inf__absorb1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_571_inf__absorb1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_572_inf_Oabsorb2,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_573_inf_Oabsorb2,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_574_inf_Oabsorb1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_575_inf_Oabsorb1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_576_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_577_le__iff__inf,axiom,
( ord_le3064231726521830256lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_578_inf__unique,axiom,
! [F: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda,X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda,Z3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z3 )
=> ( ord_le1926335974945931754lambda @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_579_inf__unique,axiom,
! [F: set_li7878742357367608272lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ( ord_le3064231726521830256lambda @ X @ Z3 )
=> ( ord_le3064231726521830256lambda @ X @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_580_inf_OorderI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A
= ( inf_in1766959559283430712lambda @ A @ B ) )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% inf.orderI
thf(fact_581_inf_OorderI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A
= ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( ord_le3064231726521830256lambda @ A @ B ) ) ).
% inf.orderI
thf(fact_582_inf_OorderE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( A
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% inf.orderE
thf(fact_583_inf_OorderE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( A
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ).
% inf.orderE
thf(fact_584_le__infI2,axiom,
! [B: set_lambda_lambda,X2: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_585_le__infI2,axiom,
! [B: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_586_le__infI1,axiom,
! [A: set_lambda_lambda,X2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_587_le__infI1,axiom,
! [A: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_588_inf__mono,axiom,
! [A: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda,D: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ( ord_le1926335974945931754lambda @ B @ D )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ ( inf_in1766959559283430712lambda @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_589_inf__mono,axiom,
! [A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ( ord_le3064231726521830256lambda @ B @ D )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ ( inf_in3354698378728726462lambda @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_590_le__infI,axiom,
! [X2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ A )
=> ( ( ord_le1926335974945931754lambda @ X2 @ B )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% le_infI
thf(fact_591_le__infI,axiom,
! [X2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ A )
=> ( ( ord_le3064231726521830256lambda @ X2 @ B )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% le_infI
thf(fact_592_le__infE,axiom,
! [X2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X2 @ A )
=> ~ ( ord_le1926335974945931754lambda @ X2 @ B ) ) ) ).
% le_infE
thf(fact_593_le__infE,axiom,
! [X2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ~ ( ( ord_le3064231726521830256lambda @ X2 @ A )
=> ~ ( ord_le3064231726521830256lambda @ X2 @ B ) ) ) ).
% le_infE
thf(fact_594_inf__le2,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_595_inf__le2,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_596_inf__le1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_597_inf__le1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_598_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_599_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_600_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_601_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_602_Int__emptyI,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A2 )
=> ~ ( member2667661554319814297lambda @ X @ B3 ) )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= bot_bo1397037886704484924lambda ) ) ).
% Int_emptyI
thf(fact_603_Int__emptyI,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A2 )
=> ~ ( member_lambda_lambda @ X @ B3 ) )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= bot_bo1271416176687173302lambda ) ) ).
% Int_emptyI
thf(fact_604_disjoint__iff,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A2 )
=> ~ ( member2667661554319814297lambda @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_605_disjoint__iff,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A2 )
=> ~ ( member_lambda_lambda @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_606_Int__empty__left,axiom,
! [B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ B3 )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_left
thf(fact_607_Int__empty__left,axiom,
! [B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B3 )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_left
thf(fact_608_Int__empty__right,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_right
thf(fact_609_Int__empty__right,axiom,
! [A2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_right
thf(fact_610_disjoint__iff__not__equal,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A2 )
=> ! [Y4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y4 @ B3 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_611_disjoint__iff__not__equal,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A2 )
=> ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ B3 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_612_Int__Collect__mono,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ ( collec621444948832755541lambda @ P ) ) @ ( inf_in1766959559283430712lambda @ B3 @ ( collec621444948832755541lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_613_Int__Collect__mono,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ ( collec8830145823237851355lambda @ P ) ) @ ( inf_in3354698378728726462lambda @ B3 @ ( collec8830145823237851355lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_614_Int__greatest,axiom,
! [C2: set_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ C2 @ B3 )
=> ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_615_Int__greatest,axiom,
! [C2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ C2 @ B3 )
=> ( ord_le3064231726521830256lambda @ C2 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_616_Int__absorb2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_617_Int__absorb2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_618_Int__absorb1,axiom,
! [B3: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_619_Int__absorb1,axiom,
! [B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_620_Int__lower2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_621_Int__lower2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_622_Int__lower1,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_623_Int__lower1,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_624_Int__mono,axiom,
! [A2: set_lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda,D2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ D2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ ( inf_in1766959559283430712lambda @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_625_Int__mono,axiom,
! [A2: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,D2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C2 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ D2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ ( inf_in3354698378728726462lambda @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_626_in__targets__iff,axiom,
! [T4: list_lambda_lambda,B3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ ( 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 ) @ B3 ) ) ) ) ) ).
% in_targets_iff
thf(fact_627_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_628_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_629_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_630_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_631_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_632_Ide__implies__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_633_Resid__Ide__Arr__ind,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_634_Resid__Arr__Ide__ind,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_635_Resid__Ide_I1_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Ide(1)
thf(fact_636_ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ).
% ide_char
thf(fact_637_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_638_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_639_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_640_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_641_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_642_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_643_Con__Ide__iff,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_644_Con__IdeI_I1_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(1)
thf(fact_645_Con__IdeI_I2_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(2)
thf(fact_646_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_647_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_648_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_649_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_650_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_651_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_652_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_653_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_654_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_655_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append_lambda_lambda @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( last_lambda_lambda @ Xs3 )
!= ( last_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_656_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_657_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_658_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_659_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_660_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_661_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_662_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_663_Resid1x__null,axiom,
! [T4: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T4 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_664_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_665_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_666_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_667_Resid1x__ide,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_3319247187298047121lambda @ lambda_resid @ A @ T4 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( paths_3319247187298047121lambda @ lambda_resid @ A @ T4 ) ) ) ) ).
% Resid1x_ide
thf(fact_668_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_669_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_670_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_671_dual__order_Orefl,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A @ A ) ).
% dual_order.refl
thf(fact_672_dual__order_Orefl,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A @ A ) ).
% dual_order.refl
thf(fact_673_order__refl,axiom,
! [X2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ X2 @ X2 ) ).
% order_refl
thf(fact_674_order__refl,axiom,
! [X2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ X2 @ X2 ) ).
% order_refl
thf(fact_675_Trgs__eqI,axiom,
! [T4: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) ) ) ).
% Trgs_eqI
thf(fact_676_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_677_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_678_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_679_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_680_Resid1x_Oelims,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( lambda_resid @ X2 @ U4 ) ) )
=> ~ ! [U4: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X2 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_692_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_693_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_694_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_695_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_696_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_697_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_698_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_699_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_700_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_701_set,axiom,
ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ u @ ua ) ) @ ( collec621444948832755541lambda @ lambda_is_App ) ).
% set
thf(fact_702_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_703_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B3 )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_704_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B3 )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_705_paths__in__rts_OTrgs_Ocong,axiom,
paths_7594662086398683701lambda = paths_7594662086398683701lambda ).
% paths_in_rts.Trgs.cong
thf(fact_706_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_707_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_708_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_709_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A ) )
=> ( ! [Z22: list_l5168835686021156086lambda] :
( A
!= ( cons_l4558873032622161958lambda @ E @ Z22 ) )
=> ~ ! [Z1: list_lambda_lambda,Z22: list_l5168835686021156086lambda] :
( ( A
= ( cons_l4558873032622161958lambda @ Z1 @ Z22 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_710_list_Oset__cases,axiom,
! [E: lambda_lambda,A: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A ) )
=> ( ! [Z22: list_lambda_lambda] :
( A
!= ( cons_lambda_lambda @ E @ Z22 ) )
=> ~ ! [Z1: lambda_lambda,Z22: list_lambda_lambda] :
( ( A
= ( cons_lambda_lambda @ Z1 @ Z22 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_711_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_712_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_713_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_714_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_715_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_716_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_717_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_718_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_719_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_720_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_lambda_lambda,X3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_721_in__set__conv__decomp__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 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_722_in__set__conv__decomp__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 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_723_in__set__conv__decomp__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 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_724_in__set__conv__decomp__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 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_725_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_726_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_727_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_728_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_729_in__set__conv__decomp,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 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_730_in__set__conv__decomp,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 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_731_append__Cons__eq__iff,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs4: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ Ys ) )
= ( append1017360404025248779lambda @ Xs4 @ ( cons_l4558873032622161958lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_732_append__Cons__eq__iff,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs4: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ Ys ) )
= ( append_lambda_lambda @ Xs4 @ ( cons_lambda_lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_733_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_734_split__list__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_735_split__list__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_736_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X6: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_737_split__list__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_738_split__list__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_739_split__list,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_740_split__list,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_741_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_742_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_743_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_744_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_745_ord__le__eq__subst,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_746_ord__le__eq__subst,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_747_ord__le__eq__subst,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_748_ord__le__eq__subst,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_749_ord__eq__le__subst,axiom,
! [A: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_750_ord__eq__le__subst,axiom,
! [A: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_751_ord__eq__le__subst,axiom,
! [A: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_752_ord__eq__le__subst,axiom,
! [A: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_753_order__eq__refl,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( X2 = Y2 )
=> ( ord_le1926335974945931754lambda @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_754_order__eq__refl,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( X2 = Y2 )
=> ( ord_le3064231726521830256lambda @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_755_order__subst2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_756_order__subst2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_757_order__subst2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_758_order__subst2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_759_order__subst1,axiom,
! [A: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_760_order__subst1,axiom,
! [A: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_761_order__subst1,axiom,
! [A: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_762_order__subst1,axiom,
! [A: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_763_order__eq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z2: set_lambda_lambda] : ( Y5 = Z2 ) )
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A6 @ B5 )
& ( ord_le1926335974945931754lambda @ B5 @ A6 ) ) ) ) ).
