TPTP Problem File: SLH0099^1.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 : Eval_FO/0005_Ailamazyan/prob_01720_063961__15736704_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1581 ( 656 unt; 330 typ; 0 def)
% Number of atoms : 3280 (2107 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 13501 ( 540 ~; 73 |; 310 &;11123 @)
% ( 0 <=>;1455 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 6 avg)
% Number of types : 61 ( 60 usr)
% Number of type conns : 865 ( 865 >; 0 *; 0 +; 0 <<)
% Number of symbols : 273 ( 270 usr; 22 con; 0-7 aty)
% Number of variables : 3887 ( 65 ^;3643 !; 179 ?;3887 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:37.727
%------------------------------------------------------------------------------
% Could-be-implicit typings (60)
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J_J,type,
sum_su5729848735222297843_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
produc1308816576071065077_a_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J_J_J,type,
sum_su7540102398289458977st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
produc4888785615639907219_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
produc2080012076104740651_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
produc6500877907478405085st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc1169793092761538656_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
produc4708264106712766336_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
produc7400749417448455946_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
produc2221362464285539356st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc8636276049245547145_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
option4961383686904876949_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc7989883672844959049_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc6930560640458034917_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc6993310590455123023_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
produc8978234371640307703st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc3388037434187558270_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc2799874090934363020_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_Pr4870381170404451655_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc5001885624171833703_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_l7568939571203725683_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc5438398690206989622st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr8517726817977820874st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_Pr914094233313105488_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc4502985402200462317_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P1195027771636113901_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Pr7343886759072863943_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc3109216143880453012st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc8199838167917776410_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_l7136637534808337348_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc7017002724195966439_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc1674160212089442510_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc5901729443898700970_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
list_P6164600145584960654at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P5056861408695629236_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr2129020469590976052at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Pr1021281732701644634_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_l1586611297644397841_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc7258583773236448510at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc5454855657571582244_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_l4703314356710769291_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_li6526943997496501093_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
option_Sum_sum_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $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__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 (270)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oadd__nth_001t__Nat__Onat,type,
add_nth_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Ailamazyan_Oadd__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
add_nt4212672348507122516_a_nat: nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oall__tuples_001t__Nat__Onat,type,
all_tuples_nat: set_nat > nat > set_list_nat ).
thf(sy_c_Ailamazyan_Oall__tuples_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
all_tu407047557562860027_a_nat: set_Sum_sum_a_nat > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlzd_001tf__a,type,
fo_nmlzd_a: set_a > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Onall__tuples_001tf__a,type,
nall_tuples_a: set_a > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Opos_001t__Nat__Onat,type,
pos_nat: nat > list_nat > option_nat ).
thf(sy_c_Ailamazyan_Opos_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
pos_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > option_nat ).
thf(sy_c_Ailamazyan_Orem__nth_001t__Nat__Onat,type,
rem_nth_nat: nat > list_nat > list_nat ).
thf(sy_c_Ailamazyan_Orem__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rem_nt658808235856662061_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Orremdups_001t__Nat__Onat,type,
rremdups_nat: list_nat > list_nat ).
thf(sy_c_Ailamazyan_Orremdups_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rremdu8304153113908149561_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_AssocList_Ozip__with__index__from_001t__Nat__Onat,type,
zip_wi3407404960461264653om_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_AssocList_Ozip__with__index__from_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_wi5053028342928268674_a_nat: nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
bNF_Gr1229660863860170270_a_nat: set_li6526943997496501093_a_nat > sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
bNF_Gr5582227268375839130_a_nat: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite6080979521523705895_a_nat: set_Sum_sum_a_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
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_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
if_lis4685338526944683083_a_nat: $o > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
append5415888156905520160_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
append338925788367110473_a_nat: list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
append2142653904031976739at_nat: list_P6164600145584960654at_nat > list_P6164600145584960654at_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
append1996214168388709506_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Oappend_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
append_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
bind_l5927154061965146638at_nat: list_l4703314356710769291_a_nat > ( list_Sum_sum_a_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
bind_n431031891494352236_a_nat: list_nat > ( nat > list_l4703314356710769291_a_nat ) > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bind_n1878750130520726888at_nat: list_nat > ( nat > list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
bind_n1484640130766398205_a_nat: list_nat > ( nat > list_P5056861408695629236_a_nat ) > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
bind_n3288368246431264471at_nat: list_nat > ( nat > list_P6164600145584960654at_nat ) > list_P6164600145584960654at_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
bind_n2905796707917019494_a_nat: list_nat > ( nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat ).
thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bind_S7309778641578091144at_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bind_S8065145433898892361at_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_P6011104703257516679at_nat ) > list_P6011104703257516679at_nat ).
thf(sy_c_List_Obind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
bind_S8497345843576593223_a_nat: list_Sum_sum_a_nat > ( sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
butlas5768530507476509265_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
concat_Sum_sum_a_nat: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Ocoset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
coset_Sum_sum_a_nat: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
distin2701893636801681144_a_nat: list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
distin3423869863095720157_a_nat: list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
drop_Sum_sum_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
enumer3164015132978342500_a_nat: nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Oextract_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
extrac3289232528736335879_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > option4961383686904876949_a_nat ).
thf(sy_c_List_Ofind_001t__Nat__Onat,type,
find_nat: ( nat > $o ) > list_nat > option_nat ).
thf(sy_c_List_Ofind_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
find_Sum_sum_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > option_Sum_sum_a_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
foldr_8827264258039228903_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
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_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
gen_le1340941697924381074_a_nat: nat > list_Sum_sum_a_nat > nat ).
thf(sy_c_List_Oinsert_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
insert_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
last_P6484183829340986144at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
last_P5310735818784388933_a_nat: list_P5056861408695629236_a_nat > produc5454855657571582244_a_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
last_P7114463934449255199at_nat: list_P6164600145584960654at_nat > produc7258583773236448510at_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
last_P17124104574654342_a_nat: list_P1195027771636113901_a_nat > produc7017002724195966439_a_nat ).
thf(sy_c_List_Olast_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
last_Sum_sum_a_nat: list_Sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
lenlex_Sum_sum_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
lex_Sum_sum_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
cons_l2563873727033190209_a_nat: list_l4703314356710769291_a_nat > list_l1586611297644397841_a_nat > list_l1586611297644397841_a_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
cons_l6787550886680756862_a_nat: list_P5056861408695629236_a_nat > list_l7136637534808337348_a_nat > list_l7136637534808337348_a_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
cons_l1779357001063338659_a_nat: list_P1195027771636113901_a_nat > list_l7568939571203725683_a_nat > list_l7568939571203725683_a_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
cons_l6604326339930385211_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
cons_P8612082756026972910_a_nat: produc5454855657571582244_a_nat > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
cons_P1192438834837063368at_nat: produc7258583773236448510at_nat > list_P6164600145584960654at_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
cons_P1525839536144884125_a_nat: produc7017002724195966439_a_nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
cons_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
nil_li2117038862230905745_a_nat: list_l1586611297644397841_a_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
nil_li1907017536197114414_a_nat: list_l7136637534808337348_a_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
nil_li4039925377676013299_a_nat: list_l7568939571203725683_a_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
nil_li1906260230833442699_a_nat: list_l4703314356710769291_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
nil_Pr237480997409426078_a_nat: list_P5056861408695629236_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
nil_Pr2041209113074292344at_nat: list_P6164600145584960654at_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
nil_Pr6585251977456444909_a_nat: list_P1195027771636113901_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
nil_Sum_sum_a_nat: list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
hd_lis3280420719747858032_a_nat: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
hd_Pro3965030861105696466_a_nat: list_P1195027771636113901_a_nat > produc7017002724195966439_a_nat ).
thf(sy_c_List_Olist_Ohd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
hd_Sum_sum_a_nat: list_Sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_li2392974972034027290_a_nat: list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ).
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__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr2196415342531872975_a_nat: list_P5056861408695629236_a_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
set_Pr4000143458196739241at_nat: list_P6164600145584960654at_nat > set_Pr2129020469590976052at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr2931239450594890620_a_nat: list_P1195027771636113901_a_nat > set_Pr7343886759072863943_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Osize__list_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
size_l7939150027356015111_a_nat: ( sum_sum_a_nat > nat ) > list_Sum_sum_a_nat > nat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
tl_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist__ex1_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
list_e7903279875808954312_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
list_u9138855634547462509_a_nat: list_Sum_sum_a_nat > nat > sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel1_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
listre5822253806113410398_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
listre3277123121685602172_a_nat: set_Pr1021281732701644634_a_nat > set_Pr914094233313105488_a_nat ).
thf(sy_c_List_Olistrel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
listre7681105055346673822at_nat: set_Pr2129020469590976052at_nat > set_Pr8517726817977820874st_nat ).
thf(sy_c_List_Olistrel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
listre7581909074878054321_a_nat: set_Pr7343886759072863943_a_nat > set_Pr4870381170404451655_a_nat ).
thf(sy_c_List_Olistrelp_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
listre4668766549960312319_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Omap__tailrec__rev_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_ta7636758496465269173_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
maps_P3994670031119939231_a_nat: ( produc7017002724195966439_a_nat > list_l4703314356710769291_a_nat ) > list_P1195027771636113901_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
maps_P5371491622303101494at_nat: ( produc7017002724195966439_a_nat > list_nat ) > list_P1195027771636113901_a_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
maps_P1432663425352833819at_nat: ( produc7017002724195966439_a_nat > list_P6011104703257516679at_nat ) > list_P1195027771636113901_a_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
maps_P3502264056893038730_a_nat: ( produc7017002724195966439_a_nat > list_P5056861408695629236_a_nat ) > list_P1195027771636113901_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
maps_P5305992172557904996at_nat: ( produc7017002724195966439_a_nat > list_P6164600145584960654at_nat ) > list_P1195027771636113901_a_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
maps_P6445506607073456153_a_nat: ( produc7017002724195966439_a_nat > list_Sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Omaps_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
maps_S8041221185523983617_a_nat: ( sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
n_list6375351914370498317_a_nat: nat > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
nth_Pr4195520319383970909_a_nat: list_P5056861408695629236_a_nat > nat > produc5454855657571582244_a_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
nth_Pr5999248435048837175at_nat: list_P6164600145584960654at_nat > nat > produc7258583773236448510at_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
nth_Pr7458973636520993902_a_nat: list_P1195027771636113901_a_nat > nat > produc7017002724195966439_a_nat ).
thf(sy_c_List_Onth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
nth_Sum_sum_a_nat: list_Sum_sum_a_nat > nat > sum_sum_a_nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc7294178216264498984_a_nat: list_nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc2474788113070794826at_nat: list_Sum_sum_a_nat > list_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc7134955936706270469_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc4105601411378645364_a_nat: list_l1586611297644397841_a_nat > list_l1586611297644397841_a_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc8430089408672153845_a_nat: list_l7136637534808337348_a_nat > list_l7136637534808337348_a_nat ).
thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc3960677393309065686_a_nat: list_l7568939571203725683_a_nat > list_l7568939571203725683_a_nat ).
thf(sy_c_List_Oproduct__lists_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc2893206433618375022_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
remdup8712921452854877243_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
replic8141442572502817605_a_nat: nat > list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
replic8955434655033810879_a_nat: nat > sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
rev_li2372797785747802603_a_nat: list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
rev_Pr6102188148953555047at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
rev_Pr1186578271699380286_a_nat: list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
rev_Pr2990306387364246552at_nat: list_P6164600145584960654at_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
rev_Pr4274687663743590477_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Orev_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rev_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rotate2765497868024679250_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rotate_Sum_sum_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sorted6245805940552704876_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
subseq4596077244143087834_a_nat: list_l4703314356710769291_a_nat > list_l1586611297644397841_a_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
subseq631295553041127183_a_nat: list_P5056861408695629236_a_nat > list_l7136637534808337348_a_nat ).
thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
subseq5198752961258583356_a_nat: list_P1195027771636113901_a_nat > list_l7568939571203725683_a_nat ).
thf(sy_c_List_Osubseqs_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
subseq8414445098004693972_a_nat: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
take_Sum_sum_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_na2013496608136855606_a_nat: list_nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
zip_Su6417478541797927256at_nat: list_Sum_sum_a_nat > list_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_Su7355543910597222519_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s5212483967078203639_a_nat: list_l4703314356710769291_a_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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s4076174644546656840_a_nat: list_P5056861408695629236_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
size_s5183913381435988258at_nat: list_P6164600145584960654at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s7247017532682008665_a_nat: list_P1195027771636113901_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
size_s5283204784079214577_a_nat: list_Sum_sum_a_nat > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
some_nat: nat > option_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
some_P7299253002876453812_a_nat: produc6993310590455123023_a_nat > option4961383686904876949_a_nat ).
thf(sy_c_Option_Ooption_OSome_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
some_Sum_sum_a_nat: sum_sum_a_nat > option_Sum_sum_a_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__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1325389633284124927_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $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_Product__Type_OPair_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc8857357843056227054_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > produc3388037434187558270_a_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc2770702104727902672_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > produc5001885624171833703_a_nat > produc1169793092761538656_a_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc2047461453323441237_a_nat: ( sum_sum_a_nat > option_nat ) > produc1674160212089442510_a_nat > produc6930560640458034917_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc5055497221108209740_a_nat: list_nat > list_Sum_sum_a_nat > produc8199838167917776410_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc4487115339913071592st_nat: list_nat > produc1828647624359046049st_nat > produc5438398690206989622st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc3967909809349605295st_nat: list_nat > produc5438398690206989622st_nat > produc8978234371640307703st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J,type,
produc92214677965515925st_nat: list_nat > produc2221362464285539356st_nat > produc6500877907478405085st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc9183987541845296142st_nat: list_Sum_sum_a_nat > list_nat > produc3109216143880453012st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc7990843422341522135_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > produc5001885624171833703_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc3577755783922481849_a_nat: list_Sum_sum_a_nat > produc5001885624171833703_a_nat > produc7989883672844959049_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc3915550414976331931_a_nat: list_Sum_sum_a_nat > produc7989883672844959049_a_nat > produc2080012076104740651_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc7578773175992006207_a_nat: list_Sum_sum_a_nat > produc4502985402200462317_a_nat > produc6993310590455123023_a_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
produc2222311365791241189_a_nat: list_Sum_sum_a_nat > produc4888785615639907219_a_nat > produc1308816576071065077_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc4502019603863884636_a_nat: nat > list_Sum_sum_a_nat > produc5901729443898700970_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc9161022033385942332_a_nat: nat > produc6930560640458034917_a_nat > produc7400749417448455946_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
produc5609319852033172942st_nat: nat > produc8978234371640307703st_nat > produc2221362464285539356st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc5571422593549375550_a_nat: nat > produc5001885624171833703_a_nat > produc2799874090934363020_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc3265382261054541654_a_nat: nat > sum_sum_a_nat > produc5454855657571582244_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
produc785221303313254649_a_nat: option_Sum_sum_a_nat > produc5001885624171833703_a_nat > produc8636276049245547145_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc2856885992714847610_a_nat: option_Sum_sum_a_nat > produc2799874090934363020_a_nat > produc4708264106712766336_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc5501371206643777598_a_nat: set_a > list_Sum_sum_a_nat > produc1674160212089442510_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc6350064662657521885_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > produc4502985402200462317_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc7669364194715613304at_nat: sum_sum_a_nat > nat > produc7258583773236448510at_nat ).
thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
produc5024437050894261379_a_nat: sum_sum_a_nat > produc2080012076104740651_a_nat > produc4888785615639907219_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc1212125651291703639_a_nat: sum_sum_a_nat > sum_sum_a_nat > produc7017002724195966439_a_nat ).
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__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
collec7555443234367654128_a_nat: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
collec7073057861543223018_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Impl_Olinorder__class_Osingle__list_001t__Nat__Onat,type,
set_li756733140798618988st_nat: nat > list_nat ).
thf(sy_c_Set__Impl_Oord_Oquicksort_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_qu7255043234445450382_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Set__Impl_Oord_Oquicksort__acc_001t__Nat__Onat,type,
set_qu8921977862092465858cc_nat: ( nat > nat > $o ) > list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord_Oquicksort__acc_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_qu7651081299428620429_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Set__Impl_Oord_Oquicksort__acc__quicksort__part__rel_001t__Nat__Onat,type,
set_qu876879003691614877el_nat: ( nat > nat > $o ) > sum_su7540102398289458977st_nat > sum_su7540102398289458977st_nat > $o ).
thf(sy_c_Set__Impl_Oord_Oquicksort__acc__quicksort__part__rel_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_qu2333772366303858162_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > sum_su5729848735222297843_a_nat > sum_su5729848735222297843_a_nat > $o ).
thf(sy_c_Set__Impl_Oord_Oquicksort__part_001t__Nat__Onat,type,
set_qu42629670029557428rt_nat: ( nat > nat > $o ) > list_nat > nat > list_nat > list_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord_Oquicksort__part_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_qu7459554806609531931_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__Nat__Onat,type,
set_or9089632773640736191rt_nat: list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort__acc_001t__Nat__Onat,type,
set_or5558937660843164036cc_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort__acc__quicksort__part__rel_001t__Nat__Onat,type,
set_or192534051960689247el_nat: sum_su7540102398289458977st_nat > sum_su7540102398289458977st_nat > $o ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort__part_001t__Nat__Onat,type,
set_or1804217446461887602rt_nat: list_nat > nat > list_nat > list_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oremdups__sorted_001t__Nat__Onat,type,
set_or6599480164596245535ed_nat: list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Osorted__list__subset_001t__Nat__Onat,type,
set_or6742139631805365739et_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J_J,type,
sum_In4817299372063595370st_nat: produc1828647624359046049st_nat > sum_su7540102398289458977st_nat ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
sum_In8892132817167105852_a_nat: produc5001885624171833703_a_nat > sum_su5729848735222297843_a_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
sum_In4791677611794227960st_nat: produc6500877907478405085st_nat > sum_su7540102398289458977st_nat ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sum_In1820214634257660838_a_nat: produc1308816576071065077_a_nat > sum_su5729848735222297843_a_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J_J_J_J,type,
accp_S1710931637807102040st_nat: ( sum_su7540102398289458977st_nat > sum_su7540102398289458977st_nat > $o ) > sum_su7540102398289458977st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J_J_J,type,
accp_S9045782574878745642_a_nat: ( sum_su5729848735222297843_a_nat > sum_su5729848735222297843_a_nat > $o ) > sum_su5729848735222297843_a_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $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__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member1159737339441260081_a_nat: produc8199838167917776410_a_nat > set_Pr914094233313105488_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member5292487352258712491st_nat: produc3109216143880453012st_nat > set_Pr8517726817977820874st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member7457213283480048528_a_nat: produc5001885624171833703_a_nat > set_Pr4870381170404451655_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member3650934779329710139_a_nat: produc5454855657571582244_a_nat > set_Pr1021281732701644634_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member5454662894994576405at_nat: produc7258583773236448510at_nat > set_Pr2129020469590976052at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member3723442691059620112_a_nat: produc7017002724195966439_a_nat > set_Pr7343886759072863943_a_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_vs,type,
vs: list_Sum_sum_a_nat ).
thf(sy_v_vs_H,type,
vs2: list_Sum_sum_a_nat ).
thf(sy_v_x____,type,
x: sum_sum_a_nat ).
thf(sy_v_xs____,type,
xs: list_Sum_sum_a_nat ).
thf(sy_v_y____,type,
y: sum_sum_a_nat ).
thf(sy_v_ys____,type,
ys: list_Sum_sum_a_nat ).
% Relevant facts (1245)
thf(fact_0_xs__ys,axiom,
xs = ys ).
% xs_ys
thf(fact_1__092_060open_062x_A_061_Ay_092_060close_062,axiom,
x = y ).
% \<open>x = y\<close>
thf(fact_2__C2_Oprems_C_I1_J,axiom,
ad_agr_list_a_nat @ ad @ ( append_Sum_sum_a_nat @ xs @ ( cons_Sum_sum_a_nat @ x @ nil_Sum_sum_a_nat ) ) @ ( append_Sum_sum_a_nat @ ys @ ( cons_Sum_sum_a_nat @ y @ nil_Sum_sum_a_nat ) ) ).
% "2.prems"(1)
thf(fact_3_append1__eq__conv,axiom,
! [Xs: list_P6164600145584960654at_nat,X: produc7258583773236448510at_nat,Ys: list_P6164600145584960654at_nat,Y: produc7258583773236448510at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ ( cons_P1192438834837063368at_nat @ X @ nil_Pr2041209113074292344at_nat ) )
= ( append2142653904031976739at_nat @ Ys @ ( cons_P1192438834837063368at_nat @ Y @ nil_Pr2041209113074292344at_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_4_append1__eq__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) )
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_5_append1__eq__conv,axiom,
! [Xs: list_l4703314356710769291_a_nat,X: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ ( cons_l6604326339930385211_a_nat @ X @ nil_li1906260230833442699_a_nat ) )
= ( append5415888156905520160_a_nat @ Ys @ ( cons_l6604326339930385211_a_nat @ Y @ nil_li1906260230833442699_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_7_append1__eq__conv,axiom,
! [Xs: list_P5056861408695629236_a_nat,X: produc5454855657571582244_a_nat,Ys: list_P5056861408695629236_a_nat,Y: produc5454855657571582244_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ ( cons_P8612082756026972910_a_nat @ X @ nil_Pr237480997409426078_a_nat ) )
= ( append338925788367110473_a_nat @ Ys @ ( cons_P8612082756026972910_a_nat @ Y @ nil_Pr237480997409426078_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_8_append1__eq__conv,axiom,
! [Xs: list_P1195027771636113901_a_nat,X: produc7017002724195966439_a_nat,Ys: list_P1195027771636113901_a_nat,Y: produc7017002724195966439_a_nat] :
( ( ( append1996214168388709506_a_nat @ Xs @ ( cons_P1525839536144884125_a_nat @ X @ nil_Pr6585251977456444909_a_nat ) )
= ( append1996214168388709506_a_nat @ Ys @ ( cons_P1525839536144884125_a_nat @ Y @ nil_Pr6585251977456444909_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_9_append1__eq__conv,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_10__C2_Oprems_C_I3_J,axiom,
fo_nmlzd_a @ ad @ ( append_Sum_sum_a_nat @ ys @ ( cons_Sum_sum_a_nat @ y @ nil_Sum_sum_a_nat ) ) ).
% "2.prems"(3)
thf(fact_11__C2_Oprems_C_I2_J,axiom,
fo_nmlzd_a @ ad @ ( append_Sum_sum_a_nat @ xs @ ( cons_Sum_sum_a_nat @ x @ nil_Sum_sum_a_nat ) ) ).
% "2.prems"(2)
thf(fact_12_append_Oright__neutral,axiom,
! [A: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ A @ nil_Pr2041209113074292344at_nat )
= A ) ).
% append.right_neutral
thf(fact_13_append_Oright__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ A @ nil_Pr5478986624290739719at_nat )
= A ) ).
% append.right_neutral
thf(fact_14_append_Oright__neutral,axiom,
! [A: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ A @ nil_li1906260230833442699_a_nat )
= A ) ).
% append.right_neutral
thf(fact_15_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_16_append_Oright__neutral,axiom,
! [A: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ A @ nil_Pr237480997409426078_a_nat )
= A ) ).
% append.right_neutral
thf(fact_17_append_Oright__neutral,axiom,
! [A: list_P1195027771636113901_a_nat] :
( ( append1996214168388709506_a_nat @ A @ nil_Pr6585251977456444909_a_nat )
= A ) ).
% append.right_neutral
thf(fact_18_append_Oright__neutral,axiom,
! [A: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ A @ nil_Sum_sum_a_nat )
= A ) ).
% append.right_neutral
thf(fact_19_append__Nil2,axiom,
! [Xs: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ Xs @ nil_Pr2041209113074292344at_nat )
= Xs ) ).
% append_Nil2
thf(fact_20_append__Nil2,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ Xs @ nil_Pr5478986624290739719at_nat )
= Xs ) ).
% append_Nil2
thf(fact_21_append__Nil2,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ Xs @ nil_li1906260230833442699_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_22_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_23_append__Nil2,axiom,
! [Xs: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ Xs @ nil_Pr237480997409426078_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_24_append__Nil2,axiom,
! [Xs: list_P1195027771636113901_a_nat] :
( ( append1996214168388709506_a_nat @ Xs @ nil_Pr6585251977456444909_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_25_append__Nil2,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ Xs @ nil_Sum_sum_a_nat )
= Xs ) ).
% append_Nil2
thf(fact_26_append__self__conv,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr2041209113074292344at_nat ) ) ).
% append_self_conv
thf(fact_27_append__self__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv
thf(fact_28_append__self__conv,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_li1906260230833442699_a_nat ) ) ).
% append_self_conv
thf(fact_29_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_30_append__self__conv,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr237480997409426078_a_nat ) ) ).
% append_self_conv
thf(fact_31_append__self__conv,axiom,
! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( ( append1996214168388709506_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr6585251977456444909_a_nat ) ) ).
% append_self_conv
thf(fact_32_append__self__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Sum_sum_a_nat ) ) ).
% append_self_conv
thf(fact_33_self__append__conv,axiom,
! [Y: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( Y
= ( append2142653904031976739at_nat @ Y @ Ys ) )
= ( Ys = nil_Pr2041209113074292344at_nat ) ) ).
