TPTP Problem File: SLH0538^1.p
View Solutions
- 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 : FOL_Seq_Calc3/0007_Fair_Stream/prob_00024_000726__11888042_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1666 ( 547 unt; 367 typ; 0 def)
% Number of atoms : 3570 ( 999 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11675 ( 294 ~; 53 |; 217 &;9368 @)
% ( 0 <=>;1743 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 56 ( 55 usr)
% Number of type conns : 1380 (1380 >; 0 *; 0 +; 0 <<)
% Number of symbols : 315 ( 312 usr; 10 con; 0-3 aty)
% Number of variables : 3911 ( 356 ^;3503 !; 52 ?;3911 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:27:18.497
%------------------------------------------------------------------------------
% Could-be-implicit typings (55)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_Pr5382606609415531783list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
stream9138423102435771216od_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
stream5653912848357171891list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
set_st6673062052851004045list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc7709606177366032167list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_Itf__a_J_M_Eo_J_Mt__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
produc5320378693649342662list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
stream1753209531944178518od_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
stream7492071314720355001list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
stream5099646356184241604_a_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
stream271976807216406061list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
stream7613764438895080353st_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_J,type,
set_st4431294094181464339list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J_J,type,
set_st8160949801704661767st_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
list_P321204300973800749list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J_J,type,
set_st1841771232946885872list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
stream8205385055505213747list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
stream5874003577662885159st_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
stream2255243159586646806list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_st990465082155492493od_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_st6729326864931668976list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
list_P4541805568828049459list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
list_P2210424090985720871st_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_Eo_J_Mt__Stream__Ostream_Itf__a_J_J,type,
produc2815237062074537146ream_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr2070066670564046349list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_Pr8962057229576493569st_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
stream_stream_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J_J,type,
set_stream_stream_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
stream6512538624140940077od_a_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
stream_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
set_stream_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_It__Set__Oset_Itf__a_J_J_J,type,
set_stream_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
produc8685980395799941037list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
produc2579390645249093025st_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J,type,
stream_stream_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stream_list_a: $tType ).
thf(ty_n_t__Stream__Ostream_It__Set__Oset_Itf__a_J_J,type,
stream_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
set_stream_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Stream__Ostream_Itf__a_J,type,
stream_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (312)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_Itf__a_J,type,
bNF_Gr7042794125918077091list_a: set_list_list_a > list_a > set_list_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_Itf__a_J,type,
bNF_Gr4634511371912843295list_a: set_list_list_a > list_list_a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Fair__Stream_Ofair_001t__List__Olist_Itf__a_J,type,
fair_fair_list_a: stream_list_a > $o ).
thf(sy_c_Fair__Stream_Ofair_001tf__a,type,
fair_fair_a: stream_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
undefi4168218097957520114list_a: stream_list_a ).
thf(sy_c_HOL_Oundefined_001t__Stream__Ostream_Itf__a_J,type,
undefined_stream_a: stream_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
if_str7505741754068378070list_a: $o > stream_list_list_a > stream_list_list_a > stream_list_list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
if_stream_list_a: $o > stream_list_a > stream_list_a > stream_list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
if_str6195518498109691379list_a: $o > stream271976807216406061list_a > stream271976807216406061list_a > stream271976807216406061list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
if_str2612623181902290925st_a_a: $o > stream5874003577662885159st_a_a > stream5874003577662885159st_a_a > stream5874003577662885159st_a_a ).
thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
if_str4944004659744619513list_a: $o > stream8205385055505213747list_a > stream8205385055505213747list_a > stream8205385055505213747list_a ).
thf(sy_c_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
if_str8987598796600344819od_a_a: $o > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a ).
thf(sy_c_If_001t__Stream__Ostream_Itf__a_J,type,
if_stream_a: $o > stream_a > stream_a > stream_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_Mt__Nat__Onat_J,type,
sup_sup_o_nat: ( $o > nat ) > ( $o > nat ) > $o > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
sup_sup_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
sup_su1566401739067690942list_a: set_list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
sup_su3048258781599657691od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
sup_su5853061255542491070list_a: set_stream_list_a > set_stream_list_a > set_stream_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
sup_sup_set_stream_a: set_stream_a > set_stream_a > set_stream_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_Itf__a_J,type,
coset_list_a: list_list_a > set_list_a ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Olast_001t__List__Olist_Itf__a_J,type,
last_list_a: list_list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olexord_001t__List__Olist_Itf__a_J,type,
lexord_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
hd_list_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
set_Pr2906193453920572092list_a: list_P321204300973800749list_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
set_Pr6156699705114756022st_a_a: list_P2210424090985720871st_a_a > set_Pr8962057229576493569st_a_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
set_Pr3039917418810828226list_a: list_P4541805568828049459list_a > set_Pr2070066670564046349list_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
set_Product_prod_a_a2: list_P1396940483166286381od_a_a > set_Product_prod_a_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__List__Olist_Itf__a_J,type,
list_update_list_a: list_list_a > nat > list_a > list_list_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
zip_list_a_list_a: list_list_a > list_list_a > list_P321204300973800749list_a ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001tf__a,type,
zip_list_a_a: list_list_a > list_a > list_P2210424090985720871st_a_a ).
thf(sy_c_List_Ozip_001tf__a_001t__List__Olist_Itf__a_J,type,
zip_a_list_a: list_a > list_list_a > list_P4541805568828049459list_a ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Set__Oset_Itf__a_J_J_J,type,
ord_le5604070792465694036_set_a: ( $o > $o > set_a ) > ( $o > $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Num__Onum_J,type,
ord_less_eq_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
ord_le354538097136858257list_a: ( $o > set_list_a ) > ( $o > set_list_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Stream__Ostream_Itf__a_J_J_J,type,
ord_le3754339485079453329ream_a: ( $o > set_stream_a ) > ( $o > set_stream_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le5542992221119063950st_a_o: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_Itf__a_M_Eo_J_J,type,
ord_le1869217262537267988_a_a_o: ( list_a > a > $o ) > ( list_a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_M_062_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_M_Eo_J_J,type,
ord_le4962707062786359566st_a_o: ( stream_list_a > stream_list_a > $o ) > ( stream_list_a > stream_list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_M_062_It__Stream__Ostream_Itf__a_J_M_Eo_J_J,type,
ord_le9064767629836747540am_a_o: ( stream_list_a > stream_a > $o ) > ( stream_list_a > stream_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Stream__Ostream_Itf__a_J_M_062_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_M_Eo_J_J,type,
ord_le4668208666631526024st_a_o: ( stream_a > stream_list_a > $o ) > ( stream_a > stream_list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Stream__Ostream_Itf__a_J_M_062_It__Stream__Ostream_Itf__a_J_M_Eo_J_J,type,
ord_le2315692933029977486am_a_o: ( stream_a > stream_a > $o ) > ( stream_a > stream_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le2844220955438509960st_a_o: ( a > list_a > $o ) > ( a > list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
ord_le2348936426438760842list_a: set_stream_list_a > set_stream_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J_J,type,
ord_le2426980790783101578ream_a: set_stream_stream_a > set_stream_stream_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
ord_le1019412967036525316ream_a: set_stream_a > set_stream_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_Itf__a_J,type,
ord_max_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_Itf__a_J,type,
ord_min_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Nat__Onat_J,type,
order_Greatest_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
order_6114237596796908690_set_a: ( ( $o > set_a ) > $o ) > $o > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
order_733672244956367037list_a: ( set_list_a > $o ) > set_list_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Stream__Ostream_Itf__a_J_J,type,
order_8427663985114061757ream_a: ( set_stream_a > $o ) > set_stream_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_Itf__a_J_M_Eo_J_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
produc8531839378509307896list_a: ( list_a > $o ) > stream_list_a > produc5320378693649342662list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_Eo_J_001t__Stream__Ostream_Itf__a_J,type,
produc8437782301311866860ream_a: ( a > $o ) > stream_a > produc2815237062074537146ream_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001tf__a,type,
produc4781227316648555537st_a_a: list_a > a > produc2579390645249093025st_a_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_Itf__a_J,type,
produc6670463072477821725list_a: a > list_a > produc8685980395799941037list_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
collec2345243735937135093list_a: ( stream_list_a > $o ) > set_stream_list_a ).
thf(sy_c_Set_OCollect_001t__Stream__Ostream_Itf__a_J,type,
collect_stream_a: ( stream_a > $o ) > set_stream_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_list_a_set_a: ( list_a > set_a ) > set_list_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Stream_Ocycle_001t__List__Olist_Itf__a_J,type,
cycle_list_a: list_list_a > stream_list_a ).
thf(sy_c_Stream_Ocycle_001tf__a,type,
cycle_a: list_a > stream_a ).
thf(sy_c_Stream_Oflat_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
flat_list_list_a: stream2255243159586646806list_a > stream_list_list_a ).
thf(sy_c_Stream_Oflat_001t__List__Olist_Itf__a_J,type,
flat_list_a: stream_list_list_a > stream_list_a ).
thf(sy_c_Stream_Oflat_001tf__a,type,
flat_a: stream_list_a > stream_a ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_Itf__a_J,type,
sdrop_list_a: nat > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osdrop_001tf__a,type,
sdrop_a: nat > stream_a > stream_a ).
thf(sy_c_Stream_Osdrop__while_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sdrop_1072776591231973358list_a: ( list_list_a > $o ) > stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Osdrop__while_001t__List__Olist_Itf__a_J,type,
sdrop_while_list_a: ( list_a > $o ) > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osdrop__while_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
sdrop_4696227070857850763list_a: ( produc9164743771328383783list_a > $o ) > stream271976807216406061list_a > stream271976807216406061list_a ).
thf(sy_c_Stream_Osdrop__while_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
sdrop_2016533438629139461st_a_a: ( produc2579390645249093025st_a_a > $o ) > stream5874003577662885159st_a_a > stream5874003577662885159st_a_a ).
thf(sy_c_Stream_Osdrop__while_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
sdrop_8123123189179987473list_a: ( produc8685980395799941037list_a > $o ) > stream8205385055505213747list_a > stream8205385055505213747list_a ).
thf(sy_c_Stream_Osdrop__while_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
sdrop_5956399235322795403od_a_a: ( product_prod_a_a > $o ) > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a ).
thf(sy_c_Stream_Osdrop__while_001tf__a,type,
sdrop_while_a: ( a > $o ) > stream_a > stream_a ).
thf(sy_c_Stream_Oshift_001t__List__Olist_Itf__a_J,type,
shift_list_a: list_list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Oshift_001tf__a,type,
shift_a: list_a > stream_a > stream_a ).
thf(sy_c_Stream_Osinterleave_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sinter664328039231274855list_a: stream_list_list_a > stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Osinterleave_001t__List__Olist_Itf__a_J,type,
sinterleave_list_a: stream_list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osinterleave_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
sinter5791379037304460804list_a: stream271976807216406061list_a > stream271976807216406061list_a > stream271976807216406061list_a ).
thf(sy_c_Stream_Osinterleave_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
sinter3949561980244396798st_a_a: stream5874003577662885159st_a_a > stream5874003577662885159st_a_a > stream5874003577662885159st_a_a ).
thf(sy_c_Stream_Osinterleave_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
sinter832779693940469002list_a: stream8205385055505213747list_a > stream8205385055505213747list_a > stream8205385055505213747list_a ).
thf(sy_c_Stream_Osinterleave_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
sinter5547950683322096900od_a_a: stream6512538624140940077od_a_a > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a ).
thf(sy_c_Stream_Osinterleave_001tf__a,type,
sinterleave_a: stream_a > stream_a > stream_a ).
thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
smap2_6771825536533895878list_a: ( list_a > list_a > list_a ) > stream_list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001tf__a,type,
smap2_3658108627635373888st_a_a: ( list_a > list_a > a ) > stream_list_a > stream_list_a > stream_a ).
thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001tf__a_001t__List__Olist_Itf__a_J,type,
smap2_5547344383464640076list_a: ( list_a > a > list_a ) > stream_list_a > stream_a > stream_list_a ).
thf(sy_c_Stream_Osmap2_001t__List__Olist_Itf__a_J_001tf__a_001tf__a,type,
smap2_list_a_a_a: ( list_a > a > a ) > stream_list_a > stream_a > stream_a ).
thf(sy_c_Stream_Osmap2_001tf__a_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
smap2_5647228111304775744list_a: ( a > list_a > list_a ) > stream_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osmap2_001tf__a_001t__List__Olist_Itf__a_J_001tf__a,type,
smap2_a_list_a_a: ( a > list_a > a ) > stream_a > stream_list_a > stream_a ).
thf(sy_c_Stream_Osmap2_001tf__a_001tf__a_001t__List__Olist_Itf__a_J,type,
smap2_a_a_list_a: ( a > a > list_a ) > stream_a > stream_a > stream_list_a ).
thf(sy_c_Stream_Osmap2_001tf__a_001tf__a_001tf__a,type,
smap2_a_a_a: ( a > a > a ) > stream_a > stream_a > stream_a ).
thf(sy_c_Stream_Osmember_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
smember_list_list_a: list_list_a > stream_list_list_a > $o ).
thf(sy_c_Stream_Osmember_001t__List__Olist_Itf__a_J,type,
smember_list_a: list_a > stream_list_a > $o ).
thf(sy_c_Stream_Osmember_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
smembe5596956825316798443list_a: produc8685980395799941037list_a > stream8205385055505213747list_a > $o ).
thf(sy_c_Stream_Osmember_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
smembe8479462165187265637od_a_a: product_prod_a_a > stream6512538624140940077od_a_a > $o ).
thf(sy_c_Stream_Osmember_001t__Set__Oset_Itf__a_J,type,
smember_set_a: set_a > stream_set_a > $o ).
thf(sy_c_Stream_Osmember_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
smembe4279752203236903368list_a: stream_list_a > stream_stream_list_a > $o ).
thf(sy_c_Stream_Osmember_001t__Stream__Ostream_Itf__a_J,type,
smember_stream_a: stream_a > stream_stream_a > $o ).
thf(sy_c_Stream_Osmember_001tf__a,type,
smember_a: a > stream_a > $o ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_Itf__a_J,type,
smerge_list_a: stream_stream_list_a > stream_list_a ).
thf(sy_c_Stream_Osmerge_001tf__a,type,
smerge_a: stream_stream_a > stream_a ).
thf(sy_c_Stream_Osnth_001t__List__Olist_Itf__a_J,type,
snth_list_a: stream_list_a > nat > list_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
snth_stream_list_a: stream_stream_list_a > nat > stream_list_a ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_Itf__a_J,type,
snth_stream_a: stream_stream_a > nat > stream_a ).
thf(sy_c_Stream_Osnth_001tf__a,type,
snth_a: stream_a > nat > a ).
thf(sy_c_Stream_Ostake_001t__List__Olist_Itf__a_J,type,
stake_list_a: nat > stream_list_a > list_list_a ).
thf(sy_c_Stream_Ostake_001tf__a,type,
stake_a: nat > stream_a > list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sCons_list_list_a: list_list_a > stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__List__Olist_Itf__a_J,type,
sCons_list_a: list_a > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
sCons_4508012570444740670list_a: produc9164743771328383783list_a > stream271976807216406061list_a > stream271976807216406061list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
sCons_1605959091327504952st_a_a: produc2579390645249093025st_a_a > stream5874003577662885159st_a_a > stream5874003577662885159st_a_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
sCons_7712548841878352964list_a: produc8685980395799941037list_a > stream8205385055505213747list_a > stream8205385055505213747list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
sCons_4259542806812723518od_a_a: product_prod_a_a > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Set__Oset_Itf__a_J,type,
sCons_set_a: set_a > stream_set_a > stream_set_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
sCons_stream_list_a: stream_list_a > stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Ostream_OSCons_001t__Stream__Ostream_Itf__a_J,type,
sCons_stream_a: stream_a > stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Ostream_OSCons_001tf__a,type,
sCons_a: a > stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
shd_list_list_list_a: stream2255243159586646806list_a > list_list_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
shd_list_list_a: stream_list_list_a > list_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_Itf__a_J,type,
shd_list_a: stream_list_a > list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
shd_Pr3168949583242555347list_a: stream271976807216406061list_a > produc9164743771328383783list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
shd_Pr7676374007274264653st_a_a: stream5874003577662885159st_a_a > produc2579390645249093025st_a_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
shd_Pr4559591720970336857list_a: stream8205385055505213747list_a > produc8685980395799941037list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
shd_Product_prod_a_a: stream6512538624140940077od_a_a > product_prod_a_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Set__Oset_Itf__a_J,type,
shd_set_a: stream_set_a > set_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
shd_stream_list_a: stream_stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Oshd_001t__Stream__Ostream_Itf__a_J,type,
shd_stream_a: stream_stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Oshd_001tf__a,type,
shd_a: stream_a > a ).
thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
smap_list_a_list_a: ( list_a > list_a ) > stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Osmap_001t__List__Olist_Itf__a_J_001tf__a,type,
smap_list_a_a: ( list_a > a ) > stream_list_a > stream_a ).
thf(sy_c_Stream_Ostream_Osmap_001tf__a_001t__List__Olist_Itf__a_J,type,
smap_a_list_a: ( a > list_a ) > stream_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Osmap_001tf__a_001tf__a,type,
smap_a_a: ( a > a ) > stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
sset_list_list_a: stream_list_list_a > set_list_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_Itf__a_J,type,
sset_list_a: stream_list_a > set_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
sset_P844431690648075579list_a: stream271976807216406061list_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
sset_P8315020979779762613st_a_a: stream5874003577662885159st_a_a > set_Pr8962057229576493569st_a_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
sset_P5198238693475834817list_a: stream8205385055505213747list_a > set_Pr2070066670564046349list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
sset_P5743994922208793403od_a_a: stream6512538624140940077od_a_a > set_Product_prod_a_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Set__Oset_Itf__a_J,type,
sset_set_a: stream_set_a > set_set_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
sset_stream_list_a: stream_stream_list_a > set_stream_list_a ).
thf(sy_c_Stream_Ostream_Osset_001t__Stream__Ostream_Itf__a_J,type,
sset_stream_a: stream_stream_a > set_stream_a ).
thf(sy_c_Stream_Ostream_Osset_001tf__a,type,
sset_a: stream_a > set_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
stl_list_list_a: stream_list_list_a > stream_list_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_Itf__a_J,type,
stl_list_a: stream_list_a > stream_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
stl_Pr427255397280820043st_a_a: stream7613764438895080353st_a_a > stream7613764438895080353st_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
stl_Pr842900660646171869list_a: stream5653912848357171891list_a > stream5653912848357171891list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
stl_Pr6107055601389525975list_a: stream271976807216406061list_a > stream271976807216406061list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
stl_Pr2324757703178138618od_a_a: stream9138423102435771216od_a_a > stream9138423102435771216od_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
stl_Pr7785550816790648401st_a_a: stream5874003577662885159st_a_a > stream5874003577662885159st_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
stl_Pr745203880014032622_a_a_a: stream5099646356184241604_a_a_a > stream5099646356184241604_a_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
stl_Pr8727924410383857251list_a: stream7492071314720355001list_a > stream7492071314720355001list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
stl_Pr4668768530486720605list_a: stream8205385055505213747list_a > stream8205385055505213747list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
stl_Pr986409416061048192od_a_a: stream1753209531944178518od_a_a > stream1753209531944178518od_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stl_Product_prod_a_a: stream6512538624140940077od_a_a > stream6512538624140940077od_a_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Set__Oset_Itf__a_J,type,
stl_set_a: stream_set_a > stream_set_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stl_stream_list_a: stream_stream_list_a > stream_stream_list_a ).
thf(sy_c_Stream_Ostream_Ostl_001t__Stream__Ostream_Itf__a_J,type,
stl_stream_a: stream_stream_a > stream_stream_a ).
thf(sy_c_Stream_Ostream_Ostl_001tf__a,type,
stl_a: stream_a > stream_a ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
stream5427457574974697293list_a: ( list_list_a > list_list_a > $o ) > stream_list_list_a > stream_list_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
stream3115889365687745985st_a_a: ( list_list_a > a > $o ) > stream_list_list_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
stream482612717156379347list_a: ( list_a > list_list_a > $o ) > stream_list_a > stream_list_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
stream4578335575074870093list_a: ( list_a > list_a > $o ) > stream_list_a > stream_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stream5366235361247201392od_a_a: ( list_a > product_prod_a_a > $o ) > stream_list_a > stream6512538624140940077od_a_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__List__Olist_Itf__a_J_001tf__a,type,
stream_all2_list_a_a: ( list_a > a > $o ) > stream_list_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
stream568159088382975326st_a_a: ( produc8685980395799941037list_a > a > $o ) > stream8205385055505213747list_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
stream2030605297770099434list_a: ( product_prod_a_a > list_a > $o ) > stream6512538624140940077od_a_a > stream_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stream4617964345477461389od_a_a: ( product_prod_a_a > product_prod_a_a > $o ) > stream6512538624140940077od_a_a > stream6512538624140940077od_a_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
stream3433837848420958564_a_a_a: ( product_prod_a_a > a > $o ) > stream6512538624140940077od_a_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
stream750952186392916301_set_a: ( set_a > set_a > $o ) > stream_set_a > stream_set_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Set__Oset_Itf__a_J_001tf__a,type,
stream_all2_set_a_a: ( set_a > a > $o ) > stream_set_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stream576332851429727821list_a: ( stream_list_a > stream_list_a > $o ) > stream_stream_list_a > stream_stream_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Stream__Ostream_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
stream2582044070471935399_set_a: ( stream_a > set_a > $o ) > stream_stream_a > stream_set_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Stream__Ostream_Itf__a_J_001t__Stream__Ostream_Itf__a_J,type,
stream8207726558890130509ream_a: ( stream_a > stream_a > $o ) > stream_stream_a > stream_stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001t__Stream__Ostream_Itf__a_J_001tf__a,type,
stream3396383680346202183am_a_a: ( stream_a > a > $o ) > stream_stream_a > stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
stream2512895734020674265list_a: ( a > list_list_a > $o ) > stream_a > stream_list_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001t__List__Olist_Itf__a_J,type,
stream_all2_a_list_a: ( a > list_a > $o ) > stream_a > stream_list_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stream7396518378111496310od_a_a: ( a > product_prod_a_a > $o ) > stream_a > stream6512538624140940077od_a_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001t__Set__Oset_Itf__a_J,type,
stream_all2_a_set_a: ( a > set_a > $o ) > stream_a > stream_set_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001t__Stream__Ostream_Itf__a_J,type,
stream3533738183241558099ream_a: ( a > stream_a > $o ) > stream_a > stream_stream_a > $o ).
thf(sy_c_Stream_Ostream_Ostream__all2_001tf__a_001tf__a,type,
stream_all2_a_a: ( a > a > $o ) > stream_a > stream_a > $o ).
thf(sy_c_Stream_Ostreams_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
streams_list_list_a: set_list_list_a > set_st6729326864931668976list_a ).
thf(sy_c_Stream_Ostreams_001t__List__Olist_Itf__a_J,type,
streams_list_a: set_list_a > set_stream_list_a ).
thf(sy_c_Stream_Ostreams_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
stream1450977367002125727list_a: set_Pr4048851178543822343list_a > set_st6673062052851004045list_a ).
thf(sy_c_Stream_Ostreams_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
stream6104187938593428505st_a_a: set_Pr8962057229576493569st_a_a > set_st8160949801704661767st_a_a ).
thf(sy_c_Stream_Ostreams_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
stream2987405652289500709list_a: set_Pr2070066670564046349list_a > set_st4431294094181464339list_a ).
thf(sy_c_Stream_Ostreams_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stream6392229770749734815od_a_a: set_Product_prod_a_a > set_st990465082155492493od_a_a ).
thf(sy_c_Stream_Ostreams_001t__Set__Oset_Itf__a_J,type,
streams_set_a: set_set_a > set_stream_set_a ).
thf(sy_c_Stream_Ostreams_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
stream6414841306446625538list_a: set_stream_list_a > set_st1841771232946885872list_a ).
thf(sy_c_Stream_Ostreams_001t__Stream__Ostream_Itf__a_J,type,
streams_stream_a: set_stream_a > set_stream_stream_a ).
thf(sy_c_Stream_Ostreams_001tf__a,type,
streams_a: set_a > set_stream_a ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
szip_list_list_a_a: stream_list_list_a > stream_a > stream7613764438895080353st_a_a ).
thf(sy_c_Stream_Oszip_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
szip_l8009862850044528992list_a: stream_list_a > stream_list_list_a > stream5653912848357171891list_a ).
thf(sy_c_Stream_Oszip_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
szip_list_a_list_a: stream_list_a > stream_list_a > stream271976807216406061list_a ).
thf(sy_c_Stream_Oszip_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
szip_l3670113457280575229od_a_a: stream_list_a > stream6512538624140940077od_a_a > stream9138423102435771216od_a_a ).
thf(sy_c_Stream_Oszip_001t__List__Olist_Itf__a_J_001tf__a,type,
szip_list_a_a: stream_list_a > stream_a > stream5874003577662885159st_a_a ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
szip_P7759488865980779121_a_a_a: stream6512538624140940077od_a_a > stream_a > stream5099646356184241604_a_a_a ).
thf(sy_c_Stream_Oszip_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
szip_a_list_list_a: stream_a > stream_list_list_a > stream7492071314720355001list_a ).
thf(sy_c_Stream_Oszip_001tf__a_001t__List__Olist_Itf__a_J,type,
szip_a_list_a: stream_a > stream_list_a > stream8205385055505213747list_a ).
thf(sy_c_Stream_Oszip_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
szip_a2498797358816541059od_a_a: stream_a > stream6512538624140940077od_a_a > stream1753209531944178518od_a_a ).
thf(sy_c_Stream_Oszip_001tf__a_001tf__a,type,
szip_a_a: stream_a > stream_a > stream6512538624140940077od_a_a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member8006451231845903178st_a_a: produc2579390645249093025st_a_a > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8003905643527612601list_a: stream_list_list_a > set_st6729326864931668976list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J,type,
member_stream_list_a: stream_list_a > set_stream_list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
member8659441389117227478list_a: stream271976807216406061list_a > set_st6673062052851004045list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
member5235206288952482128st_a_a: stream5874003577662885159st_a_a > set_st8160949801704661767st_a_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
member7566587766794810716list_a: stream8205385055505213747list_a > set_st4431294094181464339list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member262390649204803542od_a_a: stream6512538624140940077od_a_a > set_st990465082155492493od_a_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Set__Oset_Itf__a_J_J,type,
member_stream_set_a: stream_set_a > set_stream_set_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_Itf__a_J_J_J,type,
member3992053618758223289list_a: stream_stream_list_a > set_st1841771232946885872list_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_It__Stream__Ostream_Itf__a_J_J,type,
member8409225883583818419ream_a: stream_stream_a > set_stream_stream_a > $o ).
thf(sy_c_member_001t__Stream__Ostream_Itf__a_J,type,
member_stream_a: stream_a > set_stream_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_s,type,
s: stream_list_a ).
% Relevant facts (1279)
thf(fact_0_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ( member1426531477525435216od_a_a @ X @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_1_subsetI,axiom,
! [A: set_stream_list_a,B: set_stream_list_a] :
( ! [X: stream_list_a] :
( ( member_stream_list_a @ X @ A )
=> ( member_stream_list_a @ X @ B ) )
=> ( ord_le2348936426438760842list_a @ A @ B ) ) ).
% subsetI
thf(fact_2_subsetI,axiom,
! [A: set_list_list_a,B: set_list_list_a] :
( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A )
=> ( member_list_list_a @ X @ B ) )
=> ( ord_le8488217952732425610list_a @ A @ B ) ) ).
% subsetI
thf(fact_3_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_4_subsetI,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ! [X: stream_a] :
( ( member_stream_a @ X @ A )
=> ( member_stream_a @ X @ B ) )
=> ( ord_le1019412967036525316ream_a @ A @ B ) ) ).
% subsetI
thf(fact_5_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_6_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_7_subset__antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_8_subset__antisym,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( ( ord_le1019412967036525316ream_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_9_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_10_order__refl,axiom,
! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).
% order_refl
thf(fact_11_order__refl,axiom,
! [X2: $o > nat] : ( ord_less_eq_o_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_12_order__refl,axiom,
! [X2: $o > set_a] : ( ord_less_eq_o_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X2: set_stream_a] : ( ord_le1019412967036525316ream_a @ X2 @ X2 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_17_dual__order_Orefl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_18_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_20_dual__order_Orefl,axiom,
! [A2: set_stream_a] : ( ord_le1019412967036525316ream_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_24_stl__sset,axiom,
! [X2: stream_list_a,A2: stream_stream_list_a] :
( ( member_stream_list_a @ X2 @ ( sset_stream_list_a @ ( stl_stream_list_a @ A2 ) ) )
=> ( member_stream_list_a @ X2 @ ( sset_stream_list_a @ A2 ) ) ) ).