% order_eq_iff
thf(fact_764_order__eq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] : ( Y5 = Z2 ) )
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A6 @ B5 )
& ( ord_le3064231726521830256lambda @ B5 @ A6 ) ) ) ) ).
% order_eq_iff
thf(fact_765_antisym,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_766_antisym,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_767_dual__order_Otrans,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( ord_le1926335974945931754lambda @ C @ B )
=> ( ord_le1926335974945931754lambda @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_768_dual__order_Otrans,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( ord_le3064231726521830256lambda @ C @ B )
=> ( ord_le3064231726521830256lambda @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_769_dual__order_Oantisym,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( ord_le1926335974945931754lambda @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_770_dual__order_Oantisym,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( ord_le3064231726521830256lambda @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_771_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z2: set_lambda_lambda] : ( Y5 = Z2 ) )
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B5 @ A6 )
& ( ord_le1926335974945931754lambda @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_772_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] : ( Y5 = Z2 ) )
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B5 @ A6 )
& ( ord_le3064231726521830256lambda @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_773_order__trans,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ Z )
=> ( ord_le1926335974945931754lambda @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_774_order__trans,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ Z )
=> ( ord_le3064231726521830256lambda @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_775_order_Otrans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% order.trans
thf(fact_776_order_Otrans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% order.trans
thf(fact_777_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_778_order__antisym,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_779_ord__le__eq__trans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( B = C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_780_ord__le__eq__trans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( B = C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_781_ord__eq__le__trans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A = B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_782_ord__eq__le__trans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_783_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z2: set_lambda_lambda] : ( Y5 = Z2 ) )
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y4 )
& ( ord_le1926335974945931754lambda @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_784_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] : ( Y5 = Z2 ) )
= ( ^ [X3: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y4 )
& ( ord_le3064231726521830256lambda @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_785_bot_Oextremum,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% bot.extremum
thf(fact_786_bot_Oextremum,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).
% bot.extremum
thf(fact_787_bot_Oextremum__unique,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_788_bot_Oextremum__unique,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_unique
thf(fact_789_bot_Oextremum__uniqueI,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_790_bot_Oextremum__uniqueI,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
=> ( A = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_791_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_792_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_793_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_794_Resid_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ V2 @ Va ) )
=> ( ( Xa = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U4 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U4: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U4 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U4: lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U4 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U4 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U4 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_795_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_796_u,axiom,
( ( lambda_Arr @ u )
& ( lambda_is_App @ u ) ) ).
% u
thf(fact_797__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_798__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_799_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_800_Residx1_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_801_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_802_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_803_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_804_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_805_singletonI,axiom,
! [A: list_lambda_lambda] : ( member2667661554319814297lambda @ A @ ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) ).
% singletonI
thf(fact_806_singletonI,axiom,
! [A: lambda_lambda] : ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ).
% singletonI
thf(fact_807_insert__subset,axiom,
! [X2: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ X2 @ A2 ) @ B3 )
= ( ( member_lambda_lambda @ X2 @ B3 )
& ( ord_le1926335974945931754lambda @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_808_insert__subset,axiom,
! [X2: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ X2 @ A2 ) @ B3 )
= ( ( member2667661554319814297lambda @ X2 @ B3 )
& ( ord_le3064231726521830256lambda @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_809_Int__insert__left__if0,axiom,
! [A: list_lambda_lambda,C2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ B3 ) @ C2 )
= ( inf_in3354698378728726462lambda @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_810_Int__insert__left__if0,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_811_Int__insert__left__if1,axiom,
! [A: list_lambda_lambda,C2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ B3 ) @ C2 )
= ( insert8465602035467394112lambda @ A @ ( inf_in3354698378728726462lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_812_Int__insert__left__if1,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_813_insert__inter__insert,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ A2 ) @ ( insert8465602035467394112lambda @ A @ B3 ) )
= ( insert8465602035467394112lambda @ A @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_814_insert__inter__insert,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A2 ) @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_815_Int__insert__right__if0,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ ( insert8465602035467394112lambda @ A @ B3 ) )
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_816_Int__insert__right__if0,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_817_Int__insert__right__if1,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ ( insert8465602035467394112lambda @ A @ B3 ) )
= ( insert8465602035467394112lambda @ A @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_818_Int__insert__right__if1,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_819_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_820_singleton__insert__inj__eq_H,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ A2 )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( ( A = B )
& ( ord_le1926335974945931754lambda @ A2 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_821_singleton__insert__inj__eq_H,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A @ A2 )
= ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
= ( ( A = B )
& ( ord_le3064231726521830256lambda @ A2 @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_822_singleton__insert__inj__eq,axiom,
! [B: lambda_lambda,A: lambda_lambda,A2: set_lambda_lambda] :
( ( ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ A @ A2 ) )
= ( ( A = B )
& ( ord_le1926335974945931754lambda @ A2 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_823_singleton__insert__inj__eq,axiom,
! [B: list_lambda_lambda,A: list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3064231726521830256lambda @ A2 @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_824_insert__disjoint_I1_J,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ A2 ) @ B3 )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A @ B3 )
& ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= bot_bo1397037886704484924lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_825_insert__disjoint_I1_J,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A2 ) @ B3 )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= bot_bo1271416176687173302lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_826_insert__disjoint_I2_J,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ A2 ) @ B3 ) )
= ( ~ ( member2667661554319814297lambda @ A @ B3 )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_827_insert__disjoint_I2_J,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A2 ) @ B3 ) )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_828_disjoint__insert_I1_J,axiom,
! [B3: set_li7878742357367608272lambda,A: list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ B3 @ ( insert8465602035467394112lambda @ A @ A2 ) )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A @ B3 )
& ( ( inf_in3354698378728726462lambda @ B3 @ A2 )
= bot_bo1397037886704484924lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_829_disjoint__insert_I1_J,axiom,
! [B3: set_lambda_lambda,A: lambda_lambda,A2: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ B3 @ ( insert_lambda_lambda @ A @ A2 ) )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( ( inf_in1766959559283430712lambda @ B3 @ A2 )
= bot_bo1271416176687173302lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_830_disjoint__insert_I2_J,axiom,
! [A2: set_li7878742357367608272lambda,B: list_lambda_lambda,B3: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A2 @ ( insert8465602035467394112lambda @ B @ B3 ) ) )
= ( ~ ( member2667661554319814297lambda @ B @ A2 )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_831_disjoint__insert_I2_J,axiom,
! [A2: set_lambda_lambda,B: lambda_lambda,B3: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A2 @ ( insert_lambda_lambda @ B @ B3 ) ) )
= ( ~ ( member_lambda_lambda @ B @ A2 )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_832_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_833_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_834_singletonD,axiom,
! [B: list_lambda_lambda,A: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_835_singletonD,axiom,
! [B: lambda_lambda,A: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_836_singleton__iff,axiom,
! [B: list_lambda_lambda,A: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_837_singleton__iff,axiom,
! [B: lambda_lambda,A: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_838_doubleton__eq__iff,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda,D: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
= ( insert8465602035467394112lambda @ C @ ( insert8465602035467394112lambda @ D @ bot_bo1397037886704484924lambda ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_839_doubleton__eq__iff,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda,D: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D @ bot_bo1271416176687173302lambda ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_840_insert__not__empty,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( insert8465602035467394112lambda @ A @ A2 )
!= bot_bo1397037886704484924lambda ) ).
% insert_not_empty
thf(fact_841_insert__not__empty,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda] :
( ( insert_lambda_lambda @ A @ A2 )
!= bot_bo1271416176687173302lambda ) ).