% self_append_conv
thf(fact_34_self__append__conv,axiom,
! [Y: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Y @ Ys ) )
= ( Ys = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv
thf(fact_35_self__append__conv,axiom,
! [Y: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( Y
= ( append5415888156905520160_a_nat @ Y @ Ys ) )
= ( Ys = nil_li1906260230833442699_a_nat ) ) ).
% self_append_conv
thf(fact_36_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_37_self__append__conv,axiom,
! [Y: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( Y
= ( append338925788367110473_a_nat @ Y @ Ys ) )
= ( Ys = nil_Pr237480997409426078_a_nat ) ) ).
% self_append_conv
thf(fact_38_self__append__conv,axiom,
! [Y: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( Y
= ( append1996214168388709506_a_nat @ Y @ Ys ) )
= ( Ys = nil_Pr6585251977456444909_a_nat ) ) ).
% self_append_conv
thf(fact_39_self__append__conv,axiom,
! [Y: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Y
= ( append_Sum_sum_a_nat @ Y @ Ys ) )
= ( Ys = nil_Sum_sum_a_nat ) ) ).
% self_append_conv
thf(fact_40_append__self__conv2,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr2041209113074292344at_nat ) ) ).
% append_self_conv2
thf(fact_41_append__self__conv2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv2
thf(fact_42_append__self__conv2,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_li1906260230833442699_a_nat ) ) ).
% append_self_conv2
thf(fact_43_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_44_append__self__conv2,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr237480997409426078_a_nat ) ) ).
% append_self_conv2
thf(fact_45_append__self__conv2,axiom,
! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( ( append1996214168388709506_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr6585251977456444909_a_nat ) ) ).
% append_self_conv2
thf(fact_46_append__self__conv2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% append_self_conv2
thf(fact_47_self__append__conv2,axiom,
! [Y: list_P6164600145584960654at_nat,Xs: list_P6164600145584960654at_nat] :
( ( Y
= ( append2142653904031976739at_nat @ Xs @ Y ) )
= ( Xs = nil_Pr2041209113074292344at_nat ) ) ).
% self_append_conv2
thf(fact_48_self__append__conv2,axiom,
! [Y: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Xs @ Y ) )
= ( Xs = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv2
thf(fact_49_self__append__conv2,axiom,
! [Y: list_l4703314356710769291_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( Y
= ( append5415888156905520160_a_nat @ Xs @ Y ) )
= ( Xs = nil_li1906260230833442699_a_nat ) ) ).
% self_append_conv2
thf(fact_50_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_51_self__append__conv2,axiom,
! [Y: list_P5056861408695629236_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( Y
= ( append338925788367110473_a_nat @ Xs @ Y ) )
= ( Xs = nil_Pr237480997409426078_a_nat ) ) ).
% self_append_conv2
thf(fact_52_self__append__conv2,axiom,
! [Y: list_P1195027771636113901_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( Y
= ( append1996214168388709506_a_nat @ Xs @ Y ) )
= ( Xs = nil_Pr6585251977456444909_a_nat ) ) ).
% self_append_conv2
thf(fact_53_self__append__conv2,axiom,
! [Y: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( Y
= ( append_Sum_sum_a_nat @ Xs @ Y ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% self_append_conv2
thf(fact_54_Nil__is__append__conv,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( nil_Pr2041209113074292344at_nat
= ( append2142653904031976739at_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr2041209113074292344at_nat )
& ( Ys = nil_Pr2041209113074292344at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_55_Nil__is__append__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( append985823374593552924at_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_56_Nil__is__append__conv,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( nil_li1906260230833442699_a_nat
= ( append5415888156905520160_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_li1906260230833442699_a_nat )
& ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_57_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_58_Nil__is__append__conv,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( nil_Pr237480997409426078_a_nat
= ( append338925788367110473_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr237480997409426078_a_nat )
& ( Ys = nil_Pr237480997409426078_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_59_Nil__is__append__conv,axiom,
! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( nil_Pr6585251977456444909_a_nat
= ( append1996214168388709506_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Pr6585251977456444909_a_nat )
& ( Ys = nil_Pr6585251977456444909_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_60_Nil__is__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Sum_sum_a_nat )
& ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_61_append__is__Nil__conv,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Ys )
= nil_Pr2041209113074292344at_nat )
= ( ( Xs = nil_Pr2041209113074292344at_nat )
& ( Ys = nil_Pr2041209113074292344at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_62_append__is__Nil__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_Pr5478986624290739719at_nat )
& ( Ys = nil_Pr5478986624290739719at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_63_append__is__Nil__conv,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= nil_li1906260230833442699_a_nat )
= ( ( Xs = nil_li1906260230833442699_a_nat )
& ( Ys = nil_li1906260230833442699_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_64_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_65_append__is__Nil__conv,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Ys )
= nil_Pr237480997409426078_a_nat )
= ( ( Xs = nil_Pr237480997409426078_a_nat )
& ( Ys = nil_Pr237480997409426078_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_66_append__is__Nil__conv,axiom,
! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( ( append1996214168388709506_a_nat @ Xs @ Ys )
= nil_Pr6585251977456444909_a_nat )
= ( ( Xs = nil_Pr6585251977456444909_a_nat )
& ( Ys = nil_Pr6585251977456444909_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_67_append__is__Nil__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= nil_Sum_sum_a_nat )
= ( ( Xs = nil_Sum_sum_a_nat )
& ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_68_assms_I3_J,axiom,
fo_nmlzd_a @ ad @ vs2 ).
% assms(3)
thf(fact_69_assms_I2_J,axiom,
fo_nmlzd_a @ ad @ vs ).
% assms(2)
thf(fact_70_list_Oinject,axiom,
! [X21: list_Sum_sum_a_nat,X22: list_l4703314356710769291_a_nat,Y21: list_Sum_sum_a_nat,Y22: list_l4703314356710769291_a_nat] :
( ( ( cons_l6604326339930385211_a_nat @ X21 @ X22 )
= ( cons_l6604326339930385211_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_71_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_72_list_Oinject,axiom,
! [X21: produc5454855657571582244_a_nat,X22: list_P5056861408695629236_a_nat,Y21: produc5454855657571582244_a_nat,Y22: list_P5056861408695629236_a_nat] :
( ( ( cons_P8612082756026972910_a_nat @ X21 @ X22 )
= ( cons_P8612082756026972910_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_73_list_Oinject,axiom,
! [X21: produc7017002724195966439_a_nat,X22: list_P1195027771636113901_a_nat,Y21: produc7017002724195966439_a_nat,Y22: list_P1195027771636113901_a_nat] :
( ( ( cons_P1525839536144884125_a_nat @ X21 @ X22 )
= ( cons_P1525839536144884125_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_74_list_Oinject,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat,Y21: sum_sum_a_nat,Y22: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X21 @ X22 )
= ( cons_Sum_sum_a_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_75_same__append__eq,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Ys )
= ( append2142653904031976739at_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_76_same__append__eq,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Ys )
= ( append338925788367110473_a_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_77_same__append__eq,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_78_same__append__eq,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= ( append5415888156905520160_a_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_79_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_80_same__append__eq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= ( append_Sum_sum_a_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_81_append__same__eq,axiom,
! [Ys: list_P6164600145584960654at_nat,Xs: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Ys @ Xs )
= ( append2142653904031976739at_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_82_append__same__eq,axiom,
! [Ys: list_P5056861408695629236_a_nat,Xs: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Ys @ Xs )
= ( append338925788367110473_a_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_83_append__same__eq,axiom,
! [Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Ys @ Xs )
= ( append985823374593552924at_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_84_append__same__eq,axiom,
! [Ys: list_l4703314356710769291_a_nat,Xs: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Ys @ Xs )
= ( append5415888156905520160_a_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_85_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_86_append__same__eq,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Xs )
= ( append_Sum_sum_a_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_87_append__assoc,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ ( append2142653904031976739at_nat @ Xs @ Ys ) @ Zs )
= ( append2142653904031976739at_nat @ Xs @ ( append2142653904031976739at_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_88_append__assoc,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ ( append338925788367110473_a_nat @ Xs @ Ys ) @ Zs )
= ( append338925788367110473_a_nat @ Xs @ ( append338925788367110473_a_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_89_append__assoc,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ Zs )
= ( append985823374593552924at_nat @ Xs @ ( append985823374593552924at_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_90_append__assoc,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ ( append5415888156905520160_a_nat @ Xs @ Ys ) @ Zs )
= ( append5415888156905520160_a_nat @ Xs @ ( append5415888156905520160_a_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_91_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_92_append__assoc,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ Zs )
= ( append_Sum_sum_a_nat @ Xs @ ( append_Sum_sum_a_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_93_append_Oassoc,axiom,
! [A: list_P6164600145584960654at_nat,B: list_P6164600145584960654at_nat,C: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ ( append2142653904031976739at_nat @ A @ B ) @ C )
= ( append2142653904031976739at_nat @ A @ ( append2142653904031976739at_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_94_append_Oassoc,axiom,
! [A: list_P5056861408695629236_a_nat,B: list_P5056861408695629236_a_nat,C: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ ( append338925788367110473_a_nat @ A @ B ) @ C )
= ( append338925788367110473_a_nat @ A @ ( append338925788367110473_a_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_95_append_Oassoc,axiom,
! [A: list_P6011104703257516679at_nat,B: list_P6011104703257516679at_nat,C: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ A @ B ) @ C )
= ( append985823374593552924at_nat @ A @ ( append985823374593552924at_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_96_append_Oassoc,axiom,
! [A: list_l4703314356710769291_a_nat,B: list_l4703314356710769291_a_nat,C: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ ( append5415888156905520160_a_nat @ A @ B ) @ C )
= ( append5415888156905520160_a_nat @ A @ ( append5415888156905520160_a_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_97_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_98_append_Oassoc,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,C: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ A @ B ) @ C )
= ( append_Sum_sum_a_nat @ A @ ( append_Sum_sum_a_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_99__C2_OIH_C,axiom,
( ( ad_agr_list_a_nat @ ad @ xs @ ys )
=> ( ( fo_nmlzd_a @ ad @ xs )
=> ( ( fo_nmlzd_a @ ad @ ys )
=> ( xs = ys ) ) ) ) ).
% "2.IH"
thf(fact_100_norms_I1_J,axiom,
fo_nmlzd_a @ ad @ xs ).
% norms(1)
thf(fact_101_norms_I2_J,axiom,
fo_nmlzd_a @ ad @ ys ).
% norms(2)
thf(fact_102_ad__agr,axiom,
ad_agr_list_a_nat @ ad @ xs @ ys ).
% ad_agr
thf(fact_103_assms_I1_J,axiom,
ad_agr_list_a_nat @ ad @ vs @ vs2 ).
% assms(1)
thf(fact_104_ad__agr__list__comm,axiom,
! [X2: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X2 @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_105_ad__agr__list__refl,axiom,
! [X2: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X2 @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_106_ad__agr__list__trans,axiom,
! [X2: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X2 @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X2 @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_107_not__Cons__self2,axiom,
! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( cons_l6604326339930385211_a_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_108_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_109_not__Cons__self2,axiom,
! [X: produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( cons_P8612082756026972910_a_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_110_not__Cons__self2,axiom,
! [X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( cons_P1525839536144884125_a_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_111_not__Cons__self2,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( cons_Sum_sum_a_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_112_append__eq__append__conv2,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat,Ts: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Ys )
= ( append2142653904031976739at_nat @ Zs @ Ts ) )
= ( ? [Us: list_P6164600145584960654at_nat] :
( ( ( Xs
= ( append2142653904031976739at_nat @ Zs @ Us ) )
& ( ( append2142653904031976739at_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append2142653904031976739at_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append2142653904031976739at_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_113_append__eq__append__conv2,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat,Ts: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Ys )
= ( append338925788367110473_a_nat @ Zs @ Ts ) )
= ( ? [Us: list_P5056861408695629236_a_nat] :
( ( ( Xs
= ( append338925788367110473_a_nat @ Zs @ Us ) )
& ( ( append338925788367110473_a_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append338925788367110473_a_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append338925788367110473_a_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_114_append__eq__append__conv2,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ts: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Zs @ Ts ) )
= ( ? [Us: list_P6011104703257516679at_nat] :
( ( ( Xs
= ( append985823374593552924at_nat @ Zs @ Us ) )
& ( ( append985823374593552924at_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append985823374593552924at_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append985823374593552924at_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_115_append__eq__append__conv2,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat,Ts: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= ( append5415888156905520160_a_nat @ Zs @ Ts ) )
= ( ? [Us: list_l4703314356710769291_a_nat] :
( ( ( Xs
= ( append5415888156905520160_a_nat @ Zs @ Us ) )
& ( ( append5415888156905520160_a_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append5415888156905520160_a_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append5415888156905520160_a_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_116_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_117_append__eq__append__conv2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,Ts: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= ( append_Sum_sum_a_nat @ Zs @ Ts ) )
= ( ? [Us: list_Sum_sum_a_nat] :
( ( ( Xs
= ( append_Sum_sum_a_nat @ Zs @ Us ) )
& ( ( append_Sum_sum_a_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_Sum_sum_a_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_Sum_sum_a_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_118_append__eq__appendI,axiom,
! [Xs: list_P6164600145584960654at_nat,Xs1: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Us2: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append2142653904031976739at_nat @ Xs1 @ Us2 ) )
=> ( ( append2142653904031976739at_nat @ Xs @ Ys )
= ( append2142653904031976739at_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_119_append__eq__appendI,axiom,
! [Xs: list_P5056861408695629236_a_nat,Xs1: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Us2: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append338925788367110473_a_nat @ Xs1 @ Us2 ) )
=> ( ( append338925788367110473_a_nat @ Xs @ Ys )
= ( append338925788367110473_a_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_120_append__eq__appendI,axiom,
! [Xs: list_P6011104703257516679at_nat,Xs1: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Us2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append985823374593552924at_nat @ Xs1 @ Us2 ) )
=> ( ( append985823374593552924at_nat @ Xs @ Ys )
= ( append985823374593552924at_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_121_append__eq__appendI,axiom,
! [Xs: list_l4703314356710769291_a_nat,Xs1: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Us2: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append5415888156905520160_a_nat @ Xs1 @ Us2 ) )
=> ( ( append5415888156905520160_a_nat @ Xs @ Ys )
= ( append5415888156905520160_a_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_122_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_123_append__eq__appendI,axiom,
! [Xs: list_Sum_sum_a_nat,Xs1: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_Sum_sum_a_nat @ Xs1 @ Us2 ) )
=> ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= ( append_Sum_sum_a_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_124_list__nonempty__induct,axiom,
! [Xs: list_l4703314356710769291_a_nat,P: list_l4703314356710769291_a_nat > $o] :
( ( Xs != nil_li1906260230833442699_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat] : ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ nil_li1906260230833442699_a_nat ) )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( ( Xs2 != nil_li1906260230833442699_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_125_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_126_list__nonempty__induct,axiom,
! [Xs: list_P5056861408695629236_a_nat,P: list_P5056861408695629236_a_nat > $o] :
( ( Xs != nil_Pr237480997409426078_a_nat )
=> ( ! [X3: produc5454855657571582244_a_nat] : ( P @ ( cons_P8612082756026972910_a_nat @ X3 @ nil_Pr237480997409426078_a_nat ) )
=> ( ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( Xs2 != nil_Pr237480997409426078_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P8612082756026972910_a_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_127_list__nonempty__induct,axiom,
! [Xs: list_P1195027771636113901_a_nat,P: list_P1195027771636113901_a_nat > $o] :
( ( Xs != nil_Pr6585251977456444909_a_nat )
=> ( ! [X3: produc7017002724195966439_a_nat] : ( P @ ( cons_P1525839536144884125_a_nat @ X3 @ nil_Pr6585251977456444909_a_nat ) )
=> ( ! [X3: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat] :
( ( Xs2 != nil_Pr6585251977456444909_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P1525839536144884125_a_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_128_list__nonempty__induct,axiom,
! [Xs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( Xs2 != nil_Sum_sum_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_129_list__induct2_H,axiom,
! [P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat )
=> ( ! [Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] : ( P @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_130_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_131_list__induct2_H,axiom,
! [P: list_Sum_sum_a_nat > list_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( P @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_Sum_sum_a_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_132_list__induct2_H,axiom,
! [P: list_nat > list_Sum_sum_a_nat > $o,Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( P @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat )
=> ( ! [Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] : ( P @ nil_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_133_list__induct2_H,axiom,
! [P: list_l4703314356710769291_a_nat > list_nat > $o,Xs: list_l4703314356710769291_a_nat,Ys: list_nat] :
( ( P @ nil_li1906260230833442699_a_nat @ nil_nat )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] : ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_li1906260230833442699_a_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_134_list__induct2_H,axiom,
! [P: list_nat > list_l4703314356710769291_a_nat > $o,Xs: list_nat,Ys: list_l4703314356710769291_a_nat] :
( ( P @ nil_nat @ nil_li1906260230833442699_a_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_li1906260230833442699_a_nat )
=> ( ! [Y2: list_Sum_sum_a_nat,Ys2: list_l4703314356710769291_a_nat] : ( P @ nil_nat @ ( cons_l6604326339930385211_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: list_Sum_sum_a_nat,Ys2: list_l4703314356710769291_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_l6604326339930385211_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_135_list__induct2_H,axiom,
! [P: list_nat > list_P5056861408695629236_a_nat > $o,Xs: list_nat,Ys: list_P5056861408695629236_a_nat] :
( ( P @ nil_nat @ nil_Pr237480997409426078_a_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_Pr237480997409426078_a_nat )
=> ( ! [Y2: produc5454855657571582244_a_nat,Ys2: list_P5056861408695629236_a_nat] : ( P @ nil_nat @ ( cons_P8612082756026972910_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y2: produc5454855657571582244_a_nat,Ys2: list_P5056861408695629236_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P8612082756026972910_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_136_list__induct2_H,axiom,
! [P: list_P5056861408695629236_a_nat > list_nat > $o,Xs: list_P5056861408695629236_a_nat,Ys: list_nat] :
( ( P @ nil_Pr237480997409426078_a_nat @ nil_nat )
=> ( ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] : ( P @ ( cons_P8612082756026972910_a_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_Pr237480997409426078_a_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_P8612082756026972910_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_137_list__induct2_H,axiom,
! [P: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( P @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_li1906260230833442699_a_nat )
=> ( ! [Y2: list_Sum_sum_a_nat,Ys2: list_l4703314356710769291_a_nat] : ( P @ nil_Sum_sum_a_nat @ ( cons_l6604326339930385211_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat,Ys2: list_l4703314356710769291_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_l6604326339930385211_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_138_list__induct2_H,axiom,
! [P: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat] :
( ( P @ nil_li1906260230833442699_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] : ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat )
=> ( ! [Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] : ( P @ nil_li1906260230833442699_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_139_neq__Nil__conv,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( Xs != nil_li1906260230833442699_a_nat )
= ( ? [Y3: list_Sum_sum_a_nat,Ys3: list_l4703314356710769291_a_nat] :
( Xs
= ( cons_l6604326339930385211_a_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_140_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_141_neq__Nil__conv,axiom,
! [Xs: list_P5056861408695629236_a_nat] :
( ( Xs != nil_Pr237480997409426078_a_nat )
= ( ? [Y3: produc5454855657571582244_a_nat,Ys3: list_P5056861408695629236_a_nat] :
( Xs
= ( cons_P8612082756026972910_a_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_142_neq__Nil__conv,axiom,
! [Xs: list_P1195027771636113901_a_nat] :
( ( Xs != nil_Pr6585251977456444909_a_nat )
= ( ? [Y3: produc7017002724195966439_a_nat,Ys3: list_P1195027771636113901_a_nat] :
( Xs
= ( cons_P1525839536144884125_a_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_143_neq__Nil__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
= ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( Xs
= ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_144_transpose_Ocases,axiom,
! [X: list_l1586611297644397841_a_nat] :
( ( X != nil_li2117038862230905745_a_nat )
=> ( ! [Xss: list_l1586611297644397841_a_nat] :
( X
!= ( cons_l2563873727033190209_a_nat @ nil_li1906260230833442699_a_nat @ Xss ) )
=> ~ ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,Xss: list_l1586611297644397841_a_nat] :
( X
!= ( cons_l2563873727033190209_a_nat @ ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_145_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_146_transpose_Ocases,axiom,
! [X: list_l7136637534808337348_a_nat] :
( ( X != nil_li1907017536197114414_a_nat )
=> ( ! [Xss: list_l7136637534808337348_a_nat] :
( X
!= ( cons_l6787550886680756862_a_nat @ nil_Pr237480997409426078_a_nat @ Xss ) )
=> ~ ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat,Xss: list_l7136637534808337348_a_nat] :
( X
!= ( cons_l6787550886680756862_a_nat @ ( cons_P8612082756026972910_a_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_147_transpose_Ocases,axiom,
! [X: list_l7568939571203725683_a_nat] :
( ( X != nil_li4039925377676013299_a_nat )
=> ( ! [Xss: list_l7568939571203725683_a_nat] :
( X
!= ( cons_l1779357001063338659_a_nat @ nil_Pr6585251977456444909_a_nat @ Xss ) )
=> ~ ! [X3: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat,Xss: list_l7568939571203725683_a_nat] :
( X
!= ( cons_l1779357001063338659_a_nat @ ( cons_P1525839536144884125_a_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_148_transpose_Ocases,axiom,
! [X: list_l4703314356710769291_a_nat] :
( ( X != nil_li1906260230833442699_a_nat )
=> ( ! [Xss: list_l4703314356710769291_a_nat] :
( X
!= ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ Xss ) )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Xss: list_l4703314356710769291_a_nat] :
( X
!= ( cons_l6604326339930385211_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_149_mem__Collect__eq,axiom,
! [A: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_153_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X4: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_157_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_159_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_160_list_Oexhaust,axiom,
! [Y: list_l4703314356710769291_a_nat] :
( ( Y != nil_li1906260230833442699_a_nat )
=> ~ ! [X212: list_Sum_sum_a_nat,X222: list_l4703314356710769291_a_nat] :
( Y
!= ( cons_l6604326339930385211_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_161_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_162_list_Oexhaust,axiom,
! [Y: list_P5056861408695629236_a_nat] :
( ( Y != nil_Pr237480997409426078_a_nat )
=> ~ ! [X212: produc5454855657571582244_a_nat,X222: list_P5056861408695629236_a_nat] :
( Y
!= ( cons_P8612082756026972910_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_163_list_Oexhaust,axiom,
! [Y: list_P1195027771636113901_a_nat] :
( ( Y != nil_Pr6585251977456444909_a_nat )
=> ~ ! [X212: produc7017002724195966439_a_nat,X222: list_P1195027771636113901_a_nat] :
( Y
!= ( cons_P1525839536144884125_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_164_list_Oexhaust,axiom,
! [Y: list_Sum_sum_a_nat] :
( ( Y != nil_Sum_sum_a_nat )
=> ~ ! [X212: sum_sum_a_nat,X222: list_Sum_sum_a_nat] :
( Y
!= ( cons_Sum_sum_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_165_list_OdiscI,axiom,
! [List: list_l4703314356710769291_a_nat,X21: list_Sum_sum_a_nat,X22: list_l4703314356710769291_a_nat] :
( ( List
= ( cons_l6604326339930385211_a_nat @ X21 @ X22 ) )
=> ( List != nil_li1906260230833442699_a_nat ) ) ).
% list.discI
thf(fact_166_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_167_list_OdiscI,axiom,
! [List: list_P5056861408695629236_a_nat,X21: produc5454855657571582244_a_nat,X22: list_P5056861408695629236_a_nat] :
( ( List
= ( cons_P8612082756026972910_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr237480997409426078_a_nat ) ) ).
% list.discI
thf(fact_168_list_OdiscI,axiom,
! [List: list_P1195027771636113901_a_nat,X21: produc7017002724195966439_a_nat,X22: list_P1195027771636113901_a_nat] :
( ( List
= ( cons_P1525839536144884125_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr6585251977456444909_a_nat ) ) ).
% list.discI
thf(fact_169_list_OdiscI,axiom,
! [List: list_Sum_sum_a_nat,X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( List
= ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
=> ( List != nil_Sum_sum_a_nat ) ) ).