% stl_sset
thf(fact_25_stl__sset,axiom,
! [X2: stream_a,A2: stream_stream_a] :
( ( member_stream_a @ X2 @ ( sset_stream_a @ ( stl_stream_a @ A2 ) ) )
=> ( member_stream_a @ X2 @ ( sset_stream_a @ A2 ) ) ) ).
% stl_sset
thf(fact_26_stl__sset,axiom,
! [X2: set_a,A2: stream_set_a] :
( ( member_set_a @ X2 @ ( sset_set_a @ ( stl_set_a @ A2 ) ) )
=> ( member_set_a @ X2 @ ( sset_set_a @ A2 ) ) ) ).
% stl_sset
thf(fact_27_stl__sset,axiom,
! [X2: product_prod_a_a,A2: stream6512538624140940077od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( sset_P5743994922208793403od_a_a @ ( stl_Product_prod_a_a @ A2 ) ) )
=> ( member1426531477525435216od_a_a @ X2 @ ( sset_P5743994922208793403od_a_a @ A2 ) ) ) ).
% stl_sset
thf(fact_28_stl__sset,axiom,
! [X2: produc8685980395799941037list_a,A2: stream8205385055505213747list_a] :
( ( member4889668945541975382list_a @ X2 @ ( sset_P5198238693475834817list_a @ ( stl_Pr4668768530486720605list_a @ A2 ) ) )
=> ( member4889668945541975382list_a @ X2 @ ( sset_P5198238693475834817list_a @ A2 ) ) ) ).
% stl_sset
thf(fact_29_stl__sset,axiom,
! [X2: produc2579390645249093025st_a_a,A2: stream5874003577662885159st_a_a] :
( ( member8006451231845903178st_a_a @ X2 @ ( sset_P8315020979779762613st_a_a @ ( stl_Pr7785550816790648401st_a_a @ A2 ) ) )
=> ( member8006451231845903178st_a_a @ X2 @ ( sset_P8315020979779762613st_a_a @ A2 ) ) ) ).
% stl_sset
thf(fact_30_stl__sset,axiom,
! [X2: produc9164743771328383783list_a,A2: stream271976807216406061list_a] :
( ( member8191768239178080336list_a @ X2 @ ( sset_P844431690648075579list_a @ ( stl_Pr6107055601389525975list_a @ A2 ) ) )
=> ( member8191768239178080336list_a @ X2 @ ( sset_P844431690648075579list_a @ A2 ) ) ) ).
% stl_sset
thf(fact_31_stl__sset,axiom,
! [X2: list_list_a,A2: stream_list_list_a] :
( ( member_list_list_a @ X2 @ ( sset_list_list_a @ ( stl_list_list_a @ A2 ) ) )
=> ( member_list_list_a @ X2 @ ( sset_list_list_a @ A2 ) ) ) ).
% stl_sset
thf(fact_32_stl__sset,axiom,
! [X2: list_a,A2: stream_list_a] :
( ( member_list_a @ X2 @ ( sset_list_a @ ( stl_list_a @ A2 ) ) )
=> ( member_list_a @ X2 @ ( sset_list_a @ A2 ) ) ) ).
% stl_sset
thf(fact_33_stl__sset,axiom,
! [X2: a,A2: stream_a] :
( ( member_a @ X2 @ ( sset_a @ ( stl_a @ A2 ) ) )
=> ( member_a @ X2 @ ( sset_a @ A2 ) ) ) ).
% stl_sset
thf(fact_34_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_35_in__mono,axiom,
! [A: set_stream_list_a,B: set_stream_list_a,X2: stream_list_a] :
( ( ord_le2348936426438760842list_a @ A @ B )
=> ( ( member_stream_list_a @ X2 @ A )
=> ( member_stream_list_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_36_in__mono,axiom,
! [A: set_list_list_a,B: set_list_list_a,X2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A @ B )
=> ( ( member_list_list_a @ X2 @ A )
=> ( member_list_list_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_37_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_38_in__mono,axiom,
! [A: set_stream_a,B: set_stream_a,X2: stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( ( member_stream_a @ X2 @ A )
=> ( member_stream_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_39_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_40_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_41_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_42_subsetD,axiom,
! [A: set_stream_list_a,B: set_stream_list_a,C: stream_list_a] :
( ( ord_le2348936426438760842list_a @ A @ B )
=> ( ( member_stream_list_a @ C @ A )
=> ( member_stream_list_a @ C @ B ) ) ) ).
% subsetD
thf(fact_43_subsetD,axiom,
! [A: set_list_list_a,B: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A @ B )
=> ( ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ B ) ) ) ).
% subsetD
thf(fact_44_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_45_subsetD,axiom,
! [A: set_stream_a,B: set_stream_a,C: stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( ( member_stream_a @ C @ A )
=> ( member_stream_a @ C @ B ) ) ) ).
% subsetD
thf(fact_46_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% subsetD
thf(fact_47_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_48_equalityE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% equalityE
thf(fact_49_equalityE,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ( A = B )
=> ~ ( ( ord_le1019412967036525316ream_a @ A @ B )
=> ~ ( ord_le1019412967036525316ream_a @ B @ A ) ) ) ).
% equalityE
thf(fact_50_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_51_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( member1426531477525435216od_a_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_52_subset__eq,axiom,
( ord_le2348936426438760842list_a
= ( ^ [A3: set_stream_list_a,B2: set_stream_list_a] :
! [X3: stream_list_a] :
( ( member_stream_list_a @ X3 @ A3 )
=> ( member_stream_list_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_53_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B2: set_list_list_a] :
! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A3 )
=> ( member_list_list_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_54_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_55_subset__eq,axiom,
( ord_le1019412967036525316ream_a
= ( ^ [A3: set_stream_a,B2: set_stream_a] :
! [X3: stream_a] :
( ( member_stream_a @ X3 @ A3 )
=> ( member_stream_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_56_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_57_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_58_equalityD1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% equalityD1
thf(fact_59_equalityD1,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ( A = B )
=> ( ord_le1019412967036525316ream_a @ A @ B ) ) ).
% equalityD1
thf(fact_60_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_61_equalityD2,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% equalityD2
thf(fact_62_equalityD2,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ( A = B )
=> ( ord_le1019412967036525316ream_a @ B @ A ) ) ).
% equalityD2
thf(fact_63_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_64_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_65_subset__iff,axiom,
( ord_le2348936426438760842list_a
= ( ^ [A3: set_stream_list_a,B2: set_stream_list_a] :
! [T: stream_list_a] :
( ( member_stream_list_a @ T @ A3 )
=> ( member_stream_list_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_66_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B2: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A3 )
=> ( member_list_list_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_67_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_68_subset__iff,axiom,
( ord_le1019412967036525316ream_a
= ( ^ [A3: set_stream_a,B2: set_stream_a] :
! [T: stream_a] :
( ( member_stream_a @ T @ A3 )
=> ( member_stream_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_69_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A3 )
=> ( member_list_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_70_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_71_order__antisym__conv,axiom,
! [Y: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_72_order__antisym__conv,axiom,
! [Y: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( ( ord_less_eq_o_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_73_order__antisym__conv,axiom,
! [Y: $o > set_a,X2: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ X2 )
=> ( ( ord_less_eq_o_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_74_order__antisym__conv,axiom,
! [Y: set_stream_a,X2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ Y @ X2 )
=> ( ( ord_le1019412967036525316ream_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_75_order__antisym__conv,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_76_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_77_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_78_linorder__le__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_79_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_80_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A2: num,B3: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__le__eq__subst,axiom,
! [A2: num,B3: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_87_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > num,C: num] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
! [A2: num,B3: num,F: num > set_a,C: set_a] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B3: num,C: num] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B3: num,C: num] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A2: num,F: set_a > num,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A2: set_a,F: num > set_a,B3: num,C: num] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A2: set_list_a,F: nat > set_list_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_linorder__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_101_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_102_order__eq__refl,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( X2 = Y )
=> ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_103_order__eq__refl,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( X2 = Y )
=> ( ord_less_eq_o_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_104_order__eq__refl,axiom,
! [X2: $o > set_a,Y: $o > set_a] :
( ( X2 = Y )
=> ( ord_less_eq_o_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_105_order__eq__refl,axiom,
! [X2: set_stream_a,Y: set_stream_a] :
( ( X2 = Y )
=> ( ord_le1019412967036525316ream_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_106_order__eq__refl,axiom,
! [X2: num,Y: num] :
( ( X2 = Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_107_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_108_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_109_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_110_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_111_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_113_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A2: num,B3: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst2,axiom,
! [A2: num,B3: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > num,C: num] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A2: num,B3: num,F: num > set_a,C: set_a] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8861187494160871172list_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_120_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A2: nat,F: num > nat,B3: num,C: num] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A2: num,F: nat > num,B3: nat,C: nat] :
( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_order__subst1,axiom,
! [A2: num,F: num > num,B3: num,C: num] :
( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A2: set_a,F: num > set_a,B3: num,C: num] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ! [X: num,Y2: num] :
( ( ord_less_eq_num @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A2: num,F: set_a > num,B3: set_a,C: set_a] :
( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A2: nat,F: set_list_a > nat,B3: set_list_a,C: set_list_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ! [X: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > nat,B4: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B4 )
& ( ord_less_eq_o_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_131_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
= ( ^ [A4: $o > set_a,B4: $o > set_a] :
( ( ord_less_eq_o_set_a @ A4 @ B4 )
& ( ord_less_eq_o_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_132_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_stream_a,Z: set_stream_a] : ( Y3 = Z ) )
= ( ^ [A4: set_stream_a,B4: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A4 @ B4 )
& ( ord_le1019412967036525316ream_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
& ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_135_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_136_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G: $o > nat] :
! [X3: $o] : ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_137_le__fun__def,axiom,
( ord_less_eq_o_set_a
= ( ^ [F2: $o > set_a,G: $o > set_a] :
! [X3: $o] : ( ord_less_eq_set_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_138_le__funI,axiom,
! [F: $o > set_a,G2: $o > set_a] :
( ! [X: $o] : ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_set_a @ F @ G2 ) ) ).
% le_funI
thf(fact_139_le__funI,axiom,
! [F: $o > nat,G2: $o > nat] :
( ! [X: $o] : ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_140_le__funE,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_141_le__funE,axiom,
! [F: $o > set_a,G2: $o > set_a,X2: $o] :
( ( ord_less_eq_o_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_142_le__funD,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_143_le__funD,axiom,
! [F: $o > set_a,G2: $o > set_a,X2: $o] :
( ( ord_less_eq_o_set_a @ F @ G2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_144_antisym,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_145_antisym,axiom,
! [A2: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_146_antisym,axiom,
! [A2: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_o_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_147_antisym,axiom,
! [A2: set_stream_a,B3: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A2 @ B3 )
=> ( ( ord_le1019412967036525316ream_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_148_antisym,axiom,
! [A2: num,B3: num] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ord_less_eq_num @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_149_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_150_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_151_dual__order_Otrans,axiom,
! [B3: set_list_a,A2: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C @ B3 )
=> ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_152_dual__order_Otrans,axiom,
! [B3: $o > nat,A2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( ( ord_less_eq_o_nat @ C @ B3 )
=> ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_153_dual__order_Otrans,axiom,
! [B3: $o > set_a,A2: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_o_set_a @ C @ B3 )
=> ( ord_less_eq_o_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_154_dual__order_Otrans,axiom,
! [B3: set_stream_a,A2: set_stream_a,C: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ B3 @ A2 )
=> ( ( ord_le1019412967036525316ream_a @ C @ B3 )
=> ( ord_le1019412967036525316ream_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_155_dual__order_Otrans,axiom,
! [B3: num,A2: num,C: num] :
( ( ord_less_eq_num @ B3 @ A2 )
=> ( ( ord_less_eq_num @ C @ B3 )
=> ( ord_less_eq_num @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_156_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_157_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_158_dual__order_Oantisym,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_159_dual__order_Oantisym,axiom,
! [B3: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_160_dual__order_Oantisym,axiom,
! [B3: $o > set_a,A2: $o > set_a] :
( ( ord_less_eq_o_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_o_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_161_dual__order_Oantisym,axiom,
! [B3: set_stream_a,A2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ B3 @ A2 )
=> ( ( ord_le1019412967036525316ream_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_162_dual__order_Oantisym,axiom,
! [B3: num,A2: num] :
( ( ord_less_eq_num @ B3 @ A2 )
=> ( ( ord_less_eq_num @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_163_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_164_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A4 )
& ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > nat,B4: $o > nat] :
( ( ord_less_eq_o_nat @ B4 @ A4 )
& ( ord_less_eq_o_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_167_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
= ( ^ [A4: $o > set_a,B4: $o > set_a] :
( ( ord_less_eq_o_set_a @ B4 @ A4 )
& ( ord_less_eq_o_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_168_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_stream_a,Z: set_stream_a] : ( Y3 = Z ) )
= ( ^ [A4: set_stream_a,B4: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ B4 @ A4 )
& ( ord_le1019412967036525316ream_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_169_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [A4: num,B4: num] :
( ( ord_less_eq_num @ B4 @ A4 )
& ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_170_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_171_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_172_linorder__wlog,axiom,
! [P: num > num > $o,A2: num,B3: num] :
( ! [A5: num,B5: num] :
( ( ord_less_eq_num @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: num,B5: num] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_173_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_174_order__trans,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z2 )
=> ( ord_le8861187494160871172list_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_175_order__trans,axiom,
! [X2: $o > nat,Y: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z2 )
=> ( ord_less_eq_o_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_176_order__trans,axiom,
! [X2: $o > set_a,Y: $o > set_a,Z2: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ Z2 )
=> ( ord_less_eq_o_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_177_order__trans,axiom,
! [X2: set_stream_a,Y: set_stream_a,Z2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ X2 @ Y )
=> ( ( ord_le1019412967036525316ream_a @ Y @ Z2 )
=> ( ord_le1019412967036525316ream_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_178_order__trans,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_179_order__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_180_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_181_order_Otrans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_182_order_Otrans,axiom,
! [A2: $o > nat,B3: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( ( ord_less_eq_o_nat @ B3 @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_183_order_Otrans,axiom,
! [A2: $o > set_a,B3: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_o_set_a @ B3 @ C )
=> ( ord_less_eq_o_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_184_order_Otrans,axiom,
! [A2: set_stream_a,B3: set_stream_a,C: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A2 @ B3 )
=> ( ( ord_le1019412967036525316ream_a @ B3 @ C )
=> ( ord_le1019412967036525316ream_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_185_order_Otrans,axiom,
! [A2: num,B3: num,C: num] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% order.trans
thf(fact_186_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_187_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_188_order__antisym,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_189_order__antisym,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_190_order__antisym,axiom,
! [X2: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X2 @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_191_order__antisym,axiom,
! [X2: set_stream_a,Y: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ X2 @ Y )
=> ( ( ord_le1019412967036525316ream_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_192_order__antisym,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_193_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_194_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_195_ord__le__eq__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_196_ord__le__eq__trans,axiom,
! [A2: $o > nat,B3: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_197_ord__le__eq__trans,axiom,
! [A2: $o > set_a,B3: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_o_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_198_ord__le__eq__trans,axiom,
! [A2: set_stream_a,B3: set_stream_a,C: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le1019412967036525316ream_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A2: num,B3: num,C: num] :
( ( ord_less_eq_num @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_200_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_201_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( A2 = B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A2: $o > nat,B3: $o > nat,C: $o > nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_o_nat @ B3 @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A2: $o > set_a,B3: $o > set_a,C: $o > set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_o_set_a @ B3 @ C )
=> ( ord_less_eq_o_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_205_ord__eq__le__trans,axiom,
! [A2: set_stream_a,B3: set_stream_a,C: set_stream_a] :
( ( A2 = B3 )
=> ( ( ord_le1019412967036525316ream_a @ B3 @ C )
=> ( ord_le1019412967036525316ream_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_206_ord__eq__le__trans,axiom,
! [A2: num,B3: num,C: num] :
( ( A2 = B3 )
=> ( ( ord_less_eq_num @ B3 @ C )
=> ( ord_less_eq_num @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_207_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_208_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_209_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_210_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [X3: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y4 )
& ( ord_less_eq_o_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_211_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > set_a,Z: $o > set_a] : ( Y3 = Z ) )
= ( ^ [X3: $o > set_a,Y4: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y4 )
& ( ord_less_eq_o_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_212_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_stream_a,Z: set_stream_a] : ( Y3 = Z ) )
= ( ^ [X3: set_stream_a,Y4: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ X3 @ Y4 )
& ( ord_le1019412967036525316ream_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_213_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: num,Z: num] : ( Y3 = Z ) )
= ( ^ [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_214_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_215_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_216_le__cases3,axiom,
! [X2: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_217_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_218_nle__le,axiom,
! [A2: num,B3: num] :
( ( ~ ( ord_less_eq_num @ A2 @ B3 ) )
= ( ( ord_less_eq_num @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_219_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_220_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_221_Collect__mono__iff,axiom,
! [P: stream_a > $o,Q: stream_a > $o] :
( ( ord_le1019412967036525316ream_a @ ( collect_stream_a @ P ) @ ( collect_stream_a @ Q ) )
= ( ! [X3: stream_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_223_set__eq__subset,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A3: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B2 )
& ( ord_le8861187494160871172list_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_224_set__eq__subset,axiom,
( ( ^ [Y3: set_stream_a,Z: set_stream_a] : ( Y3 = Z ) )
= ( ^ [A3: set_stream_a,B2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A3 @ B2 )
& ( ord_le1019412967036525316ream_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_225_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_226_subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C2 )
=> ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_227_subset__trans,axiom,
! [A: set_stream_a,B: set_stream_a,C2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( ( ord_le1019412967036525316ream_a @ B @ C2 )
=> ( ord_le1019412967036525316ream_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_228_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_229_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_230_Collect__mono,axiom,
! [P: stream_a > $o,Q: stream_a > $o] :
( ! [X: stream_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le1019412967036525316ream_a @ ( collect_stream_a @ P ) @ ( collect_stream_a @ Q ) ) ) ).
% Collect_mono
thf(fact_231_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_232_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_233_subset__refl,axiom,
! [A: set_stream_a] : ( ord_le1019412967036525316ream_a @ A @ A ) ).
% subset_refl
thf(fact_234_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_235_Stream_Osmember__def,axiom,
( smembe8479462165187265637od_a_a
= ( ^ [X3: product_prod_a_a,S: stream6512538624140940077od_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( sset_P5743994922208793403od_a_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_236_Stream_Osmember__def,axiom,
( smembe4279752203236903368list_a
= ( ^ [X3: stream_list_a,S: stream_stream_list_a] : ( member_stream_list_a @ X3 @ ( sset_stream_list_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_237_Stream_Osmember__def,axiom,
( smember_stream_a
= ( ^ [X3: stream_a,S: stream_stream_a] : ( member_stream_a @ X3 @ ( sset_stream_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_238_Stream_Osmember__def,axiom,
( smember_list_list_a
= ( ^ [X3: list_list_a,S: stream_list_list_a] : ( member_list_list_a @ X3 @ ( sset_list_list_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_239_Stream_Osmember__def,axiom,
( smember_set_a
= ( ^ [X3: set_a,S: stream_set_a] : ( member_set_a @ X3 @ ( sset_set_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_240_Stream_Osmember__def,axiom,
( smember_a
= ( ^ [X3: a,S: stream_a] : ( member_a @ X3 @ ( sset_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_241_Stream_Osmember__def,axiom,
( smember_list_a
= ( ^ [X3: list_a,S: stream_list_a] : ( member_list_a @ X3 @ ( sset_list_a @ S ) ) ) ) ).
% Stream.smember_def
thf(fact_242_Greatest__equality,axiom,
! [P: set_list_a > $o,X2: set_list_a] :
( ( P @ X2 )
=> ( ! [Y2: set_list_a] :
( ( P @ Y2 )
=> ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
=> ( ( order_733672244956367037list_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_243_Greatest__equality,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_o_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_244_Greatest__equality,axiom,
! [P: ( $o > set_a ) > $o,X2: $o > set_a] :
( ( P @ X2 )
=> ( ! [Y2: $o > set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_set_a @ Y2 @ X2 ) )
=> ( ( order_6114237596796908690_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_245_Greatest__equality,axiom,
! [P: set_stream_a > $o,X2: set_stream_a] :
( ( P @ X2 )
=> ( ! [Y2: set_stream_a] :
( ( P @ Y2 )
=> ( ord_le1019412967036525316ream_a @ Y2 @ X2 ) )
=> ( ( order_8427663985114061757ream_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_246_Greatest__equality,axiom,
! [P: num > $o,X2: num] :
( ( P @ X2 )
=> ( ! [Y2: num] :
( ( P @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) )
=> ( ( order_Greatest_num @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_247_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_248_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_249_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_250_mem__Collect__eq,axiom,
! [A2: stream_list_a,P: stream_list_a > $o] :
( ( member_stream_list_a @ A2 @ ( collec2345243735937135093list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_251_mem__Collect__eq,axiom,
! [A2: stream_a,P: stream_a > $o] :
( ( member_stream_a @ A2 @ ( collect_stream_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_252_mem__Collect__eq,axiom,
! [A2: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_253_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_254_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_255_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_256_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_257_Collect__mem__eq,axiom,
! [A: set_stream_list_a] :
( ( collec2345243735937135093list_a
@ ^ [X3: stream_list_a] : ( member_stream_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_258_Collect__mem__eq,axiom,
! [A: set_stream_a] :
( ( collect_stream_a
@ ^ [X3: stream_a] : ( member_stream_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_259_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_260_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_261_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_262_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_263_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_264_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_265_GreatestI2__order,axiom,
! [P: set_list_a > $o,X2: set_list_a,Q: set_list_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_list_a] :
( ( P @ Y2 )
=> ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
=> ( ! [X: set_list_a] :
( ( P @ X )
=> ( ! [Y5: set_list_a] :
( ( P @ Y5 )
=> ( ord_le8861187494160871172list_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_733672244956367037list_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_266_GreatestI2__order,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat,Q: ( $o > nat ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ! [X: $o > nat] :
( ( P @ X )
=> ( ! [Y5: $o > nat] :
( ( P @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_267_GreatestI2__order,axiom,
! [P: ( $o > set_a ) > $o,X2: $o > set_a,Q: ( $o > set_a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_set_a @ Y2 @ X2 ) )
=> ( ! [X: $o > set_a] :
( ( P @ X )
=> ( ! [Y5: $o > set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_6114237596796908690_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_268_GreatestI2__order,axiom,
! [P: set_stream_a > $o,X2: set_stream_a,Q: set_stream_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_stream_a] :
( ( P @ Y2 )
=> ( ord_le1019412967036525316ream_a @ Y2 @ X2 ) )
=> ( ! [X: set_stream_a] :
( ( P @ X )
=> ( ! [Y5: set_stream_a] :
( ( P @ Y5 )
=> ( ord_le1019412967036525316ream_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_8427663985114061757ream_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_269_GreatestI2__order,axiom,
! [P: num > $o,X2: num,Q: num > $o] :
( ( P @ X2 )
=> ( ! [Y2: num] :
( ( P @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) )
=> ( ! [X: num] :
( ( P @ X )
=> ( ! [Y5: num] :
( ( P @ Y5 )
=> ( ord_less_eq_num @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_270_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_271_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_272_le__rel__bool__arg__iff,axiom,
( ord_le354538097136858257list_a
= ( ^ [X4: $o > set_list_a,Y6: $o > set_list_a] :
( ( ord_le8861187494160871172list_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8861187494160871172list_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_273_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X4: $o > $o > nat,Y6: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_274_le__rel__bool__arg__iff,axiom,
( ord_le5604070792465694036_set_a
= ( ^ [X4: $o > $o > set_a,Y6: $o > $o > set_a] :
( ( ord_less_eq_o_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_275_le__rel__bool__arg__iff,axiom,
( ord_le3754339485079453329ream_a
= ( ^ [X4: $o > set_stream_a,Y6: $o > set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le1019412967036525316ream_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_276_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_num
= ( ^ [X4: $o > num,Y6: $o > num] :
( ( ord_less_eq_num @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_num @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_277_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X4: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_278_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_279_verit__la__disequality,axiom,
! [A2: num,B3: num] :
( ( A2 = B3 )
| ~ ( ord_less_eq_num @ A2 @ B3 )
| ~ ( ord_less_eq_num @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_280_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_281_verit__comp__simplify1_I2_J,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_282_verit__comp__simplify1_I2_J,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_283_verit__comp__simplify1_I2_J,axiom,
! [A2: $o > set_a] : ( ord_less_eq_o_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_284_verit__comp__simplify1_I2_J,axiom,
! [A2: set_stream_a] : ( ord_le1019412967036525316ream_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_285_verit__comp__simplify1_I2_J,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_286_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_287_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_288_streams__iff__sset,axiom,
! [S2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) )
= ( ord_le1019412967036525316ream_a @ ( sset_stream_a @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_289_streams__iff__sset,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
= ( ord_le8861187494160871172list_a @ ( sset_list_a @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_290_streams__iff__sset,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
= ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A ) ) ).
% streams_iff_sset
thf(fact_291_streams__sset,axiom,
! [S2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) )
=> ( ord_le1019412967036525316ream_a @ ( sset_stream_a @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_292_streams__sset,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( ord_le8861187494160871172list_a @ ( sset_list_a @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_293_streams__sset,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A ) ) ).
% streams_sset
thf(fact_294_sset__streams,axiom,
! [S2: stream_stream_a,A: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( sset_stream_a @ S2 ) @ A )
=> ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) ) ) ).
% sset_streams
thf(fact_295_sset__streams,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sset_list_a @ S2 ) @ A )
=> ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) ) ) ).
% sset_streams
thf(fact_296_sset__streams,axiom,
! [S2: stream_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sset_a @ S2 ) @ A )
=> ( member_stream_a @ S2 @ ( streams_a @ A ) ) ) ).