% insert_not_empty
thf(fact_842_singleton__inject,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_843_singleton__inject,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_844_lambda__calculus_Olambda_Odisc_I11_J,axiom,
~ ( lambda_is_App @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(11)
thf(fact_845_lambda__calculus_Olambda_Odistinct__disc_I6_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(6)
thf(fact_846_lambda__calculus_Olambda_Odistinct__disc_I5_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(5)
thf(fact_847_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_848_lambda__calculus_Olambda_OdiscI_I3_J,axiom,
! [Lambda: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( Lambda
= ( lambda_App @ X41 @ X42 ) )
=> ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.discI(3)
thf(fact_849_lambda__calculus_Ois__App__def,axiom,
( lambda_is_App
= ( ^ [Lambda2: lambda_lambda] :
? [X412: lambda_lambda,X422: lambda_lambda] :
( Lambda2
= ( lambda_App @ X412 @ X422 ) ) ) ) ).
% lambda_calculus.is_App_def
thf(fact_850_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_851_Int__insert__right,axiom,
! [A: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ( member2667661554319814297lambda @ A @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ ( insert8465602035467394112lambda @ A @ B3 ) )
= ( insert8465602035467394112lambda @ A @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) )
& ( ~ ( member2667661554319814297lambda @ A @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ ( insert8465602035467394112lambda @ A @ B3 ) )
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_852_Int__insert__right,axiom,
! [A: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) )
& ( ~ ( member_lambda_lambda @ A @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_853_subset__insertI2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,B: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ord_le1926335974945931754lambda @ A2 @ ( insert_lambda_lambda @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_854_subset__insertI2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,B: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ord_le3064231726521830256lambda @ A2 @ ( insert8465602035467394112lambda @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_855_Int__insert__left,axiom,
! [A: list_lambda_lambda,C2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ( member2667661554319814297lambda @ A @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ B3 ) @ C2 )
= ( insert8465602035467394112lambda @ A @ ( inf_in3354698378728726462lambda @ B3 @ C2 ) ) ) )
& ( ~ ( member2667661554319814297lambda @ A @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A @ B3 ) @ C2 )
= ( inf_in3354698378728726462lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_856_Int__insert__left,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) )
& ( ~ ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_857_subset__insertI,axiom,
! [B3: set_lambda_lambda,A: lambda_lambda] : ( ord_le1926335974945931754lambda @ B3 @ ( insert_lambda_lambda @ A @ B3 ) ) ).
% subset_insertI
thf(fact_858_subset__insertI,axiom,
! [B3: set_li7878742357367608272lambda,A: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ B3 @ ( insert8465602035467394112lambda @ A @ B3 ) ) ).
% subset_insertI
thf(fact_859_subset__insert,axiom,
! [X2: lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ ( insert_lambda_lambda @ X2 @ B3 ) )
= ( ord_le1926335974945931754lambda @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_860_subset__insert,axiom,
! [X2: list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ A2 @ ( insert8465602035467394112lambda @ X2 @ B3 ) )
= ( ord_le3064231726521830256lambda @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_861_insert__mono,axiom,
! [C2: set_lambda_lambda,D2: set_lambda_lambda,A: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ D2 )
=> ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ A @ C2 ) @ ( insert_lambda_lambda @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_862_insert__mono,axiom,
! [C2: set_li7878742357367608272lambda,D2: set_li7878742357367608272lambda,A: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ D2 )
=> ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ A @ C2 ) @ ( insert8465602035467394112lambda @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_863_subset__singleton__iff,axiom,
! [X4: set_lambda_lambda,A: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X4 @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
= ( ( X4 = bot_bo1271416176687173302lambda )
| ( X4
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_864_subset__singleton__iff,axiom,
! [X4: set_li7878742357367608272lambda,A: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ X4 @ ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) )
= ( ( X4 = bot_bo1397037886704484924lambda )
| ( X4
= ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_865_subset__singletonD,axiom,
! [A2: set_lambda_lambda,X2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) )
=> ( ( A2 = bot_bo1271416176687173302lambda )
| ( A2
= ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singletonD
thf(fact_866_subset__singletonD,axiom,
! [A2: set_li7878742357367608272lambda,X2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) )
=> ( ( A2 = bot_bo1397037886704484924lambda )
| ( A2
= ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singletonD
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_list_Omap__disc__iff,axiom,
! [F: lambda_lambda > lambda_lambda,A: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ A )
= nil_lambda_lambda )
= ( A = nil_lambda_lambda ) ) ).
% list.map_disc_iff
thf(fact_878_map__eq__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_879_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_880_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_881_path__from__NF__is__Ide,axiom,
! [A: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U2 )
= A )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U2 ) ) ) ) ).
% path_from_NF_is_Ide
thf(fact_882_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_883_list_Omap__cong,axiom,
! [X2: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X2 = Ya )
=> ( ! [Z3: lambda_lambda] :
( ( member_lambda_lambda @ Z3 @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_884_list_Omap__cong0,axiom,
! [X2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z3: lambda_lambda] :
( ( member_lambda_lambda @ Z3 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_885_list_Oinj__map__strong,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z3: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z3 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_886_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z3 @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_887_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z3: lambda_lambda] :
( ( member_lambda_lambda @ Z3 @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_888_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_889_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_890_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_891_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_892_ex__map__conv,axiom,
! [Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ? [Xs5: list_lambda_lambda] :
( Ys
= ( map_la336867333220232516lambda @ F @ Xs5 ) ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys ) )
=> ? [Y4: lambda_lambda] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_893_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 ) )
= ( ? [Z4: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_894_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 ) )
= ( ? [Z4: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_895_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 ) )
=> ? [Z3: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_896_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 ) )
=> ? [Z3: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_897_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_898_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_899_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_900_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_901_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_902_list_Omap__sel_I1_J,axiom,
! [A: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ A ) )
= ( F @ ( hd_lambda_lambda @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_903_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_904_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_905_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_906_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_907_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_908_list_Oset__sel_I1_J,axiom,
! [A: list_l5168835686021156086lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ A ) @ ( set_li3846194476672328965lambda @ A ) ) ) ).
% list.set_sel(1)
thf(fact_909_list_Oset__sel_I1_J,axiom,
! [A: list_lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ A ) @ ( set_lambda_lambda2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_910_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append_lambda_lambda @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( hd_lambda_lambda @ Xs3 )
!= ( hd_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_911_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_912_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_913_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_914_in__sources__iff,axiom,
! [T4: list_lambda_lambda,A2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ A2 ) ) ) ) ) ).
% in_sources_iff
thf(fact_915_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_916_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_917_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_918__C9_C,axiom,
( ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ua ) ) )
= ( lambda_Trg @ ( lambda_un_App2 @ u ) ) ) ).
% "9"
thf(fact_919_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_920_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_921_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_922_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_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_OSrc__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Src_Trg
thf(fact_925_lambda__calculus_OConI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ U ) )
=> ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ) ) ).
% lambda_calculus.ConI
thf(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_936_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_937_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_938_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_939_lambda__calculus_OCoinitial__resid__resid,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ V )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ V )
!= lambda_Nil )
=> ( ( lambda_Arr @ ( lambda_resid @ T @ V ) )
& ( lambda_Arr @ ( lambda_resid @ U @ V ) )
& ( ( lambda_Src @ ( lambda_resid @ T @ V ) )
= ( lambda_Src @ ( lambda_resid @ U @ V ) ) ) ) ) ) ).
% lambda_calculus.Coinitial_resid_resid
thf(fact_940_lambda__calculus_OCoinitial__iff__Con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% lambda_calculus.Coinitial_iff_Con
thf(fact_941_lambda__calculus_OCon__implies__Coinitial3_I1_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ V )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(1)
thf(fact_942_lambda__calculus_OCon__implies__Coinitial3_I2_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ V )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(2)
thf(fact_943_lambda__calculus_OCon__implies__Coinitial3_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ U )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ U )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(3)
thf(fact_944_lambda__calculus_OCon__Arr__Src_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(1)
thf(fact_945_lambda__calculus_OCon__Arr__Src_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(2)
thf(fact_946_lambda__calculus_OSrc__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Src @ ( lambda_resid @ T @ U ) )
= ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Src_resid
thf(fact_947_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_948_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_949_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_950_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_951_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_952__C8_C,axiom,
( ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) )
= ( lambda_Trg @ ( lambda_un_App1 @ u ) ) ) ).
% "8"
thf(fact_953_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_954_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_955_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_956_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_957_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_958_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_959__C3_C,axiom,
( ( lambda_Arr @ ( lambda_un_App1 @ u ) )
& ( lambda_Arr @ ( lambda_un_App2 @ u ) ) ) ).
