TPTP Problem File: SLH0168^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0024_Small_Proof/prob_00084_002751__6832206_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1546 ( 568 unt; 274 typ; 0 def)
% Number of atoms : 3313 (1770 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11690 ( 505 ~; 91 |; 215 &;9390 @)
% ( 0 <=>;1489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 59 ( 58 usr)
% Number of type conns : 610 ( 610 >; 0 *; 0 +; 0 <<)
% Number of symbols : 219 ( 216 usr; 15 con; 0-5 aty)
% Number of variables : 3818 ( 84 ^;3538 !; 196 ?;3818 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:13.926
%------------------------------------------------------------------------------
% Could-be-implicit typings (58)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc1616951275169580055st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_Pr1190453367779242145st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
produc4326814125627636033st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
set_li6867361041382987015st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Common__Ostate_Itf__a_J_J_J_J,type,
set_Pr152097031616555065tate_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Small__Ostate_Itf__a_J_J_J_J,type,
set_Pr6052505092368253171tate_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
list_P7940050157051400743st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc254973753779126261st_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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr5046312416420021961st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr7423161166939974351list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Common__Ostate_Itf__a_J_J_J,type,
produc3537791446953558659tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Small__Ostate_Itf__a_J_J_J,type,
produc7959480069840336147tate_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__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__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_li5450038453877631591at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
produc5032551385658279741list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc432399132543013523st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_Itf__a_J_J,type,
produc1513410750981052825list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Current__Ocurrent_It__Nat__Onat_J_J,type,
produc5057560627206499557nt_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J_J,type,
set_Pr324718442235990179tate_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J_J,type,
set_Pr6306228930610421491tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Common__Ostate_It__Nat__Onat_J_J,type,
produc8402205482288377289te_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__List__Olist_I_062_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
list_list_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Current__Ocurrent_Itf__a_J_J,type,
produc7805042584321970905rent_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
set_Pr4193341848836149977_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J,type,
produc3409137331138395373tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
produc7589950997499123219tate_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $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__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
list_state_a: $tType ).
thf(ty_n_t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
list_state_a2: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Current__Ocurrent_It__Nat__Onat_J,type,
current_nat: $tType ).
thf(ty_n_t__Common__Ostate_It__Nat__Onat_J,type,
state_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Current__Ocurrent_Itf__a_J,type,
current_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Common__Ostate_Itf__a_J,type,
state_a: $tType ).
thf(ty_n_t__Stack__Ostack_Itf__a_J,type,
stack_a: $tType ).
thf(ty_n_t__Small__Ostate_Itf__a_J,type,
state_a2: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $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 (216)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
bNF_Gr1840819286107176384st_nat: set_li6867361041382987015st_nat > produc1828647624359046049st_nat > set_li6867361041382987015st_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr3130287167067265568at_nat: set_li5450038453877631591at_nat > product_prod_nat_nat > set_li5450038453877631591at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
bNF_Gr8705060421004693820st_nat: set_li6867361041382987015st_nat > list_P7940050157051400743st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr5363859321595349404at_nat: set_li5450038453877631591at_nat > list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Common_Opop_001t__Nat__Onat,type,
pop_nat: state_nat > produc8402205482288377289te_nat ).
thf(sy_c_Common_Opop_001tf__a,type,
pop_a: state_a > produc3409137331138395373tate_a ).
thf(sy_c_Common_Opush_001t__Nat__Onat,type,
push_nat: nat > state_nat > state_nat ).
thf(sy_c_Common_Opush_001tf__a,type,
push_a: a > state_a > state_a ).
thf(sy_c_Common__Aux_Olist_001t__Nat__Onat,type,
common_list_nat: state_nat > list_nat ).
thf(sy_c_Common__Aux_Olist_001tf__a,type,
common_list_a: state_a > list_a ).
thf(sy_c_Common__Aux_Olist__current_001t__Nat__Onat,type,
common8576334314191143743nt_nat: state_nat > list_nat ).
thf(sy_c_Common__Aux_Olist__current_001tf__a,type,
common1102728217005306191rent_a: state_a > list_a ).
thf(sy_c_Common__Aux_Otake__rev_001t__Nat__Onat,type,
common_take_rev_nat: nat > list_nat > list_nat ).
thf(sy_c_Common__Aux_Otake__rev_001tf__a,type,
common_take_rev_a: nat > list_a > list_a ).
thf(sy_c_Current_Ofirst_001tf__a,type,
first_a: current_a > a ).
thf(sy_c_Current_Opop_001t__Nat__Onat,type,
pop_nat2: current_nat > produc5057560627206499557nt_nat ).
thf(sy_c_Current_Opop_001tf__a,type,
pop_a2: current_a > produc7805042584321970905rent_a ).
thf(sy_c_Current_Opush_001t__Nat__Onat,type,
push_nat2: nat > current_nat > current_nat ).
thf(sy_c_Current_Opush_001tf__a,type,
push_a2: a > current_a > current_a ).
thf(sy_c_Current__Aux_Olist_001t__Nat__Onat,type,
current_list_nat: current_nat > list_nat ).
thf(sy_c_Current__Aux_Olist_001tf__a,type,
current_list_a: current_a > list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
minus_4256792079133151976st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_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_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_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__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
append2623875052807961020st_nat: list_P7940050157051400743st_nat > list_P7940050157051400743st_nat > list_P7940050157051400743st_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_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olenlex_001t__List__Olist_It__Nat__Onat_J,type,
lenlex_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__List__Olist_It__Nat__Onat_J,type,
lex_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexord_001t__List__Olist_It__Nat__Onat_J,type,
lexord_list_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_nat ).
thf(sy_c_List_Olexord_001t__Nat__Onat,type,
lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
cons_list_nat_nat: ( list_nat > nat ) > list_list_nat_nat > list_list_nat_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_nat_nat: ( nat > nat ) > list_nat_nat > list_nat_nat ).
thf(sy_c_List_Olist_OCons_001t__Common__Ostate_Itf__a_J,type,
cons_state_a: state_a > list_state_a > list_state_a ).
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_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
cons_P5007559046487125591st_nat: produc1828647624359046049st_nat > list_P7940050157051400743st_nat > list_P7940050157051400743st_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__Small__Ostate_Itf__a_J,type,
cons_state_a2: state_a2 > list_state_a2 > list_state_a2 ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
nil_list_nat_nat: list_list_nat_nat ).
thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_nat_nat: list_nat_nat ).
thf(sy_c_List_Olist_ONil_001t__Common__Ostate_Itf__a_J,type,
nil_state_a: list_state_a ).
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_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
nil_Pr8413428694792600231st_nat: list_P7940050157051400743st_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__Small__Ostate_Itf__a_J,type,
nil_state_a2: list_state_a2 ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
listre6091228620945859379st_nat: set_Pr3451248702717554689st_nat > set_Pr1190453367779242145st_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_001tf__a,type,
listrel_nat_a: set_Pr4193341848836149977_nat_a > set_Pr7423161166939974351list_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Common__Ostate_Itf__a_J,type,
listrel_a_state_a: set_Pr324718442235990179tate_a > set_Pr152097031616555065tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Nat__Onat,type,
listrel_a_nat: set_Pr4934435412358123699_a_nat > set_Pr5046312416420021961st_nat ).
thf(sy_c_List_Olistrel_001tf__a_001t__Small__Ostate_Itf__a_J,type,
listrel_a_state_a2: set_Pr6306228930610421491tate_a > set_Pr6052505092368253171tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Omeasures_001t__List__Olist_It__Nat__Onat_J,type,
measures_list_nat: list_list_nat_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
measures_nat: list_nat_nat > set_Pr1261947904930325089at_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_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Common__Ostate_Itf__a_J,type,
nth_state_a: list_state_a > nat > state_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Small__Ostate_Itf__a_J,type,
nth_state_a2: list_state_a2 > nat > state_a2 ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
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_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
take_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_It__Nat__Onat_J,type,
size_size_state_nat: state_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_Itf__a_J,type,
size_size_state_a: state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_It__Nat__Onat_J,type,
size_s5002997120254590536nt_nat: current_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_Itf__a_J,type,
size_size_current_a: current_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
size_s4663437186682225508tate_a: list_state_a > 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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
size_s8463391772401140188tate_a: list_state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_Itf__a_J,type,
size_size_state_a2: state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le3947731281898473645st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le8406513867147106209st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J_J,type,
ord_le8856296118373343491tate_a: set_Pr324718442235990179tate_a > set_Pr324718442235990179tate_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J_J,type,
ord_le926351553812985491tate_a: set_Pr6306228930610421491tate_a > set_Pr6306228930610421491tate_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Num__Onum,type,
ord_min_num: num > num > num ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc4626581765195395529st_nat: ( nat > nat ) > produc1828647624359046049st_nat > produc1616951275169580055st_nat ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
produc7129799990162260089st_nat: list_list_nat > list_list_nat > produc4326814125627636033st_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_Itf__a_J,type,
produc7723716010052024011list_a: list_nat > list_a > produc1513410750981052825list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
produc4137809223961881341tate_a: list_a > list_state_a > produc3537791446953558659tate_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4792949784200893581st_nat: list_a > list_nat > produc432399132543013523st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
produc1997082749353321475tate_a: list_a > list_state_a2 > produc7959480069840336147tate_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Common__Ostate_It__Nat__Onat_J,type,
produc7792781364869388027te_nat: nat > state_nat > produc8402205482288377289te_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_001tf__a,type,
product_Pair_nat_a: nat > a > product_prod_nat_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc8263595898873874535tate_a: a > state_a > produc3409137331138395373tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc8503237746132909001rent_a: a > current_a > produc7805042584321970905rent_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Small__Ostate_Itf__a_J,type,
produc1224139502141355779tate_a: a > state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc1382935764643595205st_nat: produc1828647624359046049st_nat > list_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Current__Ocurrent_It__Nat__Onat_J,type,
produc4614407304758426121nt_nat: produc5057560627206499557nt_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc3154331710141225339tate_a: produc3409137331138395373tate_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4952273589686483381rent_a: produc7805042584321970905rent_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Small__Ostate_Itf__a_J,type,
produc8617775573817090287tate_a: produc7589950997499123219tate_a > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc5865812112468994567st_nat: produc1828647624359046049st_nat > list_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Current__Ocurrent_It__Nat__Onat_J,type,
produc8758756790178850379nt_nat: produc5057560627206499557nt_nat > current_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc681690970763031737tate_a: produc3409137331138395373tate_a > state_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4695312889421393143rent_a: produc7805042584321970905rent_a > current_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Small__Ostate_Itf__a_J,type,
produc6287816722867230257tate_a: produc7589950997499123219tate_a > state_a2 ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Small_Opop_001tf__a,type,
pop_a3: state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opush_001tf__a,type,
push_a3: a > state_a2 > state_a2 ).
thf(sy_c_Small_Opush__rel_001tf__a,type,
push_rel_a: produc7589950997499123219tate_a > produc7589950997499123219tate_a > $o ).
thf(sy_c_Small_Ostate_OCommon_001tf__a,type,
common_a: state_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse1_001tf__a,type,
reverse1_a: current_a > stack_a > list_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse2_001tf__a,type,
reverse2_a: current_a > list_a > stack_a > list_a > nat > state_a2 ).
thf(sy_c_Small__Aux_Olist_001tf__a,type,
small_list_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Olist__current_001tf__a,type,
small_list_current_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Olist__current__rel_001tf__a,type,
small_1033803570666515670_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_Small__Aux_Osize__new__state__rel_001tf__a,type,
small_1734874108512276223_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_Small__Aux_Osize__state__rel_001tf__a,type,
small_6459853017724497003_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_It__Nat__Onat_J,type,
type_i5905290871511909543te_nat: state_nat > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_Itf__a_J,type,
type_i4669920168676019581tate_a: state_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_It__Nat__Onat_J,type,
type_i2033238430507501937nt_nat: current_nat > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_Itf__a_J,type,
type_i6141643110573041459rent_a: current_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Small__Ostate_Itf__a_J,type,
type_i464410347872898157tate_a: state_a2 > $o ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__Common__Ostate_Itf__a_J,type,
type_r2212416260012024137tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Common__Ostate_Itf__a_J,type,
type_s8424385952999958455tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Current__Ocurrent_Itf__a_J,type,
type_s933026853152659577rent_a: current_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Small__Ostate_Itf__a_J,type,
type_s6404775287138157491tate_a: state_a2 > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Common__Ostate_Itf__a_J,type,
type_s889635741254954505tate_a: state_a > state_a ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Small__Ostate_Itf__a_J,type,
type_s3703408523585882337tate_a: state_a2 > state_a2 ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
accp_P4095703987651755036tate_a: ( produc7589950997499123219tate_a > produc7589950997499123219tate_a > $o ) > produc7589950997499123219tate_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Small__Ostate_Itf__a_J,type,
accp_state_a: ( state_a2 > state_a2 > $o ) > state_a2 > $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__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
member6711608200250777424st_nat: list_P7940050157051400743st_nat > set_li6867361041382987015st_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member3067507820990806192at_nat: list_P6011104703257516679at_nat > set_li5450038453877631591at_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
member8680655010358287850st_nat: produc4326814125627636033st_nat > set_Pr1190453367779242145st_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_Itf__a_J_J,type,
member5932150393272073264list_a: produc1513410750981052825list_a > set_Pr7423161166939974351list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Common__Ostate_Itf__a_J_J_J,type,
member7736023580690228378tate_a: produc3537791446953558659tate_a > set_Pr152097031616555065tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member4851138774834033962st_nat: produc432399132543013523st_nat > set_Pr5046312416420021961st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Small__Ostate_Itf__a_J_J_J,type,
member4112945611203173692tate_a: produc7959480069840336147tate_a > set_Pr6052505092368253171tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__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_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J,type,
member7740437332958137860tate_a: produc3409137331138395373tate_a > set_Pr324718442235990179tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
member8547378267715833660tate_a: produc7589950997499123219tate_a > set_Pr6306228930610421491tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_state____,type,
state: state_a ).