% list.discI
thf(fact_170_list_Odistinct_I1_J,axiom,
! [X21: list_Sum_sum_a_nat,X22: list_l4703314356710769291_a_nat] :
( nil_li1906260230833442699_a_nat
!= ( cons_l6604326339930385211_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_171_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_172_list_Odistinct_I1_J,axiom,
! [X21: produc5454855657571582244_a_nat,X22: list_P5056861408695629236_a_nat] :
( nil_Pr237480997409426078_a_nat
!= ( cons_P8612082756026972910_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_173_list_Odistinct_I1_J,axiom,
! [X21: produc7017002724195966439_a_nat,X22: list_P1195027771636113901_a_nat] :
( nil_Pr6585251977456444909_a_nat
!= ( cons_P1525839536144884125_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_174_list_Odistinct_I1_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( nil_Sum_sum_a_nat
!= ( cons_Sum_sum_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_175_Cons__eq__appendI,axiom,
! [X: produc7258583773236448510at_nat,Xs1: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Xs: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( cons_P1192438834837063368at_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append2142653904031976739at_nat @ Xs1 @ Zs ) )
=> ( ( cons_P1192438834837063368at_nat @ X @ Xs )
= ( append2142653904031976739at_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_176_Cons__eq__appendI,axiom,
! [X: product_prod_nat_nat,Xs1: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append985823374593552924at_nat @ Xs1 @ Zs ) )
=> ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= ( append985823374593552924at_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_177_Cons__eq__appendI,axiom,
! [X: list_Sum_sum_a_nat,Xs1: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Xs: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat] :
( ( ( cons_l6604326339930385211_a_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append5415888156905520160_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_l6604326339930385211_a_nat @ X @ Xs )
= ( append5415888156905520160_a_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_178_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_179_Cons__eq__appendI,axiom,
! [X: produc5454855657571582244_a_nat,Xs1: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Xs: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( cons_P8612082756026972910_a_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append338925788367110473_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_P8612082756026972910_a_nat @ X @ Xs )
= ( append338925788367110473_a_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_180_Cons__eq__appendI,axiom,
! [X: produc7017002724195966439_a_nat,Xs1: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,Xs: list_P1195027771636113901_a_nat,Zs: list_P1195027771636113901_a_nat] :
( ( ( cons_P1525839536144884125_a_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append1996214168388709506_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_P1525839536144884125_a_nat @ X @ Xs )
= ( append1996214168388709506_a_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_181_Cons__eq__appendI,axiom,
! [X: sum_sum_a_nat,Xs1: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_Sum_sum_a_nat @ Xs1 @ Zs ) )
=> ( ( cons_Sum_sum_a_nat @ X @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_182_append__Cons,axiom,
! [X: produc7258583773236448510at_nat,Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ ( cons_P1192438834837063368at_nat @ X @ Xs ) @ Ys )
= ( cons_P1192438834837063368at_nat @ X @ ( append2142653904031976739at_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_183_append__Cons,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) @ Ys )
= ( cons_P6512896166579812791at_nat @ X @ ( append985823374593552924at_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_184_append__Cons,axiom,
! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ ( cons_l6604326339930385211_a_nat @ X @ Xs ) @ Ys )
= ( cons_l6604326339930385211_a_nat @ X @ ( append5415888156905520160_a_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_185_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_186_append__Cons,axiom,
! [X: produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ ( cons_P8612082756026972910_a_nat @ X @ Xs ) @ Ys )
= ( cons_P8612082756026972910_a_nat @ X @ ( append338925788367110473_a_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_187_append__Cons,axiom,
! [X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( append1996214168388709506_a_nat @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) @ Ys )
= ( cons_P1525839536144884125_a_nat @ X @ ( append1996214168388709506_a_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_188_append__Cons,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ Ys )
= ( cons_Sum_sum_a_nat @ X @ ( append_Sum_sum_a_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_189_eq__Nil__appendI,axiom,
! [Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append2142653904031976739at_nat @ nil_Pr2041209113074292344at_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_190_eq__Nil__appendI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_191_eq__Nil__appendI,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append5415888156905520160_a_nat @ nil_li1906260230833442699_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_192_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_193_eq__Nil__appendI,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append338925788367110473_a_nat @ nil_Pr237480997409426078_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_194_eq__Nil__appendI,axiom,
! [Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append1996214168388709506_a_nat @ nil_Pr6585251977456444909_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_195_eq__Nil__appendI,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_196_append_Oleft__neutral,axiom,
! [A: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ nil_Pr2041209113074292344at_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_197_append_Oleft__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_198_append_Oleft__neutral,axiom,
! [A: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ nil_li1906260230833442699_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_199_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_200_append_Oleft__neutral,axiom,
! [A: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ nil_Pr237480997409426078_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_201_append_Oleft__neutral,axiom,
! [A: list_P1195027771636113901_a_nat] :
( ( append1996214168388709506_a_nat @ nil_Pr6585251977456444909_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_202_append_Oleft__neutral,axiom,
! [A: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_203_append__Nil,axiom,
! [Ys: list_P6164600145584960654at_nat] :
( ( append2142653904031976739at_nat @ nil_Pr2041209113074292344at_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_204_append__Nil,axiom,
! [Ys: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ nil_Pr5478986624290739719at_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_205_append__Nil,axiom,
! [Ys: list_l4703314356710769291_a_nat] :
( ( append5415888156905520160_a_nat @ nil_li1906260230833442699_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_206_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_207_append__Nil,axiom,
! [Ys: list_P5056861408695629236_a_nat] :
( ( append338925788367110473_a_nat @ nil_Pr237480997409426078_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_208_append__Nil,axiom,
! [Ys: list_P1195027771636113901_a_nat] :
( ( append1996214168388709506_a_nat @ nil_Pr6585251977456444909_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_209_append__Nil,axiom,
! [Ys: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_210_rev__nonempty__induct,axiom,
! [Xs: list_P6164600145584960654at_nat,P: list_P6164600145584960654at_nat > $o] :
( ( Xs != nil_Pr2041209113074292344at_nat )
=> ( ! [X3: produc7258583773236448510at_nat] : ( P @ ( cons_P1192438834837063368at_nat @ X3 @ nil_Pr2041209113074292344at_nat ) )
=> ( ! [X3: produc7258583773236448510at_nat,Xs2: list_P6164600145584960654at_nat] :
( ( Xs2 != nil_Pr2041209113074292344at_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append2142653904031976739at_nat @ Xs2 @ ( cons_P1192438834837063368at_nat @ X3 @ nil_Pr2041209113074292344at_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_211_rev__nonempty__induct,axiom,
! [Xs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ! [X3: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X3: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_212_rev__nonempty__induct,axiom,
! [Xs: list_l4703314356710769291_a_nat,P: list_l4703314356710769291_a_nat > $o] :
( ( Xs != nil_li1906260230833442699_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat] : ( P @ ( cons_l6604326339930385211_a_nat @ X3 @ nil_li1906260230833442699_a_nat ) )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( ( Xs2 != nil_li1906260230833442699_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append5415888156905520160_a_nat @ Xs2 @ ( cons_l6604326339930385211_a_nat @ X3 @ nil_li1906260230833442699_a_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_213_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_214_rev__nonempty__induct,axiom,
! [Xs: list_P5056861408695629236_a_nat,P: list_P5056861408695629236_a_nat > $o] :
( ( Xs != nil_Pr237480997409426078_a_nat )
=> ( ! [X3: produc5454855657571582244_a_nat] : ( P @ ( cons_P8612082756026972910_a_nat @ X3 @ nil_Pr237480997409426078_a_nat ) )
=> ( ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( Xs2 != nil_Pr237480997409426078_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append338925788367110473_a_nat @ Xs2 @ ( cons_P8612082756026972910_a_nat @ X3 @ nil_Pr237480997409426078_a_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_215_rev__nonempty__induct,axiom,
! [Xs: list_P1195027771636113901_a_nat,P: list_P1195027771636113901_a_nat > $o] :
( ( Xs != nil_Pr6585251977456444909_a_nat )
=> ( ! [X3: produc7017002724195966439_a_nat] : ( P @ ( cons_P1525839536144884125_a_nat @ X3 @ nil_Pr6585251977456444909_a_nat ) )
=> ( ! [X3: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat] :
( ( Xs2 != nil_Pr6585251977456444909_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append1996214168388709506_a_nat @ Xs2 @ ( cons_P1525839536144884125_a_nat @ X3 @ nil_Pr6585251977456444909_a_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_216_rev__nonempty__induct,axiom,
! [Xs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat] : ( P @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( Xs2 != nil_Sum_sum_a_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_217_append__eq__Cons__conv,axiom,
! [Ys: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat,X: produc7258583773236448510at_nat,Xs: list_P6164600145584960654at_nat] :
( ( ( append2142653904031976739at_nat @ Ys @ Zs )
= ( cons_P1192438834837063368at_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Pr2041209113074292344at_nat )
& ( Zs
= ( cons_P1192438834837063368at_nat @ X @ Xs ) ) )
| ? [Ys4: list_P6164600145584960654at_nat] :
( ( Ys
= ( cons_P1192438834837063368at_nat @ X @ Ys4 ) )
& ( ( append2142653904031976739at_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_218_append__eq__Cons__conv,axiom,
! [Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Ys @ Zs )
= ( cons_P6512896166579812791at_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Pr5478986624290739719at_nat )
& ( Zs
= ( cons_P6512896166579812791at_nat @ X @ Xs ) ) )
| ? [Ys4: list_P6011104703257516679at_nat] :
( ( Ys
= ( cons_P6512896166579812791at_nat @ X @ Ys4 ) )
& ( ( append985823374593552924at_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_219_append__eq__Cons__conv,axiom,
! [Ys: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat,X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( ( append5415888156905520160_a_nat @ Ys @ Zs )
= ( cons_l6604326339930385211_a_nat @ X @ Xs ) )
= ( ( ( Ys = nil_li1906260230833442699_a_nat )
& ( Zs
= ( cons_l6604326339930385211_a_nat @ X @ Xs ) ) )
| ? [Ys4: list_l4703314356710769291_a_nat] :
( ( Ys
= ( cons_l6604326339930385211_a_nat @ X @ Ys4 ) )
& ( ( append5415888156905520160_a_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_220_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_221_append__eq__Cons__conv,axiom,
! [Ys: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat,X: produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( ( append338925788367110473_a_nat @ Ys @ Zs )
= ( cons_P8612082756026972910_a_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Pr237480997409426078_a_nat )
& ( Zs
= ( cons_P8612082756026972910_a_nat @ X @ Xs ) ) )
| ? [Ys4: list_P5056861408695629236_a_nat] :
( ( Ys
= ( cons_P8612082756026972910_a_nat @ X @ Ys4 ) )
& ( ( append338925788367110473_a_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_222_append__eq__Cons__conv,axiom,
! [Ys: list_P1195027771636113901_a_nat,Zs: list_P1195027771636113901_a_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( ( append1996214168388709506_a_nat @ Ys @ Zs )
= ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Pr6585251977456444909_a_nat )
& ( Zs
= ( cons_P1525839536144884125_a_nat @ X @ Xs ) ) )
| ? [Ys4: list_P1195027771636113901_a_nat] :
( ( Ys
= ( cons_P1525839536144884125_a_nat @ X @ Ys4 ) )
& ( ( append1996214168388709506_a_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_223_append__eq__Cons__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( ( ( Ys = nil_Sum_sum_a_nat )
& ( Zs
= ( cons_Sum_sum_a_nat @ X @ Xs ) ) )
| ? [Ys4: list_Sum_sum_a_nat] :
( ( Ys
= ( cons_Sum_sum_a_nat @ X @ Ys4 ) )
& ( ( append_Sum_sum_a_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_224_Cons__eq__append__conv,axiom,
! [X: produc7258583773236448510at_nat,Xs: list_P6164600145584960654at_nat,Ys: list_P6164600145584960654at_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( cons_P1192438834837063368at_nat @ X @ Xs )
= ( append2142653904031976739at_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr2041209113074292344at_nat )
& ( ( cons_P1192438834837063368at_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_P6164600145584960654at_nat] :
( ( ( cons_P1192438834837063368at_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append2142653904031976739at_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_225_Cons__eq__append__conv,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= ( append985823374593552924at_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr5478986624290739719at_nat )
& ( ( cons_P6512896166579812791at_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append985823374593552924at_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_226_Cons__eq__append__conv,axiom,
! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,Zs: list_l4703314356710769291_a_nat] :
( ( ( cons_l6604326339930385211_a_nat @ X @ Xs )
= ( append5415888156905520160_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_li1906260230833442699_a_nat )
& ( ( cons_l6604326339930385211_a_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_l4703314356710769291_a_nat] :
( ( ( cons_l6604326339930385211_a_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append5415888156905520160_a_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_227_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_228_Cons__eq__append__conv,axiom,
! [X: produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( cons_P8612082756026972910_a_nat @ X @ Xs )
= ( append338925788367110473_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr237480997409426078_a_nat )
& ( ( cons_P8612082756026972910_a_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_P5056861408695629236_a_nat] :
( ( ( cons_P8612082756026972910_a_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append338925788367110473_a_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_229_Cons__eq__append__conv,axiom,
! [X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat,Ys: list_P1195027771636113901_a_nat,Zs: list_P1195027771636113901_a_nat] :
( ( ( cons_P1525839536144884125_a_nat @ X @ Xs )
= ( append1996214168388709506_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr6585251977456444909_a_nat )
& ( ( cons_P1525839536144884125_a_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_P1195027771636113901_a_nat] :
( ( ( cons_P1525839536144884125_a_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append1996214168388709506_a_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_230_Cons__eq__append__conv,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_Sum_sum_a_nat )
& ( ( cons_Sum_sum_a_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_Sum_sum_a_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_231_rev__exhaust,axiom,
! [Xs: list_P6164600145584960654at_nat] :
( ( Xs != nil_Pr2041209113074292344at_nat )
=> ~ ! [Ys2: list_P6164600145584960654at_nat,Y2: produc7258583773236448510at_nat] :
( Xs
!= ( append2142653904031976739at_nat @ Ys2 @ ( cons_P1192438834837063368at_nat @ Y2 @ nil_Pr2041209113074292344at_nat ) ) ) ) ).
% rev_exhaust
thf(fact_232_rev__exhaust,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ~ ! [Ys2: list_P6011104703257516679at_nat,Y2: product_prod_nat_nat] :
( Xs
!= ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).
% rev_exhaust
thf(fact_233_rev__exhaust,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( Xs != nil_li1906260230833442699_a_nat )
=> ~ ! [Ys2: list_l4703314356710769291_a_nat,Y2: list_Sum_sum_a_nat] :
( Xs
!= ( append5415888156905520160_a_nat @ Ys2 @ ( cons_l6604326339930385211_a_nat @ Y2 @ nil_li1906260230833442699_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_234_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y2: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_235_rev__exhaust,axiom,
! [Xs: list_P5056861408695629236_a_nat] :
( ( Xs != nil_Pr237480997409426078_a_nat )
=> ~ ! [Ys2: list_P5056861408695629236_a_nat,Y2: produc5454855657571582244_a_nat] :
( Xs
!= ( append338925788367110473_a_nat @ Ys2 @ ( cons_P8612082756026972910_a_nat @ Y2 @ nil_Pr237480997409426078_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_236_rev__exhaust,axiom,
! [Xs: list_P1195027771636113901_a_nat] :
( ( Xs != nil_Pr6585251977456444909_a_nat )
=> ~ ! [Ys2: list_P1195027771636113901_a_nat,Y2: produc7017002724195966439_a_nat] :
( Xs
!= ( append1996214168388709506_a_nat @ Ys2 @ ( cons_P1525839536144884125_a_nat @ Y2 @ nil_Pr6585251977456444909_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_237_rev__exhaust,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ~ ! [Ys2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat] :
( Xs
!= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_238_rev__induct,axiom,
! [P: list_P6164600145584960654at_nat > $o,Xs: list_P6164600145584960654at_nat] :
( ( P @ nil_Pr2041209113074292344at_nat )
=> ( ! [X3: produc7258583773236448510at_nat,Xs2: list_P6164600145584960654at_nat] :
( ( P @ Xs2 )
=> ( P @ ( append2142653904031976739at_nat @ Xs2 @ ( cons_P1192438834837063368at_nat @ X3 @ nil_Pr2041209113074292344at_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_239_rev__induct,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
( ( P @ nil_Pr5478986624290739719at_nat )
=> ( ! [X3: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( P @ Xs2 )
=> ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_240_rev__induct,axiom,
! [P: list_l4703314356710769291_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
( ( P @ nil_li1906260230833442699_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( ( P @ Xs2 )
=> ( P @ ( append5415888156905520160_a_nat @ Xs2 @ ( cons_l6604326339930385211_a_nat @ X3 @ nil_li1906260230833442699_a_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_241_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_242_rev__induct,axiom,
! [P: list_P5056861408695629236_a_nat > $o,Xs: list_P5056861408695629236_a_nat] :
( ( P @ nil_Pr237480997409426078_a_nat )
=> ( ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( P @ Xs2 )
=> ( P @ ( append338925788367110473_a_nat @ Xs2 @ ( cons_P8612082756026972910_a_nat @ X3 @ nil_Pr237480997409426078_a_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_243_rev__induct,axiom,
! [P: list_P1195027771636113901_a_nat > $o,Xs: list_P1195027771636113901_a_nat] :
( ( P @ nil_Pr6585251977456444909_a_nat )
=> ( ! [X3: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat] :
( ( P @ Xs2 )
=> ( P @ ( append1996214168388709506_a_nat @ Xs2 @ ( cons_P1525839536144884125_a_nat @ X3 @ nil_Pr6585251977456444909_a_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_244_rev__induct,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ( P @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_245_exhaustive_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% exhaustive.cases
thf(fact_246_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X: list_l4703314356710769291_a_nat] :
( ( X != nil_li1906260230833442699_a_nat )
=> ~ ! [X3: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( X
!= ( cons_l6604326339930385211_a_nat @ X3 @ Xs2 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_247_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_248_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X: list_P5056861408695629236_a_nat] :
( ( X != nil_Pr237480997409426078_a_nat )
=> ~ ! [X3: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( X
!= ( cons_P8612082756026972910_a_nat @ X3 @ Xs2 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_249_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X: list_P1195027771636113901_a_nat] :
( ( X != nil_Pr6585251977456444909_a_nat )
=> ~ ! [X3: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat] :
( X
!= ( cons_P1525839536144884125_a_nat @ X3 @ Xs2 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_250_proper__intrvl_Oexhaustive_Ocases,axiom,
! [X: list_Sum_sum_a_nat] :
( ( X != nil_Sum_sum_a_nat )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) ) ) ).
% proper_intrvl.exhaustive.cases
thf(fact_251_remdups__sorted_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_sorted.cases
thf(fact_252_ord_Oremdups__sorted_Ocases,axiom,
! [X: list_l4703314356710769291_a_nat] :
( ( X != nil_li1906260230833442699_a_nat )
=> ( ! [X3: list_Sum_sum_a_nat] :
( X
!= ( cons_l6604326339930385211_a_nat @ X3 @ nil_li1906260230833442699_a_nat ) )
=> ~ ! [X3: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( X
!= ( cons_l6604326339930385211_a_nat @ X3 @ ( cons_l6604326339930385211_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_253_ord_Oremdups__sorted_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_254_ord_Oremdups__sorted_Ocases,axiom,
! [X: list_P5056861408695629236_a_nat] :
( ( X != nil_Pr237480997409426078_a_nat )
=> ( ! [X3: produc5454855657571582244_a_nat] :
( X
!= ( cons_P8612082756026972910_a_nat @ X3 @ nil_Pr237480997409426078_a_nat ) )
=> ~ ! [X3: produc5454855657571582244_a_nat,Y2: produc5454855657571582244_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( X
!= ( cons_P8612082756026972910_a_nat @ X3 @ ( cons_P8612082756026972910_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_255_ord_Oremdups__sorted_Ocases,axiom,
! [X: list_P1195027771636113901_a_nat] :
( ( X != nil_Pr6585251977456444909_a_nat )
=> ( ! [X3: produc7017002724195966439_a_nat] :
( X
!= ( cons_P1525839536144884125_a_nat @ X3 @ nil_Pr6585251977456444909_a_nat ) )
=> ~ ! [X3: produc7017002724195966439_a_nat,Y2: produc7017002724195966439_a_nat,Xs2: list_P1195027771636113901_a_nat] :
( X
!= ( cons_P1525839536144884125_a_nat @ X3 @ ( cons_P1525839536144884125_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_256_ord_Oremdups__sorted_Ocases,axiom,
! [X: list_Sum_sum_a_nat] :
( ( X != nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat] :
( X
!= ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_257__092_060open_062length_Avs_A_061_Alength_Avs_H_092_060close_062,axiom,
( ( size_s5283204784079214577_a_nat @ vs )
= ( size_s5283204784079214577_a_nat @ vs2 ) ) ).
% \<open>length vs = length vs'\<close>
thf(fact_258_to__single__list,axiom,
( cons_nat
= ( ^ [X4: nat] : ( append_nat @ ( set_li756733140798618988st_nat @ X4 ) ) ) ) ).
% to_single_list
thf(fact_259_single__list__def,axiom,
( set_li756733140798618988st_nat
= ( ^ [A3: nat] : ( cons_nat @ A3 @ nil_nat ) ) ) ).
% single_list_def
thf(fact_260_product__lists_Osimps_I1_J,axiom,
( ( produc4105601411378645364_a_nat @ nil_li2117038862230905745_a_nat )
= ( cons_l2563873727033190209_a_nat @ nil_li1906260230833442699_a_nat @ nil_li2117038862230905745_a_nat ) ) ).
% product_lists.simps(1)
thf(fact_261_product__lists_Osimps_I1_J,axiom,
( ( produc8430089408672153845_a_nat @ nil_li1907017536197114414_a_nat )
= ( cons_l6787550886680756862_a_nat @ nil_Pr237480997409426078_a_nat @ nil_li1907017536197114414_a_nat ) ) ).
% product_lists.simps(1)
thf(fact_262_product__lists_Osimps_I1_J,axiom,
( ( produc3960677393309065686_a_nat @ nil_li4039925377676013299_a_nat )
= ( cons_l1779357001063338659_a_nat @ nil_Pr6585251977456444909_a_nat @ nil_li4039925377676013299_a_nat ) ) ).
% product_lists.simps(1)
thf(fact_263_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_264_product__lists_Osimps_I1_J,axiom,
( ( produc2893206433618375022_a_nat @ nil_li1906260230833442699_a_nat )
= ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) ).
% product_lists.simps(1)
thf(fact_265_bind__simps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( bind_S8497345843576593223_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ F )
= ( append_Sum_sum_a_nat @ ( F @ X ) @ ( bind_S8497345843576593223_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_266_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_267_bind__simps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_nat] :
( ( bind_S7309778641578091144at_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_S7309778641578091144at_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_268_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_Sum_sum_a_nat] :
( ( bind_n2905796707917019494_a_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_Sum_sum_a_nat @ ( F @ X ) @ ( bind_n2905796707917019494_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_269_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_P6011104703257516679at_nat] :
( ( bind_n1878750130520726888at_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append985823374593552924at_nat @ ( F @ X ) @ ( bind_n1878750130520726888at_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_270_bind__simps_I2_J,axiom,
! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_nat] :
( ( bind_l5927154061965146638at_nat @ ( cons_l6604326339930385211_a_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_l5927154061965146638at_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_271_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_l4703314356710769291_a_nat] :
( ( bind_n431031891494352236_a_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append5415888156905520160_a_nat @ ( F @ X ) @ ( bind_n431031891494352236_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_272_bind__simps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_P6011104703257516679at_nat] :
( ( bind_S8065145433898892361at_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ F )
= ( append985823374593552924at_nat @ ( F @ X ) @ ( bind_S8065145433898892361at_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_273_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_P6164600145584960654at_nat] :
( ( bind_n3288368246431264471at_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append2142653904031976739at_nat @ ( F @ X ) @ ( bind_n3288368246431264471at_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_274_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_P5056861408695629236_a_nat] :
( ( bind_n1484640130766398205_a_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append338925788367110473_a_nat @ ( F @ X ) @ ( bind_n1484640130766398205_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_275_subseqs_Osimps_I1_J,axiom,
( ( subseq4596077244143087834_a_nat @ nil_li1906260230833442699_a_nat )
= ( cons_l2563873727033190209_a_nat @ nil_li1906260230833442699_a_nat @ nil_li2117038862230905745_a_nat ) ) ).
% subseqs.simps(1)
thf(fact_276_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_277_subseqs_Osimps_I1_J,axiom,
( ( subseq631295553041127183_a_nat @ nil_Pr237480997409426078_a_nat )
= ( cons_l6787550886680756862_a_nat @ nil_Pr237480997409426078_a_nat @ nil_li1907017536197114414_a_nat ) ) ).
% subseqs.simps(1)
thf(fact_278_subseqs_Osimps_I1_J,axiom,
( ( subseq5198752961258583356_a_nat @ nil_Pr6585251977456444909_a_nat )
= ( cons_l1779357001063338659_a_nat @ nil_Pr6585251977456444909_a_nat @ nil_li4039925377676013299_a_nat ) ) ).
% subseqs.simps(1)
thf(fact_279_subseqs_Osimps_I1_J,axiom,
( ( subseq8414445098004693972_a_nat @ nil_Sum_sum_a_nat )
= ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) ).