% "3"
thf(fact_960__C1_C,axiom,
paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ).
% "1"
thf(fact_961__C7_C,axiom,
lambda_Arr @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ).
% "7"
thf(fact_962_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_963_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_964__C5_C,axiom,
lambda_Arr @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) ).
% "5"
thf(fact_965_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_966_lambda__calculus_Olambda_Ocollapse_I3_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ( ( lambda_App @ ( lambda_un_App1 @ Lambda ) @ ( lambda_un_App2 @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(3)
thf(fact_967_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_968_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_969_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_970_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_971_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_972_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_973_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_974_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_975_red__iff,axiom,
( lambda_red
= ( ^ [A6: lambda_lambda,B5: lambda_lambda] :
? [T8: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T8 )
= A6 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T8 )
= B5 ) ) ) ) ).
% red_iff
thf(fact_976_ind,axiom,
( ( paths_366302053746739058lambda @ lambda_resid @ ua )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ua ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ua ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) )
@ ua ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ ua
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ua ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) ) ) ) ).
% ind
thf(fact_977__092_060Lambda_062_Otargets__char,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B5: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ T ) @ B5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B5 @ ( lambda_resid @ T @ T ) ) ) ) ) ) ).
% \<Lambda>.targets_char
thf(fact_978__092_060Lambda_062_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A6: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A6 )
& ( con_lambda_lambda @ lambda_resid @ T @ A6 ) ) ) ) ).
% \<Lambda>.sources_def
thf(fact_979__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_closed
thf(fact_980_sources__def,axiom,
! [T: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A6 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A6 ) ) ) ) ).
% sources_def
thf(fact_981_targets__char,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B5: 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 ) @ B5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ) ) ).
% targets_char
thf(fact_982_map__ident,axiom,
( ( map_la336867333220232516lambda
@ ^ [X3: lambda_lambda] : X3 )
= ( ^ [Xs5: list_lambda_lambda] : Xs5 ) ) ).
% map_ident
thf(fact_983_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_984_targets__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [B4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ B4 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ B4 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% targets_char\<^sub>P
thf(fact_985_singleton__conv,axiom,
! [A: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] : ( X3 = A ) )
= ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv
thf(fact_986_singleton__conv,axiom,
! [A: lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : ( X3 = A ) )
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv
thf(fact_987_singleton__conv2,axiom,
! [A: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ( ^ [Y5: list_lambda_lambda,Z2: list_lambda_lambda] : ( Y5 = Z2 )
@ A ) )
= ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv2
thf(fact_988_singleton__conv2,axiom,
! [A: lambda_lambda] :
( ( collec621444948832755541lambda
@ ( ^ [Y5: lambda_lambda,Z2: lambda_lambda] : ( Y5 = Z2 )
@ A ) )
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv2
thf(fact_989_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_990_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
@ ^ [B4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ 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 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_991_orthogonal__App__Arr__Arr,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 ) )
= ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) ) ) ).
% orthogonal_App_Arr_Arr
thf(fact_992__C6_C,axiom,
paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ).
% "6"
thf(fact_993_orthogonal__App__cong,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) ) ) ) ) ) ) ).
% orthogonal_App_cong
thf(fact_994__092_060open_062Arr_A_091_093_A_092_060Longrightarrow_062_Amap_A_I_092_060lambda_062X_O_AX_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OSrc_A_I_092_060Lambda_062_Oun__App2_A_Ihd_A_091_093_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App1_A_091_093_J_A_064_Amap_A_I_I_092_060_094bold_062_092_060circ_062_J_A_I_092_060Lambda_062_OTrg_A_I_092_060Lambda_062_Oun__App1_A_Ilast_A_091_093_J_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App2_A_091_093_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_091_093_092_060close_062,axiom,
( ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ nil_lambda_lambda ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ nil_lambda_lambda ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ nil_lambda_lambda ) ) )
@ nil_lambda_lambda ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ nil_lambda_lambda
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ nil_lambda_lambda ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ nil_lambda_lambda ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% \<open>Arr [] \<Longrightarrow> map (\<lambda>X. X \<^bold>\<circ> \<Lambda>.Src (\<Lambda>.un_App2 (hd []))) (map \<Lambda>.un_App1 []) @ map ((\<^bold>\<circ>) (\<Lambda>.Trg (\<Lambda>.un_App1 (last [])))) (map \<Lambda>.un_App2 []) \<^sup>*\<sim>\<^sup>* []\<close>
thf(fact_995__092_060open_062map_A_I_092_060lambda_062X_O_AX_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OSrc_A_I_092_060Lambda_062_Oun__App2_A_Ihd_A_Iu_A_D_AU_J_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App1_A_Iu_A_D_AU_J_J_A_064_Amap_A_I_I_092_060_094bold_062_092_060circ_062_J_A_I_092_060Lambda_062_OTrg_A_I_092_060Lambda_062_Oun__App1_A_Ilast_A_Iu_A_D_AU_J_J_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App2_A_Iu_A_D_AU_J_J_A_061_A_091_092_060Lambda_062_Oun__App1_Au_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OSrc_A_I_092_060Lambda_062_Oun__App2_Au_J_093_A_064_A_Imap_A_I_092_060lambda_062X_O_AX_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OSrc_A_I_092_060Lambda_062_Oun__App2_Au_J_J_A_Imap_A_092_060Lambda_062_Oun__App1_AU_J_A_064_A_091_092_060Lambda_062_OTrg_A_I_092_060Lambda_062_Oun__App1_A_Ilast_AU_J_J_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_Oun__App2_Au_093_J_A_064_Amap_A_I_I_092_060_094bold_062_092_060circ_062_J_A_I_092_060Lambda_062_OTrg_A_I_092_060Lambda_062_Oun__App1_A_Ilast_AU_J_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App2_AU_J_092_060close_062,axiom,
( ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
= ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) ).
% \<open>map (\<lambda>X. X \<^bold>\<circ> \<Lambda>.Src (\<Lambda>.un_App2 (hd (u # U)))) (map \<Lambda>.un_App1 (u # U)) @ map ((\<^bold>\<circ>) (\<Lambda>.Trg (\<Lambda>.un_App1 (last (u # U))))) (map \<Lambda>.un_App2 (u # U)) = [\<Lambda>.un_App1 u \<^bold>\<circ> \<Lambda>.Src (\<Lambda>.un_App2 u)] @ (map (\<lambda>X. X \<^bold>\<circ> \<Lambda>.Src (\<Lambda>.un_App2 u)) (map \<Lambda>.un_App1 U) @ [\<Lambda>.Trg (\<Lambda>.un_App1 (last U)) \<^bold>\<circ> \<Lambda>.un_App2 u]) @ map ((\<^bold>\<circ>) (\<Lambda>.Trg (\<Lambda>.un_App1 (last U)))) (map \<Lambda>.un_App2 U)\<close>
thf(fact_996__C11_C,axiom,
( ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) )
= ( append_lambda_lambda @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) ).
% "11"
thf(fact_997_calculation,axiom,
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
@ ( append_lambda_lambda @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ).
% calculation
thf(fact_998__C4_C,axiom,
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) ) ) ) ) ).
% "4"
thf(fact_999__C12_C,axiom,
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) )
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) )
@ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda )
@ ( append_lambda_lambda
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ) ) ) ).
% "12"
thf(fact_1000__092_060open_062seq_A_I_091_092_060Lambda_062_Oun__App1_Au_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OSrc_A_I_092_060Lambda_062_Oun__App2_Au_J_093_A_064_A_091_092_060Lambda_062_OSrc_A_Ihd_A_Imap_A_092_060Lambda_062_Oun__App1_AU_J_J_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_Oun__App2_Au_093_J_A_Imap_A_I_092_060lambda_062X_O_AX_A_092_060_094bold_062_092_060circ_062_A_092_060Lambda_062_OTrg_A_Ilast_A_091_092_060Lambda_062_Oun__App2_Au_093_J_J_A_Imap_A_092_060Lambda_062_Oun__App1_AU_J_A_064_Amap_A_I_I_092_060_094bold_062_092_060circ_062_J_A_I_092_060Lambda_062_OTrg_A_I_092_060Lambda_062_Oun__App1_A_Ilast_AU_J_J_J_J_A_Imap_A_092_060Lambda_062_Oun__App2_AU_J_J_092_060close_062,axiom,
( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_un_App1 @ u ) @ ( lambda_Src @ ( lambda_un_App2 @ u ) ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) ) ) @ ( lambda_un_App2 @ u ) ) @ nil_lambda_lambda ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_un_App2 @ u ) @ nil_lambda_lambda ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ua ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ua ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ ua ) ) ) ) ).