% Relevant facts (1264)
thf(fact_0__C1_C,axiom,
type_i464410347872898157tate_a @ ( common_a @ state ) ).
% "1"
thf(fact_1_step__list__common,axiom,
! [Small: state_a2,Common: state_a] :
( ( Small
= ( common_a @ Common ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_common
thf(fact_2_Small_Ostate_Oinject_I3_J,axiom,
! [X3: state_a,Y3: state_a] :
( ( ( common_a @ X3 )
= ( common_a @ Y3 ) )
= ( X3 = Y3 ) ) ).
% Small.state.inject(3)
thf(fact_3_Small__Proof_Ostep__list__current,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_current_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_current_a @ Small ) ) ) ).
% Small_Proof.step_list_current
thf(fact_4_Small__Proof_Ostep__size,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( size_size_state_a2 @ ( type_s3703408523585882337tate_a @ Small ) )
= ( size_size_state_a2 @ Small ) ) ) ).
% Small_Proof.step_size
thf(fact_5_Small_Ostep__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_s3703408523585882337tate_a @ ( common_a @ State ) )
= ( common_a @ ( type_s889635741254954505tate_a @ State ) ) ) ).
% Small.step_state.simps(1)
thf(fact_6_step__list__reverse2,axiom,
! [Small: state_a2,Current: current_a,Aux: list_a,Big: stack_a,New: list_a,Count: nat] :
( ( Small
= ( reverse2_a @ Current @ Aux @ Big @ New @ Count ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_reverse2
thf(fact_7_Small__Aux_Oinvar__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_i464410347872898157tate_a @ ( common_a @ State ) )
= ( type_i4669920168676019581tate_a @ State ) ) ).
% Small_Aux.invar_state.simps(1)
thf(fact_8_Small__Proof_Osize__new__push,axiom,
! [Small: state_a2,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( type_s6404775287138157491tate_a @ ( push_a3 @ X @ Small ) )
= ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_new_push
thf(fact_9_Small__Proof_Osize__push,axiom,
! [Small: state_a2,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( size_size_state_a2 @ ( push_a3 @ X @ Small ) )
= ( suc @ ( size_size_state_a2 @ Small ) ) ) ) ).
% Small_Proof.size_push
thf(fact_10_Small_Ostate_Odistinct_I3_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X3: state_a] :
( ( reverse1_a @ X11 @ X12 @ X13 )
!= ( common_a @ X3 ) ) ).
% Small.state.distinct(3)
thf(fact_11_Small_Ostate_Oinject_I2_J,axiom,
! [X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat,Y21: current_a,Y22: list_a,Y23: stack_a,Y24: list_a,Y25: nat] :
( ( ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 )
= ( reverse2_a @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% Small.state.inject(2)
thf(fact_12_Small_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,Y11: current_a,Y12: stack_a,Y13: list_a] :
( ( ( reverse1_a @ X11 @ X12 @ X13 )
= ( reverse1_a @ Y11 @ Y12 @ Y13 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 ) ) ) ).
% Small.state.inject(1)
thf(fact_13_Small_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat] :
( ( reverse1_a @ X11 @ X12 @ X13 )
!= ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% Small.state.distinct(1)
thf(fact_14_Small_Ostate_Oexhaust,axiom,
! [Y: state_a2] :
( ! [X112: current_a,X122: stack_a,X132: list_a] :
( Y
!= ( reverse1_a @ X112 @ X122 @ X132 ) )
=> ( ! [X212: current_a,X222: list_a,X232: stack_a,X242: list_a,X252: nat] :
( Y
!= ( reverse2_a @ X212 @ X222 @ X232 @ X242 @ X252 ) )
=> ~ ! [X32: state_a] :
( Y
!= ( common_a @ X32 ) ) ) ) ).
% Small.state.exhaust
thf(fact_15_Small_Ostep__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ( ! [Current2: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( reverse1_a @ Current2 @ Small2 @ AuxS ) )
=> ~ ! [Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ).
% Small.step_state.cases
thf(fact_16_Small__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( X
!= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( X
!= ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ).
% Small_Aux.size_new_state.cases
thf(fact_17_Small_Ostate_Odistinct_I5_J,axiom,
! [X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat,X3: state_a] :
( ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 )
!= ( common_a @ X3 ) ) ).
% Small.state.distinct(5)
thf(fact_18_invar__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( type_s889635741254954505tate_a @ Common ) ) ) ).
% invar_step
thf(fact_19_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_20_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_21_Small_Opush_Osimps_I1_J,axiom,
! [X: a,State: state_a] :
( ( push_a3 @ X @ ( common_a @ State ) )
= ( common_a @ ( push_a @ X @ State ) ) ) ).
% Small.push.simps(1)
thf(fact_22_Small__Aux_Olist__current_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( small_list_current_a @ ( common_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ).
% Small_Aux.list_current.simps(1)
thf(fact_23_Small__Aux_Olist_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( small_list_a @ ( common_a @ Common ) )
= ( common_list_a @ Common ) ) ).
% Small_Aux.list.simps(1)
thf(fact_24_Small__Aux_Osize__new__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_s6404775287138157491tate_a @ ( common_a @ State ) )
= ( type_s8424385952999958455tate_a @ State ) ) ).
% Small_Aux.size_new_state.simps(1)
thf(fact_25_Small_Opush_Osimps_I2_J,axiom,
! [X: a,Current: current_a,Small: stack_a,AuxS2: list_a] :
( ( push_a3 @ X @ ( reverse1_a @ Current @ Small @ AuxS2 ) )
= ( reverse1_a @ ( push_a2 @ X @ Current ) @ Small @ AuxS2 ) ) ).
% Small.push.simps(2)
thf(fact_26_Small_Opush_Osimps_I3_J,axiom,
! [X: a,Current: current_a,AuxS2: list_a,Big: stack_a,NewS2: list_a,Count: nat] :
( ( push_a3 @ X @ ( reverse2_a @ Current @ AuxS2 @ Big @ NewS2 @ Count ) )
= ( reverse2_a @ ( push_a2 @ X @ Current ) @ AuxS2 @ Big @ NewS2 @ Count ) ) ).
% Small.push.simps(3)
thf(fact_27_Small_Opush_Oelims,axiom,
! [X: a,Xa: state_a2,Y: state_a2] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ( ! [State2: state_a] :
( ( Xa
= ( common_a @ State2 ) )
=> ( Y
!= ( common_a @ ( push_a @ X @ State2 ) ) ) )
=> ( ! [Current2: current_a,Small2: stack_a,AuxS: list_a] :
( ( Xa
= ( reverse1_a @ Current2 @ Small2 @ AuxS ) )
=> ( Y
!= ( reverse1_a @ ( push_a2 @ X @ Current2 ) @ Small2 @ AuxS ) ) )
=> ~ ! [Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( ( Xa
= ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) )
=> ( Y
!= ( reverse2_a @ ( push_a2 @ X @ Current2 ) @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ) ).
% Small.push.elims
thf(fact_28_Small_Opush_Ocases,axiom,
! [X: produc7589950997499123219tate_a] :
( ! [X4: a,State2: state_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( common_a @ State2 ) ) )
=> ( ! [X4: a,Current2: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse1_a @ Current2 @ Small2 @ AuxS ) ) )
=> ~ ! [X4: a,Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ).
% Small.push.cases
thf(fact_29_step__size__new,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_s8424385952999958455tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( type_s8424385952999958455tate_a @ Common ) ) ) ).
% step_size_new
thf(fact_30_Common__Proof_Ostep__list__current,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common1102728217005306191rent_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ).
% Common_Proof.step_list_current
thf(fact_31_step__list,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common_list_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common_list_a @ Common ) ) ) ).
% step_list
thf(fact_32_Common__Proof_Osize__new__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_s8424385952999958455tate_a @ ( push_a @ X @ Common ) )
= ( suc @ ( type_s8424385952999958455tate_a @ Common ) ) ) ) ).
% Common_Proof.size_new_push
thf(fact_33_Common__Proof_Oinvar__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( push_a @ X @ Common ) ) ) ).
% Common_Proof.invar_push
thf(fact_34_mem__Collect__eq,axiom,
! [A: produc1828647624359046049st_nat,P: produc1828647624359046049st_nat > $o] :
( ( member7340969449405702474st_nat @ A @ ( collec1570431334306492044st_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_35_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_36_Collect__mem__eq,axiom,
! [A2: set_Pr3451248702717554689st_nat] :
( ( collec1570431334306492044st_nat
@ ^ [X5: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_37_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_38_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_39_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_40_size__neq__size__imp__neq,axiom,
! [X: state_a,Y: state_a] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_41_size__neq__size__imp__neq,axiom,
! [X: current_a,Y: current_a] :
( ( ( size_size_current_a @ X )
!= ( size_size_current_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_42_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_43_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_44_size__neq__size__imp__neq,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( size_size_state_a2 @ X )
!= ( size_size_state_a2 @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_45_old_Oprod_Oinject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_46_old_Oprod_Oinject,axiom,
! [A: list_nat,B: list_nat,A3: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_47_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_48_old_Oprod_Oinject,axiom,
! [A: a,B: state_a,A3: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_49_prod_Oinject,axiom,
! [X1: a,X2: state_a2,Y1: a,Y2: state_a2] :
( ( ( produc1224139502141355779tate_a @ X1 @ X2 )
= ( produc1224139502141355779tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_50_prod_Oinject,axiom,
! [X1: list_nat,X2: list_nat,Y1: list_nat,Y2: list_nat] :
( ( ( produc2694037385005941721st_nat @ X1 @ X2 )
= ( produc2694037385005941721st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_51_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_52_prod_Oinject,axiom,
! [X1: a,X2: state_a,Y1: a,Y2: state_a] :
( ( ( produc8263595898873874535tate_a @ X1 @ X2 )
= ( produc8263595898873874535tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_53_Small__Aux_Olist__current_Oelims,axiom,
! [X: state_a2,Y: list_a] :
( ( ( small_list_current_a @ X )
= Y )
=> ( ! [Common2: state_a] :
( ( X
= ( common_a @ Common2 ) )
=> ( Y
!= ( common1102728217005306191rent_a @ Common2 ) ) )
=> ( ! [Current2: current_a] :
( ? [Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( Y
!= ( current_list_a @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( Y
!= ( current_list_a @ Current2 ) ) ) ) ) ) ).
% Small_Aux.list_current.elims
thf(fact_54_Small__Aux_Osize__new__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6404775287138157491tate_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( type_s8424385952999958455tate_a @ State2 ) ) )
=> ( ! [Current2: current_a] :
( ? [Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ) ).
% Small_Aux.size_new_state.elims
thf(fact_55_Small_Opush_Opelims,axiom,
! [X: a,Xa: state_a2,Y: state_a2] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ( ( accp_P4095703987651755036tate_a @ push_rel_a @ ( produc1224139502141355779tate_a @ X @ Xa ) )
=> ( ! [State2: state_a] :
( ( Xa
= ( common_a @ State2 ) )
=> ( ( Y
= ( common_a @ ( push_a @ X @ State2 ) ) )
=> ~ ( accp_P4095703987651755036tate_a @ push_rel_a @ ( produc1224139502141355779tate_a @ X @ ( common_a @ State2 ) ) ) ) )
=> ( ! [Current2: current_a,Small2: stack_a,AuxS: list_a] :
( ( Xa
= ( reverse1_a @ Current2 @ Small2 @ AuxS ) )
=> ( ( Y
= ( reverse1_a @ ( push_a2 @ X @ Current2 ) @ Small2 @ AuxS ) )
=> ~ ( accp_P4095703987651755036tate_a @ push_rel_a @ ( produc1224139502141355779tate_a @ X @ ( reverse1_a @ Current2 @ Small2 @ AuxS ) ) ) ) )
=> ~ ! [Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( ( Xa
= ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) )
=> ( ( Y
= ( reverse2_a @ ( push_a2 @ X @ Current2 ) @ AuxS @ Big2 @ NewS @ Count2 ) )
=> ~ ( accp_P4095703987651755036tate_a @ push_rel_a @ ( produc1224139502141355779tate_a @ X @ ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ) ) ) ) ).