% subseqs.simps(1)
thf(fact_280_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_Sum_sum_a_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P6445506607073456153_a_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append_Sum_sum_a_nat @ ( F @ X ) @ ( maps_P6445506607073456153_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_281_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_P6164600145584960654at_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P5305992172557904996at_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append2142653904031976739at_nat @ ( F @ X ) @ ( maps_P5305992172557904996at_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_282_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_P5056861408695629236_a_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P3502264056893038730_a_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append338925788367110473_a_nat @ ( F @ X ) @ ( maps_P3502264056893038730_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_283_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_P6011104703257516679at_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P1432663425352833819at_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append985823374593552924at_nat @ ( F @ X ) @ ( maps_P1432663425352833819at_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_284_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_l4703314356710769291_a_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P3994670031119939231_a_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append5415888156905520160_a_nat @ ( F @ X ) @ ( maps_P3994670031119939231_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_285_maps__simps_I1_J,axiom,
! [F: produc7017002724195966439_a_nat > list_nat,X: produc7017002724195966439_a_nat,Xs: list_P1195027771636113901_a_nat] :
( ( maps_P5371491622303101494at_nat @ F @ ( cons_P1525839536144884125_a_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_P5371491622303101494at_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_286_maps__simps_I1_J,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( maps_S8041221185523983617_a_nat @ F @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( append_Sum_sum_a_nat @ ( F @ X ) @ ( maps_S8041221185523983617_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_287_concat__eq__append__conv,axiom,
! [Xss2: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( concat_Sum_sum_a_nat @ Xss2 )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_li1906260230833442699_a_nat )
=> ( ( Ys = nil_Sum_sum_a_nat )
& ( Zs = nil_Sum_sum_a_nat ) ) )
& ( ( Xss2 != nil_li1906260230833442699_a_nat )
=> ? [Xss1: list_l4703314356710769291_a_nat,Xs3: list_Sum_sum_a_nat,Xs4: list_Sum_sum_a_nat,Xss22: list_l4703314356710769291_a_nat] :
( ( Xss2
= ( append5415888156905520160_a_nat @ Xss1 @ ( cons_l6604326339930385211_a_nat @ ( append_Sum_sum_a_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_Sum_sum_a_nat @ ( concat_Sum_sum_a_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_Sum_sum_a_nat @ Xs4 @ ( concat_Sum_sum_a_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_288_butlast__snoc,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( butlas5768530507476509265_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_289_insert__Nil,axiom,
! [X: sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat )
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).
% insert_Nil
thf(fact_290_append__eq__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us2 )
= ( size_s5283204784079214577_a_nat @ Vs ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Us2 )
= ( append_Sum_sum_a_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_291_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us2 )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_292_bind__simps_I1_J,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( bind_S8497345843576593223_a_nat @ nil_Sum_sum_a_nat @ F )
= nil_Sum_sum_a_nat ) ).
% bind_simps(1)
thf(fact_293_concat__append,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( concat_Sum_sum_a_nat @ ( append5415888156905520160_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( concat_Sum_sum_a_nat @ Xs ) @ ( concat_Sum_sum_a_nat @ Ys ) ) ) ).
% concat_append
thf(fact_294_neq__if__length__neq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_295_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_296_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_297_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_298_butlast_Osimps_I1_J,axiom,
( ( butlas5768530507476509265_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% butlast.simps(1)
thf(fact_299_ad__agr__list__length,axiom,
! [X2: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_300_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_301_list__induct4,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_302_list__induct4,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_303_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_nat > list_nat > list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_304_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_nat > list_nat > list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5283204784079214577_a_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5283204784079214577_a_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_305_list__induct4,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_306_list__induct4,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_307_list__induct4,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5283204784079214577_a_nat @ Ws ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5283204784079214577_a_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_308_list__induct4,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,Ws: list_nat,P: list_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_309_list__induct4,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,Ws: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5283204784079214577_a_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: nat,Zs2: list_nat,W: sum_sum_a_nat,Ws2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s5283204784079214577_a_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_Sum_sum_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_310_list__induct3,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_311_list__induct3,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_312_list__induct3,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_313_list__induct3,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_nat,P: list_Sum_sum_a_nat > list_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_314_list__induct3,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_315_list__induct3,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_nat,P: list_nat > list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_316_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_Sum_sum_a_nat,P: list_nat > list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s5283204784079214577_a_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_317_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_318_list__induct2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_319_list__induct2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,P: list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_320_list__induct2,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_321_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_322_concat_Osimps_I1_J,axiom,
( ( concat_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat )
= nil_Sum_sum_a_nat ) ).
% concat.simps(1)
thf(fact_323_butlast_Osimps_I2_J,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
=> ( ( butlas5768530507476509265_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= nil_Sum_sum_a_nat ) )
& ( ( Xs != nil_Sum_sum_a_nat )
=> ( ( butlas5768530507476509265_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( butlas5768530507476509265_a_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_324_concat_Osimps_I2_J,axiom,
! [X: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( concat_Sum_sum_a_nat @ ( cons_l6604326339930385211_a_nat @ X @ Xs ) )
= ( append_Sum_sum_a_nat @ X @ ( concat_Sum_sum_a_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_325_butlast__append,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( Ys = nil_Sum_sum_a_nat )
=> ( ( butlas5768530507476509265_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( butlas5768530507476509265_a_nat @ Xs ) ) )
& ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( butlas5768530507476509265_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ Xs @ ( butlas5768530507476509265_a_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_326_maps__simps_I2_J,axiom,
! [F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( maps_S8041221185523983617_a_nat @ F @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% maps_simps(2)
thf(fact_327_same__length__different,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != Ys )
=> ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ? [Pre: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
( ( X3 != Y2 )
& ( Xs
= ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_Sum_sum_a_nat @ Pre @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_328_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y2: nat,Ys5: list_nat] :
( ( X3 != Y2 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_329_list__induct2__rev,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) @ ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2_rev
thf(fact_330_list__induct2__rev,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,P: list_Sum_sum_a_nat > list_nat > $o] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_Sum_sum_a_nat @ nil_nat )
=> ( ! [X3: sum_sum_a_nat,Y2: nat,Xs2: list_Sum_sum_a_nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_Sum_sum_a_nat @ Xs2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2_rev
thf(fact_331_list__induct2__rev,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,P: list_nat > list_Sum_sum_a_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_Sum_sum_a_nat )
=> ( ! [X3: nat,Y2: sum_sum_a_nat,Xs2: list_nat,Ys2: list_Sum_sum_a_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2_rev
thf(fact_332_list__induct2__rev,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2_rev
thf(fact_333_concat__eq__appendD,axiom,
! [Xss2: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( concat_Sum_sum_a_nat @ Xss2 )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_li1906260230833442699_a_nat )
=> ? [Xss12: list_l4703314356710769291_a_nat,Xs2: list_Sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Xss23: list_l4703314356710769291_a_nat] :
( ( Xss2
= ( append5415888156905520160_a_nat @ Xss12 @ ( cons_l6604326339930385211_a_nat @ ( append_Sum_sum_a_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_Sum_sum_a_nat @ ( concat_Sum_sum_a_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_Sum_sum_a_nat @ Xs5 @ ( concat_Sum_sum_a_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_334_nall__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( member408289922725080238_a_nat @ Vs @ ( nall_tuples_a @ AD @ N ) ) ) ) ).
% nall_tuplesI
thf(fact_335_append__butlast__last__id,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( append_Sum_sum_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) @ ( cons_Sum_sum_a_nat @ ( last_Sum_sum_a_nat @ Xs ) @ nil_Sum_sum_a_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_336_snoc__eq__iff__butlast,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= Ys )
= ( ( Ys != nil_Sum_sum_a_nat )
& ( ( butlas5768530507476509265_a_nat @ Ys )
= Xs )
& ( ( last_Sum_sum_a_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_337_list__ex1__simps_I1_J,axiom,
! [P: sum_sum_a_nat > $o] :
~ ( list_e7903279875808954312_a_nat @ P @ nil_Sum_sum_a_nat ) ).
% list_ex1_simps(1)
thf(fact_338_last__snoc,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( last_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
= X ) ).
% last_snoc
thf(fact_339_rotate1_Osimps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( rotate2765497868024679250_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_340_length__Suc__conv__rev,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ Y3 @ nil_Sum_sum_a_nat ) ) )
& ( ( size_s5283204784079214577_a_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_341_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_342_list__update__length,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) @ Y )
= ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_343_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_344_list__update__nonempty,axiom,
! [Xs: list_Sum_sum_a_nat,K: nat,X: sum_sum_a_nat] :
( ( ( list_u9138855634547462509_a_nat @ Xs @ K @ X )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% list_update_nonempty
thf(fact_345_length__list__update,axiom,
! [Xs: list_Sum_sum_a_nat,I: nat,X: sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ X ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_list_update
thf(fact_346_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_347_rotate1__is__Nil__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( rotate2765497868024679250_a_nat @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_348_length__rotate1,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rotate2765497868024679250_a_nat @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_rotate1
thf(fact_349_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_350_last__appendL,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( Ys = nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_351_last__appendR,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( Ys != nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( last_Sum_sum_a_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_352_list__update__code_I3_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,I: nat,Y: sum_sum_a_nat] :
( ( list_u9138855634547462509_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_Sum_sum_a_nat @ X @ ( list_u9138855634547462509_a_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_353_list__update__code_I1_J,axiom,
! [I: nat,Y: sum_sum_a_nat] :
( ( list_u9138855634547462509_a_nat @ nil_Sum_sum_a_nat @ I @ Y )
= nil_Sum_sum_a_nat ) ).
% list_update_code(1)
thf(fact_354_list__update_Osimps_I1_J,axiom,
! [I: nat,V: sum_sum_a_nat] :
( ( list_u9138855634547462509_a_nat @ nil_Sum_sum_a_nat @ I @ V )
= nil_Sum_sum_a_nat ) ).
% list_update.simps(1)
thf(fact_355_ord_Oquicksort__acc_Osimps_I1_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat] :
( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ nil_Sum_sum_a_nat )
= Ac ) ).
% ord.quicksort_acc.simps(1)
thf(fact_356_rotate1_Osimps_I1_J,axiom,
( ( rotate2765497868024679250_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% rotate1.simps(1)
thf(fact_357_Suc__length__conv,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( suc @ N )
= ( size_s5283204784079214577_a_nat @ Xs ) )
= ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) )
& ( ( size_s5283204784079214577_a_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_358_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_359_length__Suc__conv,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) )
& ( ( size_s5283204784079214577_a_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_360_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_361_last_Osimps,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_362_last__ConsL,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( Xs = nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_363_last__ConsR,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_364_last__append,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( Ys = nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( last_Sum_sum_a_nat @ Xs ) ) )
& ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( last_Sum_sum_a_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_365_longest__common__suffix,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
? [Ss: list_Sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Xs5 @ Ss ) )
& ( Ys
= ( append_Sum_sum_a_nat @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_Sum_sum_a_nat )
| ( Ys5 = nil_Sum_sum_a_nat )
| ( ( last_Sum_sum_a_nat @ Xs5 )
!= ( last_Sum_sum_a_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_366_ord_Oquicksort__acc_Osimps_I2_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( cons_Sum_sum_a_nat @ X @ Ac ) ) ).
% ord.quicksort_acc.simps(2)
thf(fact_367_length__append__singleton,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_368_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_369_length__Cons,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_370_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_371_length__nth__simps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_372_length__nth__simps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_373_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_374_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_375_ord_Oquicksort__part_Osimps_I1_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat] :
( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ nil_Sum_sum_a_nat )
= ( set_qu7651081299428620429_a_nat @ Less @ ( append_Sum_sum_a_nat @ Eqs @ ( cons_Sum_sum_a_nat @ X @ ( set_qu7651081299428620429_a_nat @ Less @ Ac @ Gts ) ) ) @ Lts ) ) ).
% ord.quicksort_part.simps(1)
thf(fact_376_ord_Oquicksort__part_Oelims,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Xa: sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Xc: list_Sum_sum_a_nat,Xd: list_Sum_sum_a_nat,Xe: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ Xc @ Xd @ Xe )
= Y )
=> ( ( ( Xe = nil_Sum_sum_a_nat )
=> ( Y
!= ( set_qu7651081299428620429_a_nat @ Less @ ( append_Sum_sum_a_nat @ Xc @ ( cons_Sum_sum_a_nat @ Xa @ ( set_qu7651081299428620429_a_nat @ Less @ X @ Xd ) ) ) @ Xb ) ) )
=> ~ ! [Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xe
= ( cons_Sum_sum_a_nat @ Z @ Zs2 ) )
=> ~ ( ( ( Less @ Xa @ Z )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ Xc @ ( cons_Sum_sum_a_nat @ Z @ Xd ) @ Zs2 ) ) )
& ( ~ ( Less @ Xa @ Z )
=> ( ( ( Less @ Z @ Xa )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ ( cons_Sum_sum_a_nat @ Z @ Xb ) @ Xc @ Xd @ Zs2 ) ) )
& ( ~ ( Less @ Z @ Xa )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ ( cons_Sum_sum_a_nat @ Z @ Xc ) @ Xd @ Zs2 ) ) ) ) ) ) ) ) ) ).
% ord.quicksort_part.elims
thf(fact_377_gen__length__code_I2_J,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( gen_le1340941697924381074_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( gen_le1340941697924381074_a_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_378_ord_Oquicksort__part_Osimps_I2_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,X: sum_sum_a_nat,Z2: sum_sum_a_nat,Ac: list_Sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( Less @ X @ Z2 )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ ( cons_Sum_sum_a_nat @ Z2 @ Gts ) @ Zs ) ) )
& ( ~ ( Less @ X @ Z2 )
=> ( ( ( Less @ Z2 @ X )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ ( cons_Sum_sum_a_nat @ Z2 @ Lts ) @ Eqs @ Gts @ Zs ) ) )
& ( ~ ( Less @ Z2 @ X )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ ( cons_Sum_sum_a_nat @ Z2 @ Eqs ) @ Gts @ Zs ) ) ) ) ) ) ).
% ord.quicksort_part.simps(2)
thf(fact_379_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le1340941697924381074_a_nat @ N @ nil_Sum_sum_a_nat )
= N ) ).
% gen_length_code(1)
thf(fact_380_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_381_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_382_size__neq__size__imp__neq,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ X )
!= ( size_s5283204784079214577_a_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_383_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_384_ord_Oquicksort__acc_Oelims,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( set_qu7651081299428620429_a_nat @ Less @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_nat )
=> ( Y != X ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ( Y
!= ( cons_Sum_sum_a_nat @ X3 @ X ) ) )
=> ~ ! [X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) )
=> ( Y
!= ( set_qu7459554806609531931_a_nat @ Less @ X @ X3 @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) ) ) ) ).
% ord.quicksort_acc.elims
thf(fact_385_ord_Oquicksort__acc_Osimps_I3_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,V: sum_sum_a_nat,Va2: list_Sum_sum_a_nat] :
( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ V @ Va2 ) ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V @ Va2 ) ) ) ).
% ord.quicksort_acc.simps(3)
thf(fact_386_SuccI,axiom,
! [Kl: list_Sum_sum_a_nat,K: sum_sum_a_nat,Kl2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( append_Sum_sum_a_nat @ Kl @ ( cons_Sum_sum_a_nat @ K @ nil_Sum_sum_a_nat ) ) @ Kl2 )
=> ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_387_SuccD,axiom,
! [K: sum_sum_a_nat,Kl2: set_li6526943997496501093_a_nat,Kl: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl2 @ Kl ) )
=> ( member408289922725080238_a_nat @ ( append_Sum_sum_a_nat @ Kl @ ( cons_Sum_sum_a_nat @ K @ nil_Sum_sum_a_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_388_ord_Oquicksort__def,axiom,
( set_qu7255043234445450382_a_nat
= ( ^ [Less2: sum_sum_a_nat > sum_sum_a_nat > $o] : ( set_qu7651081299428620429_a_nat @ Less2 @ nil_Sum_sum_a_nat ) ) ) ).
% ord.quicksort_def
thf(fact_389_ord_Odistinct__quicksort__part,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
= ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Ac @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) @ ( append_Sum_sum_a_nat @ Lts @ ( append_Sum_sum_a_nat @ Eqs @ ( append_Sum_sum_a_nat @ Gts @ Zs ) ) ) ) ) ) ) ).
% ord.distinct_quicksort_part
thf(fact_390_map__tailrec__rev_Oelims,axiom,
! [X: sum_sum_a_nat > sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( map_ta7636758496465269173_a_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_nat )
=> ( Y != Xb ) )
=> ~ ! [A4: sum_sum_a_nat,As: list_Sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ A4 @ As ) )
=> ( Y
!= ( map_ta7636758496465269173_a_nat @ X @ As @ ( cons_Sum_sum_a_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_391_concat__conv__foldr,axiom,
( concat_Sum_sum_a_nat
= ( ^ [Xss3: list_l4703314356710769291_a_nat] : ( foldr_8827264258039228903_a_nat @ append_Sum_sum_a_nat @ Xss3 @ nil_Sum_sum_a_nat ) ) ) ).
% concat_conv_foldr
thf(fact_392_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( rotate_Sum_sum_a_nat @ N @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_393_length__rotate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rotate_Sum_sum_a_nat @ N @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_rotate
thf(fact_394_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_395_distinct__length__2__or__more,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ ( cons_Sum_sum_a_nat @ B @ Xs ) ) )
= ( ( A != B )
& ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ Xs ) )
& ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_396_distinct_Osimps_I1_J,axiom,
distin2701893636801681144_a_nat @ nil_Sum_sum_a_nat ).
% distinct.simps(1)
thf(fact_397_distinct__singleton,axiom,
! [X: sum_sum_a_nat] : ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).
% distinct_singleton
thf(fact_398_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: sum_sum_a_nat,As2: list_Sum_sum_a_nat,Bs: list_Sum_sum_a_nat] :
( ( map_ta7636758496465269173_a_nat @ F @ ( cons_Sum_sum_a_nat @ A @ As2 ) @ Bs )
= ( map_ta7636758496465269173_a_nat @ F @ As2 @ ( cons_Sum_sum_a_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_399_ord_Odistinct__quicksort__acc,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( set_qu7651081299428620429_a_nat @ Less @ Ac @ Xs ) )
= ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Ac @ Xs ) ) ) ).
% ord.distinct_quicksort_acc
thf(fact_400_rotate__append,axiom,
! [L: list_Sum_sum_a_nat,Q: list_Sum_sum_a_nat] :
( ( rotate_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ L ) @ ( append_Sum_sum_a_nat @ L @ Q ) )
= ( append_Sum_sum_a_nat @ Q @ L ) ) ).
% rotate_append
thf(fact_401_rotate__append,axiom,
! [L: list_nat,Q: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q ) )
= ( append_nat @ Q @ L ) ) ).
% rotate_append
thf(fact_402_not__distinct__decomp,axiom,
! [Ws: list_Sum_sum_a_nat] :
( ~ ( distin2701893636801681144_a_nat @ Ws )
=> ? [Xs2: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat] :
( Ws
= ( append_Sum_sum_a_nat @ Xs2 @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) @ ( append_Sum_sum_a_nat @ Ys2 @ ( append_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ nil_Sum_sum_a_nat ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_403_empty__Shift,axiom,
! [Kl2: set_li6526943997496501093_a_nat,K: sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ nil_Sum_sum_a_nat @ Kl2 )
=> ( ( member_Sum_sum_a_nat @ K @ ( bNF_Gr5582227268375839130_a_nat @ Kl2 @ nil_Sum_sum_a_nat ) )
=> ( member408289922725080238_a_nat @ nil_Sum_sum_a_nat @ ( bNF_Gr1229660863860170270_a_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_404_Succ__Shift,axiom,
! [Kl2: set_li6526943997496501093_a_nat,K: sum_sum_a_nat,Kl: list_Sum_sum_a_nat] :
( ( bNF_Gr5582227268375839130_a_nat @ ( bNF_Gr1229660863860170270_a_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr5582227268375839130_a_nat @ Kl2 @ ( cons_Sum_sum_a_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_405_nth__append__length,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_406_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_407_listrelp_Ocases,axiom,
! [R: sum_sum_a_nat > sum_sum_a_nat > $o,A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat] :
( ( listre4668766549960312319_a_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Sum_sum_a_nat )
=> ( A22 != nil_Sum_sum_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( A1
= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_Sum_sum_a_nat] :
( ( A22
= ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre4668766549960312319_a_nat @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_408_listrelp_Osimps,axiom,
( listre4668766549960312319_a_nat
= ( ^ [R2: sum_sum_a_nat > sum_sum_a_nat > $o,A12: list_Sum_sum_a_nat,A23: list_Sum_sum_a_nat] :
( ( ( A12 = nil_Sum_sum_a_nat )
& ( A23 = nil_Sum_sum_a_nat ) )
| ? [X4: sum_sum_a_nat,Y3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( A12
= ( cons_Sum_sum_a_nat @ X4 @ Xs3 ) )
& ( A23
= ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre4668766549960312319_a_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_409_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_410_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_411_length__nth__simps_I4_J,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_412_length__nth__simps_I4_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_413_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_414_nth__Cons__Suc,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_415_listrelp_OCons,axiom,
! [R: sum_sum_a_nat > sum_sum_a_nat > $o,X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( R @ X @ Y )
=> ( ( listre4668766549960312319_a_nat @ R @ Xs @ Ys )
=> ( listre4668766549960312319_a_nat @ R @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_416_listrelp_ONil,axiom,
! [R: sum_sum_a_nat > sum_sum_a_nat > $o] : ( listre4668766549960312319_a_nat @ R @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ).
% listrelp.Nil
thf(fact_417_ShiftD,axiom,
! [Kl: list_Sum_sum_a_nat,Kl2: set_li6526943997496501093_a_nat,K: sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Kl @ ( bNF_Gr1229660863860170270_a_nat @ Kl2 @ K ) )
=> ( member408289922725080238_a_nat @ ( cons_Sum_sum_a_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_418_distinct__swap,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( distin2701893636801681144_a_nat @ ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ ( nth_Sum_sum_a_nat @ Xs @ J ) ) @ J @ ( nth_Sum_sum_a_nat @ Xs @ I ) ) )
= ( distin2701893636801681144_a_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_419_distinct__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_420_nth__list__update__eq,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_421_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_422_nth__append__length__plus,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,N: nat] :
( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) )
= ( nth_Sum_sum_a_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_423_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_424_nth__butlast,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) ) )
=> ( ( nth_Sum_sum_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_425_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_426_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_427_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_428_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_429_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_430_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_431_length__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_append
thf(fact_432_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_433_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_434_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_435_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_436_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_437_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_438_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_439_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_440_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_441_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_442_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_443_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_444_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_445_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_446_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_447_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_448_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_449_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_450_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_451_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_452_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_453_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_454_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_455_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_456_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_457_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_458_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_459_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_460_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_461_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_462_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_463_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_464_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_465_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_466_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_467_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_468_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_469_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_470_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_471_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_472_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_473_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_474_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_475_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_476_length__induct,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ! [Xs2: list_Sum_sum_a_nat] :
( ! [Ys6: list_Sum_sum_a_nat] :
( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ys6 ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_477_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_478_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_Sum_sum_a_nat,Z3: list_Sum_sum_a_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_Sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= ( size_s5283204784079214577_a_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
=> ( ( nth_Sum_sum_a_nat @ Xs3 @ I3 )
= ( nth_Sum_sum_a_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_479_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_480_Skolem__list__nth,axiom,
! [K: nat,P: nat > sum_sum_a_nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: sum_sum_a_nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Sum_sum_a_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_481_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_482_nth__equalityI,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
= ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_483_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_484_quicksort__part_Oelims,axiom,
! [X: list_nat,Xa: nat,Xb: list_nat,Xc: list_nat,Xd: list_nat,Xe: list_nat,Y: list_nat] :
( ( ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ Xc @ Xd @ Xe )
= Y )
=> ( ( ( Xe = nil_nat )
=> ( Y
!= ( set_or5558937660843164036cc_nat @ ( append_nat @ Xc @ ( cons_nat @ Xa @ ( set_or5558937660843164036cc_nat @ X @ Xd ) ) ) @ Xb ) ) )
=> ~ ! [Z: nat,Zs2: list_nat] :
( ( Xe
= ( cons_nat @ Z @ Zs2 ) )
=> ~ ( ( ( ord_less_nat @ Xa @ Z )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ Xc @ ( cons_nat @ Z @ Xd ) @ Zs2 ) ) )
& ( ~ ( ord_less_nat @ Xa @ Z )
=> ( ( ( ord_less_nat @ Z @ Xa )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ ( cons_nat @ Z @ Xb ) @ Xc @ Xd @ Zs2 ) ) )
& ( ~ ( ord_less_nat @ Z @ Xa )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ ( cons_nat @ Z @ Xc ) @ Xd @ Zs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_part.elims
thf(fact_485_quicksort__part_Osimps_I2_J,axiom,
! [X: nat,Z2: nat,Ac: list_nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( ( ord_less_nat @ X @ Z2 )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ ( cons_nat @ Z2 @ Gts ) @ Zs ) ) )
& ( ~ ( ord_less_nat @ X @ Z2 )
=> ( ( ( ord_less_nat @ Z2 @ X )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ ( cons_nat @ Z2 @ Lts ) @ Eqs @ Gts @ Zs ) ) )
& ( ~ ( ord_less_nat @ Z2 @ X )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ ( cons_nat @ Z2 @ Eqs ) @ Gts @ Zs ) ) ) ) ) ) ).