% sset_streams
thf(fact_297_sset__induct,axiom,
! [Y: stream_list_a,S2: stream_stream_list_a,P: stream_list_a > stream_stream_list_a > $o] :
( ( member_stream_list_a @ Y @ ( sset_stream_list_a @ S2 ) )
=> ( ! [S3: stream_stream_list_a] : ( P @ ( shd_stream_list_a @ S3 ) @ S3 )
=> ( ! [S3: stream_stream_list_a,Y2: stream_list_a] :
( ( member_stream_list_a @ Y2 @ ( sset_stream_list_a @ ( stl_stream_list_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_stream_list_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_298_sset__induct,axiom,
! [Y: stream_a,S2: stream_stream_a,P: stream_a > stream_stream_a > $o] :
( ( member_stream_a @ Y @ ( sset_stream_a @ S2 ) )
=> ( ! [S3: stream_stream_a] : ( P @ ( shd_stream_a @ S3 ) @ S3 )
=> ( ! [S3: stream_stream_a,Y2: stream_a] :
( ( member_stream_a @ Y2 @ ( sset_stream_a @ ( stl_stream_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_stream_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_299_sset__induct,axiom,
! [Y: set_a,S2: stream_set_a,P: set_a > stream_set_a > $o] :
( ( member_set_a @ Y @ ( sset_set_a @ S2 ) )
=> ( ! [S3: stream_set_a] : ( P @ ( shd_set_a @ S3 ) @ S3 )
=> ( ! [S3: stream_set_a,Y2: set_a] :
( ( member_set_a @ Y2 @ ( sset_set_a @ ( stl_set_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_set_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_300_sset__induct,axiom,
! [Y: product_prod_a_a,S2: stream6512538624140940077od_a_a,P: product_prod_a_a > stream6512538624140940077od_a_a > $o] :
( ( member1426531477525435216od_a_a @ Y @ ( sset_P5743994922208793403od_a_a @ S2 ) )
=> ( ! [S3: stream6512538624140940077od_a_a] : ( P @ ( shd_Product_prod_a_a @ S3 ) @ S3 )
=> ( ! [S3: stream6512538624140940077od_a_a,Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( sset_P5743994922208793403od_a_a @ ( stl_Product_prod_a_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_Product_prod_a_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_301_sset__induct,axiom,
! [Y: produc8685980395799941037list_a,S2: stream8205385055505213747list_a,P: produc8685980395799941037list_a > stream8205385055505213747list_a > $o] :
( ( member4889668945541975382list_a @ Y @ ( sset_P5198238693475834817list_a @ S2 ) )
=> ( ! [S3: stream8205385055505213747list_a] : ( P @ ( shd_Pr4559591720970336857list_a @ S3 ) @ S3 )
=> ( ! [S3: stream8205385055505213747list_a,Y2: produc8685980395799941037list_a] :
( ( member4889668945541975382list_a @ Y2 @ ( sset_P5198238693475834817list_a @ ( stl_Pr4668768530486720605list_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_Pr4668768530486720605list_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_302_sset__induct,axiom,
! [Y: produc2579390645249093025st_a_a,S2: stream5874003577662885159st_a_a,P: produc2579390645249093025st_a_a > stream5874003577662885159st_a_a > $o] :
( ( member8006451231845903178st_a_a @ Y @ ( sset_P8315020979779762613st_a_a @ S2 ) )
=> ( ! [S3: stream5874003577662885159st_a_a] : ( P @ ( shd_Pr7676374007274264653st_a_a @ S3 ) @ S3 )
=> ( ! [S3: stream5874003577662885159st_a_a,Y2: produc2579390645249093025st_a_a] :
( ( member8006451231845903178st_a_a @ Y2 @ ( sset_P8315020979779762613st_a_a @ ( stl_Pr7785550816790648401st_a_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_Pr7785550816790648401st_a_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_303_sset__induct,axiom,
! [Y: produc9164743771328383783list_a,S2: stream271976807216406061list_a,P: produc9164743771328383783list_a > stream271976807216406061list_a > $o] :
( ( member8191768239178080336list_a @ Y @ ( sset_P844431690648075579list_a @ S2 ) )
=> ( ! [S3: stream271976807216406061list_a] : ( P @ ( shd_Pr3168949583242555347list_a @ S3 ) @ S3 )
=> ( ! [S3: stream271976807216406061list_a,Y2: produc9164743771328383783list_a] :
( ( member8191768239178080336list_a @ Y2 @ ( sset_P844431690648075579list_a @ ( stl_Pr6107055601389525975list_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_Pr6107055601389525975list_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_304_sset__induct,axiom,
! [Y: list_list_a,S2: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
( ( member_list_list_a @ Y @ ( sset_list_list_a @ S2 ) )
=> ( ! [S3: stream_list_list_a] : ( P @ ( shd_list_list_a @ S3 ) @ S3 )
=> ( ! [S3: stream_list_list_a,Y2: list_list_a] :
( ( member_list_list_a @ Y2 @ ( sset_list_list_a @ ( stl_list_list_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_list_list_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_305_sset__induct,axiom,
! [Y: a,S2: stream_a,P: a > stream_a > $o] :
( ( member_a @ Y @ ( sset_a @ S2 ) )
=> ( ! [S3: stream_a] : ( P @ ( shd_a @ S3 ) @ S3 )
=> ( ! [S3: stream_a,Y2: a] :
( ( member_a @ Y2 @ ( sset_a @ ( stl_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_306_sset__induct,axiom,
! [Y: list_a,S2: stream_list_a,P: list_a > stream_list_a > $o] :
( ( member_list_a @ Y @ ( sset_list_a @ S2 ) )
=> ( ! [S3: stream_list_a] : ( P @ ( shd_list_a @ S3 ) @ S3 )
=> ( ! [S3: stream_list_a,Y2: list_a] :
( ( member_list_a @ Y2 @ ( sset_list_a @ ( stl_list_a @ S3 ) ) )
=> ( ( P @ Y2 @ ( stl_list_a @ S3 ) )
=> ( P @ Y2 @ S3 ) ) )
=> ( P @ Y @ S2 ) ) ) ) ).
% sset_induct
thf(fact_307_sinterleave_Osimps_I2_J,axiom,
! [S1: stream6512538624140940077od_a_a,S22: stream6512538624140940077od_a_a] :
( ( stl_Product_prod_a_a @ ( sinter5547950683322096900od_a_a @ S1 @ S22 ) )
= ( sinter5547950683322096900od_a_a @ S22 @ ( stl_Product_prod_a_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_308_sinterleave_Osimps_I2_J,axiom,
! [S1: stream8205385055505213747list_a,S22: stream8205385055505213747list_a] :
( ( stl_Pr4668768530486720605list_a @ ( sinter832779693940469002list_a @ S1 @ S22 ) )
= ( sinter832779693940469002list_a @ S22 @ ( stl_Pr4668768530486720605list_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_309_sinterleave_Osimps_I2_J,axiom,
! [S1: stream5874003577662885159st_a_a,S22: stream5874003577662885159st_a_a] :
( ( stl_Pr7785550816790648401st_a_a @ ( sinter3949561980244396798st_a_a @ S1 @ S22 ) )
= ( sinter3949561980244396798st_a_a @ S22 @ ( stl_Pr7785550816790648401st_a_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_310_sinterleave_Osimps_I2_J,axiom,
! [S1: stream271976807216406061list_a,S22: stream271976807216406061list_a] :
( ( stl_Pr6107055601389525975list_a @ ( sinter5791379037304460804list_a @ S1 @ S22 ) )
= ( sinter5791379037304460804list_a @ S22 @ ( stl_Pr6107055601389525975list_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_311_sinterleave_Osimps_I2_J,axiom,
! [S1: stream_list_list_a,S22: stream_list_list_a] :
( ( stl_list_list_a @ ( sinter664328039231274855list_a @ S1 @ S22 ) )
= ( sinter664328039231274855list_a @ S22 @ ( stl_list_list_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_312_sinterleave_Osimps_I2_J,axiom,
! [S1: stream_list_a,S22: stream_list_a] :
( ( stl_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
= ( sinterleave_list_a @ S22 @ ( stl_list_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_313_sinterleave_Osimps_I2_J,axiom,
! [S1: stream_a,S22: stream_a] :
( ( stl_a @ ( sinterleave_a @ S1 @ S22 ) )
= ( sinterleave_a @ S22 @ ( stl_a @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_314_sinterleave_Osimps_I1_J,axiom,
! [S1: stream_list_list_a,S22: stream_list_list_a] :
( ( shd_list_list_a @ ( sinter664328039231274855list_a @ S1 @ S22 ) )
= ( shd_list_list_a @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_315_sinterleave_Osimps_I1_J,axiom,
! [S1: stream_a,S22: stream_a] :
( ( shd_a @ ( sinterleave_a @ S1 @ S22 ) )
= ( shd_a @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_316_sinterleave_Osimps_I1_J,axiom,
! [S1: stream_list_a,S22: stream_list_a] :
( ( shd_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
= ( shd_list_a @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_317_streams__shd,axiom,
! [S2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ S2 @ ( stream6392229770749734815od_a_a @ A ) )
=> ( member1426531477525435216od_a_a @ ( shd_Product_prod_a_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_318_streams__shd,axiom,
! [S2: stream_stream_list_a,A: set_stream_list_a] :
( ( member3992053618758223289list_a @ S2 @ ( stream6414841306446625538list_a @ A ) )
=> ( member_stream_list_a @ ( shd_stream_list_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_319_streams__shd,axiom,
! [S2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) )
=> ( member_stream_a @ ( shd_stream_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_320_streams__shd,axiom,
! [S2: stream_set_a,A: set_set_a] :
( ( member_stream_set_a @ S2 @ ( streams_set_a @ A ) )
=> ( member_set_a @ ( shd_set_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_321_streams__shd,axiom,
! [S2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ S2 @ ( streams_list_list_a @ A ) )
=> ( member_list_list_a @ ( shd_list_list_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_322_streams__shd,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( member_a @ ( shd_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_323_streams__shd,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( member_list_a @ ( shd_list_a @ S2 ) @ A ) ) ).
% streams_shd
thf(fact_324_in__streams,axiom,
! [S2: stream_stream_list_a,S4: set_stream_list_a] :
( ( member3992053618758223289list_a @ ( stl_stream_list_a @ S2 ) @ ( stream6414841306446625538list_a @ S4 ) )
=> ( ( member_stream_list_a @ ( shd_stream_list_a @ S2 ) @ S4 )
=> ( member3992053618758223289list_a @ S2 @ ( stream6414841306446625538list_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_325_in__streams,axiom,
! [S2: stream_stream_a,S4: set_stream_a] :
( ( member8409225883583818419ream_a @ ( stl_stream_a @ S2 ) @ ( streams_stream_a @ S4 ) )
=> ( ( member_stream_a @ ( shd_stream_a @ S2 ) @ S4 )
=> ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_326_in__streams,axiom,
! [S2: stream_set_a,S4: set_set_a] :
( ( member_stream_set_a @ ( stl_set_a @ S2 ) @ ( streams_set_a @ S4 ) )
=> ( ( member_set_a @ ( shd_set_a @ S2 ) @ S4 )
=> ( member_stream_set_a @ S2 @ ( streams_set_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_327_in__streams,axiom,
! [S2: stream6512538624140940077od_a_a,S4: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ ( stl_Product_prod_a_a @ S2 ) @ ( stream6392229770749734815od_a_a @ S4 ) )
=> ( ( member1426531477525435216od_a_a @ ( shd_Product_prod_a_a @ S2 ) @ S4 )
=> ( member262390649204803542od_a_a @ S2 @ ( stream6392229770749734815od_a_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_328_in__streams,axiom,
! [S2: stream8205385055505213747list_a,S4: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ ( stl_Pr4668768530486720605list_a @ S2 ) @ ( stream2987405652289500709list_a @ S4 ) )
=> ( ( member4889668945541975382list_a @ ( shd_Pr4559591720970336857list_a @ S2 ) @ S4 )
=> ( member7566587766794810716list_a @ S2 @ ( stream2987405652289500709list_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_329_in__streams,axiom,
! [S2: stream5874003577662885159st_a_a,S4: set_Pr8962057229576493569st_a_a] :
( ( member5235206288952482128st_a_a @ ( stl_Pr7785550816790648401st_a_a @ S2 ) @ ( stream6104187938593428505st_a_a @ S4 ) )
=> ( ( member8006451231845903178st_a_a @ ( shd_Pr7676374007274264653st_a_a @ S2 ) @ S4 )
=> ( member5235206288952482128st_a_a @ S2 @ ( stream6104187938593428505st_a_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_330_in__streams,axiom,
! [S2: stream271976807216406061list_a,S4: set_Pr4048851178543822343list_a] :
( ( member8659441389117227478list_a @ ( stl_Pr6107055601389525975list_a @ S2 ) @ ( stream1450977367002125727list_a @ S4 ) )
=> ( ( member8191768239178080336list_a @ ( shd_Pr3168949583242555347list_a @ S2 ) @ S4 )
=> ( member8659441389117227478list_a @ S2 @ ( stream1450977367002125727list_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_331_in__streams,axiom,
! [S2: stream_list_list_a,S4: set_list_list_a] :
( ( member8003905643527612601list_a @ ( stl_list_list_a @ S2 ) @ ( streams_list_list_a @ S4 ) )
=> ( ( member_list_list_a @ ( shd_list_list_a @ S2 ) @ S4 )
=> ( member8003905643527612601list_a @ S2 @ ( streams_list_list_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_332_in__streams,axiom,
! [S2: stream_list_a,S4: set_list_a] :
( ( member_stream_list_a @ ( stl_list_a @ S2 ) @ ( streams_list_a @ S4 ) )
=> ( ( member_list_a @ ( shd_list_a @ S2 ) @ S4 )
=> ( member_stream_list_a @ S2 @ ( streams_list_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_333_in__streams,axiom,
! [S2: stream_a,S4: set_a] :
( ( member_stream_a @ ( stl_a @ S2 ) @ ( streams_a @ S4 ) )
=> ( ( member_a @ ( shd_a @ S2 ) @ S4 )
=> ( member_stream_a @ S2 @ ( streams_a @ S4 ) ) ) ) ).
% in_streams
thf(fact_334_streamsE,axiom,
! [S2: stream_stream_list_a,A: set_stream_list_a] :
( ( member3992053618758223289list_a @ S2 @ ( stream6414841306446625538list_a @ A ) )
=> ~ ( ( member_stream_list_a @ ( shd_stream_list_a @ S2 ) @ A )
=> ~ ( member3992053618758223289list_a @ ( stl_stream_list_a @ S2 ) @ ( stream6414841306446625538list_a @ A ) ) ) ) ).
% streamsE
thf(fact_335_streamsE,axiom,
! [S2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) )
=> ~ ( ( member_stream_a @ ( shd_stream_a @ S2 ) @ A )
=> ~ ( member8409225883583818419ream_a @ ( stl_stream_a @ S2 ) @ ( streams_stream_a @ A ) ) ) ) ).
% streamsE
thf(fact_336_streamsE,axiom,
! [S2: stream_set_a,A: set_set_a] :
( ( member_stream_set_a @ S2 @ ( streams_set_a @ A ) )
=> ~ ( ( member_set_a @ ( shd_set_a @ S2 ) @ A )
=> ~ ( member_stream_set_a @ ( stl_set_a @ S2 ) @ ( streams_set_a @ A ) ) ) ) ).
% streamsE
thf(fact_337_streamsE,axiom,
! [S2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ S2 @ ( stream6392229770749734815od_a_a @ A ) )
=> ~ ( ( member1426531477525435216od_a_a @ ( shd_Product_prod_a_a @ S2 ) @ A )
=> ~ ( member262390649204803542od_a_a @ ( stl_Product_prod_a_a @ S2 ) @ ( stream6392229770749734815od_a_a @ A ) ) ) ) ).
% streamsE
thf(fact_338_streamsE,axiom,
! [S2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ S2 @ ( stream2987405652289500709list_a @ A ) )
=> ~ ( ( member4889668945541975382list_a @ ( shd_Pr4559591720970336857list_a @ S2 ) @ A )
=> ~ ( member7566587766794810716list_a @ ( stl_Pr4668768530486720605list_a @ S2 ) @ ( stream2987405652289500709list_a @ A ) ) ) ) ).
% streamsE
thf(fact_339_streamsE,axiom,
! [S2: stream5874003577662885159st_a_a,A: set_Pr8962057229576493569st_a_a] :
( ( member5235206288952482128st_a_a @ S2 @ ( stream6104187938593428505st_a_a @ A ) )
=> ~ ( ( member8006451231845903178st_a_a @ ( shd_Pr7676374007274264653st_a_a @ S2 ) @ A )
=> ~ ( member5235206288952482128st_a_a @ ( stl_Pr7785550816790648401st_a_a @ S2 ) @ ( stream6104187938593428505st_a_a @ A ) ) ) ) ).
% streamsE
thf(fact_340_streamsE,axiom,
! [S2: stream271976807216406061list_a,A: set_Pr4048851178543822343list_a] :
( ( member8659441389117227478list_a @ S2 @ ( stream1450977367002125727list_a @ A ) )
=> ~ ( ( member8191768239178080336list_a @ ( shd_Pr3168949583242555347list_a @ S2 ) @ A )
=> ~ ( member8659441389117227478list_a @ ( stl_Pr6107055601389525975list_a @ S2 ) @ ( stream1450977367002125727list_a @ A ) ) ) ) ).
% streamsE
thf(fact_341_streamsE,axiom,
! [S2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ S2 @ ( streams_list_list_a @ A ) )
=> ~ ( ( member_list_list_a @ ( shd_list_list_a @ S2 ) @ A )
=> ~ ( member8003905643527612601list_a @ ( stl_list_list_a @ S2 ) @ ( streams_list_list_a @ A ) ) ) ) ).
% streamsE
thf(fact_342_streamsE,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ~ ( ( member_list_a @ ( shd_list_a @ S2 ) @ A )
=> ~ ( member_stream_list_a @ ( stl_list_a @ S2 ) @ ( streams_list_a @ A ) ) ) ) ).
% streamsE
thf(fact_343_streamsE,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ~ ( ( member_a @ ( shd_a @ S2 ) @ A )
=> ~ ( member_stream_a @ ( stl_a @ S2 ) @ ( streams_a @ A ) ) ) ) ).
% streamsE
thf(fact_344_stream_Oexpand,axiom,
! [Stream: stream6512538624140940077od_a_a,Stream2: stream6512538624140940077od_a_a] :
( ( ( ( shd_Product_prod_a_a @ Stream )
= ( shd_Product_prod_a_a @ Stream2 ) )
& ( ( stl_Product_prod_a_a @ Stream )
= ( stl_Product_prod_a_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_345_stream_Oexpand,axiom,
! [Stream: stream8205385055505213747list_a,Stream2: stream8205385055505213747list_a] :
( ( ( ( shd_Pr4559591720970336857list_a @ Stream )
= ( shd_Pr4559591720970336857list_a @ Stream2 ) )
& ( ( stl_Pr4668768530486720605list_a @ Stream )
= ( stl_Pr4668768530486720605list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_346_stream_Oexpand,axiom,
! [Stream: stream5874003577662885159st_a_a,Stream2: stream5874003577662885159st_a_a] :
( ( ( ( shd_Pr7676374007274264653st_a_a @ Stream )
= ( shd_Pr7676374007274264653st_a_a @ Stream2 ) )
& ( ( stl_Pr7785550816790648401st_a_a @ Stream )
= ( stl_Pr7785550816790648401st_a_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_347_stream_Oexpand,axiom,
! [Stream: stream271976807216406061list_a,Stream2: stream271976807216406061list_a] :
( ( ( ( shd_Pr3168949583242555347list_a @ Stream )
= ( shd_Pr3168949583242555347list_a @ Stream2 ) )
& ( ( stl_Pr6107055601389525975list_a @ Stream )
= ( stl_Pr6107055601389525975list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_348_stream_Oexpand,axiom,
! [Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( ( ( shd_list_list_a @ Stream )
= ( shd_list_list_a @ Stream2 ) )
& ( ( stl_list_list_a @ Stream )
= ( stl_list_list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_349_stream_Oexpand,axiom,
! [Stream: stream_list_a,Stream2: stream_list_a] :
( ( ( ( shd_list_a @ Stream )
= ( shd_list_a @ Stream2 ) )
& ( ( stl_list_a @ Stream )
= ( stl_list_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_350_stream_Oexpand,axiom,
! [Stream: stream_a,Stream2: stream_a] :
( ( ( ( shd_a @ Stream )
= ( shd_a @ Stream2 ) )
& ( ( stl_a @ Stream )
= ( stl_a @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_351_stream_Ocoinduct,axiom,
! [R: stream6512538624140940077od_a_a > stream6512538624140940077od_a_a > $o,Stream: stream6512538624140940077od_a_a,Stream2: stream6512538624140940077od_a_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream6512538624140940077od_a_a,Stream4: stream6512538624140940077od_a_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Product_prod_a_a @ Stream3 )
= ( shd_Product_prod_a_a @ Stream4 ) )
& ( R @ ( stl_Product_prod_a_a @ Stream3 ) @ ( stl_Product_prod_a_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_352_stream_Ocoinduct,axiom,
! [R: stream8205385055505213747list_a > stream8205385055505213747list_a > $o,Stream: stream8205385055505213747list_a,Stream2: stream8205385055505213747list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream8205385055505213747list_a,Stream4: stream8205385055505213747list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr4559591720970336857list_a @ Stream3 )
= ( shd_Pr4559591720970336857list_a @ Stream4 ) )
& ( R @ ( stl_Pr4668768530486720605list_a @ Stream3 ) @ ( stl_Pr4668768530486720605list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_353_stream_Ocoinduct,axiom,
! [R: stream5874003577662885159st_a_a > stream5874003577662885159st_a_a > $o,Stream: stream5874003577662885159st_a_a,Stream2: stream5874003577662885159st_a_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream5874003577662885159st_a_a,Stream4: stream5874003577662885159st_a_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr7676374007274264653st_a_a @ Stream3 )
= ( shd_Pr7676374007274264653st_a_a @ Stream4 ) )
& ( R @ ( stl_Pr7785550816790648401st_a_a @ Stream3 ) @ ( stl_Pr7785550816790648401st_a_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_354_stream_Ocoinduct,axiom,
! [R: stream271976807216406061list_a > stream271976807216406061list_a > $o,Stream: stream271976807216406061list_a,Stream2: stream271976807216406061list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream271976807216406061list_a,Stream4: stream271976807216406061list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr3168949583242555347list_a @ Stream3 )
= ( shd_Pr3168949583242555347list_a @ Stream4 ) )
& ( R @ ( stl_Pr6107055601389525975list_a @ Stream3 ) @ ( stl_Pr6107055601389525975list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_355_stream_Ocoinduct,axiom,
! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_list_a @ Stream3 )
= ( shd_list_list_a @ Stream4 ) )
& ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_356_stream_Ocoinduct,axiom,
! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_a @ Stream3 )
= ( shd_list_a @ Stream4 ) )
& ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_357_stream_Ocoinduct,axiom,
! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_a,Stream4: stream_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_a @ Stream3 )
= ( shd_a @ Stream4 ) )
& ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_358_stream_Ocoinduct__strong,axiom,
! [R: stream6512538624140940077od_a_a > stream6512538624140940077od_a_a > $o,Stream: stream6512538624140940077od_a_a,Stream2: stream6512538624140940077od_a_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream6512538624140940077od_a_a,Stream4: stream6512538624140940077od_a_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Product_prod_a_a @ Stream3 )
= ( shd_Product_prod_a_a @ Stream4 ) )
& ( ( R @ ( stl_Product_prod_a_a @ Stream3 ) @ ( stl_Product_prod_a_a @ Stream4 ) )
| ( ( stl_Product_prod_a_a @ Stream3 )
= ( stl_Product_prod_a_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_359_stream_Ocoinduct__strong,axiom,
! [R: stream8205385055505213747list_a > stream8205385055505213747list_a > $o,Stream: stream8205385055505213747list_a,Stream2: stream8205385055505213747list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream8205385055505213747list_a,Stream4: stream8205385055505213747list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr4559591720970336857list_a @ Stream3 )
= ( shd_Pr4559591720970336857list_a @ Stream4 ) )
& ( ( R @ ( stl_Pr4668768530486720605list_a @ Stream3 ) @ ( stl_Pr4668768530486720605list_a @ Stream4 ) )
| ( ( stl_Pr4668768530486720605list_a @ Stream3 )
= ( stl_Pr4668768530486720605list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_360_stream_Ocoinduct__strong,axiom,
! [R: stream5874003577662885159st_a_a > stream5874003577662885159st_a_a > $o,Stream: stream5874003577662885159st_a_a,Stream2: stream5874003577662885159st_a_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream5874003577662885159st_a_a,Stream4: stream5874003577662885159st_a_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr7676374007274264653st_a_a @ Stream3 )
= ( shd_Pr7676374007274264653st_a_a @ Stream4 ) )
& ( ( R @ ( stl_Pr7785550816790648401st_a_a @ Stream3 ) @ ( stl_Pr7785550816790648401st_a_a @ Stream4 ) )
| ( ( stl_Pr7785550816790648401st_a_a @ Stream3 )
= ( stl_Pr7785550816790648401st_a_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_361_stream_Ocoinduct__strong,axiom,
! [R: stream271976807216406061list_a > stream271976807216406061list_a > $o,Stream: stream271976807216406061list_a,Stream2: stream271976807216406061list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream271976807216406061list_a,Stream4: stream271976807216406061list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr3168949583242555347list_a @ Stream3 )
= ( shd_Pr3168949583242555347list_a @ Stream4 ) )
& ( ( R @ ( stl_Pr6107055601389525975list_a @ Stream3 ) @ ( stl_Pr6107055601389525975list_a @ Stream4 ) )
| ( ( stl_Pr6107055601389525975list_a @ Stream3 )
= ( stl_Pr6107055601389525975list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_362_stream_Ocoinduct__strong,axiom,
! [R: stream_list_list_a > stream_list_list_a > $o,Stream: stream_list_list_a,Stream2: stream_list_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_list_a,Stream4: stream_list_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_list_a @ Stream3 )
= ( shd_list_list_a @ Stream4 ) )
& ( ( R @ ( stl_list_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) )
| ( ( stl_list_list_a @ Stream3 )
= ( stl_list_list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_363_stream_Ocoinduct__strong,axiom,
! [R: stream_list_a > stream_list_a > $o,Stream: stream_list_a,Stream2: stream_list_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_list_a @ Stream3 )
= ( shd_list_a @ Stream4 ) )
& ( ( R @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) )
| ( ( stl_list_a @ Stream3 )
= ( stl_list_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_364_stream_Ocoinduct__strong,axiom,
! [R: stream_a > stream_a > $o,Stream: stream_a,Stream2: stream_a] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream_a,Stream4: stream_a] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd_a @ Stream3 )
= ( shd_a @ Stream4 ) )
& ( ( R @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) )
| ( ( stl_a @ Stream3 )
= ( stl_a @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_365_shd__sset,axiom,
! [A2: stream6512538624140940077od_a_a] : ( member1426531477525435216od_a_a @ ( shd_Product_prod_a_a @ A2 ) @ ( sset_P5743994922208793403od_a_a @ A2 ) ) ).
% shd_sset
thf(fact_366_shd__sset,axiom,
! [A2: stream_stream_list_a] : ( member_stream_list_a @ ( shd_stream_list_a @ A2 ) @ ( sset_stream_list_a @ A2 ) ) ).
% shd_sset
thf(fact_367_shd__sset,axiom,
! [A2: stream_stream_a] : ( member_stream_a @ ( shd_stream_a @ A2 ) @ ( sset_stream_a @ A2 ) ) ).
% shd_sset
thf(fact_368_shd__sset,axiom,
! [A2: stream_set_a] : ( member_set_a @ ( shd_set_a @ A2 ) @ ( sset_set_a @ A2 ) ) ).
% shd_sset
thf(fact_369_shd__sset,axiom,
! [A2: stream_list_list_a] : ( member_list_list_a @ ( shd_list_list_a @ A2 ) @ ( sset_list_list_a @ A2 ) ) ).
% shd_sset
thf(fact_370_shd__sset,axiom,
! [A2: stream_a] : ( member_a @ ( shd_a @ A2 ) @ ( sset_a @ A2 ) ) ).
% shd_sset
thf(fact_371_shd__sset,axiom,
! [A2: stream_list_a] : ( member_list_a @ ( shd_list_a @ A2 ) @ ( sset_list_a @ A2 ) ) ).
% shd_sset
thf(fact_372_streams__mono,axiom,
! [S2: stream_list_a,A: set_list_a,B: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( member_stream_list_a @ S2 @ ( streams_list_a @ B ) ) ) ) ).
% streams_mono
thf(fact_373_streams__mono,axiom,
! [S2: stream_stream_a,A: set_stream_a,B: set_stream_a] :
( ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) )
=> ( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ B ) ) ) ) ).
% streams_mono
thf(fact_374_streams__mono,axiom,
! [S2: stream_a,A: set_a,B: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_stream_a @ S2 @ ( streams_a @ B ) ) ) ) ).
% streams_mono
thf(fact_375_streams__mono2,axiom,
! [S4: set_list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S4 @ T2 )
=> ( ord_le2348936426438760842list_a @ ( streams_list_a @ S4 ) @ ( streams_list_a @ T2 ) ) ) ).
% streams_mono2
thf(fact_376_streams__mono2,axiom,
! [S4: set_stream_a,T2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ S4 @ T2 )
=> ( ord_le2426980790783101578ream_a @ ( streams_stream_a @ S4 ) @ ( streams_stream_a @ T2 ) ) ) ).
% streams_mono2
thf(fact_377_streams__mono2,axiom,
! [S4: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S4 @ T2 )
=> ( ord_le1019412967036525316ream_a @ ( streams_a @ S4 ) @ ( streams_a @ T2 ) ) ) ).
% streams_mono2
thf(fact_378_streams__stl,axiom,
! [S2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ S2 @ ( stream6392229770749734815od_a_a @ A ) )
=> ( member262390649204803542od_a_a @ ( stl_Product_prod_a_a @ S2 ) @ ( stream6392229770749734815od_a_a @ A ) ) ) ).
% streams_stl
thf(fact_379_streams__stl,axiom,
! [S2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ S2 @ ( stream2987405652289500709list_a @ A ) )
=> ( member7566587766794810716list_a @ ( stl_Pr4668768530486720605list_a @ S2 ) @ ( stream2987405652289500709list_a @ A ) ) ) ).
% streams_stl
thf(fact_380_streams__stl,axiom,
! [S2: stream5874003577662885159st_a_a,A: set_Pr8962057229576493569st_a_a] :
( ( member5235206288952482128st_a_a @ S2 @ ( stream6104187938593428505st_a_a @ A ) )
=> ( member5235206288952482128st_a_a @ ( stl_Pr7785550816790648401st_a_a @ S2 ) @ ( stream6104187938593428505st_a_a @ A ) ) ) ).
% streams_stl
thf(fact_381_streams__stl,axiom,
! [S2: stream271976807216406061list_a,A: set_Pr4048851178543822343list_a] :
( ( member8659441389117227478list_a @ S2 @ ( stream1450977367002125727list_a @ A ) )
=> ( member8659441389117227478list_a @ ( stl_Pr6107055601389525975list_a @ S2 ) @ ( stream1450977367002125727list_a @ A ) ) ) ).