% Small.push.pelims
thf(fact_56_Common__Proof_Osize__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( size_size_state_a @ ( push_a @ X @ Common ) )
= ( suc @ ( size_size_state_a @ Common ) ) ) ) ).
% Common_Proof.size_push
thf(fact_57_remaining__steps__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_r2212416260012024137tate_a @ ( push_a @ X @ Common ) )
= ( type_r2212416260012024137tate_a @ Common ) ) ) ).
% remaining_steps_push
thf(fact_58_Common__Proof_Ostep__size,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( size_size_state_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( size_size_state_a @ Common ) ) ) ).
% Common_Proof.step_size
thf(fact_59_Small__Aux_Osize__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( size_size_state_a2 @ ( common_a @ State ) )
= ( size_size_state_a @ State ) ) ).
% Small_Aux.size_state.simps(1)
thf(fact_60_push__list__current,axiom,
! [X: nat,Left: state_nat] :
( ( common8576334314191143743nt_nat @ ( push_nat @ X @ Left ) )
= ( cons_nat @ X @ ( common8576334314191143743nt_nat @ Left ) ) ) ).
% push_list_current
thf(fact_61_push__list__current,axiom,
! [X: a,Left: state_a] :
( ( common1102728217005306191rent_a @ ( push_a @ X @ Left ) )
= ( cons_a @ X @ ( common1102728217005306191rent_a @ Left ) ) ) ).
% push_list_current
thf(fact_62_Common__Proof_Opush__list,axiom,
! [X: nat,Common: state_nat] :
( ( common_list_nat @ ( push_nat @ X @ Common ) )
= ( cons_nat @ X @ ( common_list_nat @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_63_Common__Proof_Opush__list,axiom,
! [X: a,Common: state_a] :
( ( common_list_a @ ( push_a @ X @ Common ) )
= ( cons_a @ X @ ( common_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_64_Small__Aux_Osize__new__state_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv2: stack_a,Uw2: list_a,Ux2: nat] :
( ( type_s6404775287138157491tate_a @ ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw2 @ Ux2 ) )
= ( type_s933026853152659577rent_a @ Current ) ) ).
% Small_Aux.size_new_state.simps(2)
thf(fact_65_Small__Aux_Osize__new__state_Osimps_I3_J,axiom,
! [Current: current_a,Uy2: stack_a,Uz2: list_a] :
( ( type_s6404775287138157491tate_a @ ( reverse1_a @ Current @ Uy2 @ Uz2 ) )
= ( type_s933026853152659577rent_a @ Current ) ) ).
% Small_Aux.size_new_state.simps(3)
thf(fact_66_old_Oprod_Oexhaust,axiom,
! [Y: produc7589950997499123219tate_a] :
~ ! [A4: a,B3: state_a2] :
( Y
!= ( produc1224139502141355779tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_67_old_Oprod_Oexhaust,axiom,
! [Y: produc1828647624359046049st_nat] :
~ ! [A4: list_nat,B3: list_nat] :
( Y
!= ( produc2694037385005941721st_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_68_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_69_old_Oprod_Oexhaust,axiom,
! [Y: produc3409137331138395373tate_a] :
~ ! [A4: a,B3: state_a] :
( Y
!= ( produc8263595898873874535tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_70_surj__pair,axiom,
! [P2: produc7589950997499123219tate_a] :
? [X4: a,Y4: state_a2] :
( P2
= ( produc1224139502141355779tate_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_71_surj__pair,axiom,
! [P2: produc1828647624359046049st_nat] :
? [X4: list_nat,Y4: list_nat] :
( P2
= ( produc2694037385005941721st_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_72_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X4: nat,Y4: nat] :
( P2
= ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_73_surj__pair,axiom,
! [P2: produc3409137331138395373tate_a] :
? [X4: a,Y4: state_a] :
( P2
= ( produc8263595898873874535tate_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_74_prod__cases,axiom,
! [P: produc7589950997499123219tate_a > $o,P2: produc7589950997499123219tate_a] :
( ! [A4: a,B3: state_a2] : ( P @ ( produc1224139502141355779tate_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_75_prod__cases,axiom,
! [P: produc1828647624359046049st_nat > $o,P2: produc1828647624359046049st_nat] :
( ! [A4: list_nat,B3: list_nat] : ( P @ ( produc2694037385005941721st_nat @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_76_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_77_prod__cases,axiom,
! [P: produc3409137331138395373tate_a > $o,P2: produc3409137331138395373tate_a] :
( ! [A4: a,B3: state_a] : ( P @ ( produc8263595898873874535tate_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_78_Pair__inject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_79_Pair__inject,axiom,
! [A: list_nat,B: list_nat,A3: list_nat,B2: list_nat] :
( ( ( produc2694037385005941721st_nat @ A @ B )
= ( produc2694037385005941721st_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_80_Pair__inject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_81_Pair__inject,axiom,
! [A: a,B: state_a,A3: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_82_Small__Aux_Olist__current_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv2: stack_a,Uw2: list_a,Ux2: nat] :
( ( small_list_current_a @ ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw2 @ Ux2 ) )
= ( current_list_a @ Current ) ) ).
% Small_Aux.list_current.simps(2)
thf(fact_83_Small__Aux_Olist__current_Osimps_I3_J,axiom,
! [Current: current_a,Uy2: stack_a,Uz2: list_a] :
( ( small_list_current_a @ ( reverse1_a @ Current @ Uy2 @ Uz2 ) )
= ( current_list_a @ Current ) ) ).
% Small_Aux.list_current.simps(3)
thf(fact_84_Current__Proof_Opush__list,axiom,
! [X: nat,Current: current_nat] :
( ( current_list_nat @ ( push_nat2 @ X @ Current ) )
= ( cons_nat @ X @ ( current_list_nat @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_85_Current__Proof_Opush__list,axiom,
! [X: a,Current: current_a] :
( ( current_list_a @ ( push_a2 @ X @ Current ) )
= ( cons_a @ X @ ( current_list_a @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_86_Current__Proof_Osize__push,axiom,
! [X: a,Current: current_a] :
( ( size_size_current_a @ ( push_a2 @ X @ Current ) )
= ( suc @ ( size_size_current_a @ Current ) ) ) ).
% Current_Proof.size_push
thf(fact_87_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_88_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_89_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_90_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_91_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_92_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_93_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y5: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_94_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_95_Small__Aux_Osize__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a2 @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( size_size_state_a @ State2 ) ) )
=> ( ! [Current2: current_a] :
( ? [Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ) ) ).
% Small_Aux.size_state.elims
thf(fact_96_remaining__steps__step__0,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( type_r2212416260012024137tate_a @ Common )
= zero_zero_nat )
=> ( ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_step_0
thf(fact_97_Small__Aux_Olist__current_Opelims,axiom,
! [X: state_a2,Y: list_a] :
( ( ( small_list_current_a @ X )
= Y )
=> ( ( accp_state_a @ small_1033803570666515670_rel_a @ X )
=> ( ! [Common2: state_a] :
( ( X
= ( common_a @ Common2 ) )
=> ( ( Y
= ( common1102728217005306191rent_a @ Common2 ) )
=> ~ ( accp_state_a @ small_1033803570666515670_rel_a @ ( common_a @ Common2 ) ) ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( ( Y
= ( current_list_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1033803570666515670_rel_a @ ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) ) ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( ( Y
= ( current_list_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1033803570666515670_rel_a @ ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.list_current.pelims
thf(fact_98_Small__Aux_Osize__new__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6404775287138157491tate_a @ X )
= Y )
=> ( ( accp_state_a @ small_1734874108512276223_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( type_s8424385952999958455tate_a @ State2 ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( common_a @ State2 ) ) ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) ) ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.size_new_state.pelims
thf(fact_99_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_100_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_101_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_102_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_103_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_104_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_105_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_106_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_107_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_108_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_109_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_110_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_111_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_112_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_113_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_114_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_115_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_116_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_117_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_118_Small__Aux_Osize__state_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv2: stack_a,Uw2: list_a,Ux2: nat] :
( ( size_size_state_a2 @ ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw2 @ Ux2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ).
% Small_Aux.size_state.simps(2)
thf(fact_119_Small__Aux_Osize__state_Osimps_I3_J,axiom,
! [Current: current_a,Uy2: stack_a,Uz2: list_a] :
( ( size_size_state_a2 @ ( reverse1_a @ Current @ Uy2 @ Uz2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ).
% Small_Aux.size_state.simps(3)
thf(fact_120_Small__Aux_Osize__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a2 @ X )
= Y )
=> ( ( accp_state_a @ small_6459853017724497003_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( size_size_state_a @ State2 ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( common_a @ State2 ) ) ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( reverse2_a @ Current2 @ Uu @ Uv @ Uw @ Ux ) ) ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.size_state.pelims
thf(fact_121_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_122_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_123_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_124_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_125_remaining__steps__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r2212416260012024137tate_a @ Common ) )
=> ( ( suc @ ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) ) )
= ( type_r2212416260012024137tate_a @ Common ) ) ) ) ).
% remaining_steps_step
thf(fact_126_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_127_Cons__in__lex,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) ) @ ( lex_list_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
& ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys2 ) @ ( lex_list_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_128_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_129_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_130_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_131_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_132_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_133_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_134_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_135_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_136_min_Oabsorb3,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_min_num @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_137_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_138_min_Oabsorb4,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_min_num @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_139_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_140_min__less__iff__conj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_min_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
& ( ord_less_num @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_141_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_142_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_143_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_144_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_145_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_146_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_147_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_148_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_149_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_150_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_151_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_152_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_153_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_154_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_155_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_156_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_157_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_158_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_159_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_160_min__less__iff__disj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_min_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
| ( ord_less_num @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_161_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_162_min_Ostrict__boundedE,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ ( ord_min_num @ B @ C2 ) )
=> ~ ( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_163_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_164_min_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( A5
= ( ord_min_num @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_165_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_166_min_Ostrict__coboundedI1,axiom,
! [A: num,C2: num,B: num] :
( ( ord_less_num @ A @ C2 )
=> ( ord_less_num @ ( ord_min_num @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_167_min_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_168_min_Ostrict__coboundedI2,axiom,
! [B: num,C2: num,A: num] :
( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ ( ord_min_num @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_169_min_Ostrict__coboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_170_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_171_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_172_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_173_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_174_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_175_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_176_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_177_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_178_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_179_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_180_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_181_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_182_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_183_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_184_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I3 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_185_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_186_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_187_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_188_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_189_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_190_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_191_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_192_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_193_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_194_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_195_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_196_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_197_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_198_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_199_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_200_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_201_min_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C2 )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ).
% min.assoc
thf(fact_202_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A5: nat,B4: nat] : ( ord_min_nat @ B4 @ A5 ) ) ) ).
% min.commute
thf(fact_203_min_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C2 ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ).
% min.left_commute
thf(fact_204_Common__Proof_Osize__size__new,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s8424385952999958455tate_a @ Common ) ) ) ) ).
% Common_Proof.size_size_new
thf(fact_205_Small__Proof_Osize__size__new,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_size_new
thf(fact_206_Small__Proof_Osize__new__pop,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( suc @ ( type_s6404775287138157491tate_a @ Small3 ) )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% Small_Proof.size_new_pop
thf(fact_207_Small__Proof_Osize__pop,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( suc @ ( size_size_state_a2 @ Small3 ) )
= ( size_size_state_a2 @ Small ) ) ) ) ) ).
% Small_Proof.size_pop
thf(fact_208_Common__Proof_Opop__list,axiom,
! [Common: state_nat,X: nat,Common3: state_nat] :
( ( type_i5905290871511909543te_nat @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_nat @ Common ) )
=> ( ( ( pop_nat @ Common )
= ( produc7792781364869388027te_nat @ X @ Common3 ) )
=> ( ( cons_nat @ X @ ( common_list_nat @ Common3 ) )
= ( common_list_nat @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_209_Common__Proof_Opop__list,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( ( cons_a @ X @ ( common_list_a @ Common3 ) )
= ( common_list_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_210_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_211_Common__Proof_Osize__new__pop,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s8424385952999958455tate_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( ( suc @ ( type_s8424385952999958455tate_a @ Common3 ) )
= ( type_s8424385952999958455tate_a @ Common ) ) ) ) ) ).