% quicksort_part.simps(2)
thf(fact_486_gen__length__def,axiom,
( gen_le1340941697924381074_a_nat
= ( ^ [N2: nat,Xs3: list_Sum_sum_a_nat] : ( plus_plus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_487_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_488_distinct__conv__nth,axiom,
( distin2701893636801681144_a_nat
= ( ^ [Xs3: list_Sum_sum_a_nat] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
=> ( ( I3 != J3 )
=> ( ( nth_Sum_sum_a_nat @ Xs3 @ I3 )
!= ( nth_Sum_sum_a_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_489_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I3 != J3 )
=> ( ( nth_nat @ Xs3 @ I3 )
!= ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_490_nth__eq__iff__index__eq,axiom,
! [Xs: list_Sum_sum_a_nat,I: nat,J: nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( nth_Sum_sum_a_nat @ Xs @ I )
= ( nth_Sum_sum_a_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_491_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_492_list__update__append1,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ I @ X )
= ( append_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_493_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_494_list__update__same__conv,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( list_u9138855634547462509_a_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_Sum_sum_a_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_495_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_496_nth__list__update,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,J: nat,X: sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ X ) @ J )
= ( nth_Sum_sum_a_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_497_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_498_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( take_Sum_sum_a_nat @ ( suc @ I ) @ Xs )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ nil_Sum_sum_a_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_499_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_500_nth__rotate1,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( rotate2765497868024679250_a_nat @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_501_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_502_nth__rotate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( rotate_Sum_sum_a_nat @ M @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_503_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_504_set__swap,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( set_Sum_sum_a_nat2 @ ( list_u9138855634547462509_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ I @ ( nth_Sum_sum_a_nat @ Xs @ J ) ) @ J @ ( nth_Sum_sum_a_nat @ Xs @ I ) ) )
= ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_505_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_506_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( drop_Sum_sum_a_nat @ ( suc @ I ) @ Xs ) )
= ( drop_Sum_sum_a_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_507_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_508_remdups__sorted_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( set_or6599480164596245535ed_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X3: nat] :
( ( X
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( ord_less_nat @ X3 @ Y2 )
=> ( Y
= ( cons_nat @ X3 @ ( set_or6599480164596245535ed_nat @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) )
& ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( Y
= ( set_or6599480164596245535ed_nat @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_sorted.elims
thf(fact_509_enumerate__append__eq,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( enumer3164015132978342500_a_nat @ N @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append338925788367110473_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) @ ( enumer3164015132978342500_a_nat @ ( plus_plus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_510_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_511_list_Osize_I4_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_512_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_513_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_514_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_515_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_516_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_517_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_518_Nil__eq__concat__conv,axiom,
! [Xss2: list_l4703314356710769291_a_nat] :
( ( nil_Sum_sum_a_nat
= ( concat_Sum_sum_a_nat @ Xss2 ) )
= ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xss2 ) )
=> ( X4 = nil_Sum_sum_a_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_519_concat__eq__Nil__conv,axiom,
! [Xss2: list_l4703314356710769291_a_nat] :
( ( ( concat_Sum_sum_a_nat @ Xss2 )
= nil_Sum_sum_a_nat )
= ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xss2 ) )
=> ( X4 = nil_Sum_sum_a_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_520_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer3164015132978342500_a_nat @ N @ nil_Sum_sum_a_nat )
= nil_Pr237480997409426078_a_nat ) ).
% enumerate_simps(1)
thf(fact_521_length__enumerate,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_enumerate
thf(fact_522_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_523_length__0__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% length_0_conv
thf(fact_524_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_525_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_526_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_527_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_528_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_529_nth__Cons__0,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_530_length__nth__simps_I3_J,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% length_nth_simps(3)
thf(fact_531_length__nth__simps_I3_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% length_nth_simps(3)
thf(fact_532_take__Suc__Cons,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ ( suc @ N ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_533_take__eq__Nil2,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( take_Sum_sum_a_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Sum_sum_a_nat ) ) ) ).
% take_eq_Nil2
thf(fact_534_take__eq__Nil,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( take_Sum_sum_a_nat @ N @ Xs )
= nil_Sum_sum_a_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Sum_sum_a_nat ) ) ) ).
% take_eq_Nil
thf(fact_535_take0,axiom,
( ( take_Sum_sum_a_nat @ zero_zero_nat )
= ( ^ [Xs3: list_Sum_sum_a_nat] : nil_Sum_sum_a_nat ) ) ).
% take0
thf(fact_536_drop__Suc__Cons,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( drop_Sum_sum_a_nat @ ( suc @ N ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( drop_Sum_sum_a_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_537_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_538_append__take__drop__id,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( drop_Sum_sum_a_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_539_not__in__set__insert,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( insert_Sum_sum_a_nat @ X @ Xs )
= ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_540_length__greater__0__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) )
= ( Xs != nil_Sum_sum_a_nat ) ) ).
% length_greater_0_conv
thf(fact_541_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_542_rotate__id,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_Sum_sum_a_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_543_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_544_last__drop,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( last_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ).
% last_drop
thf(fact_545_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_546_take__0,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ zero_zero_nat @ Xs )
= nil_Sum_sum_a_nat ) ).
% take_0
thf(fact_547_append__eq__conv__conj,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_548_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_549_take__add,axiom,
! [I: nat,J: nat,Xs: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) @ ( take_Sum_sum_a_nat @ J @ ( drop_Sum_sum_a_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_550_rotate__drop__take,axiom,
( rotate_Sum_sum_a_nat
= ( ^ [N2: nat,Xs3: list_Sum_sum_a_nat] : ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( modulo_modulo_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) @ Xs3 ) @ ( take_Sum_sum_a_nat @ ( modulo_modulo_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_551_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) @ ( take_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_552_take__Nil,axiom,
! [N: nat] :
( ( take_Sum_sum_a_nat @ N @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% take_Nil
thf(fact_553_length__pos__if__in__set,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_554_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_555_drop__Nil,axiom,
! [N: nat] :
( ( drop_Sum_sum_a_nat @ N @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% drop_Nil
thf(fact_556_set__ConsD,axiom,
! [Y: sum_sum_a_nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_557_list_Oset__cases,axiom,
! [E: sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ A ) )
=> ( ! [Z22: list_Sum_sum_a_nat] :
( A
!= ( cons_Sum_sum_a_nat @ E @ Z22 ) )
=> ~ ! [Z1: sum_sum_a_nat,Z22: list_Sum_sum_a_nat] :
( ( A
= ( cons_Sum_sum_a_nat @ Z1 @ Z22 ) )
=> ~ ( member_Sum_sum_a_nat @ E @ ( set_Sum_sum_a_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_558_list_Oset__intros_I1_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X21 @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_559_list_Oset__intros_I2_J,axiom,
! [Y: sum_sum_a_nat,X22: list_Sum_sum_a_nat,X21: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ X22 ) )
=> ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_560_fo__nmlzd__take,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat,I: nat] :
( ( fo_nmlzd_a @ AD @ Xs )
=> ( fo_nmlzd_a @ AD @ ( take_Sum_sum_a_nat @ I @ Xs ) ) ) ).
% fo_nmlzd_take
thf(fact_561_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_562_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_563_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_564_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_565_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_566_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_567_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_568_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_569_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_570_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_571_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_572_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_573_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_574_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_575_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_576_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_577_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_578_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_579_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_580_forall__finite_I1_J,axiom,
! [P: nat > $o,I4: nat] :
( ( ord_less_nat @ I4 @ zero_zero_nat )
=> ( P @ I4 ) ) ).
% forall_finite(1)
thf(fact_581_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_582_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_583_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_584_nth__via__drop,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( drop_Sum_sum_a_nat @ N @ Xs )
= ( cons_Sum_sum_a_nat @ Y @ Ys ) )
=> ( ( nth_Sum_sum_a_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_585_split__list__first__prop__iff,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_Sum_sum_a_nat,X4: sum_sum_a_nat] :
( ? [Zs3: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_586_split__list__last__prop__iff,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_Sum_sum_a_nat,X4: sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_587_in__set__conv__decomp__first,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) )
& ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_588_in__set__conv__decomp__last,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) )
& ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_589_split__list__first__propE,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ? [Zs2: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_590_split__list__last__propE,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_591_split__list__first__prop,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ? [Zs2: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_592_split__list__last__prop,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ ( set_Sum_sum_a_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_593_in__set__conv__decomp,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( ? [Ys3: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys3 @ ( cons_Sum_sum_a_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_594_append__Cons__eq__iff,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xs6: list_Sum_sum_a_nat,Ys7: list_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ Ys ) )
= ( append_Sum_sum_a_nat @ Xs6 @ ( cons_Sum_sum_a_nat @ X @ Ys7 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_595_split__list__propE,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ? [Zs2: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_596_split__list__first,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ? [Ys2: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs2 ) ) )
& ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_597_split__list__prop,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ? [Zs2: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_598_split__list__last,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ? [Ys2: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs2 ) ) )
& ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_599_split__list,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ? [Ys2: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( Xs
= ( append_Sum_sum_a_nat @ Ys2 @ ( cons_Sum_sum_a_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_600_distinct_Osimps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
& ( distin2701893636801681144_a_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_601_last__in__set,axiom,
! [As2: list_Sum_sum_a_nat] :
( ( As2 != nil_Sum_sum_a_nat )
=> ( member_Sum_sum_a_nat @ ( last_Sum_sum_a_nat @ As2 ) @ ( set_Sum_sum_a_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_602_rotate__conv__mod,axiom,
( rotate_Sum_sum_a_nat
= ( ^ [N2: nat,Xs3: list_Sum_sum_a_nat] : ( rotate_Sum_sum_a_nat @ ( modulo_modulo_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_603_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_604_length__nth__simps_I1_J,axiom,
( ( size_s5283204784079214577_a_nat @ nil_Sum_sum_a_nat )
= zero_zero_nat ) ).
% length_nth_simps(1)
thf(fact_605_length__nth__simps_I1_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% length_nth_simps(1)
thf(fact_606_forall__finite_I3_J,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ ( suc @ X ) ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ X ) )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_607_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ zero_zero_nat ) )
=> ( P @ I3 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_608_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_609_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_610_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_611_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_612_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_613_Comparator__Generator_OAll__less__Suc,axiom,
! [X: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ X ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ X )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_614_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_615_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_616_in__set__butlast__appendI,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( butlas5768530507476509265_a_nat @ Xs ) ) )
| ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( butlas5768530507476509265_a_nat @ Ys ) ) ) )
=> ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( butlas5768530507476509265_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_617_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_618_Cons__in__subseqsD,axiom,
! [Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ ( cons_Sum_sum_a_nat @ Y @ Ys ) @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) )
=> ( member408289922725080238_a_nat @ Ys @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_619_List_Oinsert__def,axiom,
( insert_Sum_sum_a_nat
= ( ^ [X4: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat] : ( if_lis4685338526944683083_a_nat @ ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_Sum_sum_a_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_620_list__update__code_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( list_u9138855634547462509_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_Sum_sum_a_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_621_in__set__product__lists__length,axiom,
! [Xs: list_Sum_sum_a_nat,Xss2: list_l4703314356710769291_a_nat] :
( ( member408289922725080238_a_nat @ Xs @ ( set_li2392974972034027290_a_nat @ ( produc2893206433618375022_a_nat @ Xss2 ) ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5212483967078203639_a_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_622_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_623_id__take__nth__drop,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( Xs
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) @ ( cons_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( drop_Sum_sum_a_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_624_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_625_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,A: sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( list_u9138855634547462509_a_nat @ Xs @ I @ A )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) @ ( cons_Sum_sum_a_nat @ A @ ( drop_Sum_sum_a_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_626_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_627_take__butlast,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( take_Sum_sum_a_nat @ N @ ( butlas5768530507476509265_a_nat @ Xs ) )
= ( take_Sum_sum_a_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_628_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_629_length__code,axiom,
( size_s5283204784079214577_a_nat
= ( gen_le1340941697924381074_a_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_630_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_631_not__distinct__conv__prefix,axiom,
! [As2: list_Sum_sum_a_nat] :
( ( ~ ( distin2701893636801681144_a_nat @ As2 ) )
= ( ? [Xs3: list_Sum_sum_a_nat,Y3: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs3 ) )
& ( distin2701893636801681144_a_nat @ Xs3 )
& ( As2
= ( append_Sum_sum_a_nat @ Xs3 @ ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_632_nth__mem,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( member_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_633_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_634_list__ball__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_635_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_636_in__set__conv__nth,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( ( nth_Sum_sum_a_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_637_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_638_all__nth__imp__all__set,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,X: sum_sum_a_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) ) )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_639_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_640_all__set__conv__all__nth,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_641_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_642_remdups__sorted_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( ord_less_nat @ X @ Y )
=> ( ( set_or6599480164596245535ed_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( set_or6599480164596245535ed_nat @ ( cons_nat @ Y @ Xs ) ) ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( set_or6599480164596245535ed_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( set_or6599480164596245535ed_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ).
% remdups_sorted.simps(3)
thf(fact_643_set__update__memI,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( list_u9138855634547462509_a_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_644_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_645_distinct__Ex1,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( ( nth_Sum_sum_a_nat @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( ( nth_Sum_sum_a_nat @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_646_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X3 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_647_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_648_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( ord_less_nat @ M @ P2 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P2 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_649_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_650_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list6375351914370498317_a_nat @ N @ nil_Sum_sum_a_nat )
= ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list6375351914370498317_a_nat @ N @ nil_Sum_sum_a_nat )
= nil_li1906260230833442699_a_nat ) ) ) ).
% n_lists_Nil
thf(fact_651_in__set__simps_I2_J,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) )
= ( X = Y ) ) ).
% in_set_simps(2)
thf(fact_652_rem__nth__take__drop,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( rem_nt658808235856662061_a_nat @ I @ Xs )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) @ ( drop_Sum_sum_a_nat @ ( suc @ I ) @ Xs ) ) ) ) ).
% rem_nth_take_drop
thf(fact_653_rem__nth__take__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( rem_nth_nat @ I @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ).
% rem_nth_take_drop
thf(fact_654_rem__nth_Osimps_I1_J,axiom,
! [Uu: nat] :
( ( rem_nt658808235856662061_a_nat @ Uu @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% rem_nth.simps(1)
thf(fact_655_length__n__lists__elem,axiom,
! [Ys: list_Sum_sum_a_nat,N: nat,Xs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys @ ( set_li2392974972034027290_a_nat @ ( n_list6375351914370498317_a_nat @ N @ Xs ) ) )
=> ( ( size_s5283204784079214577_a_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_656_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_657_rem__nth_Osimps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( rem_nt658808235856662061_a_nat @ zero_zero_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= Xs ) ).
% rem_nth.simps(2)
thf(fact_658_rem__nth_Osimps_I3_J,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( rem_nt658808235856662061_a_nat @ ( suc @ N ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( rem_nt658808235856662061_a_nat @ N @ Xs ) ) ) ).
% rem_nth.simps(3)
thf(fact_659_rem__nth_Oelims,axiom,
! [X: nat,Xa: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( rem_nt658808235856662061_a_nat @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_Sum_sum_a_nat )
=> ( Y != nil_Sum_sum_a_nat ) )
=> ( ( ( X = zero_zero_nat )
=> ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) )
=> ( Y != Xs2 ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_Sum_sum_a_nat @ X3 @ ( rem_nt658808235856662061_a_nat @ N3 @ Xs2 ) ) ) ) ) ) ) ) ).
% rem_nth.elims
thf(fact_660_in__set__simps_I1_J,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Z2: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Y @ ( cons_Sum_sum_a_nat @ Z2 @ Ys ) ) ) )
= ( ( X = Y )
| ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ Z2 @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_661_in__set__simps_I3_J,axiom,
! [X: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ nil_Sum_sum_a_nat ) ) ).
% in_set_simps(3)
thf(fact_662_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_663_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_664_n__lists_Osimps_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( n_list6375351914370498317_a_nat @ zero_zero_nat @ Xs )
= ( cons_l6604326339930385211_a_nat @ nil_Sum_sum_a_nat @ nil_li1906260230833442699_a_nat ) ) ).
% n_lists.simps(1)
thf(fact_665_take__hd__drop,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) ) @ nil_Sum_sum_a_nat ) )
= ( take_Sum_sum_a_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_666_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_667_add__nth__rem__nth__self,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( add_nt4212672348507122516_a_nat @ I @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( rem_nt658808235856662061_a_nat @ I @ Xs ) )
= Xs ) ) ).
% add_nth_rem_nth_self
thf(fact_668_add__nth__rem__nth__self,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( add_nth_nat @ I @ ( nth_nat @ Xs @ I ) @ ( rem_nth_nat @ I @ Xs ) )
= Xs ) ) ).
% add_nth_rem_nth_self
thf(fact_669_zip__with__index__from__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( zip_wi5053028342928268674_a_nat @ N @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append338925788367110473_a_nat @ ( zip_wi5053028342928268674_a_nat @ N @ Xs ) @ ( zip_wi5053028342928268674_a_nat @ ( plus_plus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).
% zip_with_index_from_append
thf(fact_670_zip__with__index__from__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( zip_wi3407404960461264653om_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( zip_wi3407404960461264653om_nat @ N @ Xs ) @ ( zip_wi3407404960461264653om_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% zip_with_index_from_append
thf(fact_671_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_672_hd__append2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( hd_Sum_sum_a_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_673_zip__with__index__from__simps_I1_J,axiom,
! [N: nat] :
( ( zip_wi5053028342928268674_a_nat @ N @ nil_Sum_sum_a_nat )
= nil_Pr237480997409426078_a_nat ) ).
% zip_with_index_from_simps(1)
thf(fact_674_list_Osel_I1_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( hd_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_675_hd__concat,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( Xs != nil_li1906260230833442699_a_nat )
=> ( ( ( hd_lis3280420719747858032_a_nat @ Xs )
!= nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ ( concat_Sum_sum_a_nat @ Xs ) )
= ( hd_Sum_sum_a_nat @ ( hd_lis3280420719747858032_a_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_676_hd__in__set,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( member_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_677_list_Oset__sel_I1_J,axiom,
! [A: list_Sum_sum_a_nat] :
( ( A != nil_Sum_sum_a_nat )
=> ( member_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ A ) @ ( set_Sum_sum_a_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_678_longest__common__prefix,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
? [Ps: list_Sum_sum_a_nat,Xs5: list_Sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Ps @ Xs5 ) )
& ( Ys
= ( append_Sum_sum_a_nat @ Ps @ Ys5 ) )
& ( ( Xs5 = nil_Sum_sum_a_nat )
| ( Ys5 = nil_Sum_sum_a_nat )
| ( ( hd_Sum_sum_a_nat @ Xs5 )
!= ( hd_Sum_sum_a_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_679_hd__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( hd_Sum_sum_a_nat @ Ys ) ) )
& ( ( Xs != nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( hd_Sum_sum_a_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_680_hd__Nil__eq__last,axiom,
( ( hd_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
= ( last_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) ).
% hd_Nil_eq_last
thf(fact_681_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_682_hd__conv__nth,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ Xs )
= ( nth_Sum_sum_a_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_683_add__nth_Osimps_I1_J,axiom,
! [A: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( add_nt4212672348507122516_a_nat @ zero_zero_nat @ A @ Xs )
= ( cons_Sum_sum_a_nat @ A @ Xs ) ) ).
% add_nth.simps(1)
thf(fact_684_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( hd_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_685_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_686_hd__rotate__conv__nth,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( hd_Sum_sum_a_nat @ ( rotate_Sum_sum_a_nat @ N @ Xs ) )
= ( nth_Sum_sum_a_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_687_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_688_add__nth__take__drop,axiom,
! [I: nat,Zs: list_Sum_sum_a_nat,V: sum_sum_a_nat] :
( ( ord_less_eq_nat @ I @ ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( add_nt4212672348507122516_a_nat @ I @ V @ Zs )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Zs ) @ ( cons_Sum_sum_a_nat @ V @ ( drop_Sum_sum_a_nat @ I @ Zs ) ) ) ) ) ).
% add_nth_take_drop
thf(fact_689_add__nth__take__drop,axiom,
! [I: nat,Zs: list_nat,V: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( add_nth_nat @ I @ V @ Zs )
= ( append_nat @ ( take_nat @ I @ Zs ) @ ( cons_nat @ V @ ( drop_nat @ I @ Zs ) ) ) ) ) ).
% add_nth_take_drop
thf(fact_690_rremdups__app,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( rremdu8304153113908149561_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) )
= ( append_Sum_sum_a_nat @ ( rremdu8304153113908149561_a_nat @ Xs ) @ ( if_lis4685338526944683083_a_nat @ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ) ).
% rremdups_app
thf(fact_691_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Pr4195520319383970909_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs ) @ M )
= ( produc3265382261054541654_a_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_Sum_sum_a_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_692_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_693_distinct__adj__append__iff,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( distin3423869863095720157_a_nat @ Xs )
& ( distin3423869863095720157_a_nat @ Ys )
& ( ( Xs = nil_Sum_sum_a_nat )
| ( Ys = nil_Sum_sum_a_nat )
| ( ( last_Sum_sum_a_nat @ Xs )
!= ( hd_Sum_sum_a_nat @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_694_nth__drop,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) @ I )
= ( nth_Sum_sum_a_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_695_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_696_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_697_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_698_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_699_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_700_distinct__adj__Cons__Cons,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distin3423869863095720157_a_nat @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_701_take__all,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N )
=> ( ( take_Sum_sum_a_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_702_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_703_take__all__iff,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( take_Sum_sum_a_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_704_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_705_list__update__beyond,axiom,
! [Xs: list_Sum_sum_a_nat,I: nat,X: sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ I )
=> ( ( list_u9138855634547462509_a_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_706_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_707_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( drop_Sum_sum_a_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_708_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_709_drop__eq__Nil,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( drop_Sum_sum_a_nat @ N @ Xs )
= nil_Sum_sum_a_nat )
= ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_710_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_711_drop__all,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N )
=> ( ( drop_Sum_sum_a_nat @ N @ Xs )
= nil_Sum_sum_a_nat ) ) ).
% drop_all
thf(fact_712_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_713_enumerate__simps_I2_J,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( enumer3164015132978342500_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_P8612082756026972910_a_nat @ ( produc3265382261054541654_a_nat @ N @ X ) @ ( enumer3164015132978342500_a_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_714_zip__with__index__from__simps_I2_J,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( zip_wi5053028342928268674_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_P8612082756026972910_a_nat @ ( produc3265382261054541654_a_nat @ N @ X ) @ ( zip_wi5053028342928268674_a_nat @ ( suc @ N ) @ Xs ) ) ) ).
% zip_with_index_from_simps(2)
thf(fact_715_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_716_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_717_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_718_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_719_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_720_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_721_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_722_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_723_wlog__le,axiom,
! [P: nat > nat > $o,B: nat,A: nat] :
( ! [A4: nat,B2: nat] :
( ( P @ A4 @ B2 )
=> ( P @ B2 @ A4 ) )
=> ( ! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( P @ B @ A ) ) ) ).
% wlog_le
thf(fact_724_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_725_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_726_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_727_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_728_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_729_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_730_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_731_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_732_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_733_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_734_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_735_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_736_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_737_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_738_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_739_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_740_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_741_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_742_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_743_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_744_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_745_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_746_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_747_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_748_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_749_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_750_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z )
=> ( R3 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_751_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_752_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_753_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_754_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_755_distinct__adj__ConsD,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
=> ( distin3423869863095720157_a_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_756_distinct__adj__Nil,axiom,
distin3423869863095720157_a_nat @ nil_Sum_sum_a_nat ).
% distinct_adj_Nil
thf(fact_757_distinct__adj__appendD1,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
=> ( distin3423869863095720157_a_nat @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_758_distinct__adj__appendD2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
=> ( distin3423869863095720157_a_nat @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_759_rremdups_Osimps_I1_J,axiom,
( ( rremdu8304153113908149561_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% rremdups.simps(1)
thf(fact_760_impossible__Cons,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs
!= ( cons_Sum_sum_a_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_761_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_762_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_763_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_764_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_765_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_766_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_767_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_768_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_769_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_770_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_771_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_772_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_773_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_774_distinct__adj__singleton,axiom,
! [X: sum_sum_a_nat] : ( distin3423869863095720157_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).
% distinct_adj_singleton
thf(fact_775_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5283204784079214577_a_nat @ Xs ) )
= ( ? [X4: sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_776_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_777_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_778_add__nth__length,axiom,
! [I: nat,Zs: list_Sum_sum_a_nat,Z2: sum_sum_a_nat] :
( ( ord_less_eq_nat @ I @ ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( add_nt4212672348507122516_a_nat @ I @ Z2 @ Zs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Zs ) ) ) ) ).
% add_nth_length
thf(fact_779_add__nth__length,axiom,
! [I: nat,Zs: list_nat,Z2: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( size_size_list_nat @ ( add_nth_nat @ I @ Z2 @ Zs ) )
= ( suc @ ( size_size_list_nat @ Zs ) ) ) ) ).
% add_nth_length
thf(fact_780_rem__nth__add__nth,axiom,
! [I: nat,Zs: list_Sum_sum_a_nat,Z2: sum_sum_a_nat] :
( ( ord_less_eq_nat @ I @ ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( rem_nt658808235856662061_a_nat @ I @ ( add_nt4212672348507122516_a_nat @ I @ Z2 @ Zs ) )
= Zs ) ) ).
% rem_nth_add_nth
thf(fact_781_rem__nth__add__nth,axiom,
! [I: nat,Zs: list_nat,Z2: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( rem_nth_nat @ I @ ( add_nth_nat @ I @ Z2 @ Zs ) )
= Zs ) ) ).
% rem_nth_add_nth
thf(fact_782_distinct__adj__Cons,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin3423869863095720157_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( ( Xs = nil_Sum_sum_a_nat )
| ( ( X
!= ( hd_Sum_sum_a_nat @ Xs ) )
& ( distin3423869863095720157_a_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_783_rem__nth_Ocases,axiom,
! [X: produc5901729443898700970_a_nat] :
( ! [Uu2: nat] :
( X
!= ( produc4502019603863884636_a_nat @ Uu2 @ nil_Sum_sum_a_nat ) )
=> ( ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc4502019603863884636_a_nat @ zero_zero_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) ) )
=> ~ ! [N3: nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc4502019603863884636_a_nat @ ( suc @ N3 ) @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) ) ) ) ) ).