% streams_stl
thf(fact_382_streams__stl,axiom,
! [S2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ S2 @ ( streams_list_list_a @ A ) )
=> ( member8003905643527612601list_a @ ( stl_list_list_a @ S2 ) @ ( streams_list_list_a @ A ) ) ) ).
% streams_stl
thf(fact_383_streams__stl,axiom,
! [S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( member_stream_list_a @ ( stl_list_a @ S2 ) @ ( streams_list_a @ A ) ) ) ).
% streams_stl
thf(fact_384_streams__stl,axiom,
! [S2: stream_a,A: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( member_stream_a @ ( stl_a @ S2 ) @ ( streams_a @ A ) ) ) ).
% streams_stl
thf(fact_385_sdrop__while_Osimps,axiom,
( sdrop_5956399235322795403od_a_a
= ( ^ [P2: product_prod_a_a > $o,S: stream6512538624140940077od_a_a] : ( if_str8987598796600344819od_a_a @ ( P2 @ ( shd_Product_prod_a_a @ S ) ) @ ( sdrop_5956399235322795403od_a_a @ P2 @ ( stl_Product_prod_a_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_386_sdrop__while_Osimps,axiom,
( sdrop_8123123189179987473list_a
= ( ^ [P2: produc8685980395799941037list_a > $o,S: stream8205385055505213747list_a] : ( if_str4944004659744619513list_a @ ( P2 @ ( shd_Pr4559591720970336857list_a @ S ) ) @ ( sdrop_8123123189179987473list_a @ P2 @ ( stl_Pr4668768530486720605list_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_387_sdrop__while_Osimps,axiom,
( sdrop_2016533438629139461st_a_a
= ( ^ [P2: produc2579390645249093025st_a_a > $o,S: stream5874003577662885159st_a_a] : ( if_str2612623181902290925st_a_a @ ( P2 @ ( shd_Pr7676374007274264653st_a_a @ S ) ) @ ( sdrop_2016533438629139461st_a_a @ P2 @ ( stl_Pr7785550816790648401st_a_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_388_sdrop__while_Osimps,axiom,
( sdrop_4696227070857850763list_a
= ( ^ [P2: produc9164743771328383783list_a > $o,S: stream271976807216406061list_a] : ( if_str6195518498109691379list_a @ ( P2 @ ( shd_Pr3168949583242555347list_a @ S ) ) @ ( sdrop_4696227070857850763list_a @ P2 @ ( stl_Pr6107055601389525975list_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_389_sdrop__while_Osimps,axiom,
( sdrop_1072776591231973358list_a
= ( ^ [P2: list_list_a > $o,S: stream_list_list_a] : ( if_str7505741754068378070list_a @ ( P2 @ ( shd_list_list_a @ S ) ) @ ( sdrop_1072776591231973358list_a @ P2 @ ( stl_list_list_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_390_sdrop__while_Osimps,axiom,
( sdrop_while_list_a
= ( ^ [P2: list_a > $o,S: stream_list_a] : ( if_stream_list_a @ ( P2 @ ( shd_list_a @ S ) ) @ ( sdrop_while_list_a @ P2 @ ( stl_list_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_391_sdrop__while_Osimps,axiom,
( sdrop_while_a
= ( ^ [P2: a > $o,S: stream_a] : ( if_stream_a @ ( P2 @ ( shd_a @ S ) ) @ ( sdrop_while_a @ P2 @ ( stl_a @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_392_sinterleave_Ocode,axiom,
( sinter3949561980244396798st_a_a
= ( ^ [S12: stream5874003577662885159st_a_a,S23: stream5874003577662885159st_a_a] : ( sCons_1605959091327504952st_a_a @ ( shd_Pr7676374007274264653st_a_a @ S12 ) @ ( sinter3949561980244396798st_a_a @ S23 @ ( stl_Pr7785550816790648401st_a_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_393_sinterleave_Ocode,axiom,
( sinter5791379037304460804list_a
= ( ^ [S12: stream271976807216406061list_a,S23: stream271976807216406061list_a] : ( sCons_4508012570444740670list_a @ ( shd_Pr3168949583242555347list_a @ S12 ) @ ( sinter5791379037304460804list_a @ S23 @ ( stl_Pr6107055601389525975list_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_394_sinterleave_Ocode,axiom,
( sinter832779693940469002list_a
= ( ^ [S12: stream8205385055505213747list_a,S23: stream8205385055505213747list_a] : ( sCons_7712548841878352964list_a @ ( shd_Pr4559591720970336857list_a @ S12 ) @ ( sinter832779693940469002list_a @ S23 @ ( stl_Pr4668768530486720605list_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_395_sinterleave_Ocode,axiom,
( sinter5547950683322096900od_a_a
= ( ^ [S12: stream6512538624140940077od_a_a,S23: stream6512538624140940077od_a_a] : ( sCons_4259542806812723518od_a_a @ ( shd_Product_prod_a_a @ S12 ) @ ( sinter5547950683322096900od_a_a @ S23 @ ( stl_Product_prod_a_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_396_sinterleave_Ocode,axiom,
( sinter664328039231274855list_a
= ( ^ [S12: stream_list_list_a,S23: stream_list_list_a] : ( sCons_list_list_a @ ( shd_list_list_a @ S12 ) @ ( sinter664328039231274855list_a @ S23 @ ( stl_list_list_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_397_sinterleave_Ocode,axiom,
( sinterleave_list_a
= ( ^ [S12: stream_list_a,S23: stream_list_a] : ( sCons_list_a @ ( shd_list_a @ S12 ) @ ( sinterleave_list_a @ S23 @ ( stl_list_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_398_sinterleave_Ocode,axiom,
( sinterleave_a
= ( ^ [S12: stream_a,S23: stream_a] : ( sCons_a @ ( shd_a @ S12 ) @ ( sinterleave_a @ S23 @ ( stl_a @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_399_stream_Ocollapse,axiom,
! [Stream: stream5874003577662885159st_a_a] :
( ( sCons_1605959091327504952st_a_a @ ( shd_Pr7676374007274264653st_a_a @ Stream ) @ ( stl_Pr7785550816790648401st_a_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_400_stream_Ocollapse,axiom,
! [Stream: stream271976807216406061list_a] :
( ( sCons_4508012570444740670list_a @ ( shd_Pr3168949583242555347list_a @ Stream ) @ ( stl_Pr6107055601389525975list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_401_stream_Ocollapse,axiom,
! [Stream: stream8205385055505213747list_a] :
( ( sCons_7712548841878352964list_a @ ( shd_Pr4559591720970336857list_a @ Stream ) @ ( stl_Pr4668768530486720605list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_402_stream_Ocollapse,axiom,
! [Stream: stream6512538624140940077od_a_a] :
( ( sCons_4259542806812723518od_a_a @ ( shd_Product_prod_a_a @ Stream ) @ ( stl_Product_prod_a_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_403_stream_Ocollapse,axiom,
! [Stream: stream_list_list_a] :
( ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_404_stream_Ocollapse,axiom,
! [Stream: stream_list_a] :
( ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_405_stream_Ocollapse,axiom,
! [Stream: stream_a] :
( ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_406_sset__sinterleave,axiom,
! [S1: stream_a,S22: stream_a] :
( ( sset_a @ ( sinterleave_a @ S1 @ S22 ) )
= ( sup_sup_set_a @ ( sset_a @ S1 ) @ ( sset_a @ S22 ) ) ) ).
% sset_sinterleave
thf(fact_407_sset__sinterleave,axiom,
! [S1: stream_list_a,S22: stream_list_a] :
( ( sset_list_a @ ( sinterleave_list_a @ S1 @ S22 ) )
= ( sup_sup_set_list_a @ ( sset_list_a @ S1 ) @ ( sset_list_a @ S22 ) ) ) ).
% sset_sinterleave
thf(fact_408_flat_Osimps_I1_J,axiom,
! [Ws: stream_list_list_a] :
( ( shd_list_a @ ( flat_list_a @ Ws ) )
= ( hd_list_a @ ( shd_list_list_a @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_409_flat_Osimps_I1_J,axiom,
! [Ws: stream2255243159586646806list_a] :
( ( shd_list_list_a @ ( flat_list_list_a @ Ws ) )
= ( hd_list_list_a @ ( shd_list_list_list_a @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_410_flat_Osimps_I1_J,axiom,
! [Ws: stream_list_a] :
( ( shd_a @ ( flat_a @ Ws ) )
= ( hd_a @ ( shd_list_a @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_411_smember__code,axiom,
! [X2: produc8685980395799941037list_a,Y: produc8685980395799941037list_a,S2: stream8205385055505213747list_a] :
( ( smembe5596956825316798443list_a @ X2 @ ( sCons_7712548841878352964list_a @ Y @ S2 ) )
= ( ( X2 != Y )
=> ( smembe5596956825316798443list_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_412_smember__code,axiom,
! [X2: product_prod_a_a,Y: product_prod_a_a,S2: stream6512538624140940077od_a_a] :
( ( smembe8479462165187265637od_a_a @ X2 @ ( sCons_4259542806812723518od_a_a @ Y @ S2 ) )
= ( ( X2 != Y )
=> ( smembe8479462165187265637od_a_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_413_smember__code,axiom,
! [X2: list_list_a,Y: list_list_a,S2: stream_list_list_a] :
( ( smember_list_list_a @ X2 @ ( sCons_list_list_a @ Y @ S2 ) )
= ( ( X2 != Y )
=> ( smember_list_list_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_414_smember__code,axiom,
! [X2: list_a,Y: list_a,S2: stream_list_a] :
( ( smember_list_a @ X2 @ ( sCons_list_a @ Y @ S2 ) )
= ( ( X2 != Y )
=> ( smember_list_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_415_smember__code,axiom,
! [X2: a,Y: a,S2: stream_a] :
( ( smember_a @ X2 @ ( sCons_a @ Y @ S2 ) )
= ( ( X2 != Y )
=> ( smember_a @ X2 @ S2 ) ) ) ).
% smember_code
thf(fact_416_stream_Oexhaust__sel,axiom,
! [Stream: stream5874003577662885159st_a_a] :
( Stream
= ( sCons_1605959091327504952st_a_a @ ( shd_Pr7676374007274264653st_a_a @ Stream ) @ ( stl_Pr7785550816790648401st_a_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_417_stream_Oexhaust__sel,axiom,
! [Stream: stream271976807216406061list_a] :
( Stream
= ( sCons_4508012570444740670list_a @ ( shd_Pr3168949583242555347list_a @ Stream ) @ ( stl_Pr6107055601389525975list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_418_stream_Oexhaust__sel,axiom,
! [Stream: stream8205385055505213747list_a] :
( Stream
= ( sCons_7712548841878352964list_a @ ( shd_Pr4559591720970336857list_a @ Stream ) @ ( stl_Pr4668768530486720605list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_419_stream_Oexhaust__sel,axiom,
! [Stream: stream6512538624140940077od_a_a] :
( Stream
= ( sCons_4259542806812723518od_a_a @ ( shd_Product_prod_a_a @ Stream ) @ ( stl_Product_prod_a_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_420_stream_Oexhaust__sel,axiom,
! [Stream: stream_list_list_a] :
( Stream
= ( sCons_list_list_a @ ( shd_list_list_a @ Stream ) @ ( stl_list_list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_421_stream_Oexhaust__sel,axiom,
! [Stream: stream_list_a] :
( Stream
= ( sCons_list_a @ ( shd_list_a @ Stream ) @ ( stl_list_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_422_stream_Oexhaust__sel,axiom,
! [Stream: stream_a] :
( Stream
= ( sCons_a @ ( shd_a @ Stream ) @ ( stl_a @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_423_stream_Orel__coinduct,axiom,
! [P: stream_a > stream_a > $o,X2: stream_a,Y: stream_a,R: a > a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_a,Stream4: stream_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_a @ Stream3 ) @ ( shd_a @ Stream4 ) )
& ( P @ ( stl_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( stream_all2_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_424_stream_Orel__coinduct,axiom,
! [P: stream_list_a > stream_a > $o,X2: stream_list_a,Y: stream_a,R: list_a > a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_list_a,Stream4: stream_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_list_a @ Stream3 ) @ ( shd_a @ Stream4 ) )
& ( P @ ( stl_list_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( stream_all2_list_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_425_stream_Orel__coinduct,axiom,
! [P: stream_a > stream_list_a > $o,X2: stream_a,Y: stream_list_a,R: a > list_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_a,Stream4: stream_list_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_a @ Stream3 ) @ ( shd_list_a @ Stream4 ) )
& ( P @ ( stl_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) ) ) )
=> ( stream_all2_a_list_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_426_stream_Orel__coinduct,axiom,
! [P: stream_list_a > stream_list_a > $o,X2: stream_list_a,Y: stream_list_a,R: list_a > list_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_list_a @ Stream3 ) @ ( shd_list_a @ Stream4 ) )
& ( P @ ( stl_list_a @ Stream3 ) @ ( stl_list_a @ Stream4 ) ) ) )
=> ( stream4578335575074870093list_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_427_stream_Orel__coinduct,axiom,
! [P: stream_a > stream6512538624140940077od_a_a > $o,X2: stream_a,Y: stream6512538624140940077od_a_a,R: a > product_prod_a_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_a,Stream4: stream6512538624140940077od_a_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_a @ Stream3 ) @ ( shd_Product_prod_a_a @ Stream4 ) )
& ( P @ ( stl_a @ Stream3 ) @ ( stl_Product_prod_a_a @ Stream4 ) ) ) )
=> ( stream7396518378111496310od_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_428_stream_Orel__coinduct,axiom,
! [P: stream_a > stream_list_list_a > $o,X2: stream_a,Y: stream_list_list_a,R: a > list_list_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_a,Stream4: stream_list_list_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_a @ Stream3 ) @ ( shd_list_list_a @ Stream4 ) )
& ( P @ ( stl_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) ) ) )
=> ( stream2512895734020674265list_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_429_stream_Orel__coinduct,axiom,
! [P: stream6512538624140940077od_a_a > stream_a > $o,X2: stream6512538624140940077od_a_a,Y: stream_a,R: product_prod_a_a > a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream6512538624140940077od_a_a,Stream4: stream_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_Product_prod_a_a @ Stream3 ) @ ( shd_a @ Stream4 ) )
& ( P @ ( stl_Product_prod_a_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( stream3433837848420958564_a_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_430_stream_Orel__coinduct,axiom,
! [P: stream_list_list_a > stream_a > $o,X2: stream_list_list_a,Y: stream_a,R: list_list_a > a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_list_list_a,Stream4: stream_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_list_list_a @ Stream3 ) @ ( shd_a @ Stream4 ) )
& ( P @ ( stl_list_list_a @ Stream3 ) @ ( stl_a @ Stream4 ) ) ) )
=> ( stream3115889365687745985st_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_431_stream_Orel__coinduct,axiom,
! [P: stream_list_a > stream6512538624140940077od_a_a > $o,X2: stream_list_a,Y: stream6512538624140940077od_a_a,R: list_a > product_prod_a_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_list_a,Stream4: stream6512538624140940077od_a_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_list_a @ Stream3 ) @ ( shd_Product_prod_a_a @ Stream4 ) )
& ( P @ ( stl_list_a @ Stream3 ) @ ( stl_Product_prod_a_a @ Stream4 ) ) ) )
=> ( stream5366235361247201392od_a_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_432_stream_Orel__coinduct,axiom,
! [P: stream_list_a > stream_list_list_a > $o,X2: stream_list_a,Y: stream_list_list_a,R: list_a > list_list_a > $o] :
( ( P @ X2 @ Y )
=> ( ! [Stream3: stream_list_a,Stream4: stream_list_list_a] :
( ( P @ Stream3 @ Stream4 )
=> ( ( R @ ( shd_list_a @ Stream3 ) @ ( shd_list_list_a @ Stream4 ) )
& ( P @ ( stl_list_a @ Stream3 ) @ ( stl_list_list_a @ Stream4 ) ) ) )
=> ( stream482612717156379347list_a @ R @ X2 @ Y ) ) ) ).
% stream.rel_coinduct
thf(fact_433_stream_Orel__sel,axiom,
( stream_all2_a_a
= ( ^ [R2: a > a > $o,A4: stream_a,B4: stream_a] :
( ( R2 @ ( shd_a @ A4 ) @ ( shd_a @ B4 ) )
& ( stream_all2_a_a @ R2 @ ( stl_a @ A4 ) @ ( stl_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_434_stream_Orel__sel,axiom,
( stream_all2_list_a_a
= ( ^ [R2: list_a > a > $o,A4: stream_list_a,B4: stream_a] :
( ( R2 @ ( shd_list_a @ A4 ) @ ( shd_a @ B4 ) )
& ( stream_all2_list_a_a @ R2 @ ( stl_list_a @ A4 ) @ ( stl_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_435_stream_Orel__sel,axiom,
( stream_all2_a_list_a
= ( ^ [R2: a > list_a > $o,A4: stream_a,B4: stream_list_a] :
( ( R2 @ ( shd_a @ A4 ) @ ( shd_list_a @ B4 ) )
& ( stream_all2_a_list_a @ R2 @ ( stl_a @ A4 ) @ ( stl_list_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_436_stream_Orel__sel,axiom,
( stream4578335575074870093list_a
= ( ^ [R2: list_a > list_a > $o,A4: stream_list_a,B4: stream_list_a] :
( ( R2 @ ( shd_list_a @ A4 ) @ ( shd_list_a @ B4 ) )
& ( stream4578335575074870093list_a @ R2 @ ( stl_list_a @ A4 ) @ ( stl_list_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_437_stream_Orel__sel,axiom,
( stream7396518378111496310od_a_a
= ( ^ [R2: a > product_prod_a_a > $o,A4: stream_a,B4: stream6512538624140940077od_a_a] :
( ( R2 @ ( shd_a @ A4 ) @ ( shd_Product_prod_a_a @ B4 ) )
& ( stream7396518378111496310od_a_a @ R2 @ ( stl_a @ A4 ) @ ( stl_Product_prod_a_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_438_stream_Orel__sel,axiom,
( stream2512895734020674265list_a
= ( ^ [R2: a > list_list_a > $o,A4: stream_a,B4: stream_list_list_a] :
( ( R2 @ ( shd_a @ A4 ) @ ( shd_list_list_a @ B4 ) )
& ( stream2512895734020674265list_a @ R2 @ ( stl_a @ A4 ) @ ( stl_list_list_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_439_stream_Orel__sel,axiom,
( stream3433837848420958564_a_a_a
= ( ^ [R2: product_prod_a_a > a > $o,A4: stream6512538624140940077od_a_a,B4: stream_a] :
( ( R2 @ ( shd_Product_prod_a_a @ A4 ) @ ( shd_a @ B4 ) )
& ( stream3433837848420958564_a_a_a @ R2 @ ( stl_Product_prod_a_a @ A4 ) @ ( stl_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_440_stream_Orel__sel,axiom,
( stream3115889365687745985st_a_a
= ( ^ [R2: list_list_a > a > $o,A4: stream_list_list_a,B4: stream_a] :
( ( R2 @ ( shd_list_list_a @ A4 ) @ ( shd_a @ B4 ) )
& ( stream3115889365687745985st_a_a @ R2 @ ( stl_list_list_a @ A4 ) @ ( stl_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_441_stream_Orel__sel,axiom,
( stream5366235361247201392od_a_a
= ( ^ [R2: list_a > product_prod_a_a > $o,A4: stream_list_a,B4: stream6512538624140940077od_a_a] :
( ( R2 @ ( shd_list_a @ A4 ) @ ( shd_Product_prod_a_a @ B4 ) )
& ( stream5366235361247201392od_a_a @ R2 @ ( stl_list_a @ A4 ) @ ( stl_Product_prod_a_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_442_stream_Orel__sel,axiom,
( stream482612717156379347list_a
= ( ^ [R2: list_a > list_list_a > $o,A4: stream_list_a,B4: stream_list_list_a] :
( ( R2 @ ( shd_list_a @ A4 ) @ ( shd_list_list_a @ B4 ) )
& ( stream482612717156379347list_a @ R2 @ ( stl_list_a @ A4 ) @ ( stl_list_list_a @ B4 ) ) ) ) ) ).
% stream.rel_sel
thf(fact_443_szip_Osimps_I2_J,axiom,
! [S1: stream_a,S22: stream_a] :
( ( stl_Product_prod_a_a @ ( szip_a_a @ S1 @ S22 ) )
= ( szip_a_a @ ( stl_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_444_szip_Osimps_I2_J,axiom,
! [S1: stream_list_a,S22: stream_a] :
( ( stl_Pr7785550816790648401st_a_a @ ( szip_list_a_a @ S1 @ S22 ) )
= ( szip_list_a_a @ ( stl_list_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_445_szip_Osimps_I2_J,axiom,
! [S1: stream_a,S22: stream_list_a] :
( ( stl_Pr4668768530486720605list_a @ ( szip_a_list_a @ S1 @ S22 ) )
= ( szip_a_list_a @ ( stl_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_446_szip_Osimps_I2_J,axiom,
! [S1: stream_list_a,S22: stream_list_a] :
( ( stl_Pr6107055601389525975list_a @ ( szip_list_a_list_a @ S1 @ S22 ) )
= ( szip_list_a_list_a @ ( stl_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_447_szip_Osimps_I2_J,axiom,
! [S1: stream_a,S22: stream6512538624140940077od_a_a] :
( ( stl_Pr986409416061048192od_a_a @ ( szip_a2498797358816541059od_a_a @ S1 @ S22 ) )
= ( szip_a2498797358816541059od_a_a @ ( stl_a @ S1 ) @ ( stl_Product_prod_a_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_448_szip_Osimps_I2_J,axiom,
! [S1: stream_a,S22: stream_list_list_a] :
( ( stl_Pr8727924410383857251list_a @ ( szip_a_list_list_a @ S1 @ S22 ) )
= ( szip_a_list_list_a @ ( stl_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_449_szip_Osimps_I2_J,axiom,
! [S1: stream6512538624140940077od_a_a,S22: stream_a] :
( ( stl_Pr745203880014032622_a_a_a @ ( szip_P7759488865980779121_a_a_a @ S1 @ S22 ) )
= ( szip_P7759488865980779121_a_a_a @ ( stl_Product_prod_a_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_450_szip_Osimps_I2_J,axiom,
! [S1: stream_list_list_a,S22: stream_a] :
( ( stl_Pr427255397280820043st_a_a @ ( szip_list_list_a_a @ S1 @ S22 ) )
= ( szip_list_list_a_a @ ( stl_list_list_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_451_szip_Osimps_I2_J,axiom,
! [S1: stream_list_a,S22: stream6512538624140940077od_a_a] :
( ( stl_Pr2324757703178138618od_a_a @ ( szip_l3670113457280575229od_a_a @ S1 @ S22 ) )
= ( szip_l3670113457280575229od_a_a @ ( stl_list_a @ S1 ) @ ( stl_Product_prod_a_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_452_szip_Osimps_I2_J,axiom,
! [S1: stream_list_a,S22: stream_list_list_a] :
( ( stl_Pr842900660646171869list_a @ ( szip_l8009862850044528992list_a @ S1 @ S22 ) )
= ( szip_l8009862850044528992list_a @ ( stl_list_a @ S1 ) @ ( stl_list_list_a @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_453_Un__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
| ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_454_Un__iff,axiom,
! [C: stream_list_a,A: set_stream_list_a,B: set_stream_list_a] :
( ( member_stream_list_a @ C @ ( sup_su5853061255542491070list_a @ A @ B ) )
= ( ( member_stream_list_a @ C @ A )
| ( member_stream_list_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_455_Un__iff,axiom,
! [C: stream_a,A: set_stream_a,B: set_stream_a] :
( ( member_stream_a @ C @ ( sup_sup_set_stream_a @ A @ B ) )
= ( ( member_stream_a @ C @ A )
| ( member_stream_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_456_Un__iff,axiom,
! [C: list_list_a,A: set_list_list_a,B: set_list_list_a] :
( ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B ) )
= ( ( member_list_list_a @ C @ A )
| ( member_list_list_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_457_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_458_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_459_Un__iff,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B ) )
= ( ( member_list_a @ C @ A )
| ( member_list_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_460_UnCI,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ A ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_461_UnCI,axiom,
! [C: stream_list_a,B: set_stream_list_a,A: set_stream_list_a] :
( ( ~ ( member_stream_list_a @ C @ B )
=> ( member_stream_list_a @ C @ A ) )
=> ( member_stream_list_a @ C @ ( sup_su5853061255542491070list_a @ A @ B ) ) ) ).
% UnCI
thf(fact_462_UnCI,axiom,
! [C: stream_a,B: set_stream_a,A: set_stream_a] :
( ( ~ ( member_stream_a @ C @ B )
=> ( member_stream_a @ C @ A ) )
=> ( member_stream_a @ C @ ( sup_sup_set_stream_a @ A @ B ) ) ) ).
% UnCI
thf(fact_463_UnCI,axiom,
! [C: list_list_a,B: set_list_list_a,A: set_list_list_a] :
( ( ~ ( member_list_list_a @ C @ B )
=> ( member_list_list_a @ C @ A ) )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B ) ) ) ).
% UnCI
thf(fact_464_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_465_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_466_UnCI,axiom,
! [C: list_a,B: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C @ B )
=> ( member_list_a @ C @ A ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnCI
thf(fact_467_stream_Oinject,axiom,
! [X1: produc8685980395799941037list_a,X22: stream8205385055505213747list_a,Y1: produc8685980395799941037list_a,Y22: stream8205385055505213747list_a] :
( ( ( sCons_7712548841878352964list_a @ X1 @ X22 )
= ( sCons_7712548841878352964list_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_468_stream_Oinject,axiom,
! [X1: product_prod_a_a,X22: stream6512538624140940077od_a_a,Y1: product_prod_a_a,Y22: stream6512538624140940077od_a_a] :
( ( ( sCons_4259542806812723518od_a_a @ X1 @ X22 )
= ( sCons_4259542806812723518od_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_469_stream_Oinject,axiom,
! [X1: list_list_a,X22: stream_list_list_a,Y1: list_list_a,Y22: stream_list_list_a] :
( ( ( sCons_list_list_a @ X1 @ X22 )
= ( sCons_list_list_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_470_stream_Oinject,axiom,
! [X1: list_a,X22: stream_list_a,Y1: list_a,Y22: stream_list_a] :
( ( ( sCons_list_a @ X1 @ X22 )
= ( sCons_list_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_471_stream_Oinject,axiom,
! [X1: a,X22: stream_a,Y1: a,Y22: stream_a] :
( ( ( sCons_a @ X1 @ X22 )
= ( sCons_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_472_Un__subset__iff,axiom,
! [A: set_list_a,B: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C2 )
= ( ( ord_le8861187494160871172list_a @ A @ C2 )
& ( ord_le8861187494160871172list_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_473_Un__subset__iff,axiom,
! [A: set_stream_a,B: set_stream_a,C2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ A @ B ) @ C2 )
= ( ( ord_le1019412967036525316ream_a @ A @ C2 )
& ( ord_le1019412967036525316ream_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_474_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_475_stream_Orel__inject,axiom,
! [R: a > a > $o,X1: a,X22: stream_a,Y1: a,Y22: stream_a] :
( ( stream_all2_a_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream_all2_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_476_stream_Orel__inject,axiom,
! [R: a > list_a > $o,X1: a,X22: stream_a,Y1: list_a,Y22: stream_list_a] :
( ( stream_all2_a_list_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream_all2_a_list_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_477_stream_Orel__inject,axiom,
! [R: list_a > a > $o,X1: list_a,X22: stream_list_a,Y1: a,Y22: stream_a] :
( ( stream_all2_list_a_a @ R @ ( sCons_list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream_all2_list_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_478_stream_Orel__inject,axiom,
! [R: product_prod_a_a > a > $o,X1: product_prod_a_a,X22: stream6512538624140940077od_a_a,Y1: a,Y22: stream_a] :
( ( stream3433837848420958564_a_a_a @ R @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream3433837848420958564_a_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_479_stream_Orel__inject,axiom,
! [R: list_list_a > a > $o,X1: list_list_a,X22: stream_list_list_a,Y1: a,Y22: stream_a] :
( ( stream3115889365687745985st_a_a @ R @ ( sCons_list_list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream3115889365687745985st_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_480_stream_Orel__inject,axiom,
! [R: a > product_prod_a_a > $o,X1: a,X22: stream_a,Y1: product_prod_a_a,Y22: stream6512538624140940077od_a_a] :
( ( stream7396518378111496310od_a_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_4259542806812723518od_a_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream7396518378111496310od_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_481_stream_Orel__inject,axiom,
! [R: a > list_list_a > $o,X1: a,X22: stream_a,Y1: list_list_a,Y22: stream_list_list_a] :
( ( stream2512895734020674265list_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_list_list_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream2512895734020674265list_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_482_stream_Orel__inject,axiom,
! [R: list_a > list_a > $o,X1: list_a,X22: stream_list_a,Y1: list_a,Y22: stream_list_a] :
( ( stream4578335575074870093list_a @ R @ ( sCons_list_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream4578335575074870093list_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_483_stream_Orel__inject,axiom,
! [R: produc8685980395799941037list_a > a > $o,X1: produc8685980395799941037list_a,X22: stream8205385055505213747list_a,Y1: a,Y22: stream_a] :
( ( stream568159088382975326st_a_a @ R @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream568159088382975326st_a_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_484_stream_Orel__inject,axiom,
! [R: product_prod_a_a > list_a > $o,X1: product_prod_a_a,X22: stream6512538624140940077od_a_a,Y1: list_a,Y22: stream_list_a] :
( ( stream2030605297770099434list_a @ R @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) )
= ( ( R @ X1 @ Y1 )
& ( stream2030605297770099434list_a @ R @ X22 @ Y22 ) ) ) ).