% \<open>seq ([\<Lambda>.un_App1 u \<^bold>\<circ> \<Lambda>.Src (\<Lambda>.un_App2 u)] @ [\<Lambda>.Src (hd (map \<Lambda>.un_App1 U)) \<^bold>\<circ> \<Lambda>.un_App2 u]) (map (\<lambda>X. X \<^bold>\<circ> \<Lambda>.Trg (last [\<Lambda>.un_App2 u])) (map \<Lambda>.un_App1 U) @ map ((\<^bold>\<circ>) (\<Lambda>.Trg (\<Lambda>.un_App1 (last U)))) (map \<Lambda>.un_App2 U))\<close>
thf(fact_1001_lambda__calculus_ONF__reduct__is__trivial,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( lambda_red @ A @ B )
=> ( A = B ) ) ) ).
% lambda_calculus.NF_reduct_is_trivial
thf(fact_1002_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_1003_Collect__conv__if,axiom,
! [P: list_lambda_lambda > $o,A: list_lambda_lambda] :
( ( ( P @ A )
=> ( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A )
=> ( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if
thf(fact_1004_Collect__conv__if,axiom,
! [P: lambda_lambda > $o,A: lambda_lambda] :
( ( ( P @ A )
=> ( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A )
=> ( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if
thf(fact_1005_Collect__conv__if2,axiom,
! [P: list_lambda_lambda > $o,A: list_lambda_lambda] :
( ( ( P @ A )
=> ( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert8465602035467394112lambda @ A @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A )
=> ( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if2
thf(fact_1006_Collect__conv__if2,axiom,
! [P: lambda_lambda > $o,A: lambda_lambda] :
( ( ( P @ A )
=> ( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A )
=> ( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if2
thf(fact_1007_empty__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_1008_empty__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_1009_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ B @ C )
=> ( lambda_red @ A @ C ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_1010_lambda__calculus_Oweak__diamond,axiom,
! [A: lambda_lambda,B: lambda_lambda,B2: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ A @ B2 )
=> ~ ! [C3: lambda_lambda] :
( ( lambda_red @ B @ C3 )
=> ~ ( lambda_red @ B2 @ C3 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_1011_Collect__conj__eq,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in3354698378728726462lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1012_Collect__conj__eq,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in1766959559283430712lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1013_inf__set__def,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( collec8830145823237851355lambda
@ ( inf_in6319970306715391135mbda_o
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A5 )
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1014_inf__set__def,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
( collec621444948832755541lambda
@ ( inf_in2068736301489225125mbda_o
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A5 )
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_1015_Int__Collect,axiom,
! [X2: list_lambda_lambda,A2: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ X2 @ ( inf_in3354698378728726462lambda @ A2 @ ( collec8830145823237851355lambda @ P ) ) )
= ( ( member2667661554319814297lambda @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_1016_Int__Collect,axiom,
! [X2: lambda_lambda,A2: set_lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ X2 @ ( inf_in1766959559283430712lambda @ A2 @ ( collec621444948832755541lambda @ P ) ) )
= ( ( member_lambda_lambda @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_1017_Int__def,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A5 )
& ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1018_Int__def,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A5 )
& ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1019_Collect__subset,axiom,
! [A2: set_lambda_lambda,P: lambda_lambda > $o] :
( ord_le1926335974945931754lambda
@ ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1020_Collect__subset,axiom,
! [A2: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ord_le3064231726521830256lambda
@ ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1021_less__eq__set__def,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B4: set_lambda_lambda] :
( ord_le5488116190248881011mbda_o
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A5 )
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1022_less__eq__set__def,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ord_le4030711025213088621mbda_o
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A5 )
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1023_list_Omap__ident,axiom,
! [T: list_lambda_lambda] :
( ( map_la336867333220232516lambda
@ ^ [X3: lambda_lambda] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_1024_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_1025_lambda__calculus_Ored_Ocases,axiom,
! [A1: lambda_lambda,A22: lambda_lambda] :
( ( lambda_red @ A1 @ A22 )
=> ( ! [T3: lambda_lambda] :
( ( A1
= ( lambda_Src @ T3 ) )
=> ( ( A22
= ( lambda_Trg @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ~ ! [B6: lambda_lambda] :
( ( lambda_red @ A1 @ B6 )
=> ~ ( lambda_red @ B6 @ A22 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_1026_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T6: lambda_lambda] :
( ( A12
= ( lambda_Src @ T6 ) )
& ( A23
= ( lambda_Trg @ T6 ) )
& ( lambda_Arr @ T6 ) )
| ? [A6: lambda_lambda,B5: lambda_lambda,C4: lambda_lambda] :
( ( A12 = A6 )
& ( A23 = C4 )
& ( lambda_red @ A6 @ B5 )
& ( lambda_red @ B5 @ C4 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_1027_un__App2_Omap__Resid__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( con_li79537494059641180lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ T4
@ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4
@ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ T4 )
@ ( cons_lambda_lambda
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil )
@ nil_lambda_lambda ) ) ) ) ).
% un_App2.map_Resid_single
thf(fact_1028_un__App1_Omap__Resid__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( con_li79537494059641180lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ T4
@ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4
@ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ T4 )
@ ( cons_lambda_lambda
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil )
@ nil_lambda_lambda ) ) ) ) ).
% un_App1.map_Resid_single
thf(fact_1029__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cube
thf(fact_1030__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_1031__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ V ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ V ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_transitive
thf(fact_1032__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ 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_1033__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__backward__stable,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ A ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_backward_stable
thf(fact_1034__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ U ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_transitive
thf(fact_1035__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_symmetric
thf(fact_1036__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ 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_1037__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ V )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ V ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_reflects_con
thf(fact_1038__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_sym
thf(fact_1039__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [B5: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( member_lambda_lambda @ B5 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1040__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [A6: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( member_lambda_lambda @ A6 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1041__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.null_eqI
thf(fact_1042__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cube_ax
thf(fact_1043__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_sym_ax
thf(fact_1044__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_imp_arr_resid
thf(fact_1045__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ T
@ V ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_symmetric
thf(fact_1046__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Opartial__magma__axioms,axiom,
( partia3924075026459072775lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.partial_magma_axioms
thf(fact_1047__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_reflexive
thf(fact_1048__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__implies__arr,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_implies_arr
thf(fact_1049__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_reflexive
thf(fact_1050__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T5 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T5
@ T ) ) )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T5
@ U ) ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T5
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_subst_left(2)
thf(fact_1051__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T5 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T5
@ T ) ) )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_subst_left(1)
thf(fact_1052__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ U3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U3
@ U ) ) )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U3 ) ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U3 )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_subst_right(2)
thf(fact_1053__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ U3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U3
@ U ) ) )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U3 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_subst_right(1)
thf(fact_1054__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A4 )
& ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A4
@ T )
& ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A4
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_imp_coinitial_ax
thf(fact_1055__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__target,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ V )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_target
thf(fact_1056__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__transitive__on__ide,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ B )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ C )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ B )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ B
@ C )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ C ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_transitive_on_ide
thf(fact_1057__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OideE,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ~ ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A )
!= A ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ideE
thf(fact_1058__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__def,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
= ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A )
= A ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_def
thf(fact_1059__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__imp__con__iff__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_imp_con_iff_cong
thf(fact_1060__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oprfx__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_implies_con
thf(fact_1061__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__arr__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ A )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ A )
= T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_arr_ide
thf(fact_1062__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__ide__arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ T )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_ide_arr
thf(fact_1063__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__resid__iff__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_resid_iff_con
thf(fact_1064__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_resid
thf(fact_1065__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__def,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_def
thf(fact_1066__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OarrE,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arrE
thf(fact_1067__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__implies__arr_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_implies_arr(1)
thf(fact_1068__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__implies__arr_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_implies_arr(2)
thf(fact_1069__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otarget__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.target_is_ide
thf(fact_1070__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__are__cong,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( member_lambda_lambda @ B2
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ B
@ B2 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ B2
@ B ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_are_cong
thf(fact_1071__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__cong__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ B
@ B2 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ B2
@ B ) ) )
=> ( member_lambda_lambda @ B2
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_cong_closed
thf(fact_1072__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__cong__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A3
@ A ) ) )
=> ( member_lambda_lambda @ A3
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_cong_closed
thf(fact_1073__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__are__cong,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( member_lambda_lambda @ A3
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A3
@ A ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_are_cong
thf(fact_1074__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osource__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.source_is_ide
thf(fact_1075__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__resid__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_resid_sym
thf(fact_1076__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__are__con,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( member_lambda_lambda @ B2
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ B
@ B2 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_are_con
thf(fact_1077__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__are__con,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( member_lambda_lambda @ A3
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A3 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_are_con
thf(fact_1078__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__source__in__targets,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( member_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ T )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_source_in_targets
thf(fact_1079__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Onot__arr__null,axiom,
~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.not_arr_null
thf(fact_1080__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__un__upto__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V4 )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ V4 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V4
@ V ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_un_upto_cong
thf(fact_1081__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OconE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.conE
thf(fact_1082__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_def
thf(fact_1083__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__def,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 ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
= ( lambda_resid @ T @ U ) ) )