% Common_Proof.size_new_pop
thf(fact_212_Common__Proof_Osize__pop,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( ( suc @ ( size_size_state_a @ Common3 ) )
= ( size_size_state_a @ Common ) ) ) ) ) ).
% Common_Proof.size_pop
thf(fact_213_pop__list__current,axiom,
! [Common: state_nat,X: nat,Common3: state_nat] :
( ( type_i5905290871511909543te_nat @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_nat @ Common ) )
=> ( ( ( pop_nat @ Common )
= ( produc7792781364869388027te_nat @ X @ Common3 ) )
=> ( ( cons_nat @ X @ ( common8576334314191143743nt_nat @ Common3 ) )
= ( common8576334314191143743nt_nat @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_214_pop__list__current,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( ( cons_a @ X @ ( common1102728217005306191rent_a @ Common3 ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_215_Common__Proof_Oinvar__pop,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( type_i4669920168676019581tate_a @ Common3 ) ) ) ) ).
% Common_Proof.invar_pop
thf(fact_216_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_217_Cons__lenlex__iff,axiom,
! [M: list_nat,Ms: list_list_nat,N: list_nat,Ns: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ M @ Ms ) @ ( cons_list_nat @ N @ Ns ) ) @ ( lenlex_list_nat @ R ) )
= ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ms ) @ ( size_s3023201423986296836st_nat @ Ns ) )
| ( ( ( size_s3023201423986296836st_nat @ Ms )
= ( size_s3023201423986296836st_nat @ Ns ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ms @ Ns ) @ ( lenlex_list_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_218_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_219_lenlex__irreflexive,axiom,
! [R: set_Pr3451248702717554689st_nat,Xs: list_list_nat] :
( ! [X4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ X4 ) @ R )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Xs ) @ ( lenlex_list_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_220_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_221_remaining__steps__pop,axiom,
! [Common: state_a,X: a,Common3: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common3 ) )
=> ( ord_less_eq_nat @ ( type_r2212416260012024137tate_a @ Common3 ) @ ( type_r2212416260012024137tate_a @ Common ) ) ) ) ) ).
% remaining_steps_pop
thf(fact_222_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_223_list__current__size,axiom,
! [Common: state_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_nat @ Common ) )
=> ( ( ( common8576334314191143743nt_nat @ Common )
= nil_nat )
=> ~ ( type_i5905290871511909543te_nat @ Common ) ) ) ).
% list_current_size
thf(fact_224_list__current__size,axiom,
! [Common: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( common1102728217005306191rent_a @ Common )
= nil_a )
=> ~ ( type_i4669920168676019581tate_a @ Common ) ) ) ).
% list_current_size
thf(fact_225_Common__Proof_Olist__size,axiom,
! [Common: state_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_nat @ Common ) )
=> ( ( ( common_list_nat @ Common )
= nil_nat )
=> ~ ( type_i5905290871511909543te_nat @ Common ) ) ) ).
% Common_Proof.list_size
thf(fact_226_Common__Proof_Olist__size,axiom,
! [Common: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( common_list_a @ Common )
= nil_a )
=> ~ ( type_i4669920168676019581tate_a @ Common ) ) ) ).
% Common_Proof.list_size
thf(fact_227_in__measures_I2_J,axiom,
! [X: list_nat,Y: list_nat,F: list_nat > nat,Fs: list_list_nat_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_228_in__measures_I2_J,axiom,
! [X: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_229_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_230_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_231_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_232_Current__Proof_Osize__new__push,axiom,
! [Current: current_a,X: a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( type_s933026853152659577rent_a @ ( push_a2 @ X @ Current ) )
= ( suc @ ( type_s933026853152659577rent_a @ Current ) ) ) ) ).
% Current_Proof.size_new_push
thf(fact_233_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_234_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_235_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_236_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_237_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_238_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_239_min_Obounded__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.bounded_iff
thf(fact_240_in__measures_I1_J,axiom,
! [X: list_nat,Y: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ nil_list_nat_nat ) ) ).
% in_measures(1)
thf(fact_241_in__measures_I1_J,axiom,
! [X: nat,Y: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).
% in_measures(1)
thf(fact_242_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_243_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_244_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_245_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_246_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_247_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_248_measures__lesseq,axiom,
! [F: list_nat > nat,X: list_nat,Y: list_nat,Fs: list_list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ Fs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_249_measures__lesseq,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_250_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_251_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_252_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_253_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_254_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_255_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_256_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_257_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_258_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_259_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_260_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_261_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X4: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_262_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_263_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_264_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_265_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_266_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_267_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_268_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_269_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_270_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_271_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_272_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_273_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z2: nat] :
( ( R2 @ X4 @ Y4 )
=> ( ( R2 @ Y4 @ Z2 )
=> ( R2 @ X4 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_274_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_275_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_276_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_277_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_278_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_279_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_280_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_281_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_282_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_283_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_284_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X4: nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X4 @ nil_nat ) ) )
=> ~ ! [P3: nat > nat > $o,X4: nat,Y4: nat,Xs2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X4 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_285_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X4: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X4: a,Y4: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_286_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ~ ! [P3: nat > nat > $o,X4: nat,Ys4: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_287_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X4: a,Ys4: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_288_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_289_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_290_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_291_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_292_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_293_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_294_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X4: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X4 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_295_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X4: nat] :
( X
!= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ! [X4: nat,Y4: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X4 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_296_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y4: a,Xs2: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_297_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y5: nat,Ys: list_nat] :
( Xs
= ( cons_nat @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_298_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y5: a,Ys: list_a] :
( Xs
= ( cons_a @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_299_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X4 @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys4 ) )
=> ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_300_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys2: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X4: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P @ nil_nat @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X4: nat,Xs2: list_nat,Y4: a,Ys4: list_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_301_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys2: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs2: list_a] : ( P @ ( cons_a @ X4 @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_a @ ( cons_nat @ Y4 @ Ys4 ) )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_302_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a] : ( P @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_303_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_304_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_305_min_Omono,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C2 @ D ) ) ) ) ).
% min.mono
thf(fact_306_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_307_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_308_min_OboundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% min.boundedE
thf(fact_309_min_OboundedI,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ) ).
% min.boundedI
thf(fact_310_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( ord_min_nat @ A5 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_311_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_312_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_313_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_min_nat @ A5 @ B4 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_314_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_min_nat @ A5 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_315_min_OcoboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.coboundedI1
thf(fact_316_min_OcoboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.coboundedI2
thf(fact_317_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_318_Current__Proof_Oinvar__push,axiom,
! [Current: current_a,X: a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( type_i6141643110573041459rent_a @ ( push_a2 @ X @ Current ) ) ) ).
% Current_Proof.invar_push
thf(fact_319_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_320_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_321_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_322_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_323_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_324_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_325_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_326_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_327_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_328_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_329_impossible__Cons,axiom,
! [Xs: list_a,Ys2: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs
!= ( cons_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_330_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_331_list__induct4,axiom,
! [Xs: list_nat,Ys2: 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 @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_332_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_333_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_334_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_335_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_336_list__induct4,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_337_list__induct4,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_338_list__induct4,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_339_list__induct4,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_340_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: nat,Xs2: list_nat,Y4: a,Ys4: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_341_list__induct3,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_342_list__induct3,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_343_list__induct3,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_344_list__induct3,axiom,
! [Xs: list_a,Ys2: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_345_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X4: nat,Xs2: list_nat,Y4: a,Ys4: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_346_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat,Y4: a,Ys4: list_a,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_347_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z2: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_348_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_349_list__induct2,axiom,
! [Xs: list_a,Ys2: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_350_list__induct2,axiom,
! [Xs: list_a,Ys2: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X4: a,Xs2: list_a,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_351_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X4: nat,Xs2: list_nat,Y4: a,Ys4: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_352_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_353_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_354_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_355_Current__Proof_Olist__size,axiom,
! [Current: current_nat] :
( ( type_i2033238430507501937nt_nat @ Current )
=> ( ( ( current_list_nat @ Current )
= nil_nat )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s5002997120254590536nt_nat @ Current ) ) ) ) ).
% Current_Proof.list_size
thf(fact_356_Current__Proof_Olist__size,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( current_list_a @ Current )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) ) ) ) ).
% Current_Proof.list_size
thf(fact_357_map__tailrec__rev_Ocases,axiom,
! [X: produc1616951275169580055st_nat] :
( ! [F2: nat > nat,Bs: list_nat] :
( X
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ nil_nat @ Bs ) ) )
=> ~ ! [F2: nat > nat,A4: nat,As: list_nat,Bs: list_nat] :
( X
!= ( produc4626581765195395529st_nat @ F2 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_358_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys4 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X4: a,Xs2: list_a,Y4: a,Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_359_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X4: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_360_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys4 ) )
=> ~ ! [X4: a,Xs2: list_a,Ys4: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_361_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) )
=> ~ ! [X4: nat,Xs2: list_nat,Ys4: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_362_Nil2__notin__lex,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_363_Nil2__notin__lex,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_364_Nil__notin__lex,axiom,
! [Ys2: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) @ ( lex_a @ R ) ) ).
% Nil_notin_lex
thf(fact_365_Nil__notin__lex,axiom,
! [Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) @ ( lex_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_366_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_367_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_368_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_369_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X5: a,Ys: list_a] :
( ( Xs
= ( cons_a @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_370_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X5: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_371_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_372_measures__less,axiom,
! [F: list_nat > nat,X: list_nat,Y: list_nat,Fs: list_list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( measures_list_nat @ ( cons_list_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_373_measures__less,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_374_size__pop__suc,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( suc @ ( size_size_current_a @ Current3 ) )
= ( size_size_current_a @ Current ) ) ) ) ) ).
% size_pop_suc
thf(fact_375_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_376_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_377_min__def,axiom,
( ord_min_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% min_def
thf(fact_378_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_379_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_380_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_381_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_382_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_383_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_384_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_385_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_386_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_387_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_388_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_389_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_390_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_391_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_392_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_393_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_394_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_395_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_396_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_397_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_398_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_399_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_400_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_401_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_402_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_403_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_404_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_405_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_406_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_407_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_408_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_409_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_410_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_411_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_412_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_413_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_414_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_415_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_416_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_417_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_418_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_419_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_420_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_421_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_422_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_423_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_424_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_425_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_426_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_427_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_428_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_429_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_430_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_431_order_Ostrict__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_432_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_433_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B3: num] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_434_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_435_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P5 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_436_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_437_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_438_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_439_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_440_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_441_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_442_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_443_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_444_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_445_ord__less__eq__trans,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_446_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_447_ord__eq__less__trans,axiom,
! [A: num,B: num,C2: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_448_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_449_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_450_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_451_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_452_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_453_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_454_pinf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_455_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_456_pinf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_457_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_458_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_459_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_460_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_461_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_462_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ~ ( ord_less_num @ X6 @ T ) ) ).
% pinf(5)
thf(fact_463_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_464_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ord_less_num @ T @ X6 ) ) ).
% pinf(7)
thf(fact_465_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_466_minf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_467_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_468_minf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_469_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_470_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_471_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_472_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_473_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_474_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ord_less_num @ X6 @ T ) ) ).
% minf(5)
thf(fact_475_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_476_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ~ ( ord_less_num @ T @ X6 ) ) ).
% minf(7)
thf(fact_477_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_478_Current__Proof_Oinvar__pop,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( type_i6141643110573041459rent_a @ Current3 ) ) ) ) ).
% Current_Proof.invar_pop
thf(fact_479_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_480_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_481_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_482_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_483_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_484_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_485_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_486_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_487_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_488_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_489_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_490_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_491_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_492_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_493_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_494_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_495_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_496_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_497_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_498_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_499_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_500_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_501_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_502_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_503_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_504_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_505_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_506_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_507_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_508_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_509_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_510_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_511_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_512_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_513_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_514_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_515_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_516_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_517_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_518_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_519_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_520_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_521_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_522_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_523_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_524_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_525_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_526_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_527_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_528_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_529_order_Ostrict__trans2,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_530_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_531_order_Ostrict__trans1,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_532_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_533_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_534_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_535_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_536_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_537_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_538_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_539_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_540_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_541_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_542_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_543_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_544_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_545_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_546_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_547_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_548_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_549_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_550_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_551_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_552_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_553_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_554_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_555_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_556_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_557_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_558_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_559_Current__Proof_Osize__new__pop,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( type_s933026853152659577rent_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( suc @ ( type_s933026853152659577rent_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) )
= ( type_s933026853152659577rent_a @ Current ) ) ) ) ).
% Current_Proof.size_new_pop
thf(fact_560_Current__Proof_Osize__pop,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( suc @ ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) )
= ( size_size_current_a @ Current ) ) ) ) ).