% stream.rel_inject
thf(fact_485_stream_Orel__mono,axiom,
! [R: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R @ Ra )
=> ( ord_le2315692933029977486am_a_o @ ( stream_all2_a_a @ R ) @ ( stream_all2_a_a @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_486_stream_Orel__mono,axiom,
! [R: a > list_a > $o,Ra: a > list_a > $o] :
( ( ord_le2844220955438509960st_a_o @ R @ Ra )
=> ( ord_le4668208666631526024st_a_o @ ( stream_all2_a_list_a @ R ) @ ( stream_all2_a_list_a @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_487_stream_Orel__mono,axiom,
! [R: list_a > a > $o,Ra: list_a > a > $o] :
( ( ord_le1869217262537267988_a_a_o @ R @ Ra )
=> ( ord_le9064767629836747540am_a_o @ ( stream_all2_list_a_a @ R ) @ ( stream_all2_list_a_a @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_488_stream_Orel__mono,axiom,
! [R: list_a > list_a > $o,Ra: list_a > list_a > $o] :
( ( ord_le5542992221119063950st_a_o @ R @ Ra )
=> ( ord_le4962707062786359566st_a_o @ ( stream4578335575074870093list_a @ R ) @ ( stream4578335575074870093list_a @ Ra ) ) ) ).
% stream.rel_mono
thf(fact_489_sdrop__while__SCons,axiom,
! [P: produc8685980395799941037list_a > $o,A2: produc8685980395799941037list_a,S2: stream8205385055505213747list_a] :
( ( ( P @ A2 )
=> ( ( sdrop_8123123189179987473list_a @ P @ ( sCons_7712548841878352964list_a @ A2 @ S2 ) )
= ( sdrop_8123123189179987473list_a @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_8123123189179987473list_a @ P @ ( sCons_7712548841878352964list_a @ A2 @ S2 ) )
= ( sCons_7712548841878352964list_a @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_490_sdrop__while__SCons,axiom,
! [P: product_prod_a_a > $o,A2: product_prod_a_a,S2: stream6512538624140940077od_a_a] :
( ( ( P @ A2 )
=> ( ( sdrop_5956399235322795403od_a_a @ P @ ( sCons_4259542806812723518od_a_a @ A2 @ S2 ) )
= ( sdrop_5956399235322795403od_a_a @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_5956399235322795403od_a_a @ P @ ( sCons_4259542806812723518od_a_a @ A2 @ S2 ) )
= ( sCons_4259542806812723518od_a_a @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_491_sdrop__while__SCons,axiom,
! [P: list_list_a > $o,A2: list_list_a,S2: stream_list_list_a] :
( ( ( P @ A2 )
=> ( ( sdrop_1072776591231973358list_a @ P @ ( sCons_list_list_a @ A2 @ S2 ) )
= ( sdrop_1072776591231973358list_a @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_1072776591231973358list_a @ P @ ( sCons_list_list_a @ A2 @ S2 ) )
= ( sCons_list_list_a @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_492_sdrop__while__SCons,axiom,
! [P: list_a > $o,A2: list_a,S2: stream_list_a] :
( ( ( P @ A2 )
=> ( ( sdrop_while_list_a @ P @ ( sCons_list_a @ A2 @ S2 ) )
= ( sdrop_while_list_a @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_while_list_a @ P @ ( sCons_list_a @ A2 @ S2 ) )
= ( sCons_list_a @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_493_sdrop__while__SCons,axiom,
! [P: a > $o,A2: a,S2: stream_a] :
( ( ( P @ A2 )
=> ( ( sdrop_while_a @ P @ ( sCons_a @ A2 @ S2 ) )
= ( sdrop_while_a @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_while_a @ P @ ( sCons_a @ A2 @ S2 ) )
= ( sCons_a @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_494_Un__left__commute,axiom,
! [A: set_list_a,B: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C2 ) )
= ( sup_sup_set_list_a @ B @ ( sup_sup_set_list_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_495_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_496_Un__left__absorb,axiom,
! [A: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ A @ B ) )
= ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_497_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_498_Un__commute,axiom,
( sup_sup_set_list_a
= ( ^ [A3: set_list_a,B2: set_list_a] : ( sup_sup_set_list_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_499_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_500_Un__absorb,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_501_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_502_Un__assoc,axiom,
! [A: set_list_a,B: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A @ B ) @ C2 )
= ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_503_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_504_ball__Un,axiom,
! [A: set_list_a,B: set_list_a,P: list_a > $o] :
( ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_505_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_506_bex__Un,axiom,
! [A: set_list_a,B: set_list_a,P: list_a > $o] :
( ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: list_a] :
( ( member_list_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_507_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_508_UnI2,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_509_UnI2,axiom,
! [C: stream_list_a,B: set_stream_list_a,A: set_stream_list_a] :
( ( member_stream_list_a @ C @ B )
=> ( member_stream_list_a @ C @ ( sup_su5853061255542491070list_a @ A @ B ) ) ) ).
% UnI2
thf(fact_510_UnI2,axiom,
! [C: stream_a,B: set_stream_a,A: set_stream_a] :
( ( member_stream_a @ C @ B )
=> ( member_stream_a @ C @ ( sup_sup_set_stream_a @ A @ B ) ) ) ).
% UnI2
thf(fact_511_UnI2,axiom,
! [C: list_list_a,B: set_list_list_a,A: set_list_list_a] :
( ( member_list_list_a @ C @ B )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B ) ) ) ).
% UnI2
thf(fact_512_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_513_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_514_UnI2,axiom,
! [C: list_a,B: set_list_a,A: set_list_a] :
( ( member_list_a @ C @ B )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnI2
thf(fact_515_UnI1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_516_UnI1,axiom,
! [C: stream_list_a,A: set_stream_list_a,B: set_stream_list_a] :
( ( member_stream_list_a @ C @ A )
=> ( member_stream_list_a @ C @ ( sup_su5853061255542491070list_a @ A @ B ) ) ) ).
% UnI1
thf(fact_517_UnI1,axiom,
! [C: stream_a,A: set_stream_a,B: set_stream_a] :
( ( member_stream_a @ C @ A )
=> ( member_stream_a @ C @ ( sup_sup_set_stream_a @ A @ B ) ) ) ).
% UnI1
thf(fact_518_UnI1,axiom,
! [C: list_list_a,A: set_list_list_a,B: set_list_list_a] :
( ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B ) ) ) ).
% UnI1
thf(fact_519_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_520_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_521_UnI1,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnI1
thf(fact_522_UnE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_523_UnE,axiom,
! [C: stream_list_a,A: set_stream_list_a,B: set_stream_list_a] :
( ( member_stream_list_a @ C @ ( sup_su5853061255542491070list_a @ A @ B ) )
=> ( ~ ( member_stream_list_a @ C @ A )
=> ( member_stream_list_a @ C @ B ) ) ) ).
% UnE
thf(fact_524_UnE,axiom,
! [C: stream_a,A: set_stream_a,B: set_stream_a] :
( ( member_stream_a @ C @ ( sup_sup_set_stream_a @ A @ B ) )
=> ( ~ ( member_stream_a @ C @ A )
=> ( member_stream_a @ C @ B ) ) ) ).
% UnE
thf(fact_525_UnE,axiom,
! [C: list_list_a,A: set_list_list_a,B: set_list_list_a] :
( ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B ) )
=> ( ~ ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ B ) ) ) ).
% UnE
thf(fact_526_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_527_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_528_UnE,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B ) )
=> ( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% UnE
thf(fact_529_stream_Orel__refl,axiom,
! [Ra: a > a > $o,X2: stream_a] :
( ! [X: a] : ( Ra @ X @ X )
=> ( stream_all2_a_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl
thf(fact_530_stream_Orel__refl,axiom,
! [Ra: list_a > list_a > $o,X2: stream_list_a] :
( ! [X: list_a] : ( Ra @ X @ X )
=> ( stream4578335575074870093list_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl
thf(fact_531_stream_Orel__eq,axiom,
( ( stream_all2_a_a
@ ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [Y3: stream_a,Z: stream_a] : ( Y3 = Z ) ) ) ).
% stream.rel_eq
thf(fact_532_stream_Orel__eq,axiom,
( ( stream4578335575074870093list_a
@ ^ [Y3: list_a,Z: list_a] : ( Y3 = Z ) )
= ( ^ [Y3: stream_list_a,Z: stream_list_a] : ( Y3 = Z ) ) ) ).
% stream.rel_eq
thf(fact_533_stream_Orel__intros,axiom,
! [R: a > a > $o,X1: a,Y1: a,X22: stream_a,Y22: stream_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream_all2_a_a @ R @ X22 @ Y22 )
=> ( stream_all2_a_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_534_stream_Orel__intros,axiom,
! [R: a > list_a > $o,X1: a,Y1: list_a,X22: stream_a,Y22: stream_list_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream_all2_a_list_a @ R @ X22 @ Y22 )
=> ( stream_all2_a_list_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_535_stream_Orel__intros,axiom,
! [R: list_a > a > $o,X1: list_a,Y1: a,X22: stream_list_a,Y22: stream_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream_all2_list_a_a @ R @ X22 @ Y22 )
=> ( stream_all2_list_a_a @ R @ ( sCons_list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_536_stream_Orel__intros,axiom,
! [R: product_prod_a_a > a > $o,X1: product_prod_a_a,Y1: a,X22: stream6512538624140940077od_a_a,Y22: stream_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream3433837848420958564_a_a_a @ R @ X22 @ Y22 )
=> ( stream3433837848420958564_a_a_a @ R @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_537_stream_Orel__intros,axiom,
! [R: list_list_a > a > $o,X1: list_list_a,Y1: a,X22: stream_list_list_a,Y22: stream_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream3115889365687745985st_a_a @ R @ X22 @ Y22 )
=> ( stream3115889365687745985st_a_a @ R @ ( sCons_list_list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_538_stream_Orel__intros,axiom,
! [R: a > product_prod_a_a > $o,X1: a,Y1: product_prod_a_a,X22: stream_a,Y22: stream6512538624140940077od_a_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream7396518378111496310od_a_a @ R @ X22 @ Y22 )
=> ( stream7396518378111496310od_a_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_4259542806812723518od_a_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_539_stream_Orel__intros,axiom,
! [R: a > list_list_a > $o,X1: a,Y1: list_list_a,X22: stream_a,Y22: stream_list_list_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream2512895734020674265list_a @ R @ X22 @ Y22 )
=> ( stream2512895734020674265list_a @ R @ ( sCons_a @ X1 @ X22 ) @ ( sCons_list_list_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_540_stream_Orel__intros,axiom,
! [R: list_a > list_a > $o,X1: list_a,Y1: list_a,X22: stream_list_a,Y22: stream_list_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream4578335575074870093list_a @ R @ X22 @ Y22 )
=> ( stream4578335575074870093list_a @ R @ ( sCons_list_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_541_stream_Orel__intros,axiom,
! [R: produc8685980395799941037list_a > a > $o,X1: produc8685980395799941037list_a,Y1: a,X22: stream8205385055505213747list_a,Y22: stream_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream568159088382975326st_a_a @ R @ X22 @ Y22 )
=> ( stream568159088382975326st_a_a @ R @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) @ ( sCons_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_542_stream_Orel__intros,axiom,
! [R: product_prod_a_a > list_a > $o,X1: product_prod_a_a,Y1: list_a,X22: stream6512538624140940077od_a_a,Y22: stream_list_a] :
( ( R @ X1 @ Y1 )
=> ( ( stream2030605297770099434list_a @ R @ X22 @ Y22 )
=> ( stream2030605297770099434list_a @ R @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) @ ( sCons_list_a @ Y1 @ Y22 ) ) ) ) ).
% stream.rel_intros
thf(fact_543_stream_Orel__cases,axiom,
! [R: a > a > $o,A2: stream_a,B3: stream_a] :
( ( stream_all2_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: a,X2a: stream_a] :
( ( A2
= ( sCons_a @ X1a @ X2a ) )
=> ! [Y1a: a,Y2a: stream_a] :
( ( B3
= ( sCons_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream_all2_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_544_stream_Orel__cases,axiom,
! [R: a > list_a > $o,A2: stream_a,B3: stream_list_a] :
( ( stream_all2_a_list_a @ R @ A2 @ B3 )
=> ~ ! [X1a: a,X2a: stream_a] :
( ( A2
= ( sCons_a @ X1a @ X2a ) )
=> ! [Y1a: list_a,Y2a: stream_list_a] :
( ( B3
= ( sCons_list_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream_all2_a_list_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_545_stream_Orel__cases,axiom,
! [R: list_a > a > $o,A2: stream_list_a,B3: stream_a] :
( ( stream_all2_list_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: list_a,X2a: stream_list_a] :
( ( A2
= ( sCons_list_a @ X1a @ X2a ) )
=> ! [Y1a: a,Y2a: stream_a] :
( ( B3
= ( sCons_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream_all2_list_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_546_stream_Orel__cases,axiom,
! [R: product_prod_a_a > a > $o,A2: stream6512538624140940077od_a_a,B3: stream_a] :
( ( stream3433837848420958564_a_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: product_prod_a_a,X2a: stream6512538624140940077od_a_a] :
( ( A2
= ( sCons_4259542806812723518od_a_a @ X1a @ X2a ) )
=> ! [Y1a: a,Y2a: stream_a] :
( ( B3
= ( sCons_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream3433837848420958564_a_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_547_stream_Orel__cases,axiom,
! [R: list_list_a > a > $o,A2: stream_list_list_a,B3: stream_a] :
( ( stream3115889365687745985st_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: list_list_a,X2a: stream_list_list_a] :
( ( A2
= ( sCons_list_list_a @ X1a @ X2a ) )
=> ! [Y1a: a,Y2a: stream_a] :
( ( B3
= ( sCons_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream3115889365687745985st_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_548_stream_Orel__cases,axiom,
! [R: a > product_prod_a_a > $o,A2: stream_a,B3: stream6512538624140940077od_a_a] :
( ( stream7396518378111496310od_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: a,X2a: stream_a] :
( ( A2
= ( sCons_a @ X1a @ X2a ) )
=> ! [Y1a: product_prod_a_a,Y2a: stream6512538624140940077od_a_a] :
( ( B3
= ( sCons_4259542806812723518od_a_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream7396518378111496310od_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_549_stream_Orel__cases,axiom,
! [R: a > list_list_a > $o,A2: stream_a,B3: stream_list_list_a] :
( ( stream2512895734020674265list_a @ R @ A2 @ B3 )
=> ~ ! [X1a: a,X2a: stream_a] :
( ( A2
= ( sCons_a @ X1a @ X2a ) )
=> ! [Y1a: list_list_a,Y2a: stream_list_list_a] :
( ( B3
= ( sCons_list_list_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream2512895734020674265list_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_550_stream_Orel__cases,axiom,
! [R: list_a > list_a > $o,A2: stream_list_a,B3: stream_list_a] :
( ( stream4578335575074870093list_a @ R @ A2 @ B3 )
=> ~ ! [X1a: list_a,X2a: stream_list_a] :
( ( A2
= ( sCons_list_a @ X1a @ X2a ) )
=> ! [Y1a: list_a,Y2a: stream_list_a] :
( ( B3
= ( sCons_list_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream4578335575074870093list_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_551_stream_Orel__cases,axiom,
! [R: produc8685980395799941037list_a > a > $o,A2: stream8205385055505213747list_a,B3: stream_a] :
( ( stream568159088382975326st_a_a @ R @ A2 @ B3 )
=> ~ ! [X1a: produc8685980395799941037list_a,X2a: stream8205385055505213747list_a] :
( ( A2
= ( sCons_7712548841878352964list_a @ X1a @ X2a ) )
=> ! [Y1a: a,Y2a: stream_a] :
( ( B3
= ( sCons_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream568159088382975326st_a_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_552_stream_Orel__cases,axiom,
! [R: product_prod_a_a > list_a > $o,A2: stream6512538624140940077od_a_a,B3: stream_list_a] :
( ( stream2030605297770099434list_a @ R @ A2 @ B3 )
=> ~ ! [X1a: product_prod_a_a,X2a: stream6512538624140940077od_a_a] :
( ( A2
= ( sCons_4259542806812723518od_a_a @ X1a @ X2a ) )
=> ! [Y1a: list_a,Y2a: stream_list_a] :
( ( B3
= ( sCons_list_a @ Y1a @ Y2a ) )
=> ( ( R @ X1a @ Y1a )
=> ~ ( stream2030605297770099434list_a @ R @ X2a @ Y2a ) ) ) ) ) ).
% stream.rel_cases
thf(fact_553_stream_Oexhaust,axiom,
! [Y: stream8205385055505213747list_a] :
~ ! [X12: produc8685980395799941037list_a,X23: stream8205385055505213747list_a] :
( Y
!= ( sCons_7712548841878352964list_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_554_stream_Oexhaust,axiom,
! [Y: stream6512538624140940077od_a_a] :
~ ! [X12: product_prod_a_a,X23: stream6512538624140940077od_a_a] :
( Y
!= ( sCons_4259542806812723518od_a_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_555_stream_Oexhaust,axiom,
! [Y: stream_list_list_a] :
~ ! [X12: list_list_a,X23: stream_list_list_a] :
( Y
!= ( sCons_list_list_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_556_stream_Oexhaust,axiom,
! [Y: stream_list_a] :
~ ! [X12: list_a,X23: stream_list_a] :
( Y
!= ( sCons_list_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_557_stream_Oexhaust,axiom,
! [Y: stream_a] :
~ ! [X12: a,X23: stream_a] :
( Y
!= ( sCons_a @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_558_subset__Un__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_559_subset__Un__eq,axiom,
( ord_le1019412967036525316ream_a
= ( ^ [A3: set_stream_a,B2: set_stream_a] :
( ( sup_sup_set_stream_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_560_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_561_subset__UnE,axiom,
! [C2: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) )
=> ~ ! [A6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ A )
=> ! [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ B )
=> ( C2
!= ( sup_sup_set_list_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_562_subset__UnE,axiom,
! [C2: set_stream_a,A: set_stream_a,B: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ C2 @ ( sup_sup_set_stream_a @ A @ B ) )
=> ~ ! [A6: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A6 @ A )
=> ! [B6: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ B6 @ B )
=> ( C2
!= ( sup_sup_set_stream_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_563_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_564_Un__absorb2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( sup_sup_set_list_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_565_Un__absorb2,axiom,
! [B: set_stream_a,A: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ B @ A )
=> ( ( sup_sup_set_stream_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_566_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_567_Un__absorb1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( sup_sup_set_list_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_568_Un__absorb1,axiom,
! [A: set_stream_a,B: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ B )
=> ( ( sup_sup_set_stream_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_569_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_570_Un__upper2,axiom,
! [B: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper2
thf(fact_571_Un__upper2,axiom,
! [B: set_stream_a,A: set_stream_a] : ( ord_le1019412967036525316ream_a @ B @ ( sup_sup_set_stream_a @ A @ B ) ) ).
% Un_upper2
thf(fact_572_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_573_Un__upper1,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper1
thf(fact_574_Un__upper1,axiom,
! [A: set_stream_a,B: set_stream_a] : ( ord_le1019412967036525316ream_a @ A @ ( sup_sup_set_stream_a @ A @ B ) ) ).
% Un_upper1
thf(fact_575_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_576_Un__least,axiom,
! [A: set_list_a,C2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B @ C2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_577_Un__least,axiom,
! [A: set_stream_a,C2: set_stream_a,B: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ C2 )
=> ( ( ord_le1019412967036525316ream_a @ B @ C2 )
=> ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_578_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_579_Un__mono,axiom,
! [A: set_list_a,C2: set_list_a,B: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B @ D )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ ( sup_sup_set_list_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_580_Un__mono,axiom,
! [A: set_stream_a,C2: set_stream_a,B: set_stream_a,D: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ A @ C2 )
=> ( ( ord_le1019412967036525316ream_a @ B @ D )
=> ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ A @ B ) @ ( sup_sup_set_stream_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_581_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_582_stream_Orel__refl__strong,axiom,
! [X2: stream6512538624140940077od_a_a,Ra: product_prod_a_a > product_prod_a_a > $o] :
( ! [Z3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z3 @ ( sset_P5743994922208793403od_a_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream4617964345477461389od_a_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_583_stream_Orel__refl__strong,axiom,
! [X2: stream_stream_list_a,Ra: stream_list_a > stream_list_a > $o] :
( ! [Z3: stream_list_a] :
( ( member_stream_list_a @ Z3 @ ( sset_stream_list_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream576332851429727821list_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_584_stream_Orel__refl__strong,axiom,
! [X2: stream_stream_a,Ra: stream_a > stream_a > $o] :
( ! [Z3: stream_a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream8207726558890130509ream_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_585_stream_Orel__refl__strong,axiom,
! [X2: stream_list_list_a,Ra: list_list_a > list_list_a > $o] :
( ! [Z3: list_list_a] :
( ( member_list_list_a @ Z3 @ ( sset_list_list_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream5427457574974697293list_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_586_stream_Orel__refl__strong,axiom,
! [X2: stream_set_a,Ra: set_a > set_a > $o] :
( ! [Z3: set_a] :
( ( member_set_a @ Z3 @ ( sset_set_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream750952186392916301_set_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_587_stream_Orel__refl__strong,axiom,
! [X2: stream_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( sset_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream_all2_a_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_588_stream_Orel__refl__strong,axiom,
! [X2: stream_list_a,Ra: list_a > list_a > $o] :
( ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( stream4578335575074870093list_a @ Ra @ X2 @ X2 ) ) ).
% stream.rel_refl_strong
thf(fact_589_stream_Orel__mono__strong,axiom,
! [R: a > a > $o,X2: stream_a,Y: stream_a,Ra: a > a > $o] :
( ( stream_all2_a_a @ R @ X2 @ Y )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( sset_a @ X2 ) )
=> ( ( member_a @ Yb @ ( sset_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream_all2_a_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_590_stream_Orel__mono__strong,axiom,
! [R: a > list_a > $o,X2: stream_a,Y: stream_list_a,Ra: a > list_a > $o] :
( ( stream_all2_a_list_a @ R @ X2 @ Y )
=> ( ! [Z3: a,Yb: list_a] :
( ( member_a @ Z3 @ ( sset_a @ X2 ) )
=> ( ( member_list_a @ Yb @ ( sset_list_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream_all2_a_list_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_591_stream_Orel__mono__strong,axiom,
! [R: list_a > a > $o,X2: stream_list_a,Y: stream_a,Ra: list_a > a > $o] :
( ( stream_all2_list_a_a @ R @ X2 @ Y )
=> ( ! [Z3: list_a,Yb: a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ X2 ) )
=> ( ( member_a @ Yb @ ( sset_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream_all2_list_a_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_592_stream_Orel__mono__strong,axiom,
! [R: list_a > list_a > $o,X2: stream_list_a,Y: stream_list_a,Ra: list_a > list_a > $o] :
( ( stream4578335575074870093list_a @ R @ X2 @ Y )
=> ( ! [Z3: list_a,Yb: list_a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ X2 ) )
=> ( ( member_list_a @ Yb @ ( sset_list_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream4578335575074870093list_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_593_stream_Orel__mono__strong,axiom,
! [R: stream_a > a > $o,X2: stream_stream_a,Y: stream_a,Ra: stream_a > a > $o] :
( ( stream3396383680346202183am_a_a @ R @ X2 @ Y )
=> ( ! [Z3: stream_a,Yb: a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ X2 ) )
=> ( ( member_a @ Yb @ ( sset_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream3396383680346202183am_a_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_594_stream_Orel__mono__strong,axiom,
! [R: set_a > a > $o,X2: stream_set_a,Y: stream_a,Ra: set_a > a > $o] :
( ( stream_all2_set_a_a @ R @ X2 @ Y )
=> ( ! [Z3: set_a,Yb: a] :
( ( member_set_a @ Z3 @ ( sset_set_a @ X2 ) )
=> ( ( member_a @ Yb @ ( sset_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream_all2_set_a_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_595_stream_Orel__mono__strong,axiom,
! [R: a > stream_a > $o,X2: stream_a,Y: stream_stream_a,Ra: a > stream_a > $o] :
( ( stream3533738183241558099ream_a @ R @ X2 @ Y )
=> ( ! [Z3: a,Yb: stream_a] :
( ( member_a @ Z3 @ ( sset_a @ X2 ) )
=> ( ( member_stream_a @ Yb @ ( sset_stream_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream3533738183241558099ream_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_596_stream_Orel__mono__strong,axiom,
! [R: a > set_a > $o,X2: stream_a,Y: stream_set_a,Ra: a > set_a > $o] :
( ( stream_all2_a_set_a @ R @ X2 @ Y )
=> ( ! [Z3: a,Yb: set_a] :
( ( member_a @ Z3 @ ( sset_a @ X2 ) )
=> ( ( member_set_a @ Yb @ ( sset_set_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream_all2_a_set_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_597_stream_Orel__mono__strong,axiom,
! [R: stream_a > stream_a > $o,X2: stream_stream_a,Y: stream_stream_a,Ra: stream_a > stream_a > $o] :
( ( stream8207726558890130509ream_a @ R @ X2 @ Y )
=> ( ! [Z3: stream_a,Yb: stream_a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ X2 ) )
=> ( ( member_stream_a @ Yb @ ( sset_stream_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream8207726558890130509ream_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_598_stream_Orel__mono__strong,axiom,
! [R: stream_a > set_a > $o,X2: stream_stream_a,Y: stream_set_a,Ra: stream_a > set_a > $o] :
( ( stream2582044070471935399_set_a @ R @ X2 @ Y )
=> ( ! [Z3: stream_a,Yb: set_a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ X2 ) )
=> ( ( member_set_a @ Yb @ ( sset_set_a @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( stream2582044070471935399_set_a @ Ra @ X2 @ Y ) ) ) ).