& ( ~ ( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( lambda_Arr @ U )
& ( lambda_is_App @ U )
& ( con_lambda_lambda @ lambda_resid @ T @ U ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_def
thf(fact_1084__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__arr__self,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_arr_self
thf(fact_1085__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ W2 )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V
@ W2 )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ V )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ V )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ W2
@ V ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_resid
thf(fact_1086__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__with__join__of__iff_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ W2 )
=> ( ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ V )
& ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ W2
@ V ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_with_join_of_iff(1)
thf(fact_1087__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__with__join__of__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ W2 )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ W2
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ V )
& ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_with_join_of_iff(2)
thf(fact_1088__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_join_of(2)
thf(fact_1089__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_join_of(1)
thf(fact_1090__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_join_of(2)
thf(fact_1091__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_join_of(1)
thf(fact_1092__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ U3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U3
@ U ) ) )
=> ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ U3 ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_implies_coinitial
thf(fact_1093__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocoinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A3 )
=> ( ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A3 )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A3
@ A ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coinitial_ide_are_cong
thf(fact_1094__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__imp__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_imp_coinitial
thf(fact_1095__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T5 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T5
@ T ) ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ U3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U3
@ U ) ) )
=> ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5
@ U3 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_respects_seq
thf(fact_1096__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ U3 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U3
@ U ) ) )
=> ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U
@ U3 ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_implies_coterminal
thf(fact_1097__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcomposableD_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.composableD(2)
thf(fact_1098__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcomposableD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.composableD(1)
thf(fact_1099__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoinable__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.joinable_implies_con
thf(fact_1100__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ? [X5: lambda_lambda] :
( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ X5 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.joinable_def
thf(fact_1101__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocomposable__imp__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.composable_imp_seq
thf(fact_1102__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoinable__implies__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.joinable_implies_coinitial
thf(fact_1103__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oidentities__form__coherent__normal__sub__rts,axiom,
( cohere1690309856004283859lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( collec621444948832755541lambda
@ ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.identities_form_coherent_normal_sub_rts
thf(fact_1104__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__char,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [B5: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T )
@ B5 ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ B5
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ T ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_char
thf(fact_1105__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__con__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ B2 )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ B
@ B2 )
=> ( member_lambda_lambda @ B2
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_con_closed
thf(fact_1106__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [A6: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A6 )
& ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ A6 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_def
thf(fact_1107__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__con__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A3 )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A3 )
=> ( member_lambda_lambda @ A3
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_con_closed
thf(fact_1108__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oin__sourcesE,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ~ ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ~ ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ A ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.in_sourcesE
thf(fact_1109__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__arr__src_I2_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ A
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_arr_src(2)
thf(fact_1110__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__arr__src_I1_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ T
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_arr_src(1)
thf(fact_1111__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__iff__has__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_iff_has_target
thf(fact_1112__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__iff__has__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_iff_has_source
thf(fact_1113__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcoinitialE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ~ ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
=> ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
!= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coinitialE
thf(fact_1114__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T5 )
= ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
& ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 )
& ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coinitial_iff
thf(fact_1115__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T5 )
= ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
& ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 )
& ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coterminal_iff
thf(fact_1116__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcoterminalE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ~ ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
!= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coterminalE
thf(fact_1117__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcomposableD_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.composableD(3)
thf(fact_1118__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_eqI
thf(fact_1119__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T5 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_eqI
thf(fact_1120__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Opath__reflection,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 )
@ ( collec621444948832755541lambda
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
=> ( paths_366302053746739058lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4 ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.path_reflection
thf(fact_1121_un__App1_Opreserves__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil ) ) )
& ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil ) ) ) ) ) ).
% un_App1.preserves_cong
thf(fact_1122_un__App1_Opreserves__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ide_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
@ ( lambda_un_App1 @ A )
@ lambda_Nil ) ) ) ).
% un_App1.preserves_ide
thf(fact_1123_un__App1_Opreserves__prfx,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil ) ) ) ) ).
% un_App1.preserves_prfx
thf(fact_1124_un__App2_Opreserves__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ U
@ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil ) ) )
& ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil ) ) ) ) ) ).
% un_App2.preserves_cong
thf(fact_1125_un__App2_Opreserves__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ide_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
@ ( lambda_un_App2 @ A )
@ lambda_Nil ) ) ) ).
% un_App2.preserves_ide
thf(fact_1126_un__App2_Opreserves__prfx,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ide_lambda_lambda @ lambda_resid
@ ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil ) ) ) ) ).
% un_App2.preserves_prfx
thf(fact_1127_un__App1_Opreserves__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil ) ) ) ).
% un_App1.preserves_con
thf(fact_1128_un__App2_Opreserves__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( con_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil ) ) ) ).
% un_App2.preserves_con
thf(fact_1129_un__App1_Oextensional,axiom,
! [T: lambda_lambda] :
( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( lambda_un_App1 @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) )
& ( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( lambda_Nil
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ) ).
% un_App1.extensional
thf(fact_1130_un__App2_Oextensional,axiom,
! [T: lambda_lambda] :
( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( lambda_un_App2 @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) )
& ( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( lambda_Nil
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ) ).
% un_App2.extensional
thf(fact_1131_un__App1_Opreserves__joins,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( join_o1386658102433174083lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V )
@ ( lambda_un_App1 @ V )
@ lambda_Nil ) ) ) ).
% un_App1.preserves_joins
thf(fact_1132_un__App2_Opreserves__joins,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U
@ V )
=> ( join_o1386658102433174083lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ V )
@ ( lambda_un_App2 @ V )
@ lambda_Nil ) ) ) ).
% un_App2.preserves_joins
thf(fact_1133__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OseqE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ~ ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
=> ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
!= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.seqE
thf(fact_1134__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oseq__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
& ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
& ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.seq_def
thf(fact_1135__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__imp__common__source,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( inf_in1766959559283430712lambda
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_imp_common_source
thf(fact_1136__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocoinitial__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( inf_in1766959559283430712lambda
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coinitial_def
thf(fact_1137__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
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( inf_in1766959559283430712lambda
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coterminal_def
thf(fact_1138_un__App1_Opreserves__paths,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ T4 ) ) ) ).
% un_App1.preserves_paths
thf(fact_1139_un__App2_Opreserves__paths,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ T4 ) ) ) ).
% un_App2.preserves_paths
thf(fact_1140_un__App1_Omap__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ T4
@ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4
@ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ U2 ) ) ) ) ).
% un_App1.map_Resid
thf(fact_1141_un__App2_Omap__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ T4
@ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T4
@ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ U2 ) ) ) ) ).
% un_App2.map_Resid
thf(fact_1142__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( ide_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_char
thf(fact_1143__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( ( lambda_Arr @ T )
& ( lambda_is_App @ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arr_char
thf(fact_1144__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
= ( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( lambda_Arr @ U )
& ( lambda_is_App @ U )
& ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_char
thf(fact_1145__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Onull__char,axiom,
( ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.null_char
thf(fact_1146__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Onull__is__zero_I1_J,axiom,
! [T: lambda_lambda] :
( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ T )
= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.null_is_zero(1)
thf(fact_1147__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Onull__is__zero_I2_J,axiom,
! [T: lambda_lambda] :
( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) )
= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.null_is_zero(2)
thf(fact_1148__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OideI,axiom,
! [A: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A
@ A )
=> ( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ A
@ A )
= A )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ideI
thf(fact_1149__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OarrI,axiom,
! [T: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ T )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.arrI
thf(fact_1150__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OconI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.conI
thf(fact_1151_un__App1_Opreserves__reflects__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil ) )
= ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ).
% un_App1.preserves_reflects_arr
thf(fact_1152_un__App2_Opreserves__reflects__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil ) )
= ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ).
% un_App2.preserves_reflects_arr
thf(fact_1153__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oin__sourcesI,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ A )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ A )
=> ( member_lambda_lambda @ A
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.in_sourcesI
thf(fact_1154__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_resid
thf(fact_1155__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcoinitialI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
=> ( coinit1392077014771647542lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coinitialI
thf(fact_1156__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OcoterminalI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
=> ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coterminalI
thf(fact_1157_un__App1_Opreserves__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ( lambda_un_App1
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil ) ) ) )
& ( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( lambda_Nil
= ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App1 @ U )
@ lambda_Nil ) ) ) ) ) ) ).