% Current_Proof.size_pop
thf(fact_561_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_562_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A3: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A3 ) )
= ( ord_less_num @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_563_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A3 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_564_snd__eqD,axiom,
! [X: a,Y: state_a2,A: state_a2] :
( ( ( produc6287816722867230257tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_565_snd__eqD,axiom,
! [X: list_nat,Y: list_nat,A: list_nat] :
( ( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_566_snd__eqD,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_567_snd__eqD,axiom,
! [X: a,Y: state_a,A: state_a] :
( ( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_568_snd__eqD,axiom,
! [X: a,Y: current_a,A: current_a] :
( ( ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_569_snd__conv,axiom,
! [X1: a,X2: state_a2] :
( ( produc6287816722867230257tate_a @ ( produc1224139502141355779tate_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_570_snd__conv,axiom,
! [X1: list_nat,X2: list_nat] :
( ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_571_snd__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_572_snd__conv,axiom,
! [X1: a,X2: state_a] :
( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_573_snd__conv,axiom,
! [X1: a,X2: current_a] :
( ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_574_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_575_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_576_invar__drop__first,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( type_i6141643110573041459rent_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) ) ) ).
% invar_drop_first
thf(fact_577_Small_Opop_Osimps_I2_J,axiom,
! [Current: current_a,Small: stack_a,AuxS2: list_a] :
( ( pop_a3 @ ( reverse1_a @ Current @ Small @ AuxS2 ) )
= ( produc1224139502141355779tate_a @ ( first_a @ Current ) @ ( reverse1_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) @ Small @ AuxS2 ) ) ) ).
% Small.pop.simps(2)
thf(fact_578_Small_Opop_Osimps_I3_J,axiom,
! [Current: current_a,AuxS2: list_a,Big: stack_a,NewS2: list_a,Count: nat] :
( ( pop_a3 @ ( reverse2_a @ Current @ AuxS2 @ Big @ NewS2 @ Count ) )
= ( produc1224139502141355779tate_a @ ( first_a @ Current ) @ ( reverse2_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) @ AuxS2 @ Big @ NewS2 @ Count ) ) ) ).
% Small.pop.simps(3)
thf(fact_579_drop__first__list,axiom,
! [Current: current_nat] :
( ( type_i2033238430507501937nt_nat @ Current )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s5002997120254590536nt_nat @ Current ) )
=> ( ( current_list_nat @ ( produc8758756790178850379nt_nat @ ( pop_nat2 @ Current ) ) )
= ( tl_nat @ ( current_list_nat @ Current ) ) ) ) ) ).
% drop_first_list
thf(fact_580_drop__first__list,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( current_list_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) )
= ( tl_a @ ( current_list_a @ Current ) ) ) ) ) ).
% drop_first_list
thf(fact_581_eq__snd__iff,axiom,
! [B: state_a2,P2: produc7589950997499123219tate_a] :
( ( B
= ( produc6287816722867230257tate_a @ P2 ) )
= ( ? [A5: a] :
( P2
= ( produc1224139502141355779tate_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_582_eq__snd__iff,axiom,
! [B: list_nat,P2: produc1828647624359046049st_nat] :
( ( B
= ( produc5865812112468994567st_nat @ P2 ) )
= ( ? [A5: list_nat] :
( P2
= ( produc2694037385005941721st_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_583_eq__snd__iff,axiom,
! [B: nat,P2: product_prod_nat_nat] :
( ( B
= ( product_snd_nat_nat @ P2 ) )
= ( ? [A5: nat] :
( P2
= ( product_Pair_nat_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_584_eq__snd__iff,axiom,
! [B: state_a,P2: produc3409137331138395373tate_a] :
( ( B
= ( produc681690970763031737tate_a @ P2 ) )
= ( ? [A5: a] :
( P2
= ( produc8263595898873874535tate_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_585_eq__snd__iff,axiom,
! [B: current_a,P2: produc7805042584321970905rent_a] :
( ( B
= ( produc4695312889421393143rent_a @ P2 ) )
= ( ? [A5: a] :
( P2
= ( produc8503237746132909001rent_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_586_sndI,axiom,
! [X: produc7589950997499123219tate_a,Y: a,Z: state_a2] :
( ( X
= ( produc1224139502141355779tate_a @ Y @ Z ) )
=> ( ( produc6287816722867230257tate_a @ X )
= Z ) ) ).
% sndI
thf(fact_587_sndI,axiom,
! [X: produc1828647624359046049st_nat,Y: list_nat,Z: list_nat] :
( ( X
= ( produc2694037385005941721st_nat @ Y @ Z ) )
=> ( ( produc5865812112468994567st_nat @ X )
= Z ) ) ).
% sndI
thf(fact_588_sndI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_snd_nat_nat @ X )
= Z ) ) ).
% sndI
thf(fact_589_sndI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z ) )
=> ( ( produc681690970763031737tate_a @ X )
= Z ) ) ).
% sndI
thf(fact_590_sndI,axiom,
! [X: produc7805042584321970905rent_a,Y: a,Z: current_a] :
( ( X
= ( produc8503237746132909001rent_a @ Y @ Z ) )
=> ( ( produc4695312889421393143rent_a @ X )
= Z ) ) ).
% sndI
thf(fact_591_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_592_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_593_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_594_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_595_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X5: nat] :
( Xs
= ( cons_nat @ X5 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_596_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_597_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X5: nat] :
( Xs
= ( cons_nat @ X5 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_598_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_599_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_600_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_601_Current__Proof_Opop__list,axiom,
! [Current: current_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5002997120254590536nt_nat @ Current ) )
=> ( ( type_i2033238430507501937nt_nat @ Current )
=> ( ( cons_nat @ ( produc4614407304758426121nt_nat @ ( pop_nat2 @ Current ) ) @ ( tl_nat @ ( current_list_nat @ Current ) ) )
= ( current_list_nat @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_602_Current__Proof_Opop__list,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( cons_a @ ( produc4952273589686483381rent_a @ ( pop_a2 @ Current ) ) @ ( tl_a @ ( current_list_a @ Current ) ) )
= ( current_list_a @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_603_cons__tl,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= Ys2 )
=> ( Xs
= ( tl_nat @ Ys2 ) ) ) ).
% cons_tl
thf(fact_604_cons__tl,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys2 )
=> ( Xs
= ( tl_a @ Ys2 ) ) ) ).
% cons_tl
thf(fact_605_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_606_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_607_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_608_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_609_prod_Ocollapse,axiom,
! [Prod: produc7589950997499123219tate_a] :
( ( produc1224139502141355779tate_a @ ( produc8617775573817090287tate_a @ Prod ) @ ( produc6287816722867230257tate_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_610_prod_Ocollapse,axiom,
! [Prod: produc1828647624359046049st_nat] :
( ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_611_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_612_prod_Ocollapse,axiom,
! [Prod: produc3409137331138395373tate_a] :
( ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_613_prod_Ocollapse,axiom,
! [Prod: produc7805042584321970905rent_a] :
( ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ Prod ) @ ( produc4695312889421393143rent_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_614_prod__eq__iff,axiom,
( ( ^ [Y7: produc7805042584321970905rent_a,Z4: produc7805042584321970905rent_a] : ( Y7 = Z4 ) )
= ( ^ [S2: produc7805042584321970905rent_a,T2: produc7805042584321970905rent_a] :
( ( ( produc4952273589686483381rent_a @ S2 )
= ( produc4952273589686483381rent_a @ T2 ) )
& ( ( produc4695312889421393143rent_a @ S2 )
= ( produc4695312889421393143rent_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_615_prod__eqI,axiom,
! [P2: produc7805042584321970905rent_a,Q3: produc7805042584321970905rent_a] :
( ( ( produc4952273589686483381rent_a @ P2 )
= ( produc4952273589686483381rent_a @ Q3 ) )
=> ( ( ( produc4695312889421393143rent_a @ P2 )
= ( produc4695312889421393143rent_a @ Q3 ) )
=> ( P2 = Q3 ) ) ) ).
% prod_eqI
thf(fact_616_prod_Oexpand,axiom,
! [Prod: produc7805042584321970905rent_a,Prod2: produc7805042584321970905rent_a] :
( ( ( ( produc4952273589686483381rent_a @ Prod )
= ( produc4952273589686483381rent_a @ Prod2 ) )
& ( ( produc4695312889421393143rent_a @ Prod )
= ( produc4695312889421393143rent_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_617_fstI,axiom,
! [X: produc7589950997499123219tate_a,Y: a,Z: state_a2] :
( ( X
= ( produc1224139502141355779tate_a @ Y @ Z ) )
=> ( ( produc8617775573817090287tate_a @ X )
= Y ) ) ).
% fstI
thf(fact_618_fstI,axiom,
! [X: produc1828647624359046049st_nat,Y: list_nat,Z: list_nat] :
( ( X
= ( produc2694037385005941721st_nat @ Y @ Z ) )
=> ( ( produc1382935764643595205st_nat @ X )
= Y ) ) ).
% fstI
thf(fact_619_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_620_fstI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z ) )
=> ( ( produc3154331710141225339tate_a @ X )
= Y ) ) ).
% fstI
thf(fact_621_eq__fst__iff,axiom,
! [A: a,P2: produc7589950997499123219tate_a] :
( ( A
= ( produc8617775573817090287tate_a @ P2 ) )
= ( ? [B4: state_a2] :
( P2
= ( produc1224139502141355779tate_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_622_eq__fst__iff,axiom,
! [A: list_nat,P2: produc1828647624359046049st_nat] :
( ( A
= ( produc1382935764643595205st_nat @ P2 ) )
= ( ? [B4: list_nat] :
( P2
= ( produc2694037385005941721st_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_623_eq__fst__iff,axiom,
! [A: nat,P2: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P2 ) )
= ( ? [B4: nat] :
( P2
= ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_624_eq__fst__iff,axiom,
! [A: a,P2: produc3409137331138395373tate_a] :
( ( A
= ( produc3154331710141225339tate_a @ P2 ) )
= ( ? [B4: state_a] :
( P2
= ( produc8263595898873874535tate_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_625_fst__conv,axiom,
! [X1: a,X2: state_a2] :
( ( produc8617775573817090287tate_a @ ( produc1224139502141355779tate_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_626_fst__conv,axiom,
! [X1: list_nat,X2: list_nat] :
( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_627_fst__conv,axiom,
! [X1: nat,X2: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_628_fst__conv,axiom,
! [X1: a,X2: state_a] :
( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_629_fst__eqD,axiom,
! [X: a,Y: state_a2,A: a] :
( ( ( produc8617775573817090287tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_630_fst__eqD,axiom,
! [X: list_nat,Y: list_nat,A: list_nat] :
( ( ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_631_fst__eqD,axiom,
! [X: nat,Y: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_632_fst__eqD,axiom,
! [X: a,Y: state_a,A: a] :
( ( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_633_prod_Oexhaust__sel,axiom,
! [Prod: produc7589950997499123219tate_a] :
( Prod
= ( produc1224139502141355779tate_a @ ( produc8617775573817090287tate_a @ Prod ) @ ( produc6287816722867230257tate_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_634_prod_Oexhaust__sel,axiom,
! [Prod: produc1828647624359046049st_nat] :
( Prod
= ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ Prod ) @ ( produc5865812112468994567st_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_635_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_636_prod_Oexhaust__sel,axiom,
! [Prod: produc3409137331138395373tate_a] :
( Prod
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_637_prod_Oexhaust__sel,axiom,
! [Prod: produc7805042584321970905rent_a] :
( Prod
= ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ Prod ) @ ( produc4695312889421393143rent_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_638_surjective__pairing,axiom,
! [T: produc7589950997499123219tate_a] :
( T
= ( produc1224139502141355779tate_a @ ( produc8617775573817090287tate_a @ T ) @ ( produc6287816722867230257tate_a @ T ) ) ) ).
% surjective_pairing
thf(fact_639_surjective__pairing,axiom,
! [T: produc1828647624359046049st_nat] :
( T
= ( produc2694037385005941721st_nat @ ( produc1382935764643595205st_nat @ T ) @ ( produc5865812112468994567st_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_640_surjective__pairing,axiom,
! [T: product_prod_nat_nat] :
( T
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).
% surjective_pairing
thf(fact_641_surjective__pairing,axiom,
! [T: produc3409137331138395373tate_a] :
( T
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ T ) @ ( produc681690970763031737tate_a @ T ) ) ) ).
% surjective_pairing
thf(fact_642_surjective__pairing,axiom,
! [T: produc7805042584321970905rent_a] :
( T
= ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ T ) @ ( produc4695312889421393143rent_a @ T ) ) ) ).