% stream.rel_mono_strong
thf(fact_599_stream_Orel__cong,axiom,
! [X2: stream_a,Ya: stream_a,Y: stream_a,Xa: stream_a,R: a > a > $o,Ra: a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( sset_a @ Ya ) )
=> ( ( member_a @ Yb @ ( sset_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream_all2_a_a @ R @ X2 @ Y )
= ( stream_all2_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_600_stream_Orel__cong,axiom,
! [X2: stream_a,Ya: stream_a,Y: stream_list_a,Xa: stream_list_a,R: a > list_a > $o,Ra: a > list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: list_a] :
( ( member_a @ Z3 @ ( sset_a @ Ya ) )
=> ( ( member_list_a @ Yb @ ( sset_list_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream_all2_a_list_a @ R @ X2 @ Y )
= ( stream_all2_a_list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_601_stream_Orel__cong,axiom,
! [X2: stream_list_a,Ya: stream_list_a,Y: stream_a,Xa: stream_a,R: list_a > a > $o,Ra: list_a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: list_a,Yb: a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ Ya ) )
=> ( ( member_a @ Yb @ ( sset_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream_all2_list_a_a @ R @ X2 @ Y )
= ( stream_all2_list_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_602_stream_Orel__cong,axiom,
! [X2: stream_list_a,Ya: stream_list_a,Y: stream_list_a,Xa: stream_list_a,R: list_a > list_a > $o,Ra: list_a > list_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: list_a,Yb: list_a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ Ya ) )
=> ( ( member_list_a @ Yb @ ( sset_list_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream4578335575074870093list_a @ R @ X2 @ Y )
= ( stream4578335575074870093list_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_603_stream_Orel__cong,axiom,
! [X2: stream_stream_a,Ya: stream_stream_a,Y: stream_a,Xa: stream_a,R: stream_a > a > $o,Ra: stream_a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: stream_a,Yb: a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ Ya ) )
=> ( ( member_a @ Yb @ ( sset_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream3396383680346202183am_a_a @ R @ X2 @ Y )
= ( stream3396383680346202183am_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_604_stream_Orel__cong,axiom,
! [X2: stream_set_a,Ya: stream_set_a,Y: stream_a,Xa: stream_a,R: set_a > a > $o,Ra: set_a > a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: set_a,Yb: a] :
( ( member_set_a @ Z3 @ ( sset_set_a @ Ya ) )
=> ( ( member_a @ Yb @ ( sset_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream_all2_set_a_a @ R @ X2 @ Y )
= ( stream_all2_set_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_605_stream_Orel__cong,axiom,
! [X2: stream_a,Ya: stream_a,Y: stream_stream_a,Xa: stream_stream_a,R: a > stream_a > $o,Ra: a > stream_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: stream_a] :
( ( member_a @ Z3 @ ( sset_a @ Ya ) )
=> ( ( member_stream_a @ Yb @ ( sset_stream_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream3533738183241558099ream_a @ R @ X2 @ Y )
= ( stream3533738183241558099ream_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_606_stream_Orel__cong,axiom,
! [X2: stream_a,Ya: stream_a,Y: stream_set_a,Xa: stream_set_a,R: a > set_a > $o,Ra: a > set_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: set_a] :
( ( member_a @ Z3 @ ( sset_a @ Ya ) )
=> ( ( member_set_a @ Yb @ ( sset_set_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream_all2_a_set_a @ R @ X2 @ Y )
= ( stream_all2_a_set_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_607_stream_Orel__cong,axiom,
! [X2: stream_stream_a,Ya: stream_stream_a,Y: stream_stream_a,Xa: stream_stream_a,R: stream_a > stream_a > $o,Ra: stream_a > stream_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: stream_a,Yb: stream_a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ Ya ) )
=> ( ( member_stream_a @ Yb @ ( sset_stream_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream8207726558890130509ream_a @ R @ X2 @ Y )
= ( stream8207726558890130509ream_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_608_stream_Orel__cong,axiom,
! [X2: stream_stream_a,Ya: stream_stream_a,Y: stream_set_a,Xa: stream_set_a,R: stream_a > set_a > $o,Ra: stream_a > set_a > $o] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: stream_a,Yb: set_a] :
( ( member_stream_a @ Z3 @ ( sset_stream_a @ Ya ) )
=> ( ( member_set_a @ Yb @ ( sset_set_a @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( stream2582044070471935399_set_a @ R @ X2 @ Y )
= ( stream2582044070471935399_set_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% stream.rel_cong
thf(fact_609_stream_Osel_I2_J,axiom,
! [X1: produc2579390645249093025st_a_a,X22: stream5874003577662885159st_a_a] :
( ( stl_Pr7785550816790648401st_a_a @ ( sCons_1605959091327504952st_a_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_610_stream_Osel_I2_J,axiom,
! [X1: produc9164743771328383783list_a,X22: stream271976807216406061list_a] :
( ( stl_Pr6107055601389525975list_a @ ( sCons_4508012570444740670list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_611_stream_Osel_I2_J,axiom,
! [X1: produc8685980395799941037list_a,X22: stream8205385055505213747list_a] :
( ( stl_Pr4668768530486720605list_a @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_612_stream_Osel_I2_J,axiom,
! [X1: product_prod_a_a,X22: stream6512538624140940077od_a_a] :
( ( stl_Product_prod_a_a @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_613_stream_Osel_I2_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] :
( ( stl_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_614_stream_Osel_I2_J,axiom,
! [X1: list_a,X22: stream_list_a] :
( ( stl_list_a @ ( sCons_list_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_615_stream_Osel_I2_J,axiom,
! [X1: a,X22: stream_a] :
( ( stl_a @ ( sCons_a @ X1 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_616_stream_Osel_I1_J,axiom,
! [X1: produc8685980395799941037list_a,X22: stream8205385055505213747list_a] :
( ( shd_Pr4559591720970336857list_a @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_617_stream_Osel_I1_J,axiom,
! [X1: product_prod_a_a,X22: stream6512538624140940077od_a_a] :
( ( shd_Product_prod_a_a @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_618_stream_Osel_I1_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] :
( ( shd_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_619_stream_Osel_I1_J,axiom,
! [X1: list_a,X22: stream_list_a] :
( ( shd_list_a @ ( sCons_list_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_620_stream_Osel_I1_J,axiom,
! [X1: a,X22: stream_a] :
( ( shd_a @ ( sCons_a @ X1 @ X22 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_621_stream_Oset__induct,axiom,
! [X2: stream_list_a,A2: stream_stream_list_a,P: stream_list_a > stream_stream_list_a > $o] :
( ( member_stream_list_a @ X2 @ ( sset_stream_list_a @ A2 ) )
=> ( ! [Z1: stream_list_a,Z22: stream_stream_list_a] : ( P @ Z1 @ ( sCons_stream_list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: stream_list_a,Z22: stream_stream_list_a,Xa2: stream_list_a] :
( ( member_stream_list_a @ Xa2 @ ( sset_stream_list_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_stream_list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_622_stream_Oset__induct,axiom,
! [X2: stream_a,A2: stream_stream_a,P: stream_a > stream_stream_a > $o] :
( ( member_stream_a @ X2 @ ( sset_stream_a @ A2 ) )
=> ( ! [Z1: stream_a,Z22: stream_stream_a] : ( P @ Z1 @ ( sCons_stream_a @ Z1 @ Z22 ) )
=> ( ! [Z1: stream_a,Z22: stream_stream_a,Xa2: stream_a] :
( ( member_stream_a @ Xa2 @ ( sset_stream_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_stream_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_623_stream_Oset__induct,axiom,
! [X2: set_a,A2: stream_set_a,P: set_a > stream_set_a > $o] :
( ( member_set_a @ X2 @ ( sset_set_a @ A2 ) )
=> ( ! [Z1: set_a,Z22: stream_set_a] : ( P @ Z1 @ ( sCons_set_a @ Z1 @ Z22 ) )
=> ( ! [Z1: set_a,Z22: stream_set_a,Xa2: set_a] :
( ( member_set_a @ Xa2 @ ( sset_set_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_set_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_624_stream_Oset__induct,axiom,
! [X2: produc8685980395799941037list_a,A2: stream8205385055505213747list_a,P: produc8685980395799941037list_a > stream8205385055505213747list_a > $o] :
( ( member4889668945541975382list_a @ X2 @ ( sset_P5198238693475834817list_a @ A2 ) )
=> ( ! [Z1: produc8685980395799941037list_a,Z22: stream8205385055505213747list_a] : ( P @ Z1 @ ( sCons_7712548841878352964list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: produc8685980395799941037list_a,Z22: stream8205385055505213747list_a,Xa2: produc8685980395799941037list_a] :
( ( member4889668945541975382list_a @ Xa2 @ ( sset_P5198238693475834817list_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_7712548841878352964list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_625_stream_Oset__induct,axiom,
! [X2: product_prod_a_a,A2: stream6512538624140940077od_a_a,P: product_prod_a_a > stream6512538624140940077od_a_a > $o] :
( ( member1426531477525435216od_a_a @ X2 @ ( sset_P5743994922208793403od_a_a @ A2 ) )
=> ( ! [Z1: product_prod_a_a,Z22: stream6512538624140940077od_a_a] : ( P @ Z1 @ ( sCons_4259542806812723518od_a_a @ Z1 @ Z22 ) )
=> ( ! [Z1: product_prod_a_a,Z22: stream6512538624140940077od_a_a,Xa2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa2 @ ( sset_P5743994922208793403od_a_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_4259542806812723518od_a_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_626_stream_Oset__induct,axiom,
! [X2: list_list_a,A2: stream_list_list_a,P: list_list_a > stream_list_list_a > $o] :
( ( member_list_list_a @ X2 @ ( sset_list_list_a @ A2 ) )
=> ( ! [Z1: list_list_a,Z22: stream_list_list_a] : ( P @ Z1 @ ( sCons_list_list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: list_list_a,Z22: stream_list_list_a,Xa2: list_list_a] :
( ( member_list_list_a @ Xa2 @ ( sset_list_list_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_list_list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_627_stream_Oset__induct,axiom,
! [X2: a,A2: stream_a,P: a > stream_a > $o] :
( ( member_a @ X2 @ ( sset_a @ A2 ) )
=> ( ! [Z1: a,Z22: stream_a] : ( P @ Z1 @ ( sCons_a @ Z1 @ Z22 ) )
=> ( ! [Z1: a,Z22: stream_a,Xa2: a] :
( ( member_a @ Xa2 @ ( sset_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_628_stream_Oset__induct,axiom,
! [X2: list_a,A2: stream_list_a,P: list_a > stream_list_a > $o] :
( ( member_list_a @ X2 @ ( sset_list_a @ A2 ) )
=> ( ! [Z1: list_a,Z22: stream_list_a] : ( P @ Z1 @ ( sCons_list_a @ Z1 @ Z22 ) )
=> ( ! [Z1: list_a,Z22: stream_list_a,Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( sset_list_a @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons_list_a @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_629_stream_Oset__cases,axiom,
! [E: stream_list_a,A2: stream_stream_list_a] :
( ( member_stream_list_a @ E @ ( sset_stream_list_a @ A2 ) )
=> ( ! [Z22: stream_stream_list_a] :
( A2
!= ( sCons_stream_list_a @ E @ Z22 ) )
=> ~ ! [Z1: stream_list_a,Z22: stream_stream_list_a] :
( ( A2
= ( sCons_stream_list_a @ Z1 @ Z22 ) )
=> ~ ( member_stream_list_a @ E @ ( sset_stream_list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_630_stream_Oset__cases,axiom,
! [E: stream_a,A2: stream_stream_a] :
( ( member_stream_a @ E @ ( sset_stream_a @ A2 ) )
=> ( ! [Z22: stream_stream_a] :
( A2
!= ( sCons_stream_a @ E @ Z22 ) )
=> ~ ! [Z1: stream_a,Z22: stream_stream_a] :
( ( A2
= ( sCons_stream_a @ Z1 @ Z22 ) )
=> ~ ( member_stream_a @ E @ ( sset_stream_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_631_stream_Oset__cases,axiom,
! [E: set_a,A2: stream_set_a] :
( ( member_set_a @ E @ ( sset_set_a @ A2 ) )
=> ( ! [Z22: stream_set_a] :
( A2
!= ( sCons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: stream_set_a] :
( ( A2
= ( sCons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E @ ( sset_set_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_632_stream_Oset__cases,axiom,
! [E: produc8685980395799941037list_a,A2: stream8205385055505213747list_a] :
( ( member4889668945541975382list_a @ E @ ( sset_P5198238693475834817list_a @ A2 ) )
=> ( ! [Z22: stream8205385055505213747list_a] :
( A2
!= ( sCons_7712548841878352964list_a @ E @ Z22 ) )
=> ~ ! [Z1: produc8685980395799941037list_a,Z22: stream8205385055505213747list_a] :
( ( A2
= ( sCons_7712548841878352964list_a @ Z1 @ Z22 ) )
=> ~ ( member4889668945541975382list_a @ E @ ( sset_P5198238693475834817list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_633_stream_Oset__cases,axiom,
! [E: product_prod_a_a,A2: stream6512538624140940077od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( sset_P5743994922208793403od_a_a @ A2 ) )
=> ( ! [Z22: stream6512538624140940077od_a_a] :
( A2
!= ( sCons_4259542806812723518od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: stream6512538624140940077od_a_a] :
( ( A2
= ( sCons_4259542806812723518od_a_a @ Z1 @ Z22 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( sset_P5743994922208793403od_a_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_634_stream_Oset__cases,axiom,
! [E: list_list_a,A2: stream_list_list_a] :
( ( member_list_list_a @ E @ ( sset_list_list_a @ A2 ) )
=> ( ! [Z22: stream_list_list_a] :
( A2
!= ( sCons_list_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_list_a,Z22: stream_list_list_a] :
( ( A2
= ( sCons_list_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_list_a @ E @ ( sset_list_list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_635_stream_Oset__cases,axiom,
! [E: a,A2: stream_a] :
( ( member_a @ E @ ( sset_a @ A2 ) )
=> ( ! [Z22: stream_a] :
( A2
!= ( sCons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: stream_a] :
( ( A2
= ( sCons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( sset_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_636_stream_Oset__cases,axiom,
! [E: list_a,A2: stream_list_a] :
( ( member_list_a @ E @ ( sset_list_a @ A2 ) )
=> ( ! [Z22: stream_list_a] :
( A2
!= ( sCons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: stream_list_a] :
( ( A2
= ( sCons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( sset_list_a @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_637_stream_Oset__intros_I1_J,axiom,
! [X1: stream_list_a,X22: stream_stream_list_a] : ( member_stream_list_a @ X1 @ ( sset_stream_list_a @ ( sCons_stream_list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_638_stream_Oset__intros_I1_J,axiom,
! [X1: stream_a,X22: stream_stream_a] : ( member_stream_a @ X1 @ ( sset_stream_a @ ( sCons_stream_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_639_stream_Oset__intros_I1_J,axiom,
! [X1: set_a,X22: stream_set_a] : ( member_set_a @ X1 @ ( sset_set_a @ ( sCons_set_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_640_stream_Oset__intros_I1_J,axiom,
! [X1: produc8685980395799941037list_a,X22: stream8205385055505213747list_a] : ( member4889668945541975382list_a @ X1 @ ( sset_P5198238693475834817list_a @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_641_stream_Oset__intros_I1_J,axiom,
! [X1: product_prod_a_a,X22: stream6512538624140940077od_a_a] : ( member1426531477525435216od_a_a @ X1 @ ( sset_P5743994922208793403od_a_a @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_642_stream_Oset__intros_I1_J,axiom,
! [X1: list_list_a,X22: stream_list_list_a] : ( member_list_list_a @ X1 @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_643_stream_Oset__intros_I1_J,axiom,
! [X1: a,X22: stream_a] : ( member_a @ X1 @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_644_stream_Oset__intros_I1_J,axiom,
! [X1: list_a,X22: stream_list_a] : ( member_list_a @ X1 @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ).
% stream.set_intros(1)
thf(fact_645_stream_Oset__intros_I2_J,axiom,
! [Y: stream_list_a,X22: stream_stream_list_a,X1: stream_list_a] :
( ( member_stream_list_a @ Y @ ( sset_stream_list_a @ X22 ) )
=> ( member_stream_list_a @ Y @ ( sset_stream_list_a @ ( sCons_stream_list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_646_stream_Oset__intros_I2_J,axiom,
! [Y: stream_a,X22: stream_stream_a,X1: stream_a] :
( ( member_stream_a @ Y @ ( sset_stream_a @ X22 ) )
=> ( member_stream_a @ Y @ ( sset_stream_a @ ( sCons_stream_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_647_stream_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: stream_set_a,X1: set_a] :
( ( member_set_a @ Y @ ( sset_set_a @ X22 ) )
=> ( member_set_a @ Y @ ( sset_set_a @ ( sCons_set_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_648_stream_Oset__intros_I2_J,axiom,
! [Y: produc8685980395799941037list_a,X22: stream8205385055505213747list_a,X1: produc8685980395799941037list_a] :
( ( member4889668945541975382list_a @ Y @ ( sset_P5198238693475834817list_a @ X22 ) )
=> ( member4889668945541975382list_a @ Y @ ( sset_P5198238693475834817list_a @ ( sCons_7712548841878352964list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_649_stream_Oset__intros_I2_J,axiom,
! [Y: product_prod_a_a,X22: stream6512538624140940077od_a_a,X1: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( sset_P5743994922208793403od_a_a @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y @ ( sset_P5743994922208793403od_a_a @ ( sCons_4259542806812723518od_a_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_650_stream_Oset__intros_I2_J,axiom,
! [Y: list_list_a,X22: stream_list_list_a,X1: list_list_a] :
( ( member_list_list_a @ Y @ ( sset_list_list_a @ X22 ) )
=> ( member_list_list_a @ Y @ ( sset_list_list_a @ ( sCons_list_list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_651_stream_Oset__intros_I2_J,axiom,
! [Y: a,X22: stream_a,X1: a] :
( ( member_a @ Y @ ( sset_a @ X22 ) )
=> ( member_a @ Y @ ( sset_a @ ( sCons_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_652_stream_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: stream_list_a,X1: list_a] :
( ( member_list_a @ Y @ ( sset_list_a @ X22 ) )
=> ( member_list_a @ Y @ ( sset_list_a @ ( sCons_list_a @ X1 @ X22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_653_streams__Stream,axiom,
! [X2: stream_list_a,S2: stream_stream_list_a,A: set_stream_list_a] :
( ( member3992053618758223289list_a @ ( sCons_stream_list_a @ X2 @ S2 ) @ ( stream6414841306446625538list_a @ A ) )
= ( ( member_stream_list_a @ X2 @ A )
& ( member3992053618758223289list_a @ S2 @ ( stream6414841306446625538list_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_654_streams__Stream,axiom,
! [X2: stream_a,S2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ ( sCons_stream_a @ X2 @ S2 ) @ ( streams_stream_a @ A ) )
= ( ( member_stream_a @ X2 @ A )
& ( member8409225883583818419ream_a @ S2 @ ( streams_stream_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_655_streams__Stream,axiom,
! [X2: set_a,S2: stream_set_a,A: set_set_a] :
( ( member_stream_set_a @ ( sCons_set_a @ X2 @ S2 ) @ ( streams_set_a @ A ) )
= ( ( member_set_a @ X2 @ A )
& ( member_stream_set_a @ S2 @ ( streams_set_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_656_streams__Stream,axiom,
! [X2: produc8685980395799941037list_a,S2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ ( sCons_7712548841878352964list_a @ X2 @ S2 ) @ ( stream2987405652289500709list_a @ A ) )
= ( ( member4889668945541975382list_a @ X2 @ A )
& ( member7566587766794810716list_a @ S2 @ ( stream2987405652289500709list_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_657_streams__Stream,axiom,
! [X2: product_prod_a_a,S2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ ( sCons_4259542806812723518od_a_a @ X2 @ S2 ) @ ( stream6392229770749734815od_a_a @ A ) )
= ( ( member1426531477525435216od_a_a @ X2 @ A )
& ( member262390649204803542od_a_a @ S2 @ ( stream6392229770749734815od_a_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_658_streams__Stream,axiom,
! [X2: list_list_a,S2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ ( sCons_list_list_a @ X2 @ S2 ) @ ( streams_list_list_a @ A ) )
= ( ( member_list_list_a @ X2 @ A )
& ( member8003905643527612601list_a @ S2 @ ( streams_list_list_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_659_streams__Stream,axiom,
! [X2: a,S2: stream_a,A: set_a] :
( ( member_stream_a @ ( sCons_a @ X2 @ S2 ) @ ( streams_a @ A ) )
= ( ( member_a @ X2 @ A )
& ( member_stream_a @ S2 @ ( streams_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_660_streams__Stream,axiom,
! [X2: list_a,S2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ ( sCons_list_a @ X2 @ S2 ) @ ( streams_list_a @ A ) )
= ( ( member_list_a @ X2 @ A )
& ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) ) ) ) ).
% streams_Stream
thf(fact_661_streams_Ocoinduct,axiom,
! [X5: stream_stream_list_a > $o,X2: stream_stream_list_a,A: set_stream_list_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_stream_list_a] :
( ( X5 @ X )
=> ? [A7: stream_list_a,S5: stream_stream_list_a] :
( ( X
= ( sCons_stream_list_a @ A7 @ S5 ) )
& ( member_stream_list_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member3992053618758223289list_a @ S5 @ ( stream6414841306446625538list_a @ A ) ) ) ) )
=> ( member3992053618758223289list_a @ X2 @ ( stream6414841306446625538list_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_662_streams_Ocoinduct,axiom,
! [X5: stream_stream_a > $o,X2: stream_stream_a,A: set_stream_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_stream_a] :
( ( X5 @ X )
=> ? [A7: stream_a,S5: stream_stream_a] :
( ( X
= ( sCons_stream_a @ A7 @ S5 ) )
& ( member_stream_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member8409225883583818419ream_a @ S5 @ ( streams_stream_a @ A ) ) ) ) )
=> ( member8409225883583818419ream_a @ X2 @ ( streams_stream_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_663_streams_Ocoinduct,axiom,
! [X5: stream_set_a > $o,X2: stream_set_a,A: set_set_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_set_a] :
( ( X5 @ X )
=> ? [A7: set_a,S5: stream_set_a] :
( ( X
= ( sCons_set_a @ A7 @ S5 ) )
& ( member_set_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member_stream_set_a @ S5 @ ( streams_set_a @ A ) ) ) ) )
=> ( member_stream_set_a @ X2 @ ( streams_set_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_664_streams_Ocoinduct,axiom,
! [X5: stream8205385055505213747list_a > $o,X2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( X5 @ X2 )
=> ( ! [X: stream8205385055505213747list_a] :
( ( X5 @ X )
=> ? [A7: produc8685980395799941037list_a,S5: stream8205385055505213747list_a] :
( ( X
= ( sCons_7712548841878352964list_a @ A7 @ S5 ) )
& ( member4889668945541975382list_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member7566587766794810716list_a @ S5 @ ( stream2987405652289500709list_a @ A ) ) ) ) )
=> ( member7566587766794810716list_a @ X2 @ ( stream2987405652289500709list_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_665_streams_Ocoinduct,axiom,
! [X5: stream6512538624140940077od_a_a > $o,X2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( X5 @ X2 )
=> ( ! [X: stream6512538624140940077od_a_a] :
( ( X5 @ X )
=> ? [A7: product_prod_a_a,S5: stream6512538624140940077od_a_a] :
( ( X
= ( sCons_4259542806812723518od_a_a @ A7 @ S5 ) )
& ( member1426531477525435216od_a_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member262390649204803542od_a_a @ S5 @ ( stream6392229770749734815od_a_a @ A ) ) ) ) )
=> ( member262390649204803542od_a_a @ X2 @ ( stream6392229770749734815od_a_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_666_streams_Ocoinduct,axiom,
! [X5: stream_list_list_a > $o,X2: stream_list_list_a,A: set_list_list_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_list_list_a] :
( ( X5 @ X )
=> ? [A7: list_list_a,S5: stream_list_list_a] :
( ( X
= ( sCons_list_list_a @ A7 @ S5 ) )
& ( member_list_list_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member8003905643527612601list_a @ S5 @ ( streams_list_list_a @ A ) ) ) ) )
=> ( member8003905643527612601list_a @ X2 @ ( streams_list_list_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_667_streams_Ocoinduct,axiom,
! [X5: stream_a > $o,X2: stream_a,A: set_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_a] :
( ( X5 @ X )
=> ? [A7: a,S5: stream_a] :
( ( X
= ( sCons_a @ A7 @ S5 ) )
& ( member_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member_stream_a @ S5 @ ( streams_a @ A ) ) ) ) )
=> ( member_stream_a @ X2 @ ( streams_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_668_streams_Ocoinduct,axiom,
! [X5: stream_list_a > $o,X2: stream_list_a,A: set_list_a] :
( ( X5 @ X2 )
=> ( ! [X: stream_list_a] :
( ( X5 @ X )
=> ? [A7: list_a,S5: stream_list_a] :
( ( X
= ( sCons_list_a @ A7 @ S5 ) )
& ( member_list_a @ A7 @ A )
& ( ( X5 @ S5 )
| ( member_stream_list_a @ S5 @ ( streams_list_a @ A ) ) ) ) )
=> ( member_stream_list_a @ X2 @ ( streams_list_a @ A ) ) ) ) ).
% streams.coinduct
thf(fact_669_streams_Osimps,axiom,
! [A2: stream_stream_list_a,A: set_stream_list_a] :
( ( member3992053618758223289list_a @ A2 @ ( stream6414841306446625538list_a @ A ) )
= ( ? [A4: stream_list_a,S: stream_stream_list_a] :
( ( A2
= ( sCons_stream_list_a @ A4 @ S ) )
& ( member_stream_list_a @ A4 @ A )
& ( member3992053618758223289list_a @ S @ ( stream6414841306446625538list_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_670_streams_Osimps,axiom,
! [A2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ A2 @ ( streams_stream_a @ A ) )
= ( ? [A4: stream_a,S: stream_stream_a] :
( ( A2
= ( sCons_stream_a @ A4 @ S ) )
& ( member_stream_a @ A4 @ A )
& ( member8409225883583818419ream_a @ S @ ( streams_stream_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_671_streams_Osimps,axiom,
! [A2: stream_set_a,A: set_set_a] :
( ( member_stream_set_a @ A2 @ ( streams_set_a @ A ) )
= ( ? [A4: set_a,S: stream_set_a] :
( ( A2
= ( sCons_set_a @ A4 @ S ) )
& ( member_set_a @ A4 @ A )
& ( member_stream_set_a @ S @ ( streams_set_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_672_streams_Osimps,axiom,
! [A2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ A2 @ ( stream2987405652289500709list_a @ A ) )
= ( ? [A4: produc8685980395799941037list_a,S: stream8205385055505213747list_a] :
( ( A2
= ( sCons_7712548841878352964list_a @ A4 @ S ) )
& ( member4889668945541975382list_a @ A4 @ A )
& ( member7566587766794810716list_a @ S @ ( stream2987405652289500709list_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_673_streams_Osimps,axiom,
! [A2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ A2 @ ( stream6392229770749734815od_a_a @ A ) )
= ( ? [A4: product_prod_a_a,S: stream6512538624140940077od_a_a] :
( ( A2
= ( sCons_4259542806812723518od_a_a @ A4 @ S ) )
& ( member1426531477525435216od_a_a @ A4 @ A )
& ( member262390649204803542od_a_a @ S @ ( stream6392229770749734815od_a_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_674_streams_Osimps,axiom,
! [A2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ A2 @ ( streams_list_list_a @ A ) )
= ( ? [A4: list_list_a,S: stream_list_list_a] :
( ( A2
= ( sCons_list_list_a @ A4 @ S ) )
& ( member_list_list_a @ A4 @ A )
& ( member8003905643527612601list_a @ S @ ( streams_list_list_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_675_streams_Osimps,axiom,
! [A2: stream_a,A: set_a] :
( ( member_stream_a @ A2 @ ( streams_a @ A ) )
= ( ? [A4: a,S: stream_a] :
( ( A2
= ( sCons_a @ A4 @ S ) )
& ( member_a @ A4 @ A )
& ( member_stream_a @ S @ ( streams_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_676_streams_Osimps,axiom,
! [A2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ A2 @ ( streams_list_a @ A ) )
= ( ? [A4: list_a,S: stream_list_a] :
( ( A2
= ( sCons_list_a @ A4 @ S ) )
& ( member_list_a @ A4 @ A )
& ( member_stream_list_a @ S @ ( streams_list_a @ A ) ) ) ) ) ).