% rem_nth.cases
thf(fact_784_nth__take__lemma,axiom,
! [K: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ K @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
= ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) )
=> ( ( take_Sum_sum_a_nat @ K @ Xs )
= ( take_Sum_sum_a_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_785_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_786_append__eq__append__conv__if,axiom,
! [Xs_1: list_Sum_sum_a_nat,Xs_2: list_Sum_sum_a_nat,Ys_1: list_Sum_sum_a_nat,Ys_2: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs_1 @ Xs_2 )
= ( append_Sum_sum_a_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ ( size_s5283204784079214577_a_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs_1 ) @ ( size_s5283204784079214577_a_nat @ Ys_1 ) )
=> ( ( ( take_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( size_s5283204784079214577_a_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_787_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_788_nth__equal__first__eq,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_789_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_790_lex__take__index,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( take_Sum_sum_a_nat @ I2 @ Xs )
= ( take_Sum_sum_a_nat @ I2 @ Ys ) )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) @ ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_791_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_792_nth__zip,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( nth_Pr7458973636520993902_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) @ I )
= ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( nth_Sum_sum_a_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_793_nth__zip,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr5999248435048837175at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) @ I )
= ( produc7669364194715613304at_nat @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_794_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( nth_Pr4195520319383970909_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) @ I )
= ( produc3265382261054541654_a_nat @ ( nth_nat @ Xs @ I ) @ ( nth_Sum_sum_a_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_795_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_796_listrel1__iff__update,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
= ( ? [Y3: sum_sum_a_nat,N2: nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( Ys
= ( list_u9138855634547462509_a_nat @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_797_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y3: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_798_zip__eq__Nil__iff,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
= nil_Pr6585251977456444909_a_nat )
= ( ( Xs = nil_Sum_sum_a_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_799_Nil__eq__zip__iff,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( nil_Pr6585251977456444909_a_nat
= ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_Sum_sum_a_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_800_zip__Cons__Cons,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( zip_Su7355543910597222519_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) )
= ( cons_P1525839536144884125_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_801_Cons__listrel1__Cons,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( listre5822253806113410398_a_nat @ R ) )
= ( ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_802_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us2 ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us2 @ Vs ) )
= ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Us2 ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_803_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us2: list_nat,Ys: list_Sum_sum_a_nat,Vs: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Us2 ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_nat @ Us2 @ Vs ) )
= ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Us2 ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_804_zip__append,axiom,
! [Xs: list_nat,Us2: list_Sum_sum_a_nat,Ys: list_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us2 ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us2 @ Vs ) )
= ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Us2 ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_805_zip__append,axiom,
! [Xs: list_nat,Us2: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us2 ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us2 @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us2 ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_806_Cons__in__lex,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( lex_Sum_sum_a_nat @ R ) )
= ( ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R )
& ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_807_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_808_append__listrel1I,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Us2: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Us2 @ Vs ) @ ( listre5822253806113410398_a_nat @ R ) ) ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Us2 ) @ ( append_Sum_sum_a_nat @ Ys @ Vs ) ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ).
% append_listrel1I
thf(fact_809_lex__append__leftI,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Zs ) @ ( lex_Sum_sum_a_nat @ R ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Xs @ Zs ) ) @ ( lex_Sum_sum_a_nat @ R ) ) ) ).
% lex_append_leftI
thf(fact_810_listrel1I2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,X: sum_sum_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ X @ Ys ) ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ).
% listrel1I2
thf(fact_811_lex__append__rightI,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Vs: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( lex_Sum_sum_a_nat @ R ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Vs )
= ( size_s5283204784079214577_a_nat @ Us2 ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Us2 ) @ ( append_Sum_sum_a_nat @ Ys @ Vs ) ) @ ( lex_Sum_sum_a_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_812_lex__append__rightI,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_813_proper__intrvl_Oproper__interval__Compl__set__aux_Ocases,axiom,
! [X: produc8636276049245547145_a_nat] :
( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,Uv: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Uv ) ) )
=> ~ ! [Ao: option_Sum_sum_a_nat,Uu2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Uu2 @ nil_Sum_sum_a_nat ) ) ) ) ) ).
% proper_intrvl.proper_interval_Compl_set_aux.cases
thf(fact_814_proper__intrvl_Oset__less__eq__aux__Compl_Ocases,axiom,
! [X: produc8636276049245547145_a_nat] :
( ! [Ao: option_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ Xs2 @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ) ).
% proper_intrvl.set_less_eq_aux_Compl.cases
thf(fact_815_proper__intrvl_OCompl__set__less__eq__aux_Ocases,axiom,
! [X: produc8636276049245547145_a_nat] :
( ! [Ao: option_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [Ao: option_Sum_sum_a_nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc785221303313254649_a_nat @ Ao @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ) ) ).
% proper_intrvl.Compl_set_less_eq_aux.cases
thf(fact_816_ord_Osorted__list__subset_Ocases,axiom,
! [X: produc1169793092761538656_a_nat] :
( ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) ) )
=> ( ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [Eq: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc2770702104727902672_a_nat @ Eq @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ) ).
% ord.sorted_list_subset.cases
thf(fact_817_listrel1__eq__len,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_818_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_819_not__listrel1__Nil,axiom,
! [Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) @ ( listre5822253806113410398_a_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_820_not__Nil__listrel1,axiom,
! [Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xs ) @ ( listre5822253806113410398_a_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_821_ord_Oquicksort__part_Ocases,axiom,
! [X: produc1308816576071065077_a_nat] :
( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat] :
( X
!= ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ nil_Sum_sum_a_nat ) ) ) ) ) )
=> ~ ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( X
!= ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) ) ) ) ).
% ord.quicksort_part.cases
thf(fact_822_proper__intrvl_Oproper__interval__set__Compl__aux_Ocases,axiom,
! [X: produc4708264106712766336_a_nat] :
( ! [Ao: option_Sum_sum_a_nat,N3: nat] :
( X
!= ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N3 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,N3: nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N3 @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) )
=> ( ! [Ao: option_Sum_sum_a_nat,N3: nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N3 @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ nil_Sum_sum_a_nat ) ) ) )
=> ~ ! [Ao: option_Sum_sum_a_nat,N3: nat,X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc2856885992714847610_a_nat @ Ao @ ( produc5571422593549375550_a_nat @ N3 @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% proper_intrvl.proper_interval_set_Compl_aux.cases
thf(fact_823_Nil2__notin__lex,axiom,
! [Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) @ ( lex_Sum_sum_a_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_824_Nil__notin__lex,axiom,
! [Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys ) @ ( lex_Sum_sum_a_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_825_ad__agr__list__mono,axiom,
! [X2: set_a,Y6: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X2 @ Y6 )
=> ( ( ad_agr_list_a_nat @ Y6 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X2 @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_826_zip__eq__ConsE,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xy: produc7017002724195966439_a_nat,Xys: list_P1195027771636113901_a_nat] :
( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
= ( cons_P1525839536144884125_a_nat @ Xy @ Xys ) )
=> ~ ! [X3: sum_sum_a_nat,Xs5: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ X3 @ Xs5 ) )
=> ! [Y2: sum_sum_a_nat,Ys5: list_Sum_sum_a_nat] :
( ( Ys
= ( cons_Sum_sum_a_nat @ Y2 @ Ys5 ) )
=> ( ( Xy
= ( produc1212125651291703639_a_nat @ X3 @ Y2 ) )
=> ( Xys
!= ( zip_Su7355543910597222519_a_nat @ Xs5 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_827_successively_Ocases,axiom,
! [X: produc3388037434187558270_a_nat] :
( ! [P3: sum_sum_a_nat > sum_sum_a_nat > $o] :
( X
!= ( produc8857357843056227054_a_nat @ P3 @ nil_Sum_sum_a_nat ) )
=> ( ! [P3: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat] :
( X
!= ( produc8857357843056227054_a_nat @ P3 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [P3: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc8857357843056227054_a_nat @ P3 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_828_sorted__wrt_Ocases,axiom,
! [X: produc3388037434187558270_a_nat] :
( ! [P3: sum_sum_a_nat > sum_sum_a_nat > $o] :
( X
!= ( produc8857357843056227054_a_nat @ P3 @ nil_Sum_sum_a_nat ) )
=> ~ ! [P3: sum_sum_a_nat > sum_sum_a_nat > $o,X3: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc8857357843056227054_a_nat @ P3 @ ( cons_Sum_sum_a_nat @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_829_proper__intrvl_Oexhaustive__above_Ocases,axiom,
! [X: produc4502985402200462317_a_nat] :
( ! [X3: sum_sum_a_nat] :
( X
!= ( produc6350064662657521885_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc6350064662657521885_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ).
% proper_intrvl.exhaustive_above.cases
thf(fact_830_shuffles_Ocases,axiom,
! [X: produc5001885624171833703_a_nat] :
( ! [Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
=> ( ! [Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ Xs2 @ nil_Sum_sum_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_831_splice_Ocases,axiom,
! [X: produc5001885624171833703_a_nat] :
( ! [Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ys2 ) )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_832_proper__intrvl_Oproper__interval__set__aux_Ocases,axiom,
! [X: produc5001885624171833703_a_nat] :
( ! [Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ Xs2 @ nil_Sum_sum_a_nat ) )
=> ( ! [Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) @ ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% proper_intrvl.proper_interval_set_aux.cases
thf(fact_833_ord_Oquicksort__acc_Ocases,axiom,
! [X: produc5001885624171833703_a_nat] :
( ! [Ac2: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ Ac2 @ nil_Sum_sum_a_nat ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
( X
!= ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) ) ) ).
% ord.quicksort_acc.cases
thf(fact_834_set__subset__Cons,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_835_listrel1I1,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ).
% listrel1I1
thf(fact_836_Cons__listrel1E1,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
=> ( ! [Y2: sum_sum_a_nat] :
( ( Ys
= ( cons_Sum_sum_a_nat @ Y2 @ Xs ) )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_Sum_sum_a_nat] :
( ( Ys
= ( cons_Sum_sum_a_nat @ X @ Zs2 ) )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Zs2 ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_837_Cons__listrel1E2,axiom,
! [Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( listre5822253806113410398_a_nat @ R ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ X3 @ Ys ) )
=> ~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y @ Zs2 ) )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Zs2 @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_838_in__set__impl__in__set__zip2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ~ ! [X3: sum_sum_a_nat] :
~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_839_in__set__impl__in__set__zip2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Y: nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: sum_sum_a_nat] :
~ ( member5454662894994576405at_nat @ ( produc7669364194715613304at_nat @ X3 @ Y ) @ ( set_Pr4000143458196739241at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_840_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ~ ! [X3: nat] :
~ ( member3650934779329710139_a_nat @ ( produc3265382261054541654_a_nat @ X3 @ Y ) @ ( set_Pr2196415342531872975_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_841_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys: list_nat,Y: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
=> ~ ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_842_in__set__impl__in__set__zip1,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ~ ! [Y2: sum_sum_a_nat] :
~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y2 ) @ ( set_Pr2931239450594890620_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_843_in__set__impl__in__set__zip1,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,X: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ~ ! [Y2: nat] :
~ ( member5454662894994576405at_nat @ ( produc7669364194715613304at_nat @ X @ Y2 ) @ ( set_Pr4000143458196739241at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_844_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ! [Y2: sum_sum_a_nat] :
~ ( member3650934779329710139_a_nat @ ( produc3265382261054541654_a_nat @ X @ Y2 ) @ ( set_Pr2196415342531872975_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_845_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ! [Y2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_846_lex__append__left__iff,axiom,
! [R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ X3 ) @ R )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Xs @ Zs ) ) @ ( lex_Sum_sum_a_nat @ R ) )
= ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Zs ) @ ( lex_Sum_sum_a_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_847_lex__append__leftD,axiom,
! [R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
~ ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ X3 ) @ R )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Xs @ Zs ) ) @ ( lex_Sum_sum_a_nat @ R ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Zs ) @ ( lex_Sum_sum_a_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_848_hd__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( hd_Pro3965030861105696466_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= ( produc1212125651291703639_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( hd_Sum_sum_a_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_849_listrel1I,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat,Us2: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R )
=> ( ( Xs
= ( append_Sum_sum_a_nat @ Us2 @ ( cons_Sum_sum_a_nat @ X @ Vs ) ) )
=> ( ( Ys
= ( append_Sum_sum_a_nat @ Us2 @ ( cons_Sum_sum_a_nat @ Y @ Vs ) ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_850_listrel1E,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y2 ) @ R )
=> ! [Us3: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Us3 @ ( cons_Sum_sum_a_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_Sum_sum_a_nat @ Us3 @ ( cons_Sum_sum_a_nat @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_851_last__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( last_P17124104574654342_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= ( produc1212125651291703639_a_nat @ ( last_Sum_sum_a_nat @ Xs ) @ ( last_Sum_sum_a_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_852_last__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P7114463934449255199at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= ( produc7669364194715613304at_nat @ ( last_Sum_sum_a_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_853_last__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( last_P5310735818784388933_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= ( produc3265382261054541654_a_nat @ ( last_nat @ Xs ) @ ( last_Sum_sum_a_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_854_last__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_855_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) @ ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) ) @ ( listre5822253806113410398_a_nat @ R ) )
= ( ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre5822253806113410398_a_nat @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_856_Cons__lenlex__iff,axiom,
! [M: sum_sum_a_nat,Ms: list_Sum_sum_a_nat,N: sum_sum_a_nat,Ns: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ M @ Ms ) @ ( cons_Sum_sum_a_nat @ N @ Ns ) ) @ ( lenlex_Sum_sum_a_nat @ R ) )
= ( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ms ) @ ( size_s5283204784079214577_a_nat @ Ns ) )
| ( ( ( size_s5283204784079214577_a_nat @ Ms )
= ( size_s5283204784079214577_a_nat @ Ns ) )
& ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ms @ Ns ) @ ( lenlex_Sum_sum_a_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_857_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_858_listrel__iff__nth,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre7581909074878054321_a_nat @ R ) )
= ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N2 ) @ ( nth_Sum_sum_a_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_859_listrel__iff__nth,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,R: set_Pr2129020469590976052at_nat] :
( ( member5292487352258712491st_nat @ ( produc9183987541845296142st_nat @ Xs @ Ys ) @ ( listre7681105055346673822at_nat @ R ) )
= ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( member5454662894994576405at_nat @ ( produc7669364194715613304at_nat @ ( nth_Sum_sum_a_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_860_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,R: set_Pr1021281732701644634_a_nat] :
( ( member1159737339441260081_a_nat @ ( produc5055497221108209740_a_nat @ Xs @ Ys ) @ ( listre3277123121685602172_a_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member3650934779329710139_a_nat @ ( produc3265382261054541654_a_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_Sum_sum_a_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_861_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_862_rotate1__hd__tl,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( rotate2765497868024679250_a_nat @ Xs )
= ( append_Sum_sum_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) @ ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ nil_Sum_sum_a_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_863_tl__append2,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( tl_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_864_Nil__lenlex__iff1,axiom,
! [Ns: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Ns ) @ ( lenlex_Sum_sum_a_nat @ R ) )
= ( Ns != nil_Sum_sum_a_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_865_hd__Cons__tl,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( tl_Sum_sum_a_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_866_list_Ocollapse,axiom,
! [List: list_Sum_sum_a_nat] :
( ( List != nil_Sum_sum_a_nat )
=> ( ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ List ) @ ( tl_Sum_sum_a_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_867_list_Osel_I3_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( tl_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_868_list_Osel_I2_J,axiom,
( ( tl_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% list.sel(2)
thf(fact_869_Nil__tl,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( tl_Sum_sum_a_nat @ Xs ) )
= ( ( Xs = nil_Sum_sum_a_nat )
| ? [X4: sum_sum_a_nat] :
( Xs
= ( cons_Sum_sum_a_nat @ X4 @ nil_Sum_sum_a_nat ) ) ) ) ).
% Nil_tl
thf(fact_870_tl__Nil,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( tl_Sum_sum_a_nat @ Xs )
= nil_Sum_sum_a_nat )
= ( ( Xs = nil_Sum_sum_a_nat )
| ? [X4: sum_sum_a_nat] :
( Xs
= ( cons_Sum_sum_a_nat @ X4 @ nil_Sum_sum_a_nat ) ) ) ) ).
% tl_Nil
thf(fact_871_list_Oset__sel_I2_J,axiom,
! [A: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( A != nil_Sum_sum_a_nat )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( tl_Sum_sum_a_nat @ A ) ) )
=> ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_872_tl__append__if,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
=> ( ( tl_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( tl_Sum_sum_a_nat @ Ys ) ) )
& ( ( Xs != nil_Sum_sum_a_nat )
=> ( ( tl_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_873_list_Oexpand,axiom,
! [List: list_Sum_sum_a_nat,List2: list_Sum_sum_a_nat] :
( ( ( List = nil_Sum_sum_a_nat )
= ( List2 = nil_Sum_sum_a_nat ) )
=> ( ( ( List != nil_Sum_sum_a_nat )
=> ( ( List2 != nil_Sum_sum_a_nat )
=> ( ( ( hd_Sum_sum_a_nat @ List )
= ( hd_Sum_sum_a_nat @ List2 ) )
& ( ( tl_Sum_sum_a_nat @ List )
= ( tl_Sum_sum_a_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_874_last__tl,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
| ( ( tl_Sum_sum_a_nat @ Xs )
!= nil_Sum_sum_a_nat ) )
=> ( ( last_Sum_sum_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ).
% last_tl
thf(fact_875_listrel_ONil,axiom,
! [R: set_Pr7343886759072863943_a_nat] : ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) @ ( listre7581909074878054321_a_nat @ R ) ) ).
% listrel.Nil
thf(fact_876_listrel__Nil1,axiom,
! [Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ nil_Sum_sum_a_nat @ Xs ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ( Xs = nil_Sum_sum_a_nat ) ) ).
% listrel_Nil1
thf(fact_877_listrel__Nil2,axiom,
! [Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ nil_Sum_sum_a_nat ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ( Xs = nil_Sum_sum_a_nat ) ) ).
% listrel_Nil2
thf(fact_878_listrel__eq__len,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_879_listrel__eq__len,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,R: set_Pr2129020469590976052at_nat] :
( ( member5292487352258712491st_nat @ ( produc9183987541845296142st_nat @ Xs @ Ys ) @ ( listre7681105055346673822at_nat @ R ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_880_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,R: set_Pr1021281732701644634_a_nat] :
( ( member1159737339441260081_a_nat @ ( produc5055497221108209740_a_nat @ Xs @ Ys ) @ ( listre3277123121685602172_a_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_881_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_882_list_Oexhaust__sel,axiom,
! [List: list_Sum_sum_a_nat] :
( ( List != nil_Sum_sum_a_nat )
=> ( List
= ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ List ) @ ( tl_Sum_sum_a_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_883_Nil__lenlex__iff2,axiom,
! [Ns: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ns @ nil_Sum_sum_a_nat ) @ ( lenlex_Sum_sum_a_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_884_listrel__Cons2,axiom,
! [Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ~ ! [X3: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y ) @ R )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs2 @ Ys ) @ ( listre7581909074878054321_a_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_885_listrel__Cons1,axiom,
! [Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ Y @ Ys ) @ Xs ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ~ ! [Y2: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ Y @ Y2 ) @ R )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ys @ Ys2 ) @ ( listre7581909074878054321_a_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_886_listrel_OCons,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X @ Y ) @ R )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs @ Ys ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( cons_Sum_sum_a_nat @ Y @ Ys ) ) @ ( listre7581909074878054321_a_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_887_nth__tl,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) ) )
=> ( ( nth_Sum_sum_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_888_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_889_lenlex__length,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Ms @ Ns ) @ ( lenlex_Sum_sum_a_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Ms ) @ ( size_s5283204784079214577_a_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_890_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_891_lenlex__append1,axiom,
! [Us2: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,R3: set_Pr7343886759072863943_a_nat,Vs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Us2 @ Xs ) @ ( lenlex_Sum_sum_a_nat @ R3 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Vs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ ( append_Sum_sum_a_nat @ Us2 @ Vs ) @ ( append_Sum_sum_a_nat @ Xs @ Ys ) ) @ ( lenlex_Sum_sum_a_nat @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_892_lenlex__append1,axiom,
! [Us2: list_nat,Xs: list_nat,R3: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us2 @ Xs ) @ ( lenlex_nat @ R3 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_893_listrel_Ocases,axiom,
! [A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A1 @ A22 ) @ ( listre7581909074878054321_a_nat @ R ) )
=> ( ( ( A1 = nil_Sum_sum_a_nat )
=> ( A22 != nil_Sum_sum_a_nat ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( A1
= ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_Sum_sum_a_nat] :
( ( A22
= ( cons_Sum_sum_a_nat @ Y2 @ Ys2 ) )
=> ( ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X3 @ Y2 ) @ R )
=> ~ ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs2 @ Ys2 ) @ ( listre7581909074878054321_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_894_listrel_Osimps,axiom,
! [A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat,R: set_Pr7343886759072863943_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A1 @ A22 ) @ ( listre7581909074878054321_a_nat @ R ) )
= ( ( ( A1 = nil_Sum_sum_a_nat )
& ( A22 = nil_Sum_sum_a_nat ) )
| ? [X4: sum_sum_a_nat,Y3: sum_sum_a_nat,Xs3: list_Sum_sum_a_nat,Ys3: list_Sum_sum_a_nat] :
( ( A1
= ( cons_Sum_sum_a_nat @ X4 @ Xs3 ) )
& ( A22
= ( cons_Sum_sum_a_nat @ Y3 @ Ys3 ) )
& ( member3723442691059620112_a_nat @ ( produc1212125651291703639_a_nat @ X4 @ Y3 ) @ R )
& ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ Xs3 @ Ys3 ) @ ( listre7581909074878054321_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_895_take__Suc,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( take_Sum_sum_a_nat @ ( suc @ N ) @ Xs )
= ( cons_Sum_sum_a_nat @ ( hd_Sum_sum_a_nat @ Xs ) @ ( take_Sum_sum_a_nat @ N @ ( tl_Sum_sum_a_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_896_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s5283204784079214577_a_nat
= ( ^ [Xs3: list_Sum_sum_a_nat] : ( if_nat @ ( Xs3 = nil_Sum_sum_a_nat ) @ zero_zero_nat @ ( suc @ ( size_s5283204784079214577_a_nat @ ( tl_Sum_sum_a_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_897_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs3: list_nat] : ( if_nat @ ( Xs3 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_898_Nitpick_Osize__list__simp_I1_J,axiom,
( size_l7939150027356015111_a_nat
= ( ^ [F2: sum_sum_a_nat > nat,Xs3: list_Sum_sum_a_nat] : ( if_nat @ ( Xs3 = nil_Sum_sum_a_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_Sum_sum_a_nat @ Xs3 ) ) @ ( size_l7939150027356015111_a_nat @ F2 @ ( tl_Sum_sum_a_nat @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_899_all__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,Xs: set_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs )
=> ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_900_all__tuplesI,axiom,
! [Vs: list_nat,N: nat,Xs: set_nat] :
( ( ( size_size_list_nat @ Vs )
= N )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs )
=> ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_901_all__tuplesD,axiom,
! [Vs: list_Sum_sum_a_nat,Xs: set_Sum_sum_a_nat,N: nat] :
( ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
& ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_902_all__tuplesD,axiom,
! [Vs: list_nat,Xs: set_nat,N: nat] :
( ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) )
=> ( ( ( size_size_list_nat @ Vs )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_903_size__list__append,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( size_l7939150027356015111_a_nat @ F @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_l7939150027356015111_a_nat @ F @ Xs ) @ ( size_l7939150027356015111_a_nat @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_904_list_Osize__gen_I1_J,axiom,
! [X: sum_sum_a_nat > nat] :
( ( size_l7939150027356015111_a_nat @ X @ nil_Sum_sum_a_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_905_remdups__adj__length__ge1,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s5283204784079214577_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_906_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_907_fo__nmlzd__app__Inr,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,N4: nat,AD: set_a] :
( ~ ( member_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N ) @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ~ ( member_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N4 ) @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( fo_nmlzd_a @ AD @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N ) @ nil_Sum_sum_a_nat ) ) )
=> ( ( fo_nmlzd_a @ AD @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N4 ) @ nil_Sum_sum_a_nat ) ) )
=> ( N = N4 ) ) ) ) ) ).