% surjective_pairing
thf(fact_643_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > state_a2 > $o,X: a,Y: state_a2,A: produc7589950997499123219tate_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc1224139502141355779tate_a @ X @ Y ) )
=> ( P @ ( produc8617775573817090287tate_a @ A ) @ ( produc6287816722867230257tate_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_644_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_nat > list_nat > $o,X: list_nat,Y: list_nat,A: produc1828647624359046049st_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( produc2694037385005941721st_nat @ X @ Y ) )
=> ( P @ ( produc1382935764643595205st_nat @ A ) @ ( produc5865812112468994567st_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_645_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_646_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > state_a > $o,X: a,Y: state_a,A: produc3409137331138395373tate_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8263595898873874535tate_a @ X @ Y ) )
=> ( P @ ( produc3154331710141225339tate_a @ A ) @ ( produc681690970763031737tate_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_647_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > current_a > $o,X: a,Y: current_a,A: produc7805042584321970905rent_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8503237746132909001rent_a @ X @ Y ) )
=> ( P @ ( produc4952273589686483381rent_a @ A ) @ ( produc4695312889421393143rent_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_648_exI__realizer,axiom,
! [P: state_a2 > a > $o,Y: state_a2,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc6287816722867230257tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) ) @ ( produc8617775573817090287tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_649_exI__realizer,axiom,
! [P: list_nat > list_nat > $o,Y: list_nat,X: list_nat] :
( ( P @ Y @ X )
=> ( P @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_650_exI__realizer,axiom,
! [P: nat > nat > $o,Y: nat,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_651_exI__realizer,axiom,
! [P: state_a > a > $o,Y: state_a,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) ) @ ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_652_exI__realizer,axiom,
! [P: current_a > a > $o,Y: current_a,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) ) @ ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_653_conjI__realizer,axiom,
! [P: a > $o,P2: a,Q: state_a2 > $o,Q3: state_a2] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc8617775573817090287tate_a @ ( produc1224139502141355779tate_a @ P2 @ Q3 ) ) )
& ( Q @ ( produc6287816722867230257tate_a @ ( produc1224139502141355779tate_a @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_654_conjI__realizer,axiom,
! [P: list_nat > $o,P2: list_nat,Q: list_nat > $o,Q3: list_nat] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc1382935764643595205st_nat @ ( produc2694037385005941721st_nat @ P2 @ Q3 ) ) )
& ( Q @ ( produc5865812112468994567st_nat @ ( produc2694037385005941721st_nat @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_655_conjI__realizer,axiom,
! [P: nat > $o,P2: nat,Q: nat > $o,Q3: nat] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q3 ) ) )
& ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_656_conjI__realizer,axiom,
! [P: a > $o,P2: a,Q: state_a > $o,Q3: state_a] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ P2 @ Q3 ) ) )
& ( Q @ ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_657_conjI__realizer,axiom,
! [P: a > $o,P2: a,Q: current_a > $o,Q3: current_a] :
( ( P @ P2 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ P2 @ Q3 ) ) )
& ( Q @ ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ P2 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_658_remdups__adj__length__ge1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_659_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_660_Suc__min,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( suc @ ( ord_min_nat @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( minus_minus_nat @ Y @ ( suc @ zero_zero_nat ) ) ) )
= ( ord_min_nat @ X @ Y ) ) ) ) ).
% Suc_min
thf(fact_661_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_662_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_663_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_664_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_665_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_666_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_667_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_668_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_669_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_670_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_671_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_672_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_673_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_674_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_675_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_676_remdups__adj__Cons__alt,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) )
= ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_677_remdups__adj__Cons__alt,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_678_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_679_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_680_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_681_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_682_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_683_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_684_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_685_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_686_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_687_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_688_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_689_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_690_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_691_remdups__adj_Osimps_I3_J,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( ( X = Y )
=> ( ( remdups_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_692_remdups__adj_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_693_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_694_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_695_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_696_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_697_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_698_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_699_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_700_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_701_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_702_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_703_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_704_remdups__adj_Oelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ( ! [X4: a] :
( ( X
= ( cons_a @ X4 @ nil_a ) )
=> ( Y
!= ( cons_a @ X4 @ nil_a ) ) )
=> ~ ! [X4: a,Y4: a,Xs2: list_a] :
( ( X
= ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs2 ) ) )
=> ~ ( ( ( X4 = Y4 )
=> ( Y
= ( remdups_adj_a @ ( cons_a @ X4 @ Xs2 ) ) ) )
& ( ( X4 != Y4 )
=> ( Y
= ( cons_a @ X4 @ ( remdups_adj_a @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_705_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X4: nat] :
( ( X
= ( cons_nat @ X4 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X4 @ nil_nat ) ) )
=> ~ ! [X4: nat,Y4: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X4 @ ( cons_nat @ Y4 @ Xs2 ) ) )
=> ~ ( ( ( X4 = Y4 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X4 @ Xs2 ) ) ) )
& ( ( X4 != Y4 )
=> ( Y
= ( cons_nat @ X4 @ ( remdups_adj_nat @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_706_remdups__adj_Osimps_I2_J,axiom,
! [X: a] :
( ( remdups_adj_a @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_707_remdups__adj_Osimps_I2_J,axiom,
! [X: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_708_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_709_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_710_Current__Proof_Osize__pop__sub,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( size_size_current_a @ Current3 )
= ( minus_minus_nat @ ( size_size_current_a @ Current ) @ one_one_nat ) ) ) ) ) ).
% Current_Proof.size_pop_sub
thf(fact_711_size__drop__first__sub,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) )
= ( minus_minus_nat @ ( size_size_current_a @ Current ) @ one_one_nat ) ) ) ) ).
% size_drop_first_sub
thf(fact_712_remaining__steps__step__sub,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( minus_minus_nat @ ( type_r2212416260012024137tate_a @ Common ) @ one_one_nat ) ) ) ).
% remaining_steps_step_sub
thf(fact_713_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_714_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_715_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_716_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_717_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_718_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_719_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_720_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_721_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_722_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_723_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_724_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_725_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_726_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_727_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_728_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_729_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_730_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_731_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_732_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_733_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_734_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_735_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_736_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_737_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_738_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_739_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_740_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_741_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_742_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_743_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_744_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_745_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_746_list__eq__iff__nth__eq,axiom,
( ( ^ [Y7: list_nat,Z4: list_nat] : ( Y7 = Z4 ) )
= ( ^ [Xs3: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_747_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X8: nat] : ( P @ I2 @ X8 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_748_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_749_nth__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_750_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_751_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_752_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_753_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_754_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_755_lex__take__index,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys2 ) @ ( lex_list_nat @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( take_list_nat @ I3 @ Xs )
= ( take_list_nat @ I3 @ Ys2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Ys2 @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_756_lex__take__index,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_757_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_state_a2,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s8463391772401140188tate_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ ( nth_a @ Xs @ N4 ) @ ( nth_state_a2 @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_758_listrel__iff__nth,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys2 ) @ ( listre6091228620945859379st_nat @ R ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ Xs @ N4 ) @ ( nth_list_nat @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_759_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s4663437186682225508tate_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ ( nth_a @ Xs @ N4 ) @ ( nth_state_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_760_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ ( nth_nat @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_761_take__take,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( take_nat @ N @ ( take_nat @ M @ Xs ) )
= ( take_nat @ ( ord_min_nat @ N @ M ) @ Xs ) ) ).
% take_take
thf(fact_762_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_763_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_764_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_765_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_766_take__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_767_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_768_take__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_769_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_770_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_771_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_772_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_773_length__take,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( take_nat @ N @ Xs ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_take
thf(fact_774_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_775_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_776_take__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ).
% take_equalityI
thf(fact_777_listrel__mono,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R @ S )
=> ( ord_le8406513867147106209st_nat @ ( listrel_nat_nat @ R ) @ ( listrel_nat_nat @ S ) ) ) ).
% listrel_mono
thf(fact_778_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_779_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_780_take__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_781_listrel__Nil2,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_782_listrel__Nil2,axiom,
! [Xs: list_nat,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ nil_a ) @ ( listrel_nat_a @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_783_listrel__Nil2,axiom,
! [Xs: list_a,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ nil_nat ) @ ( listrel_a_nat @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_784_listrel__Nil2,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_785_listrel__Nil1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_786_listrel__Nil1,axiom,
! [Xs: list_nat,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ nil_a @ Xs ) @ ( listrel_a_nat @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_787_listrel__Nil1,axiom,
! [Xs: list_a,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ nil_nat @ Xs ) @ ( listrel_nat_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_788_listrel__Nil1,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_789_listrel_ONil,axiom,
! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).
% listrel.Nil
thf(fact_790_listrel_ONil,axiom,
! [R: set_Pr4934435412358123699_a_nat] : ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ nil_a @ nil_nat ) @ ( listrel_a_nat @ R ) ) ).
% listrel.Nil
thf(fact_791_listrel_ONil,axiom,
! [R: set_Pr4193341848836149977_nat_a] : ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ nil_nat @ nil_a ) @ ( listrel_nat_a @ R ) ) ).
% listrel.Nil
thf(fact_792_listrel_ONil,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).
% listrel.Nil
thf(fact_793_listrel__eq__len,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_794_take__1,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ zero_zero_nat @ X )
& ( ord_less_nat @ zero_zero_nat @ Y ) )
=> ( ( ( take_nat @ X @ Xs )
= ( take_nat @ Y @ Ys2 ) )
=> ( ( take_nat @ one_one_nat @ Xs )
= ( take_nat @ one_one_nat @ Ys2 ) ) ) ) ).
% take_1
thf(fact_795_tl__take,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs ) ) ) ).
% tl_take
thf(fact_796_listrel_OCons,axiom,
! [X: nat,Y: a,R: set_Pr4193341848836149977_nat_a,Xs: list_nat,Ys2: list_a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ R )
=> ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys2 ) @ ( listrel_nat_a @ R ) )
=> ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ ( cons_nat @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_nat_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_797_listrel_OCons,axiom,
! [X: a,Y: nat,R: set_Pr4934435412358123699_a_nat,Xs: list_a,Ys2: list_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
=> ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys2 ) @ ( listrel_a_nat @ R ) )
=> ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_a_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_798_listrel_OCons,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Ys2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_799_listrel_OCons,axiom,
! [X: a,Y: state_a2,R: set_Pr6306228930610421491tate_a,Xs: list_a,Ys2: list_state_a2] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) @ R )
=> ( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a2 @ R ) )
=> ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a2 @ Y @ Ys2 ) ) @ ( listrel_a_state_a2 @ R ) ) ) ) ).