% un_App1.preserves_resid
thf(fact_1158_un__App2_Opreserves__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U )
=> ( ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( ( lambda_un_App2
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
= ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil ) ) ) )
& ( ~ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) )
@ T
@ U ) )
=> ( lambda_Nil
= ( lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil )
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
@ ( lambda_un_App2 @ U )
@ lambda_Nil ) ) ) ) ) ) ).
% un_App2.preserves_resid
thf(fact_1159__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_OseqI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
=> ( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U )
=> ( ( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
= ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ U ) )
=> ( seq_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.seqI
thf(fact_1160_sub__rts_Oresid_Ocong,axiom,
sub_re469828149216679723lambda = sub_re469828149216679723lambda ).
% sub_rts.resid.cong
thf(fact_1161_un__App2_Olifts__to__paths,axiom,
( simula3069223088535412739lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T8: list_lambda_lambda] :
( if_lis7195930416592267574lambda
@ ( paths_366302053746739058lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T8 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ T8 )
@ nil_lambda_lambda ) ) ).
% un_App2.lifts_to_paths
thf(fact_1162_un__App1_Olifts__to__paths,axiom,
( simula3069223088535412739lambda
@ ( paths_8607357831768207192lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) ) )
@ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T8: list_lambda_lambda] :
( if_lis7195930416592267574lambda
@ ( paths_366302053746739058lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T8 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ T8 )
@ nil_lambda_lambda ) ) ).
% un_App1.lifts_to_paths
thf(fact_1163_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1164_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,A: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1165_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,A: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1166_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B3 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1167_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1168_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1169_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1170_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1171_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1172_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1173_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1174_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B3 @ ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1175_simulation_Opreserves__resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1176_simulation_Opreserves__resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1177_simulation_Opreserves__resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1178_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1179_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1180_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1181_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1182_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ B3 @ ( F2 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1183_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ B3 @ ( F2 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1184_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( arr_lambda_lambda @ B3 @ ( F2 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1185_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( arr_lambda_lambda @ B3 @ ( F2 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1186_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1187_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1188_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1189_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B3 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1190_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia6873046715171720854lambda @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1191_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia6873046715171720854lambda @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1192_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia1461432607296028304lambda @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1193_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia1461432607296028304lambda @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1194_simulation_Opreserves__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( paths_2015741907369688696lambda @ A2 @ T4 )
=> ( paths_2015741907369688696lambda @ B3 @ ( map_li385728805257398596lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1195_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_2015741907369688696lambda @ B3 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1196_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_366302053746739058lambda @ B3 @ ( map_la336867333220232516lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1197_simulation_Olifts__to__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( simula1230958359699143427lambda @ ( paths_3312059784463760222lambda @ A2 ) @ ( paths_3312059784463760222lambda @ B3 )
@ ^ [T8: list_l5168835686021156086lambda] : ( if_lis4115203008988617916lambda @ ( paths_2015741907369688696lambda @ A2 @ T8 ) @ ( map_li385728805257398596lambda @ F2 @ T8 ) @ nil_li8379291187143545974lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1198_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( simula3219528783544830217lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_3312059784463760222lambda @ B3 )
@ ^ [T8: list_lambda_lambda] : ( if_lis4115203008988617916lambda @ ( paths_366302053746739058lambda @ A2 @ T8 ) @ ( map_la6434941737452557130lambda @ F2 @ T8 ) @ nil_li8379291187143545974lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1199_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( simula3069223088535412739lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_8607357831768207192lambda @ B3 )
@ ^ [T8: list_lambda_lambda] : ( if_lis7195930416592267574lambda @ ( paths_366302053746739058lambda @ A2 @ T8 ) @ ( map_la336867333220232516lambda @ F2 @ T8 ) @ nil_lambda_lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1200_simulation_Omap__Resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_li385728805257398596lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ U2 ) )
= ( paths_3312059784463760222lambda @ B3 @ ( map_li385728805257398596lambda @ F2 @ T4 ) @ ( map_li385728805257398596lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1201_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la6434941737452557130lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_3312059784463760222lambda @ B3 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) @ ( map_la6434941737452557130lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1202_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ B3 @ ( map_la336867333220232516lambda @ F2 @ T4 ) @ ( map_la336867333220232516lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1203_simulation_Omap__Resid__single,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
=> ( ( map_li385728805257398596lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) ) )
= ( paths_3312059784463760222lambda @ B3 @ ( map_li385728805257398596lambda @ F2 @ T4 ) @ ( cons_l4558873032622161958lambda @ ( F2 @ U ) @ nil_li8379291187143545974lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1204_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la6434941737452557130lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_3312059784463760222lambda @ B3 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) @ ( cons_l4558873032622161958lambda @ ( F2 @ U ) @ nil_li8379291187143545974lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1205_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la336867333220232516lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_8607357831768207192lambda @ B3 @ ( map_la336867333220232516lambda @ F2 @ T4 ) @ ( cons_lambda_lambda @ ( F2 @ U ) @ nil_lambda_lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1206_un__App2_Opreserves__sources,axiom,
! [T: lambda_lambda] :
( ord_le1926335974945931754lambda
@ ( image_2356535832909760517lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
@ ( source5627170679743844886lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil ) ) ) ).
% un_App2.preserves_sources
thf(fact_1207_un__App1_Opreserves__sources,axiom,
! [T: lambda_lambda] :
( ord_le1926335974945931754lambda
@ ( image_2356535832909760517lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
@ ( source5627170679743844886lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil ) ) ) ).
% un_App1.preserves_sources
thf(fact_1208__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oinclusion__is__simulation,axiom,
( simula5235733974575307523lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ lambda_resid
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ T6
@ ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.inclusion_is_simulation
thf(fact_1209_incl__is__simulation,axiom,
( simula1175017789036123529lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T6: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T6 ) @ ( cons_lambda_lambda @ T6 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% incl_is_simulation
thf(fact_1210_un__App1_Osimulation__axioms,axiom,
( simula5235733974575307523lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ lambda_resid
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil ) ) ).
% un_App1.simulation_axioms
thf(fact_1211_un__App2_Osimulation__axioms,axiom,
( simula5235733974575307523lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ lambda_resid
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil ) ) ).
% un_App2.simulation_axioms
thf(fact_1212_image__empty,axiom,
! [F: list_lambda_lambda > list_lambda_lambda] :
( ( image_5886195720093285125lambda @ F @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_1213_image__empty,axiom,
! [F: list_lambda_lambda > lambda_lambda] :
( ( image_2458965899662700159lambda @ F @ bot_bo1397037886704484924lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_1214_image__empty,axiom,
! [F: lambda_lambda > list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F @ bot_bo1271416176687173302lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_1215_image__empty,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( image_2356535832909760517lambda @ F @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_1216_empty__is__image,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_5886195720093285125lambda @ F @ A2 ) )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_1217_empty__is__image,axiom,
! [F: lambda_lambda > list_lambda_lambda,A2: set_lambda_lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_3692823078739859851lambda @ F @ A2 ) )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_1218_empty__is__image,axiom,
! [F: list_lambda_lambda > lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2458965899662700159lambda @ F @ A2 ) )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_1219_empty__is__image,axiom,
! [F: lambda_lambda > lambda_lambda,A2: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2356535832909760517lambda @ F @ A2 ) )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_1220_image__is__empty,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( ( image_5886195720093285125lambda @ F @ A2 )
= bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_1221_image__is__empty,axiom,
! [F: lambda_lambda > list_lambda_lambda,A2: set_lambda_lambda] :
( ( ( image_3692823078739859851lambda @ F @ A2 )
= bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_1222_image__is__empty,axiom,
! [F: list_lambda_lambda > lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( ( image_2458965899662700159lambda @ F @ A2 )
= bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_1223_image__is__empty,axiom,
! [F: lambda_lambda > lambda_lambda,A2: set_lambda_lambda] :
( ( ( image_2356535832909760517lambda @ F @ A2 )
= bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_1224_un__App1_Opreserves__targets,axiom,
! [T: lambda_lambda] :
( ord_le1926335974945931754lambda
@ ( image_2356535832909760517lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App1 @ T6 )
@ lambda_Nil )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
@ ( target5490931411331770124lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App1 @ T )
@ lambda_Nil ) ) ) ).
% un_App1.preserves_targets
thf(fact_1225_un__App2_Opreserves__targets,axiom,
! [T: lambda_lambda] :
( ord_le1926335974945931754lambda
@ ( image_2356535832909760517lambda
@ ^ [T6: lambda_lambda] :
( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) ) )
@ T6 )
@ ( lambda_un_App2 @ T6 )
@ lambda_Nil )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T ) )
@ ( target5490931411331770124lambda @ lambda_resid
@ ( if_lambda_lambda
@ ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T6: lambda_lambda] :
( ( lambda_Arr @ T6 )
& ( lambda_is_App @ T6 ) ) )
@ T )
@ ( lambda_un_App2 @ T )
@ lambda_Nil ) ) ) ).