% fo_nmlzd_app_Inr
thf(fact_908_remdups__adj__Nil__iff,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( remdup8712921452854877243_a_nat @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_909_remdups__adj__Cons__alt,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( cons_Sum_sum_a_nat @ X @ ( tl_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) )
= ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_910_remdups__adj_Osimps_I3_J,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( X = Y )
=> ( ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) )
= ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) )
= ( cons_Sum_sum_a_nat @ X @ ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_911_remdups__adj_Osimps_I1_J,axiom,
( ( remdup8712921452854877243_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% remdups_adj.simps(1)
thf(fact_912_remdups__adj_Oelims,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( remdup8712921452854877243_a_nat @ X )
= Y )
=> ( ( ( X = nil_Sum_sum_a_nat )
=> ( Y != nil_Sum_sum_a_nat ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( X
= ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ( Y
!= ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
=> ~ ! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( X
= ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y2 )
=> ( Y
= ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y2 )
=> ( Y
= ( cons_Sum_sum_a_nat @ X3 @ ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_913_remdups__adj_Osimps_I2_J,axiom,
! [X: sum_sum_a_nat] :
( ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_914_remdups__adj__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_915_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_916_remdups__adj__append__two,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( remdup8712921452854877243_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) ) )
= ( append_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) @ ( if_lis4685338526944683083_a_nat @ ( X = Y ) @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_917_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_s5283204784079214577_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) ) )
=> ( ( nth_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) @ I )
!= ( nth_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_918_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_919_remdups__adj__append,axiom,
! [Xs_1: list_Sum_sum_a_nat,X: sum_sum_a_nat,Xs_2: list_Sum_sum_a_nat] :
( ( remdup8712921452854877243_a_nat @ ( append_Sum_sum_a_nat @ Xs_1 @ ( cons_Sum_sum_a_nat @ X @ Xs_2 ) ) )
= ( append_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ ( append_Sum_sum_a_nat @ Xs_1 @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) @ ( tl_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_920_remdups__adj__append_H,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( Xs = nil_Sum_sum_a_nat )
| ( Ys = nil_Sum_sum_a_nat )
| ( ( last_Sum_sum_a_nat @ Xs )
!= ( hd_Sum_sum_a_nat @ Ys ) ) )
=> ( ( remdup8712921452854877243_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) @ ( remdup8712921452854877243_a_nat @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_921_remdups__adj__singleton__iff,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ ( remdup8712921452854877243_a_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_Sum_sum_a_nat )
& ( Xs
= ( replic8955434655033810879_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( hd_Sum_sum_a_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_922_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_923_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_Sum_sum_a_nat @ ( replic8141442572502817605_a_nat @ I @ nil_Sum_sum_a_nat ) )
= nil_Sum_sum_a_nat ) ).
% concat_replicate_trivial
thf(fact_924_length__replicate,axiom,
! [N: nat,X: sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_925_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_926_empty__replicate,axiom,
! [N: nat,X: sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( replic8955434655033810879_a_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_927_replicate__empty,axiom,
! [N: nat,X: sum_sum_a_nat] :
( ( ( replic8955434655033810879_a_nat @ N @ X )
= nil_Sum_sum_a_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_928_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_929_replicate__app__Cons__same,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_930_append__replicate__commute,axiom,
! [N: nat,X: sum_sum_a_nat,K: nat] :
( ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) @ ( replic8955434655033810879_a_nat @ K @ X ) )
= ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ K @ X ) @ ( replic8955434655033810879_a_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_931_replicate__0,axiom,
! [X: sum_sum_a_nat] :
( ( replic8955434655033810879_a_nat @ zero_zero_nat @ X )
= nil_Sum_sum_a_nat ) ).
% replicate_0
thf(fact_932_replicate__Suc,axiom,
! [N: nat,X: sum_sum_a_nat] :
( ( replic8955434655033810879_a_nat @ ( suc @ N ) @ X )
= ( cons_Sum_sum_a_nat @ X @ ( replic8955434655033810879_a_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_933_replicate__eqI,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat,X: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= N )
=> ( ! [Y2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replic8955434655033810879_a_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_934_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_935_replicate__length__same,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replic8955434655033810879_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_936_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_937_replicate__add,axiom,
! [N: nat,M: nat,X: sum_sum_a_nat] :
( ( replic8955434655033810879_a_nat @ ( plus_plus_nat @ N @ M ) @ X )
= ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) @ ( replic8955434655033810879_a_nat @ M @ X ) ) ) ).
% replicate_add
thf(fact_938_comm__append__are__replicate,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= ( append_Sum_sum_a_nat @ Ys @ Xs ) )
=> ? [M5: nat,N3: nat,Zs2: list_Sum_sum_a_nat] :
( ( ( concat_Sum_sum_a_nat @ ( replic8141442572502817605_a_nat @ M5 @ Zs2 ) )
= Xs )
& ( ( concat_Sum_sum_a_nat @ ( replic8141442572502817605_a_nat @ N3 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_939_replicate__append__same,axiom,
! [I: nat,X: sum_sum_a_nat] :
( ( append_Sum_sum_a_nat @ ( replic8955434655033810879_a_nat @ I @ X ) @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( cons_Sum_sum_a_nat @ X @ ( replic8955434655033810879_a_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_940_remdups__adj__replicate,axiom,
! [N: nat,X: sum_sum_a_nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdup8712921452854877243_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) )
= nil_Sum_sum_a_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdup8712921452854877243_a_nat @ ( replic8955434655033810879_a_nat @ N @ X ) )
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_941_remdups__adj__singleton,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( remdup8712921452854877243_a_nat @ Xs )
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
=> ( Xs
= ( replic8955434655033810879_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_942_remdups__adj__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_943_quicksort__part_Opelims,axiom,
! [X: list_nat,Xa: nat,Xb: list_nat,Xc: list_nat,Xd: list_nat,Xe: list_nat,Y: list_nat] :
( ( ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ Xc @ Xd @ Xe )
= Y )
=> ( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ X @ ( produc5609319852033172942st_nat @ Xa @ ( produc3967909809349605295st_nat @ Xb @ ( produc4487115339913071592st_nat @ Xc @ ( produc2694037385005941721st_nat @ Xd @ Xe ) ) ) ) ) ) )
=> ( ( ( Xe = nil_nat )
=> ( ( Y
= ( set_or5558937660843164036cc_nat @ ( append_nat @ Xc @ ( cons_nat @ Xa @ ( set_or5558937660843164036cc_nat @ X @ Xd ) ) ) @ Xb ) )
=> ~ ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ X @ ( produc5609319852033172942st_nat @ Xa @ ( produc3967909809349605295st_nat @ Xb @ ( produc4487115339913071592st_nat @ Xc @ ( produc2694037385005941721st_nat @ Xd @ nil_nat ) ) ) ) ) ) ) ) )
=> ~ ! [Z: nat,Zs2: list_nat] :
( ( Xe
= ( cons_nat @ Z @ Zs2 ) )
=> ( ( ( ( ord_less_nat @ Xa @ Z )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ Xc @ ( cons_nat @ Z @ Xd ) @ Zs2 ) ) )
& ( ~ ( ord_less_nat @ Xa @ Z )
=> ( ( ( ord_less_nat @ Z @ Xa )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ ( cons_nat @ Z @ Xb ) @ Xc @ Xd @ Zs2 ) ) )
& ( ~ ( ord_less_nat @ Z @ Xa )
=> ( Y
= ( set_or1804217446461887602rt_nat @ X @ Xa @ Xb @ ( cons_nat @ Z @ Xc ) @ Xd @ Zs2 ) ) ) ) ) )
=> ~ ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ X @ ( produc5609319852033172942st_nat @ Xa @ ( produc3967909809349605295st_nat @ Xb @ ( produc4487115339913071592st_nat @ Xc @ ( produc2694037385005941721st_nat @ Xd @ ( cons_nat @ Z @ Zs2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% quicksort_part.pelims
thf(fact_944_card__set__1__iff__replicate,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_Sum_sum_a_nat )
& ? [X4: sum_sum_a_nat] :
( Xs
= ( replic8955434655033810879_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_945_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X4: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_946_card__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_length
thf(fact_947_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_948_distinct__card,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_949_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_950_card__distinct,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Xs ) ) ).
% card_distinct
thf(fact_951_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_952_length__rremdups,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rremdu8304153113908149561_a_nat @ Xs ) )
= ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_953_length__rremdups,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rremdups_nat @ Xs ) )
= ( finite_card_nat @ ( set_nat2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_954_quicksort__part_Opsimps_I2_J,axiom,
! [Ac: list_nat,X: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Z2: nat,Zs: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac @ ( produc5609319852033172942st_nat @ X @ ( produc3967909809349605295st_nat @ Lts @ ( produc4487115339913071592st_nat @ Eqs @ ( produc2694037385005941721st_nat @ Gts @ ( cons_nat @ Z2 @ Zs ) ) ) ) ) ) ) )
=> ( ( ( ord_less_nat @ X @ Z2 )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ ( cons_nat @ Z2 @ Gts ) @ Zs ) ) )
& ( ~ ( ord_less_nat @ X @ Z2 )
=> ( ( ( ord_less_nat @ Z2 @ X )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ ( cons_nat @ Z2 @ Lts ) @ Eqs @ Gts @ Zs ) ) )
& ( ~ ( ord_less_nat @ Z2 @ X )
=> ( ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_nat @ Z2 @ Zs ) )
= ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ ( cons_nat @ Z2 @ Eqs ) @ Gts @ Zs ) ) ) ) ) ) ) ).
% quicksort_part.psimps(2)
thf(fact_955_ord_Oquicksort__part_Opsimps_I1_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac @ ( produc5024437050894261379_a_nat @ X @ ( produc3915550414976331931_a_nat @ Lts @ ( produc3577755783922481849_a_nat @ Eqs @ ( produc7990843422341522135_a_nat @ Gts @ nil_Sum_sum_a_nat ) ) ) ) ) ) )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ nil_Sum_sum_a_nat )
= ( set_qu7651081299428620429_a_nat @ Less @ ( append_Sum_sum_a_nat @ Eqs @ ( cons_Sum_sum_a_nat @ X @ ( set_qu7651081299428620429_a_nat @ Less @ Ac @ Gts ) ) ) @ Lts ) ) ) ).
% ord.quicksort_part.psimps(1)
thf(fact_956_ord_Oquicksort__part_Opelims,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Xa: sum_sum_a_nat,Xb: list_Sum_sum_a_nat,Xc: list_Sum_sum_a_nat,Xd: list_Sum_sum_a_nat,Xe: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ Xc @ Xd @ Xe )
= Y )
=> ( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ X @ ( produc5024437050894261379_a_nat @ Xa @ ( produc3915550414976331931_a_nat @ Xb @ ( produc3577755783922481849_a_nat @ Xc @ ( produc7990843422341522135_a_nat @ Xd @ Xe ) ) ) ) ) ) )
=> ( ( ( Xe = nil_Sum_sum_a_nat )
=> ( ( Y
= ( set_qu7651081299428620429_a_nat @ Less @ ( append_Sum_sum_a_nat @ Xc @ ( cons_Sum_sum_a_nat @ Xa @ ( set_qu7651081299428620429_a_nat @ Less @ X @ Xd ) ) ) @ Xb ) )
=> ~ ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ X @ ( produc5024437050894261379_a_nat @ Xa @ ( produc3915550414976331931_a_nat @ Xb @ ( produc3577755783922481849_a_nat @ Xc @ ( produc7990843422341522135_a_nat @ Xd @ nil_Sum_sum_a_nat ) ) ) ) ) ) ) ) )
=> ~ ! [Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( Xe
= ( cons_Sum_sum_a_nat @ Z @ Zs2 ) )
=> ( ( ( ( Less @ Xa @ Z )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ Xc @ ( cons_Sum_sum_a_nat @ Z @ Xd ) @ Zs2 ) ) )
& ( ~ ( Less @ Xa @ Z )
=> ( ( ( Less @ Z @ Xa )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ ( cons_Sum_sum_a_nat @ Z @ Xb ) @ Xc @ Xd @ Zs2 ) ) )
& ( ~ ( Less @ Z @ Xa )
=> ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ Xa @ Xb @ ( cons_Sum_sum_a_nat @ Z @ Xc ) @ Xd @ Zs2 ) ) ) ) ) )
=> ~ ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ X @ ( produc5024437050894261379_a_nat @ Xa @ ( produc3915550414976331931_a_nat @ Xb @ ( produc3577755783922481849_a_nat @ Xc @ ( produc7990843422341522135_a_nat @ Xd @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ord.quicksort_part.pelims
thf(fact_957_ord_Oquicksort__part_Opsimps_I2_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat,Z2: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac @ ( produc5024437050894261379_a_nat @ X @ ( produc3915550414976331931_a_nat @ Lts @ ( produc3577755783922481849_a_nat @ Eqs @ ( produc7990843422341522135_a_nat @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) ) ) ) ) ) ) )
=> ( ( ( Less @ X @ Z2 )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ ( cons_Sum_sum_a_nat @ Z2 @ Gts ) @ Zs ) ) )
& ( ~ ( Less @ X @ Z2 )
=> ( ( ( Less @ Z2 @ X )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ ( cons_Sum_sum_a_nat @ Z2 @ Lts ) @ Eqs @ Gts @ Zs ) ) )
& ( ~ ( Less @ Z2 @ X )
=> ( ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ ( cons_Sum_sum_a_nat @ Z2 @ Zs ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ ( cons_Sum_sum_a_nat @ Z2 @ Eqs ) @ Gts @ Zs ) ) ) ) ) ) ) ).
% ord.quicksort_part.psimps(2)
thf(fact_958_quicksort__acc__quicksort__part_Opinduct_I1_J,axiom,
! [A0: list_nat,A1: list_nat,P: list_nat > list_nat > $o,Q3: list_nat > nat > list_nat > list_nat > list_nat > list_nat > $o] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ A0 @ A1 ) ) )
=> ( ! [Ac2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ nil_nat ) ) )
=> ( P @ Ac2 @ nil_nat ) )
=> ( ! [Ac2: list_nat,X3: nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Ac2 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ( ! [Ac2: list_nat,X3: nat,V2: nat,Va: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va ) ) ) ) )
=> ( ( Q3 @ Ac2 @ X3 @ nil_nat @ nil_nat @ nil_nat @ ( cons_nat @ V2 @ Va ) )
=> ( P @ Ac2 @ ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va ) ) ) ) )
=> ( ! [Ac2: list_nat,X3: nat,Lts2: list_nat,Eqs2: list_nat,Gts2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac2 @ ( produc5609319852033172942st_nat @ X3 @ ( produc3967909809349605295st_nat @ Lts2 @ ( produc4487115339913071592st_nat @ Eqs2 @ ( produc2694037385005941721st_nat @ Gts2 @ nil_nat ) ) ) ) ) ) )
=> ( ( P @ Ac2 @ Gts2 )
=> ( ( P @ ( append_nat @ Eqs2 @ ( cons_nat @ X3 @ ( set_or5558937660843164036cc_nat @ Ac2 @ Gts2 ) ) ) @ Lts2 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ nil_nat ) ) ) )
=> ( ! [Ac2: list_nat,X3: nat,Lts2: list_nat,Eqs2: list_nat,Gts2: list_nat,Z: nat,Zs2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac2 @ ( produc5609319852033172942st_nat @ X3 @ ( produc3967909809349605295st_nat @ Lts2 @ ( produc4487115339913071592st_nat @ Eqs2 @ ( produc2694037385005941721st_nat @ Gts2 @ ( cons_nat @ Z @ Zs2 ) ) ) ) ) ) ) )
=> ( ( ( ord_less_nat @ X3 @ Z )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ ( cons_nat @ Z @ Gts2 ) @ Zs2 ) )
=> ( ( ~ ( ord_less_nat @ X3 @ Z )
=> ( ( ord_less_nat @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ ( cons_nat @ Z @ Lts2 ) @ Eqs2 @ Gts2 @ Zs2 ) ) )
=> ( ( ~ ( ord_less_nat @ X3 @ Z )
=> ( ~ ( ord_less_nat @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ ( cons_nat @ Z @ Eqs2 ) @ Gts2 @ Zs2 ) ) )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ ( cons_nat @ Z @ Zs2 ) ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% quicksort_acc_quicksort_part.pinduct(1)
thf(fact_959_quicksort__acc__quicksort__part_Opinduct_I2_J,axiom,
! [A22: list_nat,A32: nat,A42: list_nat,A5: list_nat,A6: list_nat,A7: list_nat,P: list_nat > list_nat > $o,Q3: list_nat > nat > list_nat > list_nat > list_nat > list_nat > $o] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ A22 @ ( produc5609319852033172942st_nat @ A32 @ ( produc3967909809349605295st_nat @ A42 @ ( produc4487115339913071592st_nat @ A5 @ ( produc2694037385005941721st_nat @ A6 @ A7 ) ) ) ) ) ) )
=> ( ! [Ac2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ nil_nat ) ) )
=> ( P @ Ac2 @ nil_nat ) )
=> ( ! [Ac2: list_nat,X3: nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Ac2 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ( ! [Ac2: list_nat,X3: nat,V2: nat,Va: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac2 @ ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va ) ) ) ) )
=> ( ( Q3 @ Ac2 @ X3 @ nil_nat @ nil_nat @ nil_nat @ ( cons_nat @ V2 @ Va ) )
=> ( P @ Ac2 @ ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va ) ) ) ) )
=> ( ! [Ac2: list_nat,X3: nat,Lts2: list_nat,Eqs2: list_nat,Gts2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac2 @ ( produc5609319852033172942st_nat @ X3 @ ( produc3967909809349605295st_nat @ Lts2 @ ( produc4487115339913071592st_nat @ Eqs2 @ ( produc2694037385005941721st_nat @ Gts2 @ nil_nat ) ) ) ) ) ) )
=> ( ( P @ Ac2 @ Gts2 )
=> ( ( P @ ( append_nat @ Eqs2 @ ( cons_nat @ X3 @ ( set_or5558937660843164036cc_nat @ Ac2 @ Gts2 ) ) ) @ Lts2 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ nil_nat ) ) ) )
=> ( ! [Ac2: list_nat,X3: nat,Lts2: list_nat,Eqs2: list_nat,Gts2: list_nat,Z: nat,Zs2: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac2 @ ( produc5609319852033172942st_nat @ X3 @ ( produc3967909809349605295st_nat @ Lts2 @ ( produc4487115339913071592st_nat @ Eqs2 @ ( produc2694037385005941721st_nat @ Gts2 @ ( cons_nat @ Z @ Zs2 ) ) ) ) ) ) ) )
=> ( ( ( ord_less_nat @ X3 @ Z )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ ( cons_nat @ Z @ Gts2 ) @ Zs2 ) )
=> ( ( ~ ( ord_less_nat @ X3 @ Z )
=> ( ( ord_less_nat @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ ( cons_nat @ Z @ Lts2 ) @ Eqs2 @ Gts2 @ Zs2 ) ) )
=> ( ( ~ ( ord_less_nat @ X3 @ Z )
=> ( ~ ( ord_less_nat @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ ( cons_nat @ Z @ Eqs2 ) @ Gts2 @ Zs2 ) ) )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ ( cons_nat @ Z @ Zs2 ) ) ) ) ) )
=> ( Q3 @ A22 @ A32 @ A42 @ A5 @ A6 @ A7 ) ) ) ) ) ) ) ).
% quicksort_acc_quicksort_part.pinduct(2)
thf(fact_960_ad__agr__list__set,axiom,
! [X2: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: a] :
( ( ad_agr_list_a_nat @ X2 @ Xs @ Ys )
=> ( ( member_a @ Y @ X2 )
=> ( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% ad_agr_list_set
thf(fact_961_ord_Oquicksort__acc_Opsimps_I1_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac @ nil_Sum_sum_a_nat ) ) )
=> ( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ nil_Sum_sum_a_nat )
= Ac ) ) ).
% ord.quicksort_acc.psimps(1)
thf(fact_962_ord_Oquicksort__acc_Opsimps_I2_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) )
=> ( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( cons_Sum_sum_a_nat @ X @ Ac ) ) ) ).
% ord.quicksort_acc.psimps(2)
thf(fact_963_length__quicksort__accp,axiom,
! [Ac: list_nat,Xs: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac @ Xs ) ) )
=> ( ( size_size_list_nat @ ( set_or5558937660843164036cc_nat @ Ac @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Ac ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% length_quicksort_accp
thf(fact_964_ord_Olength__quicksort__accp,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac @ Xs ) ) )
=> ( ( size_s5283204784079214577_a_nat @ ( set_qu7651081299428620429_a_nat @ Less @ Ac @ Xs ) )
= ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Ac ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ).
% ord.length_quicksort_accp
thf(fact_965_ord_Olength__quicksort__accp,axiom,
! [Less: nat > nat > $o,Ac: list_nat,Xs: list_nat] :
( ( accp_S1710931637807102040st_nat @ ( set_qu876879003691614877el_nat @ Less ) @ ( sum_In4817299372063595370st_nat @ ( produc2694037385005941721st_nat @ Ac @ Xs ) ) )
=> ( ( size_size_list_nat @ ( set_qu8921977862092465858cc_nat @ Less @ Ac @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Ac ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% ord.length_quicksort_accp
thf(fact_966_ord_Oquicksort__acc_Opelims,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( set_qu7651081299428620429_a_nat @ Less @ X @ Xa )
= Y )
=> ( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ X @ Xa ) ) )
=> ( ( ( Xa = nil_Sum_sum_a_nat )
=> ( ( Y = X )
=> ~ ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ) )
=> ( ! [X3: sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) )
=> ( ( Y
= ( cons_Sum_sum_a_nat @ X3 @ X ) )
=> ~ ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ X @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) ) ) )
=> ~ ! [X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
( ( Xa
= ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) )
=> ( ( Y
= ( set_qu7459554806609531931_a_nat @ Less @ X @ X3 @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) )
=> ~ ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ X @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% ord.quicksort_acc.pelims
thf(fact_967_ord_Oquicksort__acc_Opsimps_I3_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,V: sum_sum_a_nat,Va2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ V @ Va2 ) ) ) ) )
=> ( ( set_qu7651081299428620429_a_nat @ Less @ Ac @ ( cons_Sum_sum_a_nat @ X @ ( cons_Sum_sum_a_nat @ V @ Va2 ) ) )
= ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V @ Va2 ) ) ) ) ).
% ord.quicksort_acc.psimps(3)
thf(fact_968_ord_Oquicksort__acc__quicksort__part_Opinduct_I1_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,A0: list_Sum_sum_a_nat,A1: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q3: list_Sum_sum_a_nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ A0 @ A1 ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ nil_Sum_sum_a_nat ) ) )
=> ( P @ Ac2 @ nil_Sum_sum_a_nat ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) )
=> ( P @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) )
=> ( ( Q3 @ Ac2 @ X3 @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V2 @ Va ) )
=> ( P @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ nil_Sum_sum_a_nat ) ) ) ) ) ) )
=> ( ( P @ Ac2 @ Gts2 )
=> ( ( P @ ( append_Sum_sum_a_nat @ Eqs2 @ ( cons_Sum_sum_a_nat @ X3 @ ( set_qu7651081299428620429_a_nat @ Less @ Ac2 @ Gts2 ) ) ) @ Lts2 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ nil_Sum_sum_a_nat ) ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) ) ) )
=> ( ( ( Less @ X3 @ Z )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ ( cons_Sum_sum_a_nat @ Z @ Gts2 ) @ Zs2 ) )
=> ( ( ~ ( Less @ X3 @ Z )
=> ( ( Less @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ ( cons_Sum_sum_a_nat @ Z @ Lts2 ) @ Eqs2 @ Gts2 @ Zs2 ) ) )
=> ( ( ~ ( Less @ X3 @ Z )
=> ( ~ ( Less @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ ( cons_Sum_sum_a_nat @ Z @ Eqs2 ) @ Gts2 @ Zs2 ) ) )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% ord.quicksort_acc_quicksort_part.pinduct(1)
thf(fact_969_ord_Oquicksort__acc__quicksort__part_Opinduct_I2_J,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,A22: list_Sum_sum_a_nat,A32: sum_sum_a_nat,A42: list_Sum_sum_a_nat,A5: list_Sum_sum_a_nat,A6: list_Sum_sum_a_nat,A7: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q3: list_Sum_sum_a_nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ A22 @ ( produc5024437050894261379_a_nat @ A32 @ ( produc3915550414976331931_a_nat @ A42 @ ( produc3577755783922481849_a_nat @ A5 @ ( produc7990843422341522135_a_nat @ A6 @ A7 ) ) ) ) ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ nil_Sum_sum_a_nat ) ) )
=> ( P @ Ac2 @ nil_Sum_sum_a_nat ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) ) )
=> ( P @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ nil_Sum_sum_a_nat ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,V2: sum_sum_a_nat,Va: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In8892132817167105852_a_nat @ ( produc7990843422341522135_a_nat @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) )
=> ( ( Q3 @ Ac2 @ X3 @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ nil_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ V2 @ Va ) )
=> ( P @ Ac2 @ ( cons_Sum_sum_a_nat @ X3 @ ( cons_Sum_sum_a_nat @ V2 @ Va ) ) ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ nil_Sum_sum_a_nat ) ) ) ) ) ) )
=> ( ( P @ Ac2 @ Gts2 )
=> ( ( P @ ( append_Sum_sum_a_nat @ Eqs2 @ ( cons_Sum_sum_a_nat @ X3 @ ( set_qu7651081299428620429_a_nat @ Less @ Ac2 @ Gts2 ) ) ) @ Lts2 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ nil_Sum_sum_a_nat ) ) ) )
=> ( ! [Ac2: list_Sum_sum_a_nat,X3: sum_sum_a_nat,Lts2: list_Sum_sum_a_nat,Eqs2: list_Sum_sum_a_nat,Gts2: list_Sum_sum_a_nat,Z: sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac2 @ ( produc5024437050894261379_a_nat @ X3 @ ( produc3915550414976331931_a_nat @ Lts2 @ ( produc3577755783922481849_a_nat @ Eqs2 @ ( produc7990843422341522135_a_nat @ Gts2 @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) ) ) )
=> ( ( ( Less @ X3 @ Z )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ ( cons_Sum_sum_a_nat @ Z @ Gts2 ) @ Zs2 ) )
=> ( ( ~ ( Less @ X3 @ Z )
=> ( ( Less @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ ( cons_Sum_sum_a_nat @ Z @ Lts2 ) @ Eqs2 @ Gts2 @ Zs2 ) ) )
=> ( ( ~ ( Less @ X3 @ Z )
=> ( ~ ( Less @ Z @ X3 )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ ( cons_Sum_sum_a_nat @ Z @ Eqs2 ) @ Gts2 @ Zs2 ) ) )
=> ( Q3 @ Ac2 @ X3 @ Lts2 @ Eqs2 @ Gts2 @ ( cons_Sum_sum_a_nat @ Z @ Zs2 ) ) ) ) ) )
=> ( Q3 @ A22 @ A32 @ A42 @ A5 @ A6 @ A7 ) ) ) ) ) ) ) ).