% listrel.Cons
thf(fact_800_listrel_OCons,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys2 ) @ ( listre6091228620945859379st_nat @ R ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) ) @ ( listre6091228620945859379st_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_801_listrel_OCons,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_802_listrel_OCons,axiom,
! [X: a,Y: state_a,R: set_Pr324718442235990179tate_a,Xs: list_a,Ys2: list_state_a] :
( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) @ R )
=> ( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a @ R ) )
=> ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a @ Y @ Ys2 ) ) @ ( listrel_a_state_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_803_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs: list_a,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ ( cons_nat @ Y @ Ys2 ) @ Xs ) @ ( listrel_nat_a @ R ) )
=> ~ ! [Y4: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ Y @ Y4 ) @ R )
=> ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Ys2 @ Ys4 ) @ ( listrel_nat_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_804_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_nat,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_nat @ R ) )
=> ~ ! [Y4: nat,Ys4: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Y4 ) @ R )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Ys2 @ Ys4 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_805_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y4: a,Ys4: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y4 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Ys4 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_806_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_state_a2,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [Y4: state_a2,Ys4: list_state_a2] :
( ( Xs
= ( cons_state_a2 @ Y4 @ Ys4 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ Y @ Y4 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Ys2 @ Ys4 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_807_listrel__Cons1,axiom,
! [Y: list_nat,Ys2: list_list_nat,Xs: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ Y @ Ys2 ) @ Xs ) @ ( listre6091228620945859379st_nat @ R ) )
=> ~ ! [Y4: list_nat,Ys4: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y4 @ Ys4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ Y4 ) @ R )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys2 @ Ys4 ) @ ( listre6091228620945859379st_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_808_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [Y4: state_a,Ys4: list_state_a] :
( ( Xs
= ( cons_state_a @ Y4 @ Ys4 ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ Y @ Y4 ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Ys2 @ Ys4 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_809_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys2 ) @ Xs ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [Y4: nat,Ys4: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y4 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_810_listrel__Cons2,axiom,
! [Xs: list_a,Y: nat,Ys2: list_nat,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_a_nat @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y ) @ R )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys2 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_811_listrel__Cons2,axiom,
! [Xs: list_nat,Y: a,Ys2: list_a,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_nat_a @ R ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Xs2 ) )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y ) @ R )
=> ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs2 @ Ys2 ) @ ( listrel_nat_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_812_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_813_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a2,Ys2: list_state_a2,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ ( cons_state_a2 @ Y @ Ys2 ) ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys2 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_814_listrel__Cons2,axiom,
! [Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( cons_list_nat @ Y @ Ys2 ) ) @ ( listre6091228620945859379st_nat @ R ) )
=> ~ ! [X4: list_nat,Xs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ X4 @ Xs2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y ) @ R )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys2 ) @ ( listre6091228620945859379st_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_815_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a,Ys2: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs @ ( cons_state_a @ Y @ Ys2 ) ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X4 @ Y ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ Ys2 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_816_listrel__Cons2,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [X4: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_817_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_818_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_a,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ A1 @ A22 ) @ ( listrel_nat_a @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_a ) )
=> ~ ! [X4: nat,Y4: a,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs2 ) )
=> ! [Ys4: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y4 ) @ R )
=> ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs2 @ Ys4 ) @ ( listrel_nat_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_819_listrel_Ocases,axiom,
! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X4: a,Y4: nat,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys4: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y4 ) @ R )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys4 ) @ ( listrel_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_820_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X4: a,Y4: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys4: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys4 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_821_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a2 ) )
=> ~ ! [X4: a,Y4: state_a2,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys4: list_state_a2] :
( ( A22
= ( cons_state_a2 @ Y4 @ Ys4 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y4 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys4 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_822_listrel_Ocases,axiom,
! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( listre6091228620945859379st_nat @ R ) )
=> ( ( ( A1 = nil_list_nat )
=> ( A22 != nil_list_nat ) )
=> ~ ! [X4: list_nat,Y4: list_nat,Xs2: list_list_nat] :
( ( A1
= ( cons_list_nat @ X4 @ Xs2 ) )
=> ! [Ys4: list_list_nat] :
( ( A22
= ( cons_list_nat @ Y4 @ Ys4 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y4 ) @ R )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs2 @ Ys4 ) @ ( listre6091228620945859379st_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_823_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X4: a,Y4: state_a,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys4: list_state_a] :
( ( A22
= ( cons_state_a @ Y4 @ Ys4 ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X4 @ Y4 ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ Ys4 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_824_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X4: nat,Y4: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs2 ) )
=> ! [Ys4: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys4 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_825_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_a,R: set_Pr4193341848836149977_nat_a] :
( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ A1 @ A22 ) @ ( listrel_nat_a @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_a ) )
| ? [X5: nat,Y5: a,Xs3: list_nat,Ys: list_a] :
( ( A1
= ( cons_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_a @ Y5 @ Ys ) )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X5 @ Y5 ) @ R )
& ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs3 @ Ys ) @ ( listrel_nat_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_826_listrel_Osimps,axiom,
! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_nat ) )
| ? [X5: a,Y5: nat,Xs3: list_a,Ys: list_nat] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y5 @ Ys ) )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X5 @ Y5 ) @ R )
& ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys ) @ ( listrel_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_827_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X5: a,Y5: a,Xs3: list_a,Ys: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_a @ Y5 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_828_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a2 ) )
| ? [X5: a,Y5: state_a2,Xs3: list_a,Ys: list_state_a2] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_state_a2 @ Y5 @ Ys ) )
& ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y5 ) @ R )
& ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs3 @ Ys ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_829_listrel_Osimps,axiom,
! [A1: list_list_nat,A22: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ A1 @ A22 ) @ ( listre6091228620945859379st_nat @ R ) )
= ( ( ( A1 = nil_list_nat )
& ( A22 = nil_list_nat ) )
| ? [X5: list_nat,Y5: list_nat,Xs3: list_list_nat,Ys: list_list_nat] :
( ( A1
= ( cons_list_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_list_nat @ Y5 @ Ys ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X5 @ Y5 ) @ R )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs3 @ Ys ) @ ( listre6091228620945859379st_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_830_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a ) )
| ? [X5: a,Y5: state_a,Xs3: list_a,Ys: list_state_a] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y5 @ Ys ) )
& ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X5 @ Y5 ) @ R )
& ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs3 @ Ys ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_831_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X5: nat,Y5: nat,Xs3: list_nat,Ys: list_nat] :
( ( A1
= ( cons_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y5 @ Ys ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_832_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_833_take__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_834_lexord__take__index__conv,axiom,
! [X: list_list_nat,Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ X ) @ ( size_s3023201423986296836st_nat @ Y ) )
& ( ( take_list_nat @ ( size_s3023201423986296836st_nat @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_s3023201423986296836st_nat @ X ) @ ( size_s3023201423986296836st_nat @ Y ) ) )
& ( ( take_list_nat @ I2 @ X )
= ( take_list_nat @ I2 @ Y ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( nth_list_nat @ X @ I2 ) @ ( nth_list_nat @ Y @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_835_lexord__take__index__conv,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
& ( ( take_nat @ ( size_size_list_nat @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) ) )
& ( ( take_nat @ I2 @ X )
= ( take_nat @ I2 @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X @ I2 ) @ ( nth_nat @ Y @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_836_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_837_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_838_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_839_take__hd,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ).
% take_hd
thf(fact_840_take__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% take_hd
thf(fact_841_same__append__eq,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_842_append__same__eq,axiom,
! [Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys2 @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_843_append__assoc,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_844_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
= ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_845_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_846_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_847_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_848_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_849_append__self__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_a ) ) ).
% append_self_conv
thf(fact_850_append__self__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_851_self__append__conv,axiom,
! [Y: list_a,Ys2: list_a] :
( ( Y
= ( append_a @ Y @ Ys2 ) )
= ( Ys2 = nil_a ) ) ).
% self_append_conv
thf(fact_852_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_853_append__self__conv2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_854_append__self__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_855_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_856_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_857_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_858_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_859_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_860_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_861_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_862_hd__remdups__adj,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( remdups_adj_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% hd_remdups_adj
thf(fact_863_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_864_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_865_hd__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_866_hd__append2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_867_tl__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_868_tl__append2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_869_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys2: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys2 ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_870_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_871_take__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_872_hd__take,axiom,
! [J2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( hd_nat @ ( take_nat @ J2 @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_873_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_874_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_875_lexord__cons__cons,axiom,
! [A: a,X: list_a,B: a,Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_876_lexord__cons__cons,axiom,
! [A: list_nat,X: list_list_nat,B: list_nat,Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ A @ X ) @ ( cons_list_nat @ B @ Y ) ) @ ( lexord_list_nat @ R ) )
= ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_877_lexord__cons__cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_878_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_879_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_880_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_881_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_882_lexord__Nil__left,axiom,
! [Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R ) )
= ( ? [A5: a,X5: list_a] :
( Y
= ( cons_a @ A5 @ X5 ) ) ) ) ).
% lexord_Nil_left
thf(fact_883_lexord__Nil__left,axiom,
! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
= ( ? [A5: nat,X5: list_nat] :
( Y
= ( cons_nat @ A5 @ X5 ) ) ) ) ).
% lexord_Nil_left
thf(fact_884_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_885_lexord__sufI,axiom,
! [U: list_nat,W2: list_nat,R: set_Pr1261947904930325089at_nat,V: list_nat,Z: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V ) @ ( append_nat @ W2 @ Z ) ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_sufI
thf(fact_886_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_887_longest__common__prefix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ps: list_a,Xs4: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys2
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs4 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_888_longest__common__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ps: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs4 ) )
& ( Ys2
= ( append_nat @ Ps @ Ys5 ) )
& ( ( Xs4 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( hd_nat @ Xs4 )
!= ( hd_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_889_butlast__append,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( ( Ys2 = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys2 != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ Xs @ ( butlast_a @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_890_butlast__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_891_hd__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_892_hd__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Ys2 ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_893_lexord__append__leftI,axiom,
! [U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_leftI
thf(fact_894_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_895_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_896_append__Nil,axiom,
! [Ys2: list_a] :
( ( append_a @ nil_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_897_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_898_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_899_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_900_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_a @ nil_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_901_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_902_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_903_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys2: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_904_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_905_append__Cons,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys2 )
= ( cons_a @ X @ ( append_a @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_906_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_907_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_908_lexord__append__rightI,axiom,
! [Y: list_a,X: list_a,R: set_Product_prod_a_a] :
( ? [B5: a,Z3: list_a] :
( Y
= ( cons_a @ B5 @ Z3 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ ( append_a @ X @ Y ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_rightI
thf(fact_909_lexord__append__rightI,axiom,
! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
( ? [B5: nat,Z3: list_nat] :
( Y
= ( cons_nat @ B5 @ Z3 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_rightI
thf(fact_910_lexord__append__leftD,axiom,
! [X: list_list_nat,U: list_list_nat,V: list_list_nat,R: set_Pr3451248702717554689st_nat] :
( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ X @ U ) @ ( append_list_nat @ X @ V ) ) @ ( lexord_list_nat @ R ) )
=> ( ! [A4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ A4 ) @ R )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ U @ V ) @ ( lexord_list_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_911_lexord__append__leftD,axiom,
! [X: list_nat,U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) )
=> ( ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_912_lexord__sufE,axiom,
! [Xs: list_nat,Zs: list_nat,Ys2: list_nat,Qs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys2 @ Qs ) ) @ ( lexord_nat @ R ) )
=> ( ( Xs != Ys2 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Qs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lexord_nat @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_913_cons__hd,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= Ys2 )
=> ( X
= ( hd_nat @ Ys2 ) ) ) ).
% cons_hd
thf(fact_914_cons__hd,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys2 )
=> ( X
= ( hd_a @ Ys2 ) ) ) ).
% cons_hd
thf(fact_915_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_916_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_917_lexord__append__left__rightI,axiom,
! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_918_lexord__append__left__rightI,axiom,
! [A: list_nat,B: list_nat,R: set_Pr3451248702717554689st_nat,U: list_list_nat,X: list_list_nat,Y: list_list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ U @ ( cons_list_nat @ A @ X ) ) @ ( append_list_nat @ U @ ( cons_list_nat @ B @ Y ) ) ) @ ( lexord_list_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_919_lexord__append__left__rightI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_920_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_921_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_922_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y4: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_923_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys4: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys4 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_924_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X @ Ys6 )
= Ys2 )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_925_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys2 )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_926_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys6: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_927_append__eq__Cons__conv,axiom,
! [Ys2: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys2 @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys2 = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys2
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_928_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_929_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_930_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_931_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_932_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_933_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_934_tl__append__if,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( tl_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_935_tl__append__if,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( tl_nat @ Ys2 ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_936_lex__append__leftI,axiom,
! [Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) ) ) ).
% lex_append_leftI
thf(fact_937_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_938_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_939_lexord__linear,axiom,
! [R: set_Pr3451248702717554689st_nat,X: list_list_nat,Y: list_list_nat] :
( ! [A4: list_nat,B3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B3 @ A4 ) @ R ) )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) )
| ( X = Y )
| ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Y @ X ) @ ( lexord_list_nat @ R ) ) ) ) ).
% lexord_linear
thf(fact_940_lexord__linear,axiom,
! [R: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
( ! [A4: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A4 ) @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
| ( X = Y )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_linear
thf(fact_941_lexord__irreflexive,axiom,
! [R: set_Pr3451248702717554689st_nat,Xs: list_list_nat] :
( ! [X4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ X4 ) @ R )
=> ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Xs ) @ ( lexord_list_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_942_lexord__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_943_lexord__Nil__right,axiom,
! [X: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ nil_a ) @ ( lexord_a @ R ) ) ).
% lexord_Nil_right
thf(fact_944_lexord__Nil__right,axiom,
! [X: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ nil_nat ) @ ( lexord_nat @ R ) ) ).
% lexord_Nil_right
thf(fact_945_same__length__different,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ? [Pre: list_a,X4: a,Xs4: list_a,Y4: a,Ys5: list_a] :
( ( X4 != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_946_same__length__different,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X4: nat,Xs4: list_nat,Y4: nat,Ys5: list_nat] :
( ( X4 != Y4 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_947_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_948_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_949_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_950_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_951_remdups__adj__append__two,axiom,
! [Xs: list_a,X: a,Y: a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X @ ( cons_a @ Y @ nil_a ) ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) @ ( if_list_a @ ( X = Y ) @ nil_a @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_952_remdups__adj__append__two,axiom,
! [Xs: list_nat,X: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) @ ( if_list_nat @ ( X = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_953_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_954_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_955_lex__append__leftD,axiom,
! [R: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ! [X4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ X4 ) @ R )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs @ Ys2 ) @ ( append_list_nat @ Xs @ Zs ) ) @ ( lex_list_nat @ R ) )
=> ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys2 @ Zs ) @ ( lex_list_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_956_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_957_lex__append__left__iff,axiom,
! [R: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ! [X4: list_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ X4 ) @ R )
=> ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs @ Ys2 ) @ ( append_list_nat @ Xs @ Zs ) ) @ ( lex_list_nat @ R ) )
= ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Ys2 @ Zs ) @ ( lex_list_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_958_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_959_lex__append__rightI,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_960_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys2 ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_961_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_962_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_963_lexord__lex,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R ) )
= ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
& ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ) ).