% un_App2.preserves_targets
thf(fact_1226_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_1227_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_1228_simulation_Opreserves__sources,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F2 @ ( source6119337877621565212lambda @ A2 @ T ) ) @ ( source6119337877621565212lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1229_simulation_Opreserves__sources,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F2 @ ( source6119337877621565212lambda @ A2 @ T ) ) @ ( source5627170679743844886lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1230_simulation_Opreserves__sources,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ ( source5627170679743844886lambda @ A2 @ T ) ) @ ( source6119337877621565212lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1231_simulation_Opreserves__sources,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F2 @ ( source5627170679743844886lambda @ A2 @ T ) ) @ ( source5627170679743844886lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1232_simulation_Opreserves__targets,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B3 @ F2 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F2 @ ( target5490931411331770124lambda @ A2 @ T ) ) @ ( target5490931411331770124lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1233_simulation_Opreserves__targets,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B3 @ F2 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ ( target5490931411331770124lambda @ A2 @ T ) ) @ ( target4026644918049360146lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1234_simulation_Opreserves__targets,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B3 @ F2 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F2 @ ( target4026644918049360146lambda @ A2 @ T ) ) @ ( target5490931411331770124lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1235_simulation_Opreserves__targets,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B3: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B3 @ F2 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F2 @ ( target4026644918049360146lambda @ A2 @ T ) ) @ ( target4026644918049360146lambda @ B3 @ ( F2 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1236_image__constant,axiom,
! [X2: list_lambda_lambda,A2: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ A2 )
=> ( ( image_5886195720093285125lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_1237_image__constant,axiom,
! [X2: lambda_lambda,A2: set_lambda_lambda,C: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A2 )
=> ( ( image_3692823078739859851lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_1238_image__constant,axiom,
! [X2: list_lambda_lambda,A2: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ A2 )
=> ( ( image_2458965899662700159lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_1239_image__constant,axiom,
! [X2: lambda_lambda,A2: set_lambda_lambda,C: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A2 )
=> ( ( image_2356535832909760517lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_1240_image__constant__conv,axiom,
! [A2: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ( A2 = bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= bot_bo1397037886704484924lambda ) )
& ( ( A2 != bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1241_image__constant__conv,axiom,
! [A2: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( ( A2 = bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= bot_bo1271416176687173302lambda ) )
& ( ( A2 != bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X3: list_lambda_lambda] : C
@ A2 )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1242_image__constant__conv,axiom,
! [A2: set_lambda_lambda,C: list_lambda_lambda] :
( ( ( A2 = bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= bot_bo1397037886704484924lambda ) )
& ( ( A2 != bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1243_image__constant__conv,axiom,
! [A2: set_lambda_lambda,C: lambda_lambda] :
( ( ( A2 = bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= bot_bo1271416176687173302lambda ) )
& ( ( A2 != bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X3: lambda_lambda] : C
@ A2 )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1244_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_1245_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_1246_image__Int__subset,axiom,
! [F: list_lambda_lambda > lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) @ ( inf_in1766959559283430712lambda @ ( image_2458965899662700159lambda @ F @ A2 ) @ ( image_2458965899662700159lambda @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1247_image__Int__subset,axiom,
! [F: lambda_lambda > lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) @ ( inf_in1766959559283430712lambda @ ( image_2356535832909760517lambda @ F @ A2 ) @ ( image_2356535832909760517lambda @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1248_image__Int__subset,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) @ ( inf_in3354698378728726462lambda @ ( image_5886195720093285125lambda @ F @ A2 ) @ ( image_5886195720093285125lambda @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1249_image__Int__subset,axiom,
! [F: lambda_lambda > list_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) @ ( inf_in3354698378728726462lambda @ ( image_3692823078739859851lambda @ F @ A2 ) @ ( image_3692823078739859851lambda @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1250_image__mono,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ A2 ) @ ( image_2356535832909760517lambda @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1251_image__mono,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ A2 ) @ ( image_3692823078739859851lambda @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1252_image__mono,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F: list_lambda_lambda > lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ A2 ) @ ( image_2458965899662700159lambda @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1253_image__mono,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ A2 ) @ ( image_5886195720093285125lambda @ F @ B3 ) ) ) ).
% image_mono
thf(fact_1254_image__subsetI,axiom,
! [A2: set_li7878742357367608272lambda,F: list_lambda_lambda > lambda_lambda,B3: set_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A2 )
=> ( member_lambda_lambda @ ( F @ X ) @ B3 ) )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1255_image__subsetI,axiom,
! [A2: set_lambda_lambda,F: lambda_lambda > lambda_lambda,B3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A2 )
=> ( member_lambda_lambda @ ( F @ X ) @ B3 ) )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1256_image__subsetI,axiom,
! [A2: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda,B3: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A2 )
=> ( member2667661554319814297lambda @ ( F @ X ) @ B3 ) )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1257_image__subsetI,axiom,
! [A2: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda,B3: set_li7878742357367608272lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A2 )
=> ( member2667661554319814297lambda @ ( F @ X ) @ B3 ) )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_1258_subset__imageE,axiom,
! [B3: set_li7878742357367608272lambda,F: lambda_lambda > list_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ ( image_3692823078739859851lambda @ F @ A2 ) )
=> ~ ! [C5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C5 @ A2 )
=> ( B3
!= ( image_3692823078739859851lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1259_subset__imageE,axiom,
! [B3: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ ( image_5886195720093285125lambda @ F @ A2 ) )
=> ~ ! [C5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C5 @ A2 )
=> ( B3
!= ( image_5886195720093285125lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1260_lambda__calculus_OApp__is__simulation1,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T6: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T6 ) @ ( lambda_App @ T6 @ A ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation1
thf(fact_1261_lambda__calculus_OApp__is__simulation2,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T6: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T6 ) @ ( lambda_App @ A @ T6 ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation2
thf(fact_1262_map__App__map__un__App1,axiom,
! [U2: list_lambda_lambda,B: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ B )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App2 @ ( set_lambda_lambda2 @ U2 ) ) @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ B )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
= U2 ) ) ) ) ) ).
% map_App_map_un_App1
thf(fact_1263_map__App__map__un__App2,axiom,
! [U2: list_lambda_lambda,A: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ A )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App1 @ ( set_lambda_lambda2 @ U2 ) ) @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) )
= U2 ) ) ) ) ) ).
% map_App_map_un_App2
thf(fact_1264_Arr__map__App2,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ T4 ) ) ) ) ).
% Arr_map_App2
thf(fact_1265_Arr__map__App1,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ B )
@ T4 ) ) ) ) ).
% Arr_map_App1
thf(fact_1266_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_1267_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_1268_Resid__single__ide_I1_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
@ ( image_3692823078739859851lambda
@ ^ [B5: lambda_lambda] : ( cons_lambda_lambda @ B5 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Resid_single_ide(1)
thf(fact_1269_cong__map__App2,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ T4 ) ) ) ) ) ) ).
% cong_map_App2
thf(fact_1270_map__App2__Resid,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ U2 ) ) ) ) ) ).
% map_App2_Resid
thf(fact_1271_img__Resid__Srcs,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A6: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A6 @ nil_lambda_lambda ) @ T4 )
@ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
@ ( image_3692823078739859851lambda
@ ^ [B5: lambda_lambda] : ( cons_lambda_lambda @ B5 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ).
% img_Resid_Srcs
thf(fact_1272_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_1273_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_1274_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_1275_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_1276_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_1277_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
% Helper facts (7)
thf(help_If_2_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X2: lambda_lambda,Y2: lambda_lambda] :
( ( if_lambda_lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X2: lambda_lambda,Y2: lambda_lambda] :
( ( if_lambda_lambda @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X2: list_l5168835686021156086lambda,Y2: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X2: list_l5168835686021156086lambda,Y2: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (8)
thf(conj_0,hypothesis,
paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ x41 @ x42 ) @ ua ) ).
thf(conj_1,hypothesis,
$true ).
thf(conj_2,hypothesis,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( T2 != lambda_Nil ) ) ).
thf(conj_3,hypothesis,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ T2 )
= ( lambda_Trg @ T2 ) ) ) ).
thf(conj_4,hypothesis,
( u
= ( lambda_App @ x41 @ x42 ) ) ).
thf(conj_5,hypothesis,
lambda_Arr @ x41 ).
thf(conj_6,hypothesis,
lambda_Arr @ x42 ).
thf(conj_7,conjecture,
ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ x41 ) @ ( lambda_Trg @ x42 ) ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ x41 ) @ ( lambda_Trg @ x42 ) ) @ nil_lambda_lambda ) ) ).
%------------------------------------------------------------------------------