% streams.simps
thf(fact_677_streams_Ocases,axiom,
! [A2: stream_stream_list_a,A: set_stream_list_a] :
( ( member3992053618758223289list_a @ A2 @ ( stream6414841306446625538list_a @ A ) )
=> ~ ! [A5: stream_list_a,S3: stream_stream_list_a] :
( ( A2
= ( sCons_stream_list_a @ A5 @ S3 ) )
=> ( ( member_stream_list_a @ A5 @ A )
=> ~ ( member3992053618758223289list_a @ S3 @ ( stream6414841306446625538list_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_678_streams_Ocases,axiom,
! [A2: stream_stream_a,A: set_stream_a] :
( ( member8409225883583818419ream_a @ A2 @ ( streams_stream_a @ A ) )
=> ~ ! [A5: stream_a,S3: stream_stream_a] :
( ( A2
= ( sCons_stream_a @ A5 @ S3 ) )
=> ( ( member_stream_a @ A5 @ A )
=> ~ ( member8409225883583818419ream_a @ S3 @ ( streams_stream_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_679_streams_Ocases,axiom,
! [A2: stream_set_a,A: set_set_a] :
( ( member_stream_set_a @ A2 @ ( streams_set_a @ A ) )
=> ~ ! [A5: set_a,S3: stream_set_a] :
( ( A2
= ( sCons_set_a @ A5 @ S3 ) )
=> ( ( member_set_a @ A5 @ A )
=> ~ ( member_stream_set_a @ S3 @ ( streams_set_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_680_streams_Ocases,axiom,
! [A2: stream8205385055505213747list_a,A: set_Pr2070066670564046349list_a] :
( ( member7566587766794810716list_a @ A2 @ ( stream2987405652289500709list_a @ A ) )
=> ~ ! [A5: produc8685980395799941037list_a,S3: stream8205385055505213747list_a] :
( ( A2
= ( sCons_7712548841878352964list_a @ A5 @ S3 ) )
=> ( ( member4889668945541975382list_a @ A5 @ A )
=> ~ ( member7566587766794810716list_a @ S3 @ ( stream2987405652289500709list_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_681_streams_Ocases,axiom,
! [A2: stream6512538624140940077od_a_a,A: set_Product_prod_a_a] :
( ( member262390649204803542od_a_a @ A2 @ ( stream6392229770749734815od_a_a @ A ) )
=> ~ ! [A5: product_prod_a_a,S3: stream6512538624140940077od_a_a] :
( ( A2
= ( sCons_4259542806812723518od_a_a @ A5 @ S3 ) )
=> ( ( member1426531477525435216od_a_a @ A5 @ A )
=> ~ ( member262390649204803542od_a_a @ S3 @ ( stream6392229770749734815od_a_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_682_streams_Ocases,axiom,
! [A2: stream_list_list_a,A: set_list_list_a] :
( ( member8003905643527612601list_a @ A2 @ ( streams_list_list_a @ A ) )
=> ~ ! [A5: list_list_a,S3: stream_list_list_a] :
( ( A2
= ( sCons_list_list_a @ A5 @ S3 ) )
=> ( ( member_list_list_a @ A5 @ A )
=> ~ ( member8003905643527612601list_a @ S3 @ ( streams_list_list_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_683_streams_Ocases,axiom,
! [A2: stream_a,A: set_a] :
( ( member_stream_a @ A2 @ ( streams_a @ A ) )
=> ~ ! [A5: a,S3: stream_a] :
( ( A2
= ( sCons_a @ A5 @ S3 ) )
=> ( ( member_a @ A5 @ A )
=> ~ ( member_stream_a @ S3 @ ( streams_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_684_streams_Ocases,axiom,
! [A2: stream_list_a,A: set_list_a] :
( ( member_stream_list_a @ A2 @ ( streams_list_a @ A ) )
=> ~ ! [A5: list_a,S3: stream_list_a] :
( ( A2
= ( sCons_list_a @ A5 @ S3 ) )
=> ( ( member_list_a @ A5 @ A )
=> ~ ( member_stream_list_a @ S3 @ ( streams_list_a @ A ) ) ) ) ) ).
% streams.cases
thf(fact_685_sinterleave__code,axiom,
! [X2: produc8685980395799941037list_a,S1: stream8205385055505213747list_a,S22: stream8205385055505213747list_a] :
( ( sinter832779693940469002list_a @ ( sCons_7712548841878352964list_a @ X2 @ S1 ) @ S22 )
= ( sCons_7712548841878352964list_a @ X2 @ ( sinter832779693940469002list_a @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_686_sinterleave__code,axiom,
! [X2: product_prod_a_a,S1: stream6512538624140940077od_a_a,S22: stream6512538624140940077od_a_a] :
( ( sinter5547950683322096900od_a_a @ ( sCons_4259542806812723518od_a_a @ X2 @ S1 ) @ S22 )
= ( sCons_4259542806812723518od_a_a @ X2 @ ( sinter5547950683322096900od_a_a @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_687_sinterleave__code,axiom,
! [X2: list_list_a,S1: stream_list_list_a,S22: stream_list_list_a] :
( ( sinter664328039231274855list_a @ ( sCons_list_list_a @ X2 @ S1 ) @ S22 )
= ( sCons_list_list_a @ X2 @ ( sinter664328039231274855list_a @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_688_sinterleave__code,axiom,
! [X2: list_a,S1: stream_list_a,S22: stream_list_a] :
( ( sinterleave_list_a @ ( sCons_list_a @ X2 @ S1 ) @ S22 )
= ( sCons_list_a @ X2 @ ( sinterleave_list_a @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_689_sinterleave__code,axiom,
! [X2: a,S1: stream_a,S22: stream_a] :
( ( sinterleave_a @ ( sCons_a @ X2 @ S1 ) @ S22 )
= ( sCons_a @ X2 @ ( sinterleave_a @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_690_sup_Obounded__iff,axiom,
! [B3: set_list_a,C: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A2 )
= ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
& ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_691_sup_Obounded__iff,axiom,
! [B3: $o > nat,C: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_o_nat @ B3 @ A2 )
& ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_692_sup_Obounded__iff,axiom,
! [B3: $o > set_a,C: $o > set_a,A2: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( sup_sup_o_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_o_set_a @ B3 @ A2 )
& ( ord_less_eq_o_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_693_sup_Obounded__iff,axiom,
! [B3: set_stream_a,C: set_stream_a,A2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ B3 @ C ) @ A2 )
= ( ( ord_le1019412967036525316ream_a @ B3 @ A2 )
& ( ord_le1019412967036525316ream_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_694_sup_Obounded__iff,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_695_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_696_le__sup__iff,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X2 @ Y ) @ Z2 )
= ( ( ord_le8861187494160871172list_a @ X2 @ Z2 )
& ( ord_le8861187494160871172list_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_697_le__sup__iff,axiom,
! [X2: $o > nat,Y: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_o_nat @ X2 @ Z2 )
& ( ord_less_eq_o_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_698_le__sup__iff,axiom,
! [X2: $o > set_a,Y: $o > set_a,Z2: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( sup_sup_o_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_o_set_a @ X2 @ Z2 )
& ( ord_less_eq_o_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_699_le__sup__iff,axiom,
! [X2: set_stream_a,Y: set_stream_a,Z2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ X2 @ Y ) @ Z2 )
= ( ( ord_le1019412967036525316ream_a @ X2 @ Z2 )
& ( ord_le1019412967036525316ream_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_700_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X2 @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_701_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_702_sup_Oidem,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_703_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_704_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_705_sup__idem,axiom,
! [X2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_706_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_707_sup__idem,axiom,
! [X2: nat] :
( ( sup_sup_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_708_sup_Oleft__idem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ A2 @ B3 ) )
= ( sup_sup_set_list_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_709_sup_Oleft__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) )
= ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_710_sup_Oleft__idem,axiom,
! [A2: nat,B3: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) )
= ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_711_sup__left__idem,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ X2 @ Y ) )
= ( sup_sup_set_list_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_712_sup__left__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_713_sup__left__idem,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_714_sup_Oright__idem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A2 @ B3 ) @ B3 )
= ( sup_sup_set_list_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_715_sup_Oright__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ B3 )
= ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_716_sup_Oright__idem,axiom,
! [A2: nat,B3: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B3 ) @ B3 )
= ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_717_sup_OcoboundedI2,axiom,
! [C: set_list_a,B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ B3 )
=> ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_718_sup_OcoboundedI2,axiom,
! [C: $o > nat,B3: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ C @ B3 )
=> ( ord_less_eq_o_nat @ C @ ( sup_sup_o_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_719_sup_OcoboundedI2,axiom,
! [C: $o > set_a,B3: $o > set_a,A2: $o > set_a] :
( ( ord_less_eq_o_set_a @ C @ B3 )
=> ( ord_less_eq_o_set_a @ C @ ( sup_sup_o_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_720_sup_OcoboundedI2,axiom,
! [C: set_stream_a,B3: set_stream_a,A2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ C @ B3 )
=> ( ord_le1019412967036525316ream_a @ C @ ( sup_sup_set_stream_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_721_sup_OcoboundedI2,axiom,
! [C: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_722_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_723_sup_OcoboundedI1,axiom,
! [C: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ A2 )
=> ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_724_sup_OcoboundedI1,axiom,
! [C: $o > nat,A2: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ C @ A2 )
=> ( ord_less_eq_o_nat @ C @ ( sup_sup_o_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_725_sup_OcoboundedI1,axiom,
! [C: $o > set_a,A2: $o > set_a,B3: $o > set_a] :
( ( ord_less_eq_o_set_a @ C @ A2 )
=> ( ord_less_eq_o_set_a @ C @ ( sup_sup_o_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_726_sup_OcoboundedI1,axiom,
! [C: set_stream_a,A2: set_stream_a,B3: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ C @ A2 )
=> ( ord_le1019412967036525316ream_a @ C @ ( sup_sup_set_stream_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_727_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_728_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_729_sup__left__commute,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) )
= ( sup_sup_set_list_a @ Y @ ( sup_sup_set_list_a @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_730_sup__left__commute,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_731_sup__left__commute,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_732_sup_Oleft__commute,axiom,
! [B3: set_list_a,A2: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ B3 @ ( sup_sup_set_list_a @ A2 @ C ) )
= ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_733_sup_Oleft__commute,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_734_sup_Oleft__commute,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( sup_sup_nat @ B3 @ ( sup_sup_nat @ A2 @ C ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_735_sup__commute,axiom,
( sup_sup_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] : ( sup_sup_set_list_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_736_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_737_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_738_sup_Ocommute,axiom,
( sup_sup_set_list_a
= ( ^ [A4: set_list_a,B4: set_list_a] : ( sup_sup_set_list_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_739_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_740_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A4: nat,B4: nat] : ( sup_sup_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_741_sup__assoc,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_742_sup__assoc,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_743_sup__assoc,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_744_sup_Oassoc,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A2 @ B3 ) @ C )
= ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_745_sup_Oassoc,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_746_sup_Oassoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B3 ) @ C )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_747_inf__sup__aci_I5_J,axiom,
( sup_sup_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] : ( sup_sup_set_list_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_748_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_749_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_750_inf__sup__aci_I6_J,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_751_inf__sup__aci_I6_J,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_752_inf__sup__aci_I6_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_753_inf__sup__aci_I7_J,axiom,
! [X2: set_list_a,Y: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y @ Z2 ) )
= ( sup_sup_set_list_a @ Y @ ( sup_sup_set_list_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_754_inf__sup__aci_I7_J,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_755_inf__sup__aci_I7_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_756_inf__sup__aci_I8_J,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( sup_sup_set_list_a @ X2 @ Y ) )
= ( sup_sup_set_list_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_757_inf__sup__aci_I8_J,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_758_inf__sup__aci_I8_J,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_759_inf__sup__ord_I4_J,axiom,
! [Y: set_list_a,X2: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_760_inf__sup__ord_I4_J,axiom,
! [Y: $o > nat,X2: $o > nat] : ( ord_less_eq_o_nat @ Y @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_761_inf__sup__ord_I4_J,axiom,
! [Y: $o > set_a,X2: $o > set_a] : ( ord_less_eq_o_set_a @ Y @ ( sup_sup_o_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_762_inf__sup__ord_I4_J,axiom,
! [Y: set_stream_a,X2: set_stream_a] : ( ord_le1019412967036525316ream_a @ Y @ ( sup_sup_set_stream_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_763_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_764_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_765_inf__sup__ord_I3_J,axiom,
! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ ( sup_sup_set_list_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_766_inf__sup__ord_I3_J,axiom,
! [X2: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ X2 @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_767_inf__sup__ord_I3_J,axiom,
! [X2: $o > set_a,Y: $o > set_a] : ( ord_less_eq_o_set_a @ X2 @ ( sup_sup_o_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_768_inf__sup__ord_I3_J,axiom,
! [X2: set_stream_a,Y: set_stream_a] : ( ord_le1019412967036525316ream_a @ X2 @ ( sup_sup_set_stream_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_769_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_770_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_771_le__supE,axiom,
! [A2: set_stream_a,B3: set_stream_a,X2: set_stream_a] :
( ( ord_le1019412967036525316ream_a @ ( sup_sup_set_stream_a @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_le1019412967036525316ream_a @ A2 @ X2 )
=> ~ ( ord_le1019412967036525316ream_a @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_772_le__supE,axiom,
! [A2: set_a,B3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
=> ~ ( ord_less_eq_set_a @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_773_le__supE,axiom,
! [A2: nat,B3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).
% le_supE
thf(fact_774_le__supI,axiom,
! [A2: set_a,X2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ( ord_less_eq_set_a @ B3 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_775_le__supI,axiom,
! [A2: nat,X2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% le_supI
thf(fact_776_sup__ge1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_777_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_778_sup__ge2,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_779_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_780_le__supI1,axiom,
! [X2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_781_le__supI1,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_782_le__supI2,axiom,
! [X2: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X2 @ B3 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_783_le__supI2,axiom,
! [X2: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_784_sup_Omono,axiom,
! [C: set_a,A2: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_785_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_786_sup__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_787_sup__mono,axiom,
! [A2: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_788_sup__least,axiom,
! [Y: set_a,X2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_789_sup__least,axiom,
! [Y: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_790_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_791_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_792_sup_OorderE,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_793_sup_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_794_sup_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_795_sup_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_796_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z3 @ X )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_797_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_798_sup_Oabsorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_799_sup_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_800_sup_Oabsorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_801_sup_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_802_sup__absorb1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_803_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_804_sup__absorb2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( sup_sup_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_805_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_806_sup_OboundedE,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_807_sup_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_808_sup_OboundedI,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_809_sup_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_810_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_811_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_812_sup_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_813_sup_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_814_sup_Ocobounded2,axiom,
! [B3: set_a,A2: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_815_sup_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_816_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_817_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_818_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_819_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_820_szip_Ocode,axiom,
( szip_list_a_list_a
= ( ^ [S12: stream_list_a,S23: stream_list_a] : ( sCons_4508012570444740670list_a @ ( produc6837034575241423639list_a @ ( shd_list_a @ S12 ) @ ( shd_list_a @ S23 ) ) @ ( szip_list_a_list_a @ ( stl_list_a @ S12 ) @ ( stl_list_a @ S23 ) ) ) ) ) ).
% szip.code
thf(fact_821_szip_Ocode,axiom,
( szip_list_a_a
= ( ^ [S12: stream_list_a,S23: stream_a] : ( sCons_1605959091327504952st_a_a @ ( produc4781227316648555537st_a_a @ ( shd_list_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( szip_list_a_a @ ( stl_list_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).
% szip.code
thf(fact_822_szip_Ocode,axiom,
( szip_a_list_a
= ( ^ [S12: stream_a,S23: stream_list_a] : ( sCons_7712548841878352964list_a @ ( produc6670463072477821725list_a @ ( shd_a @ S12 ) @ ( shd_list_a @ S23 ) ) @ ( szip_a_list_a @ ( stl_a @ S12 ) @ ( stl_list_a @ S23 ) ) ) ) ) ).
% szip.code
thf(fact_823_szip_Ocode,axiom,
( szip_a_a
= ( ^ [S12: stream_a,S23: stream_a] : ( sCons_4259542806812723518od_a_a @ ( product_Pair_a_a @ ( shd_a @ S12 ) @ ( shd_a @ S23 ) ) @ ( szip_a_a @ ( stl_a @ S12 ) @ ( stl_a @ S23 ) ) ) ) ) ).
% szip.code
thf(fact_824_flat_Ocode,axiom,
( flat_a
= ( ^ [Ws2: stream_list_a] :
( sCons_a @ ( hd_a @ ( shd_list_a @ Ws2 ) )
@ ( flat_a
@ ( if_stream_list_a
@ ( ( tl_a @ ( shd_list_a @ Ws2 ) )
= nil_a )
@ ( stl_list_a @ Ws2 )
@ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws2 ) ) @ ( stl_list_a @ Ws2 ) ) ) ) ) ) ) ).
% flat.code
thf(fact_825_stream_Omap__sel_I2_J,axiom,
! [F: list_a > list_a,A2: stream_list_a] :
( ( stl_list_a @ ( smap_list_a_list_a @ F @ A2 ) )
= ( smap_list_a_list_a @ F @ ( stl_list_a @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_826_stream_Omap__sel_I2_J,axiom,
! [F: a > list_a,A2: stream_a] :
( ( stl_list_a @ ( smap_a_list_a @ F @ A2 ) )
= ( smap_a_list_a @ F @ ( stl_a @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_827_stream_Omap__sel_I2_J,axiom,
! [F: list_a > a,A2: stream_list_a] :
( ( stl_a @ ( smap_list_a_a @ F @ A2 ) )
= ( smap_list_a_a @ F @ ( stl_list_a @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_828_stream_Omap__sel_I2_J,axiom,
! [F: a > a,A2: stream_a] :
( ( stl_a @ ( smap_a_a @ F @ A2 ) )
= ( smap_a_a @ F @ ( stl_a @ A2 ) ) ) ).
% stream.map_sel(2)
thf(fact_829_stream_Omap__ident__strong,axiom,
! [T3: stream_a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( sset_a @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( smap_a_a @ F @ T3 )
= T3 ) ) ).
% stream.map_ident_strong
thf(fact_830_stream_Omap__ident__strong,axiom,
! [T3: stream_list_a,F: list_a > list_a] :
( ! [Z3: list_a] :
( ( member_list_a @ Z3 @ ( sset_list_a @ T3 ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( smap_list_a_list_a @ F @ T3 )
= T3 ) ) ).
% stream.map_ident_strong
thf(fact_831_smap__streams,axiom,
! [S2: stream_a,A: set_a,F: a > a,B: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_stream_a @ ( smap_a_a @ F @ S2 ) @ ( streams_a @ B ) ) ) ) ).
% smap_streams
thf(fact_832_smap__streams,axiom,
! [S2: stream_a,A: set_a,F: a > list_a,B: set_list_a] :
( ( member_stream_a @ S2 @ ( streams_a @ A ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B ) )
=> ( member_stream_list_a @ ( smap_a_list_a @ F @ S2 ) @ ( streams_list_a @ B ) ) ) ) ).
% smap_streams
thf(fact_833_smap__streams,axiom,
! [S2: stream_list_a,A: set_list_a,F: list_a > a,B: set_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_stream_a @ ( smap_list_a_a @ F @ S2 ) @ ( streams_a @ B ) ) ) ) ).
% smap_streams
thf(fact_834_smap__streams,axiom,
! [S2: stream_list_a,A: set_list_a,F: list_a > list_a,B: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ A ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B ) )
=> ( member_stream_list_a @ ( smap_list_a_list_a @ F @ S2 ) @ ( streams_list_a @ B ) ) ) ) ).
% smap_streams
thf(fact_835_smap2_Osimps_I2_J,axiom,
! [F: list_a > list_a > list_a,S1: stream_list_a,S22: stream_list_a] :
( ( stl_list_a @ ( smap2_6771825536533895878list_a @ F @ S1 @ S22 ) )
= ( smap2_6771825536533895878list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_836_smap2_Osimps_I2_J,axiom,
! [F: list_a > a > list_a,S1: stream_list_a,S22: stream_a] :
( ( stl_list_a @ ( smap2_5547344383464640076list_a @ F @ S1 @ S22 ) )
= ( smap2_5547344383464640076list_a @ F @ ( stl_list_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_837_smap2_Osimps_I2_J,axiom,
! [F: a > list_a > list_a,S1: stream_a,S22: stream_list_a] :
( ( stl_list_a @ ( smap2_5647228111304775744list_a @ F @ S1 @ S22 ) )
= ( smap2_5647228111304775744list_a @ F @ ( stl_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_838_smap2_Osimps_I2_J,axiom,
! [F: a > a > list_a,S1: stream_a,S22: stream_a] :
( ( stl_list_a @ ( smap2_a_a_list_a @ F @ S1 @ S22 ) )
= ( smap2_a_a_list_a @ F @ ( stl_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_839_smap2_Osimps_I2_J,axiom,
! [F: list_a > list_a > a,S1: stream_list_a,S22: stream_list_a] :
( ( stl_a @ ( smap2_3658108627635373888st_a_a @ F @ S1 @ S22 ) )
= ( smap2_3658108627635373888st_a_a @ F @ ( stl_list_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_840_smap2_Osimps_I2_J,axiom,
! [F: list_a > a > a,S1: stream_list_a,S22: stream_a] :
( ( stl_a @ ( smap2_list_a_a_a @ F @ S1 @ S22 ) )
= ( smap2_list_a_a_a @ F @ ( stl_list_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_841_smap2_Osimps_I2_J,axiom,
! [F: a > list_a > a,S1: stream_a,S22: stream_list_a] :
( ( stl_a @ ( smap2_a_list_a_a @ F @ S1 @ S22 ) )
= ( smap2_a_list_a_a @ F @ ( stl_a @ S1 ) @ ( stl_list_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_842_smap2_Osimps_I2_J,axiom,
! [F: a > a > a,S1: stream_a,S22: stream_a] :
( ( stl_a @ ( smap2_a_a_a @ F @ S1 @ S22 ) )
= ( smap2_a_a_a @ F @ ( stl_a @ S1 ) @ ( stl_a @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_843_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_list_a] :
( ( stl_list_a @ ( flat_list_a @ Ws ) )
= ( flat_list_a
@ ( if_str7505741754068378070list_a
@ ( ( tl_list_a @ ( shd_list_list_a @ Ws ) )
= nil_list_a )
@ ( stl_list_list_a @ Ws )
@ ( sCons_list_list_a @ ( tl_list_a @ ( shd_list_list_a @ Ws ) ) @ ( stl_list_list_a @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_844_flat_Osimps_I2_J,axiom,
! [Ws: stream_list_a] :
( ( stl_a @ ( flat_a @ Ws ) )
= ( flat_a
@ ( if_stream_list_a
@ ( ( tl_a @ ( shd_list_a @ Ws ) )
= nil_a )
@ ( stl_list_a @ Ws )
@ ( sCons_list_a @ ( tl_a @ ( shd_list_a @ Ws ) ) @ ( stl_list_a @ Ws ) ) ) ) ) ).
% flat.simps(2)
thf(fact_845_flat__unfold,axiom,
! [Ws: stream_list_a] :
( ( ( shd_list_a @ Ws )
!= nil_a )
=> ( ( flat_a @ Ws )
= ( shift_a @ ( shd_list_a @ Ws ) @ ( flat_a @ ( stl_list_a @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_846_flat__Cons,axiom,
! [X2: a,Xs: list_a,Ws: stream_list_a] :
( ( flat_a @ ( sCons_list_a @ ( cons_a @ X2 @ Xs ) @ Ws ) )
= ( sCons_a @ X2 @ ( flat_a @ ( if_stream_list_a @ ( Xs = nil_a ) @ Ws @ ( sCons_list_a @ Xs @ Ws ) ) ) ) ) ).
% flat_Cons
thf(fact_847_flat__Stream,axiom,
! [Xs: list_a,Ws: stream_list_a] :
( ( Xs != nil_a )
=> ( ( flat_a @ ( sCons_list_a @ Xs @ Ws ) )
= ( shift_a @ Xs @ ( flat_a @ Ws ) ) ) ) ).
% flat_Stream
thf(fact_848_shift__simps_I2_J,axiom,
! [Xs: list_list_a,S2: stream_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( stl_list_a @ ( shift_list_a @ Xs @ S2 ) )
= ( stl_list_a @ S2 ) ) )
& ( ( Xs != nil_list_a )
=> ( ( stl_list_a @ ( shift_list_a @ Xs @ S2 ) )
= ( shift_list_a @ ( tl_list_a @ Xs ) @ S2 ) ) ) ) ).
% shift_simps(2)
thf(fact_849_shift__simps_I2_J,axiom,
! [Xs: list_a,S2: stream_a] :
( ( ( Xs = nil_a )
=> ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
= ( stl_a @ S2 ) ) )
& ( ( Xs != nil_a )
=> ( ( stl_a @ ( shift_a @ Xs @ S2 ) )
= ( shift_a @ ( tl_a @ Xs ) @ S2 ) ) ) ) ).
% shift_simps(2)
thf(fact_850_Cons__in__shuffles__iff,axiom,
! [Z2: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z2 )
& ( member_list_a @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z2 )
& ( member_list_a @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_851_Nil__in__shuffles,axiom,
! [Xs: list_a,Ys: list_a] :
( ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_in_shuffles
thf(fact_852_Nil__in__shufflesI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = nil_a )
=> ( ( Ys = nil_a )
=> ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_853_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X @ Xs2 ) )
=> ! [Z3: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs2 ) )
=> ( ( X = Z3 )
=> ~ ( member_list_a @ Zs2 @ ( shuffles_a @ Xs2 @ Ys ) ) ) ) )
=> ~ ! [Y2: a,Ys2: list_a] :
( ( Ys
= ( cons_a @ Y2 @ Ys2 ) )
=> ! [Z3: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs2 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member_list_a @ Zs2 @ ( shuffles_a @ Xs @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_854_sset__shift,axiom,
! [Xs: list_a,S2: stream_a] :
( ( sset_a @ ( shift_a @ Xs @ S2 ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( sset_a @ S2 ) ) ) ).
% sset_shift
thf(fact_855_sset__shift,axiom,
! [Xs: list_list_a,S2: stream_list_a] :
( ( sset_list_a @ ( shift_list_a @ Xs @ S2 ) )
= ( sup_sup_set_list_a @ ( set_list_a2 @ Xs ) @ ( sset_list_a @ S2 ) ) ) ).
% sset_shift
thf(fact_856_sdrop__while_Oraw__induct,axiom,
! [Pa: produc5320378693649342662list_a > stream_list_a > $o,P: list_a > $o,S2: stream_list_a,Y: stream_list_a] :
( ! [Sdrop_while: ( list_a > $o ) > stream_list_a > stream_list_a] :
( ! [S5: list_a > $o,B7: stream_list_a] :
( ( ( Sdrop_while @ S5 @ B7 )
!= undefi4168218097957520114list_a )
=> ( Pa @ ( produc8531839378509307896list_a @ S5 @ B7 ) @ ( Sdrop_while @ S5 @ B7 ) ) )
=> ! [P3: list_a > $o,S3: stream_list_a,Pa2: stream_list_a] :
( ( ( ( P3 @ ( shd_list_a @ S3 ) )
=> ( ( Sdrop_while @ P3 @ ( stl_list_a @ S3 ) )
= Pa2 ) )
& ( ~ ( P3 @ ( shd_list_a @ S3 ) )
=> ( S3 = Pa2 ) ) )
=> ( ( Pa2 != undefi4168218097957520114list_a )
=> ( Pa @ ( produc8531839378509307896list_a @ P3 @ S3 ) @ Pa2 ) ) ) )
=> ( ( ( sdrop_while_list_a @ P @ S2 )
= Y )
=> ( ( Y != undefi4168218097957520114list_a )
=> ( Pa @ ( produc8531839378509307896list_a @ P @ S2 ) @ Y ) ) ) ) ).
% sdrop_while.raw_induct
thf(fact_857_sdrop__while_Oraw__induct,axiom,
! [Pa: produc2815237062074537146ream_a > stream_a > $o,P: a > $o,S2: stream_a,Y: stream_a] :
( ! [Sdrop_while: ( a > $o ) > stream_a > stream_a] :
( ! [S5: a > $o,B7: stream_a] :
( ( ( Sdrop_while @ S5 @ B7 )
!= undefined_stream_a )
=> ( Pa @ ( produc8437782301311866860ream_a @ S5 @ B7 ) @ ( Sdrop_while @ S5 @ B7 ) ) )
=> ! [P3: a > $o,S3: stream_a,Pa2: stream_a] :
( ( ( ( P3 @ ( shd_a @ S3 ) )
=> ( ( Sdrop_while @ P3 @ ( stl_a @ S3 ) )
= Pa2 ) )
& ( ~ ( P3 @ ( shd_a @ S3 ) )
=> ( S3 = Pa2 ) ) )
=> ( ( Pa2 != undefined_stream_a )
=> ( Pa @ ( produc8437782301311866860ream_a @ P3 @ S3 ) @ Pa2 ) ) ) )
=> ( ( ( sdrop_while_a @ P @ S2 )
= Y )
=> ( ( Y != undefined_stream_a )
=> ( Pa @ ( produc8437782301311866860ream_a @ P @ S2 ) @ Y ) ) ) ) ).
% sdrop_while.raw_induct
thf(fact_858_subset__code_I1_J,axiom,
! [Xs: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_859_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_860_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_861_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_862_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_863_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_864_list_Oset__sel_I1_J,axiom,
! [A2: list_list_a] :
( ( A2 != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A2 ) @ ( set_list_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_865_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X2: a] :
( ( A2 != nil_a )
=> ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_866_list_Oset__sel_I2_J,axiom,
! [A2: list_list_a,X2: list_a] :
( ( A2 != nil_list_a )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ ( tl_list_a @ A2 ) ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_867_set__shuffles,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( set_a2 @ Zs )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_868_lexord__partial__trans,axiom,
! [Xs: list_a,R3: set_Product_prod_a_a,Ys: list_a,Zs: list_a] :
( ! [X: a,Y2: a,Z3: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ Z3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z3 ) @ R3 ) ) ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lexord_a @ R3 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R3 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs ) @ ( lexord_a @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_869_lexord__partial__trans,axiom,
! [Xs: list_list_a,R3: set_Pr4048851178543822343list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X: list_a,Y2: list_a,Z3: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y2 ) @ R3 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y2 @ Z3 ) @ R3 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Z3 ) @ R3 ) ) ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lexord_list_a @ R3 ) )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lexord_list_a @ R3 ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs ) @ ( lexord_list_a @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_870_cycle_Osimps_I2_J,axiom,
! [Xs: list_list_a] :
( ( stl_list_a @ ( cycle_list_a @ Xs ) )
= ( cycle_list_a @ ( append_list_a @ ( tl_list_a @ Xs ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) ) ) ) ).