% lexord_lex
thf(fact_964_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_965_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_966_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y5 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_967_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y5 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_968_length__one__hd,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% length_one_hd
thf(fact_969_length__one__hd,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ).
% length_one_hd
thf(fact_970_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_971_remdups__adj__append,axiom,
! [Xs_1: list_a,X: a,Xs_2: list_a] :
( ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X @ Xs_2 ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X @ nil_a ) ) ) @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_972_remdups__adj__append,axiom,
! [Xs_1: list_nat,X: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_973_take__hd_H,axiom,
! [Ys2: list_a,X: a,Xs: list_a] :
( ( Ys2 != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys2 ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys2 ) )
=> ( ( hd_a @ Ys2 )
= X ) ) ) ).
% take_hd'
thf(fact_974_take__hd_H,axiom,
! [Ys2: list_nat,X: nat,Xs: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys2 ) @ ( cons_nat @ X @ Xs ) )
= ( take_nat @ ( suc @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) )
=> ( ( hd_nat @ Ys2 )
= X ) ) ) ).
% take_hd'
thf(fact_975_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_976_take__Suc,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_977_take__Suc,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_978_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys2 )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Ys2 ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_979_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_nat )
=> ( ( append_nat @ ( common_take_rev_nat @ N @ Xs ) @ Ys2 )
= ( append_nat @ ( common_take_rev_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_nat @ Xs ) ) @ ( cons_nat @ ( hd_nat @ Xs ) @ Ys2 ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_980_take__rev__step,axiom,
! [Xs: list_a,N: nat,Acc: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Acc ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_981_take__rev__step,axiom,
! [Xs: list_nat,N: nat,Acc: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( common_take_rev_nat @ N @ ( tl_nat @ Xs ) ) @ ( cons_nat @ ( hd_nat @ Xs ) @ Acc ) )
= ( append_nat @ ( common_take_rev_nat @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_982_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_a @ N @ nil_a )
= nil_a ) ).
% take_rev_empty
thf(fact_983_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_nat @ N @ nil_nat )
= nil_nat ) ).
% take_rev_empty
thf(fact_984_psubset__imp__ex__mem,axiom,
! [A2: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat] :
( ( ord_le3947731281898473645st_nat @ A2 @ B6 )
=> ? [B3: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ B3 @ ( minus_4256792079133151976st_nat @ B6 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_985_psubset__imp__ex__mem,axiom,
! [A2: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B6 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B6 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_986_psubsetD,axiom,
! [A2: set_Pr3451248702717554689st_nat,B6: set_Pr3451248702717554689st_nat,C2: produc1828647624359046049st_nat] :
( ( ord_le3947731281898473645st_nat @ A2 @ B6 )
=> ( ( member7340969449405702474st_nat @ C2 @ A2 )
=> ( member7340969449405702474st_nat @ C2 @ B6 ) ) ) ).
% psubsetD
thf(fact_987_psubsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B6 )
=> ( ( member8440522571783428010at_nat @ C2 @ A2 )
=> ( member8440522571783428010at_nat @ C2 @ B6 ) ) ) ).
% psubsetD
thf(fact_988_take__rev__nth,axiom,
! [N: nat,Xs: list_a,X: a,Ys2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( X
= ( nth_a @ Xs @ N ) )
=> ( ( cons_a @ X @ ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys2 ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Ys2 ) ) ) ) ).
% take_rev_nth
thf(fact_989_take__rev__nth,axiom,
! [N: nat,Xs: list_nat,X: nat,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( X
= ( nth_nat @ Xs @ N ) )
=> ( ( cons_nat @ X @ ( append_nat @ ( common_take_rev_nat @ N @ Xs ) @ Ys2 ) )
= ( append_nat @ ( common_take_rev_nat @ ( suc @ N ) @ Xs ) @ Ys2 ) ) ) ) ).
% take_rev_nth
thf(fact_990_take__hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_991_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_992_subrelI,axiom,
! [R: set_Pr6306228930610421491tate_a,S: set_Pr6306228930610421491tate_a] :
( ! [X4: a,Y4: state_a2] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y4 ) @ R )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y4 ) @ S ) )
=> ( ord_le926351553812985491tate_a @ R @ S ) ) ).
% subrelI
thf(fact_993_subrelI,axiom,
! [R: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ! [X4: list_nat,Y4: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y4 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y4 ) @ S ) )
=> ( ord_le8406513867147106209st_nat @ R @ S ) ) ).
% subrelI
thf(fact_994_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X4: nat,Y4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_995_subrelI,axiom,
! [R: set_Pr324718442235990179tate_a,S: set_Pr324718442235990179tate_a] :
( ! [X4: a,Y4: state_a] :
( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X4 @ Y4 ) @ R )
=> ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X4 @ Y4 ) @ S ) )
=> ( ord_le8856296118373343491tate_a @ R @ S ) ) ).
% subrelI
thf(fact_996_SuccD,axiom,
! [K: produc1828647624359046049st_nat,Kl: set_li6867361041382987015st_nat,Kl2: list_P7940050157051400743st_nat] :
( ( member7340969449405702474st_nat @ K @ ( bNF_Gr8705060421004693820st_nat @ Kl @ Kl2 ) )
=> ( member6711608200250777424st_nat @ ( append2623875052807961020st_nat @ Kl2 @ ( cons_P5007559046487125591st_nat @ K @ nil_Pr8413428694792600231st_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_997_SuccD,axiom,
! [K: product_prod_nat_nat,Kl: set_li5450038453877631591at_nat,Kl2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl @ Kl2 ) )
=> ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl2 @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_998_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_999_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_1000_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X5: list_nat] : X5 ) ) ).
% drop0
thf(fact_1001_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1002_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1003_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_1004_append__take__drop__id,axiom,
! [N: nat,Xs: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs ) @ ( drop_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1005_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_1006_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_1007_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1008_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_1009_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1010_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_1011_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_1012_hd__drop__1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( drop_nat @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_1013_hd__drop__1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_1014_hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ ( drop_a @ ( suc @ N ) @ Xs ) )
= ( drop_a @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_1015_hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ ( drop_nat @ ( suc @ N ) @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_1016_tl__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% tl_drop
thf(fact_1017_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_1018_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_1019_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_1020_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_1021_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys2: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys2 ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1022_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1023_drop__take,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_1024_tl__drop__2,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ ( suc @ N ) @ Xs ) ) ).
% tl_drop_2
thf(fact_1025_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_1026_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1027_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1028_drop__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1029_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_1030_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_1031_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1032_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_1033_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1034_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_1035_SuccI,axiom,
! [Kl2: list_P7940050157051400743st_nat,K: produc1828647624359046049st_nat,Kl: set_li6867361041382987015st_nat] :
( ( member6711608200250777424st_nat @ ( append2623875052807961020st_nat @ Kl2 @ ( cons_P5007559046487125591st_nat @ K @ nil_Pr8413428694792600231st_nat ) ) @ Kl )
=> ( member7340969449405702474st_nat @ K @ ( bNF_Gr8705060421004693820st_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1036_SuccI,axiom,
! [Kl2: list_P6011104703257516679at_nat,K: product_prod_nat_nat,Kl: set_li5450038453877631591at_nat] :
( ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl2 @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl )
=> ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1037_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1038_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1039_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1040_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_1041_empty__Shift,axiom,
! [Kl: set_li6867361041382987015st_nat,K: produc1828647624359046049st_nat] :
( ( member6711608200250777424st_nat @ nil_Pr8413428694792600231st_nat @ Kl )
=> ( ( member7340969449405702474st_nat @ K @ ( bNF_Gr8705060421004693820st_nat @ Kl @ nil_Pr8413428694792600231st_nat ) )
=> ( member6711608200250777424st_nat @ nil_Pr8413428694792600231st_nat @ ( bNF_Gr1840819286107176384st_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1042_empty__Shift,axiom,
! [Kl: set_li5450038453877631591at_nat,K: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ Kl )
=> ( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl @ nil_Pr5478986624290739719at_nat ) )
=> ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1043_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1044_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1045_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_1046_Succ__Shift,axiom,
! [Kl: set_list_a,K: a,Kl2: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ_a @ Kl @ ( cons_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_1047_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_1048_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_1049_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_1050_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_1051_nth__list__update__neq,axiom,
! [I: nat,J2: nat,Xs: list_nat,X: nat] :
( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_1052_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1053_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_1054_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1055_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( drop_nat @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_1056_list__update__length,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys2 ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_1057_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_1058_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_1059_list__update_Osimps_I1_J,axiom,
! [I: nat,V: a] :
( ( list_update_a @ nil_a @ I @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_1060_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_1061_list__update__code_I1_J,axiom,
! [I: nat,Y: a] :
( ( list_update_a @ nil_a @ I @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_1062_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_1063_list__update__swap,axiom,
! [I: nat,I5: nat,Xs: list_nat,X: nat,X9: nat] :
( ( I != I5 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I5 @ X9 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I5 @ X9 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_1064_take__update__swap,axiom,
! [M: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1065_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1066_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1067_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1068_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1069_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_1070_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J2: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1071_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_1072_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1073_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1074_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_1075_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
= ( ? [Y5: nat,N4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ Y5 ) @ R )
& ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
& ( Ys2
= ( list_update_nat @ Xs @ N4 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1076_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_1077_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_1078_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1079_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_1080_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_1081_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_1082_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_1083_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_1084_take__last__length,axiom,
! [Xs: list_nat] :
( ( ( take_nat @ ( suc @ zero_zero_nat ) @ ( rev_nat @ Xs ) )
= ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ Xs ) ) ) ).
% take_last_length
thf(fact_1085_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_take__rev__drop,axiom,
! [N: nat,Xs: list_nat,Acc: list_nat] :
( ( append_nat @ ( common_take_rev_nat @ N @ Xs ) @ Acc )
= ( append_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ ( rev_nat @ Xs ) ) @ Acc ) ) ).
% take_rev_drop
thf(fact_1093_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1094_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1095_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1096_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_1097_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_1098_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1099_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1100_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_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_1106_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1107_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_1108_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_1109_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1110_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_1111_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P5: nat > nat > $o,Xs3: list_nat] :
! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P5 @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1112_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1113_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_1114_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_1115_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1116_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1117_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1118_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1119_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I2 ) ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1120_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_1121_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_1122_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_1123_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1124_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1125_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1126_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1127_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1128_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1129_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1130_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1131_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1132_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1133_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1134_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1135_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1136_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1137_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1138_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1139_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1140_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_1141_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1142_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_1143_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_1144_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_1145_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1146_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1147_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1148_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1149_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1150_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1151_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1152_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1153_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1154_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1155_min__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) ) ) ).
% min_number_of(1)
thf(fact_1156_min__0__1_I4_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(4)
thf(fact_1157_min__0__1_I3_J,axiom,
! [X: num] :
( ( ord_min_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= zero_zero_nat ) ).
% min_0_1(3)
thf(fact_1158_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_1159_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= one_one_nat ) ).
% min_0_1(6)
thf(fact_1160_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= one_one_nat ) ).
% min_0_1(5)
thf(fact_1161_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1162_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1163_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1164_length__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_1165_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1166_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1167_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys2: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_1168_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_1169_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1170_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_1171_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_1172_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1173_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1174_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1175_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1176_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1177_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1178_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1179_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1180_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1181_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1182_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1183_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_1184_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_1185_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1186_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1187_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1188_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1189_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1190_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1191_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1192_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1193_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1194_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1195_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1196_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1197_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1198_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1199_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1200_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1201_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1202_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1203_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1204_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1205_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1206_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1207_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_1208_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1209_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_1210_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_1211_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1212_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1213_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1214_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1215_group__cancel_Oadd2,axiom,
! [B6: nat,K: nat,B: nat,A: nat] :
( ( B6
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1216_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_1217_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1218_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_1219_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_1220_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_1221_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% min_add_distrib_left
thf(fact_1222_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% min_add_distrib_right
thf(fact_1223_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1224_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1225_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1226_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1227_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1228_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1229_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1230_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_1231_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1232_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1233_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1234_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1235_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1236_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1237_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1238_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1239_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1240_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1241_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A5 @ C ) ) ) ) ).
% le_iff_add
thf(fact_1242_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1243_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1244_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1245_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1246_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1247_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1248_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1249_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1250_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1251_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1252_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1253_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1254_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1255_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_1256_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_1257_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1258_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1259_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_1260_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1261_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1262_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1263_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
% Helper facts (7)
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_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
type_i464410347872898157tate_a @ ( type_s3703408523585882337tate_a @ ( common_a @ state ) ) ).
%------------------------------------------------------------------------------