% ord.quicksort_acc_quicksort_part.pinduct(2)
thf(fact_970_length__quicksort__partp,axiom,
! [Ac: list_nat,X: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( accp_S1710931637807102040st_nat @ set_or192534051960689247el_nat @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac @ ( produc5609319852033172942st_nat @ X @ ( produc3967909809349605295st_nat @ Lts @ ( produc4487115339913071592st_nat @ Eqs @ ( produc2694037385005941721st_nat @ Gts @ Zs ) ) ) ) ) ) )
=> ( ( size_size_list_nat @ ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_list_nat @ Ac ) @ one_one_nat ) @ ( size_size_list_nat @ Lts ) ) @ ( size_size_list_nat @ Eqs ) ) @ ( size_size_list_nat @ Gts ) ) @ ( size_size_list_nat @ Zs ) ) ) ) ).
% length_quicksort_partp
thf(fact_971_ord_Olength__quicksort__partp,axiom,
! [Less: sum_sum_a_nat > sum_sum_a_nat > $o,Ac: list_Sum_sum_a_nat,X: sum_sum_a_nat,Lts: list_Sum_sum_a_nat,Eqs: list_Sum_sum_a_nat,Gts: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( accp_S9045782574878745642_a_nat @ ( set_qu2333772366303858162_a_nat @ Less ) @ ( sum_In1820214634257660838_a_nat @ ( produc2222311365791241189_a_nat @ Ac @ ( produc5024437050894261379_a_nat @ X @ ( produc3915550414976331931_a_nat @ Lts @ ( produc3577755783922481849_a_nat @ Eqs @ ( produc7990843422341522135_a_nat @ Gts @ Zs ) ) ) ) ) ) )
=> ( ( size_s5283204784079214577_a_nat @ ( set_qu7459554806609531931_a_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Ac ) @ one_one_nat ) @ ( size_s5283204784079214577_a_nat @ Lts ) ) @ ( size_s5283204784079214577_a_nat @ Eqs ) ) @ ( size_s5283204784079214577_a_nat @ Gts ) ) @ ( size_s5283204784079214577_a_nat @ Zs ) ) ) ) ).
% ord.length_quicksort_partp
thf(fact_972_ord_Olength__quicksort__partp,axiom,
! [Less: nat > nat > $o,Ac: list_nat,X: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( accp_S1710931637807102040st_nat @ ( set_qu876879003691614877el_nat @ Less ) @ ( sum_In4791677611794227960st_nat @ ( produc92214677965515925st_nat @ Ac @ ( produc5609319852033172942st_nat @ X @ ( produc3967909809349605295st_nat @ Lts @ ( produc4487115339913071592st_nat @ Eqs @ ( produc2694037385005941721st_nat @ Gts @ Zs ) ) ) ) ) ) )
=> ( ( size_size_list_nat @ ( set_qu42629670029557428rt_nat @ Less @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_list_nat @ Ac ) @ one_one_nat ) @ ( size_size_list_nat @ Lts ) ) @ ( size_size_list_nat @ Eqs ) ) @ ( size_size_list_nat @ Gts ) ) @ ( size_size_list_nat @ Zs ) ) ) ) ).
% ord.length_quicksort_partp
thf(fact_973_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_974_rotate__length01,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_Sum_sum_a_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_975_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_976_rotate1__length01,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat )
=> ( ( rotate2765497868024679250_a_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_977_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_978_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_979_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_980_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_981_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_982_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_983_fo__nmlz__rec_Ocases,axiom,
! [X: produc7400749417448455946_a_nat] :
( ! [I2: nat,M5: sum_sum_a_nat > option_nat,AD2: set_a] :
( X
!= ( produc9161022033385942332_a_nat @ I2 @ ( produc2047461453323441237_a_nat @ M5 @ ( produc5501371206643777598_a_nat @ AD2 @ nil_Sum_sum_a_nat ) ) ) )
=> ( ! [I2: nat,M5: sum_sum_a_nat > option_nat,AD2: set_a,X3: a,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc9161022033385942332_a_nat @ I2 @ ( produc2047461453323441237_a_nat @ M5 @ ( produc5501371206643777598_a_nat @ AD2 @ ( cons_Sum_sum_a_nat @ ( sum_Inl_a_nat @ X3 ) @ Xs2 ) ) ) ) )
=> ~ ! [I2: nat,M5: sum_sum_a_nat > option_nat,AD2: set_a,N3: nat,Xs2: list_Sum_sum_a_nat] :
( X
!= ( produc9161022033385942332_a_nat @ I2 @ ( produc2047461453323441237_a_nat @ M5 @ ( produc5501371206643777598_a_nat @ AD2 @ ( cons_Sum_sum_a_nat @ ( sum_Inr_nat_a @ N3 ) @ Xs2 ) ) ) ) ) ) ) ).
% fo_nmlz_rec.cases
thf(fact_984_rotate1__fixpoint__card,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( rotate2765497868024679250_a_nat @ Xs )
= Xs )
=> ( ( Xs = nil_Sum_sum_a_nat )
| ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= one_one_nat ) ) ) ).
% rotate1_fixpoint_card
thf(fact_985_comm__append__is__replicate,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( Ys != nil_Sum_sum_a_nat )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Ys )
= ( append_Sum_sum_a_nat @ Ys @ Xs ) )
=> ? [N3: nat,Zs2: list_Sum_sum_a_nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
& ( ( concat_Sum_sum_a_nat @ ( replic8141442572502817605_a_nat @ N3 @ Zs2 ) )
= ( append_Sum_sum_a_nat @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_986_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_987_nth__Cons__pos,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_988_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_989_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_990_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_991_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_992_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_993_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_994_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_995_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_996_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_997_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_998_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_999_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1000_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1001_length__drop,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1002_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1003_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1004_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1005_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1006_take__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ N @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) @ ( take_Sum_sum_a_nat @ ( minus_minus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1007_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1008_drop__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( drop_Sum_sum_a_nat @ N @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ N @ Xs ) @ ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1009_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1010_length__tl,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( tl_Sum_sum_a_nat @ Xs ) )
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1011_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1012_length__butlast,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) )
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1013_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1014_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1015_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1016_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1017_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1018_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1019_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1020_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1021_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1022_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1023_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1024_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1025_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1026_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1027_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1028_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1029_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1030_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1031_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1032_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1033_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1034_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1035_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1036_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1037_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1038_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1039_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1040_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1041_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1042_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1043_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1044_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1045_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1046_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1047_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1048_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1049_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_1050_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1051_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1052_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1053_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1054_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1055_nth__Cons_H,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1056_nth__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N )
= ( nth_Sum_sum_a_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1057_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1058_drop__Cons_H,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_Sum_sum_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_Sum_sum_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1059_list__update__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N @ X )
= ( append_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( list_u9138855634547462509_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N @ X )
= ( append_Sum_sum_a_nat @ Xs @ ( list_u9138855634547462509_a_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1060_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1061_butlast__conv__take,axiom,
( butlas5768530507476509265_a_nat
= ( ^ [Xs3: list_Sum_sum_a_nat] : ( take_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_1062_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs3: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_1063_rem__nth__length,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( rem_nt658808235856662061_a_nat @ I @ Xs ) )
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ) ).
% rem_nth_length
thf(fact_1064_rem__nth__length,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( rem_nth_nat @ I @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% rem_nth_length
thf(fact_1065_butlast__list__update,axiom,
! [K: nat,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( K
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlas5768530507476509265_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
= ( butlas5768530507476509265_a_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlas5768530507476509265_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
= ( list_u9138855634547462509_a_nat @ ( butlas5768530507476509265_a_nat @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1066_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1067_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1068_nth__non__equal__first__eq,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1069_take__Cons_H,axiom,
! [N: nat,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_Sum_sum_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= nil_Sum_sum_a_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_Sum_sum_a_nat @ N @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( take_Sum_sum_a_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1070_Cons__replicate__eq,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat,Y: sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X @ Xs )
= ( replic8955434655033810879_a_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replic8955434655033810879_a_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1071_last__conv__nth,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( last_Sum_sum_a_nat @ Xs )
= ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1072_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1073_last__list__update,axiom,
! [Xs: list_Sum_sum_a_nat,K: nat,X: sum_sum_a_nat] :
( ( Xs != nil_Sum_sum_a_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ X ) )
= ( last_Sum_sum_a_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1074_last__list__update,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1075_butlast__take,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( butlas5768530507476509265_a_nat @ ( take_Sum_sum_a_nat @ N @ Xs ) )
= ( take_Sum_sum_a_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1076_butlast__take,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1077_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1078_drop__Cons__numeral,axiom,
! [V: num,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( drop_Sum_sum_a_nat @ ( numeral_numeral_nat @ V ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1079_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1080_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1081_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1082_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1083_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1084_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1085_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1086_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1087_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1088_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1089_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1090_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1091_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1092_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1093_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1094_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1095_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1096_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1097_nth__Cons__numeral,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,V: num] :
( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1098_take__Cons__numeral,axiom,
! [V: num,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ ( numeral_numeral_nat @ V ) @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( cons_Sum_sum_a_nat @ X @ ( take_Sum_sum_a_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1099_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1100_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1101_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1102_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1103_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1104_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1105_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1106_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1107_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1108_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1109_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1110_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1111_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1112_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1113_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1114_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1115_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1116_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1117_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1118_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1119_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1120_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1121_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1122_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1123_length__product,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s7247017532682008665_a_nat @ ( produc7134955936706270469_a_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_product
thf(fact_1124_length__product,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( size_s5183913381435988258at_nat @ ( produc2474788113070794826at_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_1125_length__product,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( produc7294178216264498984_a_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_product
thf(fact_1126_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_1127_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1128_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1129_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1130_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1131_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1132_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1133_product__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
=> ( ( nth_Pr7458973636520993902_a_nat @ ( produc7134955936706270469_a_nat @ Xs @ Ys ) @ N )
= ( produc1212125651291703639_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) @ ( nth_Sum_sum_a_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1134_product__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5999248435048837175at_nat @ ( produc2474788113070794826at_nat @ Xs @ Ys ) @ N )
= ( produc7669364194715613304at_nat @ ( nth_Sum_sum_a_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1135_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
=> ( ( nth_Pr4195520319383970909_a_nat @ ( produc7294178216264498984_a_nat @ Xs @ Ys ) @ N )
= ( produc3265382261054541654_a_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) @ ( nth_Sum_sum_a_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1136_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1137_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1138_subset__code_I3_J,axiom,
~ ( ord_le1325389633284124927_a_nat @ ( coset_Sum_sum_a_nat @ nil_Sum_sum_a_nat ) @ ( set_Sum_sum_a_nat2 @ nil_Sum_sum_a_nat ) ) ).
% subset_code(3)
thf(fact_1139_extract__Some__iff,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( extrac3289232528736335879_a_nat @ P @ Xs )
= ( some_P7299253002876453812_a_nat @ ( produc7578773175992006207_a_nat @ Ys @ ( produc6350064662657521885_a_nat @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Ys ) )
& ( P @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_1140_extract__SomeE,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( extrac3289232528736335879_a_nat @ P @ Xs )
= ( some_P7299253002876453812_a_nat @ ( produc7578773175992006207_a_nat @ Ys @ ( produc6350064662657521885_a_nat @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append_Sum_sum_a_nat @ Ys @ ( cons_Sum_sum_a_nat @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X6: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X6 @ ( set_Sum_sum_a_nat2 @ Ys ) )
& ( P @ X6 ) ) ) ) ).
% extract_SomeE
thf(fact_1141_find__Some__iff2,axiom,
! [X: sum_sum_a_nat,P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ( ( some_Sum_sum_a_nat @ X )
= ( find_Sum_sum_a_nat @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( P @ ( nth_Sum_sum_a_nat @ Xs @ I3 ) )
& ( X
= ( nth_Sum_sum_a_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_Sum_sum_a_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1142_find__Some__iff2,axiom,
! [X: nat,P: nat > $o,Xs: list_nat] :
( ( ( some_nat @ X )
= ( find_nat @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I3 ) )
& ( X
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_1143_find_Osimps_I2_J,axiom,
! [P: sum_sum_a_nat > $o,X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( P @ X )
=> ( ( find_Sum_sum_a_nat @ P @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( some_Sum_sum_a_nat @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_Sum_sum_a_nat @ P @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( find_Sum_sum_a_nat @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1144_find__Some__iff,axiom,
! [P: sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( find_Sum_sum_a_nat @ P @ Xs )
= ( some_Sum_sum_a_nat @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( P @ ( nth_Sum_sum_a_nat @ Xs @ I3 ) )
& ( X
= ( nth_Sum_sum_a_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_Sum_sum_a_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1145_find__Some__iff,axiom,
! [P: nat > $o,Xs: list_nat,X: nat] :
( ( ( find_nat @ P @ Xs )
= ( some_nat @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I3 ) )
& ( X
= ( nth_nat @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_1146_pos__sound,axiom,
! [A: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,I: nat] :
( ( ( pos_Sum_sum_a_nat @ A @ Xs )
= ( some_nat @ I ) )
=> ( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( ( nth_Sum_sum_a_nat @ Xs @ I )
= A ) ) ) ).
% pos_sound
thf(fact_1147_pos__sound,axiom,
! [A: nat,Xs: list_nat,I: nat] :
( ( ( pos_nat @ A @ Xs )
= ( some_nat @ I ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= A ) ) ) ).
% pos_sound
thf(fact_1148_pos__length,axiom,
! [A: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,I: nat] :
( ( ( pos_Sum_sum_a_nat @ A @ Xs )
= ( some_nat @ I ) )
=> ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% pos_length
thf(fact_1149_pos__length,axiom,
! [A: nat,Xs: list_nat,I: nat] :
( ( ( pos_nat @ A @ Xs )
= ( some_nat @ I ) )
=> ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) ) ) ).
% pos_length
thf(fact_1150_rev__update,axiom,
! [K: nat,Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ord_less_nat @ K @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( rev_Sum_sum_a_nat @ ( list_u9138855634547462509_a_nat @ Xs @ K @ Y ) )
= ( list_u9138855634547462509_a_nat @ ( rev_Sum_sum_a_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1151_rev__update,axiom,
! [K: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1152_rev__is__Nil__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( rev_Sum_sum_a_nat @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% rev_is_Nil_conv
thf(fact_1153_Nil__is__rev__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( rev_Sum_sum_a_nat @ Xs ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% Nil_is_rev_conv
thf(fact_1154_length__rev,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rev_Sum_sum_a_nat @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_rev
thf(fact_1155_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_1156_rev__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( rev_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( rev_Sum_sum_a_nat @ Ys ) @ ( rev_Sum_sum_a_nat @ Xs ) ) ) ).
% rev_append
thf(fact_1157_length__concat__rev,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( concat_Sum_sum_a_nat @ ( rev_li2372797785747802603_a_nat @ Xs ) ) )
= ( size_s5283204784079214577_a_nat @ ( concat_Sum_sum_a_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_1158_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_1159_singleton__rev__conv,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat )
= ( rev_Sum_sum_a_nat @ Xs ) )
= ( ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_1160_rev__singleton__conv,axiom,
! [Xs: list_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( rev_Sum_sum_a_nat @ Xs )
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) )
= ( Xs
= ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ).
% rev_singleton_conv
thf(fact_1161_rev__eq__Cons__iff,axiom,
! [Xs: list_Sum_sum_a_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( rev_Sum_sum_a_nat @ Xs )
= ( cons_Sum_sum_a_nat @ Y @ Ys ) )
= ( Xs
= ( append_Sum_sum_a_nat @ ( rev_Sum_sum_a_nat @ Ys ) @ ( cons_Sum_sum_a_nat @ Y @ nil_Sum_sum_a_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_1162_rev_Osimps_I2_J,axiom,
! [X: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( rev_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X @ Xs ) )
= ( append_Sum_sum_a_nat @ ( rev_Sum_sum_a_nat @ Xs ) @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ) ).
% rev.simps(2)
thf(fact_1163_rev_Osimps_I1_J,axiom,
( ( rev_Sum_sum_a_nat @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% rev.simps(1)
thf(fact_1164_zip__rev,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( rev_Sum_sum_a_nat @ Xs ) @ ( rev_Sum_sum_a_nat @ Ys ) )
= ( rev_Pr4274687663743590477_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1165_zip__rev,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( rev_Sum_sum_a_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr2990306387364246552at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1166_zip__rev,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( rev_nat @ Xs ) @ ( rev_Sum_sum_a_nat @ Ys ) )
= ( rev_Pr1186578271699380286_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1167_zip__rev,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1168_drop__rev,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( drop_Sum_sum_a_nat @ N @ ( rev_Sum_sum_a_nat @ Xs ) )
= ( rev_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1169_drop__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1170_rev__drop,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( rev_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ I @ Xs ) )
= ( take_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ I ) @ ( rev_Sum_sum_a_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_1171_rev__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( drop_nat @ I @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_1172_rev__take,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( rev_Sum_sum_a_nat @ ( take_Sum_sum_a_nat @ I @ Xs ) )
= ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ I ) @ ( rev_Sum_sum_a_nat @ Xs ) ) ) ).
% rev_take
thf(fact_1173_rev__take,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( take_nat @ I @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_take
thf(fact_1174_take__rev,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( take_Sum_sum_a_nat @ N @ ( rev_Sum_sum_a_nat @ Xs ) )
= ( rev_Sum_sum_a_nat @ ( drop_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_1175_take__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_1176_rotate__rev,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( rotate_Sum_sum_a_nat @ N @ ( rev_Sum_sum_a_nat @ Xs ) )
= ( rev_Sum_sum_a_nat @ ( rotate_Sum_sum_a_nat @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1177_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1178_rev__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( rev_Sum_sum_a_nat @ Xs ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1179_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1180_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1181_set__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( set_nat2 @ ( set_or6599480164596245535ed_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ) ).
% set_remdups_sorted
thf(fact_1182_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1183_sorted__wrt01,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat )
=> ( sorted6245805940552704876_a_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1184_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1185_sorted__wrt__nth__less,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,I: nat,J: nat] :
( ( sorted6245805940552704876_a_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( nth_Sum_sum_a_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1186_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1187_sorted__wrt__iff__nth__less,axiom,
( sorted6245805940552704876_a_nat
= ( ^ [P4: sum_sum_a_nat > sum_sum_a_nat > $o,Xs3: list_Sum_sum_a_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
=> ( P4 @ ( nth_Sum_sum_a_nat @ Xs3 @ I3 ) @ ( nth_Sum_sum_a_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1188_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P4: nat > nat > $o,Xs3: list_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P4 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1189_distinct__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( distinct_nat @ ( set_or6599480164596245535ed_nat @ Xs ) ) ) ).
% distinct_remdups_sorted
thf(fact_1190_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1191_sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_1192_strict__sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1193_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1194_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 )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).
% sorted_append
thf(fact_1195_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1196_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q3: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y2 )
=> ( Q3 @ X3 @ Y2 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q3 @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1197_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_1198_sorted__wrt__append,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( sorted6245805940552704876_a_nat @ P @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( sorted6245805940552704876_a_nat @ P @ Xs )
& ( sorted6245805940552704876_a_nat @ P @ Ys )
& ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ! [Y3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( P @ X4 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1199_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
=> ( P @ X4 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1200_sorted__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or6599480164596245535ed_nat @ Xs ) ) ) ).
% sorted_remdups_sorted
thf(fact_1201_sorted__wrt__drop,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_1202_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_1203_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_1204_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_1205_sorted__drop,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_drop
thf(fact_1206_sorted__take,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_take
thf(fact_1207_sorted__wrt1,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o,X: sum_sum_a_nat] : ( sorted6245805940552704876_a_nat @ P @ ( cons_Sum_sum_a_nat @ X @ nil_Sum_sum_a_nat ) ) ).
% sorted_wrt1
thf(fact_1208_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1209_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1210_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1211_sorted2,axiom,
! [X: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1212_sorted__wrt_Osimps_I1_J,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o] : ( sorted6245805940552704876_a_nat @ P @ nil_Sum_sum_a_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1213_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1214_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_1215_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1216_sorted__quicksort,axiom,
! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or9089632773640736191rt_nat @ Xs ) ) ).
% sorted_quicksort
thf(fact_1217_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_1218_sorted__last,axiom,
! [Xs: list_nat,X: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X @ ( last_nat @ Xs ) ) ) ) ).
% sorted_last
thf(fact_1219_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_1220_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1221_sorted__quicksort__acc,axiom,
! [Ac: list_nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Ac )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Ac ) )
=> ( ord_less_nat @ X3 @ Xa3 ) ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or5558937660843164036cc_nat @ Ac @ Xs ) ) ) ) ).
% sorted_quicksort_acc
thf(fact_1222_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1223_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1224_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1225_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I3 ) ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1226_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1227_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_1228_sorted__list__subset__correct,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( set_or6742139631805365739et_nat
@ ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 )
@ Xs
@ Ys )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ) ) ) ) ).
% sorted_list_subset_correct
thf(fact_1229_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1230_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1231_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1232_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1233_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1234_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1235_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1236_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_1237_nat__add__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q ) ) ) ).
% nat_add_max_right
thf(fact_1238_nat__add__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q ) @ ( plus_plus_nat @ N @ Q ) ) ) ).
% nat_add_max_left
thf(fact_1239_sorted__list__subset_Osimps_I3_J,axiom,
! [Eq2: nat > nat > $o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( set_or6742139631805365739et_nat @ Eq2 @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ( Eq2 @ X @ Y )
=> ( set_or6742139631805365739et_nat @ Eq2 @ Xs @ Ys ) )
& ( ~ ( Eq2 @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
& ( set_or6742139631805365739et_nat @ Eq2 @ ( cons_nat @ X @ Xs ) @ Ys ) ) ) ) ) ).
% sorted_list_subset.simps(3)
thf(fact_1240_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q ) ) ) ).
% nat_mult_max_right
thf(fact_1241_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q )
= ( ord_max_nat @ ( times_times_nat @ M @ Q ) @ ( times_times_nat @ N @ Q ) ) ) ).
% nat_mult_max_left
thf(fact_1242_sorted__list__subset_Oelims_I3_J,axiom,
! [X: nat > nat > $o,Xa: list_nat,Xb: list_nat] :
( ~ ( set_or6742139631805365739et_nat @ X @ Xa @ Xb )
=> ( ( ? [X3: nat,Xs2: list_nat] :
( Xa
= ( cons_nat @ X3 @ Xs2 ) )
=> ( Xb != nil_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( ( X @ X3 @ Y2 )
=> ( set_or6742139631805365739et_nat @ X @ Xs2 @ Ys2 ) )
& ( ~ ( X @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
& ( set_or6742139631805365739et_nat @ X @ ( cons_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ).
% sorted_list_subset.elims(3)
thf(fact_1243_sorted__list__subset_Oelims_I2_J,axiom,
! [X: nat > nat > $o,Xa: list_nat,Xb: list_nat] :
( ( set_or6742139631805365739et_nat @ X @ Xa @ Xb )
=> ( ( Xa != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( ( ( X @ X3 @ Y2 )
=> ( set_or6742139631805365739et_nat @ X @ Xs2 @ Ys2 ) )
& ( ~ ( X @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
& ( set_or6742139631805365739et_nat @ X @ ( cons_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ).
% sorted_list_subset.elims(2)
thf(fact_1244_sorted__list__subset_Oelims_I1_J,axiom,
! [X: nat > nat > $o,Xa: list_nat,Xb: list_nat,Y: $o] :
( ( ( set_or6742139631805365739et_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ~ Y )
=> ( ( ? [X3: nat,Xs2: list_nat] :
( Xa
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( Xb = nil_nat )
=> Y ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y2: nat,Ys2: list_nat] :
( ( Xb
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( Y
= ( ~ ( ( ( X @ X3 @ Y2 )
=> ( set_or6742139631805365739et_nat @ X @ Xs2 @ Ys2 ) )
& ( ~ ( X @ X3 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X3 )
& ( set_or6742139631805365739et_nat @ X @ ( cons_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_subset.elims(1)
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( append_Sum_sum_a_nat @ xs @ ( cons_Sum_sum_a_nat @ x @ nil_Sum_sum_a_nat ) )
= ( append_Sum_sum_a_nat @ ys @ ( cons_Sum_sum_a_nat @ y @ nil_Sum_sum_a_nat ) ) ) ).
%------------------------------------------------------------------------------