% cycle.simps(2)
thf(fact_871_cycle_Osimps_I2_J,axiom,
! [Xs: list_a] :
( ( stl_a @ ( cycle_a @ Xs ) )
= ( cycle_a @ ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% cycle.simps(2)
thf(fact_872_sset__cycle,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( sset_a @ ( cycle_a @ Xs ) )
= ( set_a2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_873_sset__cycle,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( sset_list_a @ ( cycle_list_a @ Xs ) )
= ( set_list_a2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_874_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_875_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_876_last__in__set,axiom,
! [As: list_list_a] :
( ( As != nil_list_a )
=> ( member_list_a @ ( last_list_a @ As ) @ ( set_list_a2 @ As ) ) ) ).
% last_in_set
thf(fact_877_subset__code_I2_J,axiom,
! [A: set_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( coset_list_a @ Ys ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Ys ) )
=> ~ ( member_list_a @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_878_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ~ ( member_a @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_879_subset__subseqs,axiom,
! [X5: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ X5 @ ( set_a2 @ Xs ) )
=> ( member_set_a @ X5 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_880_image__eqI,axiom,
! [B3: a,F: a > a,X2: a,A: set_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B3 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_881_image__eqI,axiom,
! [B3: list_a,F: a > list_a,X2: a,A: set_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_list_a @ B3 @ ( image_a_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_882_image__eqI,axiom,
! [B3: a,F: list_a > a,X2: list_a,A: set_list_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_list_a @ X2 @ A )
=> ( member_a @ B3 @ ( image_list_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_883_image__eqI,axiom,
! [B3: list_a,F: list_a > list_a,X2: list_a,A: set_list_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_list_a @ X2 @ A )
=> ( member_list_a @ B3 @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_884_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_885_image__subsetI,axiom,
! [A: set_a,F: a > list_a,B: set_list_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_886_image__subsetI,axiom,
! [A: set_list_a,F: list_a > list_a,B: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B ) )
=> ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_887_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_888_image__subsetI,axiom,
! [A: set_list_a,F: list_a > a,B: set_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_889_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_890_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_891_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_a @ B3 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_892_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: list_a,F: a > list_a] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_list_a @ B3 @ ( image_a_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_893_rev__image__eqI,axiom,
! [X2: list_a,A: set_list_a,B3: a,F: list_a > a] :
( ( member_list_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_a @ B3 @ ( image_list_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_894_rev__image__eqI,axiom,
! [X2: list_a,A: set_list_a,B3: list_a,F: list_a > list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_list_a @ B3 @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_895_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_896_imageI,axiom,
! [X2: a,A: set_a,F: a > list_a] :
( ( member_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( image_a_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_897_imageI,axiom,
! [X2: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_list_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_898_imageI,axiom,
! [X2: list_a,A: set_list_a,F: list_a > list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_899_stream_Oset__map,axiom,
! [F: a > a,V: stream_a] :
( ( sset_a @ ( smap_a_a @ F @ V ) )
= ( image_a_a @ F @ ( sset_a @ V ) ) ) ).
% stream.set_map
thf(fact_900_stream_Oset__map,axiom,
! [F: list_a > a,V: stream_list_a] :
( ( sset_a @ ( smap_list_a_a @ F @ V ) )
= ( image_list_a_a @ F @ ( sset_list_a @ V ) ) ) ).
% stream.set_map
thf(fact_901_stream_Oset__map,axiom,
! [F: a > list_a,V: stream_a] :
( ( sset_list_a @ ( smap_a_list_a @ F @ V ) )
= ( image_a_list_a @ F @ ( sset_a @ V ) ) ) ).
% stream.set_map
thf(fact_902_stream_Oset__map,axiom,
! [F: list_a > list_a,V: stream_list_a] :
( ( sset_list_a @ ( smap_list_a_list_a @ F @ V ) )
= ( image_list_a_list_a @ F @ ( sset_list_a @ V ) ) ) ).
% stream.set_map
thf(fact_903_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_904_less__supI1,axiom,
! [X2: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% less_supI1
thf(fact_905_less__supI2,axiom,
! [X2: nat,B3: nat,A2: nat] :
( ( ord_less_nat @ X2 @ B3 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% less_supI2
thf(fact_906_sup_Oabsorb3,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb3
thf(fact_907_sup_Oabsorb4,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_908_sup_Ostrict__boundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_909_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_910_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_911_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_912_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_913_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_914_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_915_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_916_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_917_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_918_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_919_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_920_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_921_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_922_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N: nat] :
( ( P2 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P2 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_923_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_924_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_925_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_926_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_927_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_928_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_929_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_930_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_931_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_932_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_933_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_934_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_935_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_936_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_937_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_938_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_939_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_940_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_941_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_942_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_943_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_944_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_945_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_946_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_947_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_948_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_949_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_950_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_951_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_952_order__less__le__subst1,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_953_order__less__le__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( ord_less_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_954_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_955_order__le__less__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_956_order__le__less__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_957_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_958_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_959_order__le__less__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_960_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_961_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_962_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_963_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_964_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_965_order__neq__le__trans,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_966_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_967_order__le__neq__trans,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_968_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_969_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_970_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_971_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_972_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_973_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_974_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_975_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_set_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_976_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_977_dual__order_Ostrict__implies__order,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_set_a @ B3 @ A2 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_978_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_979_order_Ostrict__implies__order,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_980_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_981_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_982_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_983_dual__order_Ostrict__trans2,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_984_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_985_dual__order_Ostrict__trans1,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_set_a @ C @ B3 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_986_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_987_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_988_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_989_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_990_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_991_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_992_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_993_order_Ostrict__trans2,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_994_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_995_order_Ostrict__trans1,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_996_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_997_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_998_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_999_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1000_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1001_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1002_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1003_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1004_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1005_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1006_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1007_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1008_nless__le,axiom,
! [A2: set_a,B3: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_1009_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_1010_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_1011_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_1012_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_1013_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A8 ) )
= ( ord_less_nat @ A8 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1014_sdrop__stl,axiom,
! [N2: nat,S2: stream_list_a] :
( ( sdrop_list_a @ N2 @ ( stl_list_a @ S2 ) )
= ( stl_list_a @ ( sdrop_list_a @ N2 @ S2 ) ) ) ).
% sdrop_stl
thf(fact_1015_sdrop__stl,axiom,
! [N2: nat,S2: stream_a] :
( ( sdrop_a @ N2 @ ( stl_a @ S2 ) )
= ( stl_a @ ( sdrop_a @ N2 @ S2 ) ) ) ).
% sdrop_stl
thf(fact_1016_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B3 )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1017_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T3 ) ) ).
% pinf(6)
thf(fact_1018_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T3 @ X7 ) ) ).
% pinf(8)
thf(fact_1019_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T3 ) ) ).
% minf(6)
thf(fact_1020_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_1021_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_set_a @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1022_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1023_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ~ ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1024_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1025_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1026_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_1027_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_1028_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X7 ) ) ).
% minf(8)
thf(fact_1029_stake_Osimps_I2_J,axiom,
! [N2: nat,S2: stream_list_a] :
( ( stake_list_a @ ( suc @ N2 ) @ S2 )
= ( cons_list_a @ ( shd_list_a @ S2 ) @ ( stake_list_a @ N2 @ ( stl_list_a @ S2 ) ) ) ) ).
% stake.simps(2)
thf(fact_1030_stake_Osimps_I2_J,axiom,
! [N2: nat,S2: stream_a] :
( ( stake_a @ ( suc @ N2 ) @ S2 )
= ( cons_a @ ( shd_a @ S2 ) @ ( stake_a @ N2 @ ( stl_a @ S2 ) ) ) ) ).
% stake.simps(2)
thf(fact_1031_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_list_a] :
( ( stl_list_a @ ( sdrop_list_a @ N2 @ S2 ) )
= ( sdrop_list_a @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_1032_sdrop__simps_I2_J,axiom,
! [N2: nat,S2: stream_a] :
( ( stl_a @ ( sdrop_a @ N2 @ S2 ) )
= ( sdrop_a @ ( suc @ N2 ) @ S2 ) ) ).
% sdrop_simps(2)
thf(fact_1033_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1034_psubsetD,axiom,
! [A: set_list_a,B: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1035_set__take__subset,axiom,
! [N2: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ N2 @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_take_subset
thf(fact_1036_sdrop_Osimps_I2_J,axiom,
! [N2: nat,S2: stream_list_a] :
( ( sdrop_list_a @ ( suc @ N2 ) @ S2 )
= ( sdrop_list_a @ N2 @ ( stl_list_a @ S2 ) ) ) ).
% sdrop.simps(2)
thf(fact_1037_sdrop_Osimps_I2_J,axiom,
! [N2: nat,S2: stream_a] :
( ( sdrop_a @ ( suc @ N2 ) @ S2 )
= ( sdrop_a @ N2 @ ( stl_a @ S2 ) ) ) ).
% sdrop.simps(2)
thf(fact_1038_set__take__subset__set__take,axiom,
! [M2: nat,N2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ M2 @ Xs ) ) @ ( set_a2 @ ( take_a @ N2 @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1039_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1040_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1041_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1042_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1043_set__drop__subset,axiom,
! [N2: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ N2 @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1044_set__drop__subset__set__drop,axiom,
! [N2: nat,M2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ M2 @ Xs ) ) @ ( set_a2 @ ( drop_a @ N2 @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1045_SuccI,axiom,
! [Kl: list_list_a,K: list_a,Kl2: set_list_list_a] :
( ( member_list_list_a @ ( append_list_a @ Kl @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl2 )
=> ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_1046_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_1047_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_1048_SuccD,axiom,
! [K: list_a,Kl2: set_list_list_a,Kl: list_list_a] :
( ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ Kl ) )
=> ( member_list_list_a @ ( append_list_a @ Kl @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_1049_min_Obounded__iff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B3 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.bounded_iff
thf(fact_1050_min_Oabsorb2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_min_nat @ A2 @ B3 )
= B3 ) ) ).
% min.absorb2
thf(fact_1051_min_Oabsorb1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_min_nat @ A2 @ B3 )
= A2 ) ) ).
% min.absorb1
thf(fact_1052_min__less__iff__conj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Z2 @ X2 )
& ( ord_less_nat @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_1053_min_Oabsorb4,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_min_nat @ A2 @ B3 )
= B3 ) ) ).
% min.absorb4
thf(fact_1054_min_Oabsorb3,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_min_nat @ A2 @ B3 )
= A2 ) ) ).
% min.absorb3
thf(fact_1055_min_Omono,axiom,
! [A2: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B3 ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_1056_min_OorderE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2
= ( ord_min_nat @ A2 @ B3 ) ) ) ).
% min.orderE
thf(fact_1057_min_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% min.orderI
thf(fact_1058_min_OboundedE,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.boundedE
thf(fact_1059_min_OboundedI,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B3 @ C ) ) ) ) ).
% min.boundedI
thf(fact_1060_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( ord_min_nat @ A4 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_1061_min_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B3 ) @ A2 ) ).
% min.cobounded1
thf(fact_1062_min_Ocobounded2,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B3 ) @ B3 ) ).
% min.cobounded2
thf(fact_1063_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_min_nat @ A4 @ B4 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_1064_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_1065_min_OcoboundedI1,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B3 ) @ C ) ) ).
% min.coboundedI1
thf(fact_1066_min_OcoboundedI2,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B3 ) @ C ) ) ).
% min.coboundedI2
thf(fact_1067_min__le__iff__disj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
| ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_1068_min__def,axiom,
( ord_min_set_a
= ( ^ [A4: set_a,B4: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% min_def
thf(fact_1069_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ A4 @ B4 ) ) ) ).
% min_def
thf(fact_1070_min__absorb1,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_min_set_a @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1071_min__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_min_nat @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1072_min__absorb2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_min_set_a @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1073_min__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_min_nat @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1074_min_Ostrict__coboundedI2,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A2 @ B3 ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_1075_min_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A2 @ B3 ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_1076_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( A4
= ( ord_min_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1077_min_Ostrict__boundedE,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( ord_min_nat @ B3 @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% min.strict_boundedE
thf(fact_1078_min__less__iff__disj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_nat @ X2 @ Z2 )
| ( ord_less_nat @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_1079_set__update__subsetI,axiom,
! [Xs: list_list_a,A: set_list_a,X2: list_a,I: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
=> ( ( member_list_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( list_update_list_a @ Xs @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1080_set__update__subsetI,axiom,
! [Xs: list_a,A: set_a,X2: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
=> ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1081_empty__Shift,axiom,
! [Kl2: set_list_list_a,K: list_a] :
( ( member_list_list_a @ nil_list_a @ Kl2 )
=> ( ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl2 @ nil_list_a ) )
=> ( member_list_list_a @ nil_list_a @ ( bNF_Gr7042794125918077091list_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_1082_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_1083_snth__in,axiom,
! [S2: stream_a,X5: set_a,N2: nat] :
( ( member_stream_a @ S2 @ ( streams_a @ X5 ) )
=> ( member_a @ ( snth_a @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_1084_snth__in,axiom,
! [S2: stream_list_a,X5: set_list_a,N2: nat] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ X5 ) )
=> ( member_list_a @ ( snth_list_a @ S2 @ N2 ) @ X5 ) ) ).
% snth_in
thf(fact_1085_streams__iff__snth,axiom,
! [S2: stream_a,X5: set_a] :
( ( member_stream_a @ S2 @ ( streams_a @ X5 ) )
= ( ! [N: nat] : ( member_a @ ( snth_a @ S2 @ N ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_1086_streams__iff__snth,axiom,
! [S2: stream_list_a,X5: set_list_a] :
( ( member_stream_list_a @ S2 @ ( streams_list_a @ X5 ) )
= ( ! [N: nat] : ( member_list_a @ ( snth_list_a @ S2 @ N ) @ X5 ) ) ) ).
% streams_iff_snth
thf(fact_1087_snth__sset,axiom,
! [S2: stream_a,N2: nat] : ( member_a @ ( snth_a @ S2 @ N2 ) @ ( sset_a @ S2 ) ) ).
% snth_sset
thf(fact_1088_snth__sset,axiom,
! [S2: stream_list_a,N2: nat] : ( member_list_a @ ( snth_list_a @ S2 @ N2 ) @ ( sset_list_a @ S2 ) ) ).
% snth_sset
thf(fact_1089_snth_Osimps_I2_J,axiom,
! [S2: stream_list_a,N2: nat] :
( ( snth_list_a @ S2 @ ( suc @ N2 ) )
= ( snth_list_a @ ( stl_list_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_1090_snth_Osimps_I2_J,axiom,
! [S2: stream_a,N2: nat] :
( ( snth_a @ S2 @ ( suc @ N2 ) )
= ( snth_a @ ( stl_a @ S2 ) @ N2 ) ) ).
% snth.simps(2)
thf(fact_1091_flat__snth,axiom,
! [S2: stream_list_a,N2: nat] :
( ! [X: list_a] :
( ( member_list_a @ X @ ( sset_list_a @ S2 ) )
=> ( X != nil_a ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) )
=> ( ( snth_a @ ( flat_a @ S2 ) @ N2 )
= ( nth_a @ ( shd_list_a @ S2 ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) )
=> ( ( snth_a @ ( flat_a @ S2 ) @ N2 )
= ( snth_a @ ( flat_a @ ( stl_list_a @ S2 ) ) @ ( minus_minus_nat @ N2 @ ( size_size_list_a @ ( shd_list_a @ S2 ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_1092_fair__def,axiom,
( fair_fair_a
= ( ^ [S: stream_a] :
! [X3: a] :
( ( member_a @ X3 @ ( sset_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_1093_fair__def,axiom,
( fair_fair_list_a
= ( ^ [S: stream_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ ( sset_list_a @ S ) )
=> ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( ( snth_list_a @ S @ N )
= X3 ) ) ) ) ) ).
% fair_def
thf(fact_1094_snth__sset__smerge,axiom,
! [Ss: stream_stream_a,N2: nat,M2: nat] : ( member_a @ ( snth_a @ ( snth_stream_a @ Ss @ N2 ) @ M2 ) @ ( sset_a @ ( smerge_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_1095_snth__sset__smerge,axiom,
! [Ss: stream_stream_list_a,N2: nat,M2: nat] : ( member_list_a @ ( snth_list_a @ ( snth_stream_list_a @ Ss @ N2 ) @ M2 ) @ ( sset_list_a @ ( smerge_list_a @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_1096_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1097_psubset__imp__ex__mem,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ? [B5: list_a] : ( member_list_a @ B5 @ ( minus_646659088055828811list_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1098_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1099_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1100_Diff__mono,axiom,
! [A: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1101_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1102_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1103_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1104_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B ) )
= ( ( member_list_a @ C @ A )
& ~ ( member_list_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1105_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1106_DiffI,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ~ ( member_list_a @ C @ B )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1107_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1108_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_1109_DiffD2,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( member_list_a @ C @ B ) ) ).
% DiffD2
thf(fact_1110_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_1111_DiffD1,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ( member_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_1112_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1113_DiffE,axiom,
! [C: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B ) )
=> ~ ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1114_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1115_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1116_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1117_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1118_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1119_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1120_div__less__mono,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( modulo_modulo_nat @ A @ N2 )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B @ N2 )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A @ N2 ) @ ( divide_divide_nat @ B @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1121_verit__le__mono__div,axiom,
! [A: nat,B: nat,N2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A @ N2 )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B @ N2 )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1122_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1123_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_1124_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_1125_add__le__same__cancel1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1126_add__le__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1127_le__add__same__cancel1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1128_le__add__same__cancel2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
thf(fact_1129_le__add__diff__inverse,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1130_le__add__diff__inverse2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1131_image__add__0,axiom,
! [S4: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S4 )
= S4 ) ).
% image_add_0
thf(fact_1132_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N2: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1133_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1134_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_1135_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1136_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1137_add__nonpos__nonpos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1138_add__nonneg__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1139_add__increasing2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1140_add__decreasing2,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1141_add__increasing,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_1142_add__decreasing,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1143_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1144_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1145_add__le__less__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1146_add__less__le__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1147_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A2 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1148_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1149_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1150_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1151_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1152_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1153_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1154_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1155_le__add__diff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1156_diff__add,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
= B3 ) ) ).
% diff_add
thf(fact_1157_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1158_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1159_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1160_add__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1161_add__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_1162_less__eqE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ! [C4: nat] :
( B3
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_1163_add__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_1164_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_1165_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_1166_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_1167_sorted__take,axiom,
! [Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N2 @ Xs ) ) ) ).
% sorted_take
thf(fact_1168_sorted__nths,axiom,
! [Xs: list_nat,I3: set_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( nths_nat @ Xs @ I3 ) ) ) ).
% sorted_nths
thf(fact_1169_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1170_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1171_sorted2,axiom,
! [X2: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1172_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1173_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_1174_sorted__tl,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs ) ) ) ).
% sorted_tl
thf(fact_1175_sorted__drop,axiom,
! [Xs: list_nat,N2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N2 @ Xs ) ) ) ).
% sorted_drop
thf(fact_1176_sorted__replicate,axiom,
! [N2: nat,X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N2 @ X2 ) ) ).
% sorted_replicate
thf(fact_1177_set__nths__subset,axiom,
! [Xs: list_a,I3: set_nat] : ( ord_less_eq_set_a @ ( set_a2 @ ( nths_a @ Xs @ I3 ) ) @ ( set_a2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1178_sorted1,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted1
thf(fact_1179_sorted__simps_I2_J,axiom,
! [X2: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X2 @ X3 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_1180_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ).
% sorted_append
thf(fact_1181_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1182_add__neg__nonpos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1183_add__nonneg__pos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1184_add__nonpos__neg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1185_add__pos__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1186_add__strict__increasing,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1187_add__strict__increasing2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1188_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_1189_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1190_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1191_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1192_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1193_in__set__zipE,axiom,
! [X2: a,Y: a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1194_in__set__zipE,axiom,
! [X2: a,Y: list_a,Xs: list_a,Ys: list_list_a] :
( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X2 @ Y ) @ ( set_Pr3039917418810828226list_a @ ( zip_a_list_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ~ ( member_list_a @ Y @ ( set_list_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1195_in__set__zipE,axiom,
! [X2: list_a,Y: a,Xs: list_list_a,Ys: list_a] :
( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X2 @ Y ) @ ( set_Pr6156699705114756022st_a_a @ ( zip_list_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1196_in__set__zipE,axiom,
! [X2: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y ) @ ( set_Pr2906193453920572092list_a @ ( zip_list_a_list_a @ Xs @ Ys ) ) )
=> ~ ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ~ ( member_list_a @ Y @ ( set_list_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1197_zip__same,axiom,
! [A2: a,B3: a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
= ( ( member_a @ A2 @ ( set_a2 @ Xs ) )
& ( A2 = B3 ) ) ) ).
% zip_same
thf(fact_1198_zip__same,axiom,
! [A2: list_a,B3: list_a,Xs: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A2 @ B3 ) @ ( set_Pr2906193453920572092list_a @ ( zip_list_a_list_a @ Xs @ Xs ) ) )
= ( ( member_list_a @ A2 @ ( set_list_a2 @ Xs ) )
& ( A2 = B3 ) ) ) ).
% zip_same
thf(fact_1199_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I2 ) ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1200_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1201_foldr__max__sorted,axiom,
! [Xs: list_nat,Y: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ( Xs = nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y )
= Y ) )
& ( ( Xs != nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y )
= ( ord_max_nat @ ( nth_nat @ Xs @ zero_zero_nat ) @ Y ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_1202_max_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% max.bounded_iff
thf(fact_1203_max_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_max_nat @ A2 @ B3 )
= B3 ) ) ).
% max.absorb2
thf(fact_1204_max_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_max_nat @ A2 @ B3 )
= A2 ) ) ).
% max.absorb1
thf(fact_1205_max__less__iff__conj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_nat @ X2 @ Z2 )
& ( ord_less_nat @ Y @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_1206_max_Oabsorb4,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_max_nat @ A2 @ B3 )
= B3 ) ) ).
% max.absorb4
thf(fact_1207_max_Oabsorb3,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_max_nat @ A2 @ B3 )
= A2 ) ) ).
% max.absorb3
thf(fact_1208_mult__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1209_mult__mono_H,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1210_mult__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_1211_mult__right__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1212_split__mult__neg__le,axiom,
! [A2: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1213_mult__nonneg__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1214_mult__nonneg__nonpos,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1215_mult__nonpos__nonneg,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1216_mult__nonneg__nonpos2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1217_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1218_max_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D2 ) @ ( ord_max_nat @ A2 @ B3 ) ) ) ) ).
% max.mono
thf(fact_1219_max_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( ord_max_nat @ A2 @ B3 ) ) ) ).
% max.orderE
thf(fact_1220_max_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( ord_max_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% max.orderI
thf(fact_1221_max_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% max.boundedE
thf(fact_1222_max_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 ) ) ) ).
% max.boundedI
thf(fact_1223_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% max.order_iff
thf(fact_1224_max_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% max.cobounded1
thf(fact_1225_max_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% max.cobounded2
thf(fact_1226_le__max__iff__disj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X2 @ Y ) )
= ( ( ord_less_eq_nat @ Z2 @ X2 )
| ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_1227_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B4 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_1228_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_max_nat @ A4 @ B4 )
= B4 ) ) ) ).
% max.absorb_iff2
thf(fact_1229_max_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% max.coboundedI1
thf(fact_1230_max_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% max.coboundedI2
thf(fact_1231_max__def,axiom,
( ord_max_set_a
= ( ^ [A4: set_a,B4: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% max_def
thf(fact_1232_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% max_def
thf(fact_1233_max__absorb1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_max_set_a @ X2 @ Y )
= X2 ) ) ).
% max_absorb1
thf(fact_1234_max__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_max_nat @ X2 @ Y )
= X2 ) ) ).
% max_absorb1
thf(fact_1235_max__absorb2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_max_set_a @ X2 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1236_max__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_max_nat @ X2 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_1237_max_Ostrict__coboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% max.strict_coboundedI2
thf(fact_1238_max_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% max.strict_coboundedI1
thf(fact_1239_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_1240_max_Ostrict__boundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% max.strict_boundedE
thf(fact_1241_less__max__iff__disj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Z2 @ X2 )
| ( ord_less_nat @ Z2 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_1242_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1243_mult__strict__mono,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1244_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1245_mult__strict__mono_H,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1246_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1247_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1248_mult__le__less__imp__less,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1249_mult__less__le__imp__less,axiom,
! [A2: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1250_mult__le__one,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1251_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_1252_div__mod__decomp,axiom,
! [A: nat,N2: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_1253_power__decreasing__iff,axiom,
! [B3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M2 ) @ ( power_power_nat @ B3 @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1254_power__increasing__iff,axiom,
! [B3: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1255_power__mono__iff,axiom,
! [A2: nat,B3: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) )
= ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1256_zero__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% zero_le_power
thf(fact_1257_one__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_1258_power__mono,axiom,
! [A2: nat,B3: nat,N2: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ) ).
% power_mono
thf(fact_1259_power__less__imp__less__base,axiom,
! [A2: nat,N2: nat,B3: nat] :
( ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_1260_power__le__one,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1261_power__le__imp__le__base,axiom,
! [A2: nat,N2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ ( power_power_nat @ B3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_1262_power__inject__base,axiom,
! [A2: nat,N2: nat,B3: nat] :
( ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
= ( power_power_nat @ B3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_1263_power__increasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1264_power__Suc__le__self,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_1265_power__decreasing,axiom,
! [N2: nat,N5: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N5 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_1266_power__le__imp__le__exp,axiom,
! [A2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M2 ) @ ( power_power_nat @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_1267_power__eq__imp__eq__base,axiom,
! [A2: nat,N2: nat,B3: nat] :
( ( ( power_power_nat @ A2 @ N2 )
= ( power_power_nat @ B3 @ N2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( A2 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1268_power__eq__iff__eq__base,axiom,
! [N2: nat,A2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ( power_power_nat @ A2 @ N2 )
= ( power_power_nat @ B3 @ N2 ) )
= ( A2 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1269_self__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_1270_power__strict__mono,axiom,
! [A2: nat,B3: nat,N2: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ) ) ).
% power_strict_mono
thf(fact_1271_numeral__le__iff,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
= ( ord_less_eq_num @ M2 @ N2 ) ) ).
% numeral_le_iff
thf(fact_1272_min__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) ) ) ).
% min_number_of(1)
thf(fact_1273_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_1274_one__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% one_le_numeral
thf(fact_1275_zero__le__numeral,axiom,
! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% zero_le_numeral
thf(fact_1276_not__numeral__le__zero,axiom,
! [N2: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1277_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_1278_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
% Helper facts (19)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y: set_a] :
( ( if_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y: set_a] :
( ( if_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_Itf__a_J_T,axiom,
! [X2: stream_a,Y: stream_a] :
( ( if_stream_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_Itf__a_J_T,axiom,
! [X2: stream_a,Y: stream_a] :
( ( if_stream_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: stream_list_a,Y: stream_list_a] :
( ( if_stream_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_Itf__a_J_J_T,axiom,
! [X2: stream_list_a,Y: stream_list_a] :
( ( if_stream_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream_list_list_a,Y: stream_list_list_a] :
( ( if_str7505741754068378070list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream_list_list_a,Y: stream_list_list_a] :
( ( if_str7505741754068378070list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X2: stream6512538624140940077od_a_a,Y: stream6512538624140940077od_a_a] :
( ( if_str8987598796600344819od_a_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
! [X2: stream6512538624140940077od_a_a,Y: stream6512538624140940077od_a_a] :
( ( if_str8987598796600344819od_a_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J_T,axiom,
! [X2: stream5874003577662885159st_a_a,Y: stream5874003577662885159st_a_a] :
( ( if_str2612623181902290925st_a_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J_J_T,axiom,
! [X2: stream5874003577662885159st_a_a,Y: stream5874003577662885159st_a_a] :
( ( if_str2612623181902290925st_a_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream8205385055505213747list_a,Y: stream8205385055505213747list_a] :
( ( if_str4944004659744619513list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream8205385055505213747list_a,Y: stream8205385055505213747list_a] :
( ( if_str4944004659744619513list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream271976807216406061list_a,Y: stream271976807216406061list_a] :
( ( if_str6195518498109691379list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_T,axiom,
! [X2: stream271976807216406061list_a,Y: stream271976807216406061list_a] :
( ( if_str6195518498109691379list_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_a @ ( sset_a @ ( flat_a @ ( stl_list_a @ s ) ) ) @ ( sset_a @ ( flat_a @ s ) ) ).
%------------------------------------------------------------------------------