TPTP Problem File: SLH0950^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0026_RealTimeDeque_Dequeue_Proof/prob_00021_000707__7082622_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1637 ( 634 unt; 364 typ; 0 def)
% Number of atoms : 3319 (2133 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 12380 ( 659 ~; 90 |; 255 &;10003 @)
% ( 0 <=>;1373 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 6 avg)
% Number of types : 61 ( 60 usr)
% Number of type conns : 787 ( 787 >; 0 *; 0 +; 0 <<)
% Number of symbols : 307 ( 304 usr; 23 con; 0-3 aty)
% Number of variables : 3748 ( 48 ^;3522 !; 178 ?;3748 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:54.702
%------------------------------------------------------------------------------
% Could-be-implicit typings (60)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
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__Product____Type__Oprod_I_062_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc564279554677168641st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__RealTimeDeque__Odeque_Itf__a_J_J_J_J,type,
set_Pr775331825050515926eque_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Idle__Oidle_Itf__a_J_J_J_J,type,
set_Pr8671745777326831373idle_a: $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__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__RealTimeDeque__Odeque_Itf__a_J_J_J,type,
produc4537397550861375862eque_a: $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__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Idle__Oidle_Itf__a_J_J_J,type,
produc2967520842975235501idle_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__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__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J_J,type,
list_P9016464757445826172eque_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J_J,type,
set_Pr3011860922491142998eque_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__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J_J,type,
list_P474335768696175027idle_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J_J,type,
set_Pr7225968907286947725idle_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Idle__Oidle_It__Nat__Onat_J_J,type,
produc6105442644609652625le_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__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
produc7037199475535947254eque_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
list_P3592885314253461005_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
list_P2851791750731487283_nat_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__Idle__Oidle_Itf__a_J_J,type,
produc7590564867095724333idle_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__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__List__Olist_It__RealTimeDeque__Odeque_Itf__a_J_J,type,
list_deque_a: $tType ).
thf(ty_n_t__Set__Oset_It__RealTimeDeque__Odeque_Itf__a_J_J,type,
set_deque_a: $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__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__List__Olist_It__Idle__Oidle_Itf__a_J_J,type,
list_idle_a: $tType ).
thf(ty_n_t__RealTimeDeque__Odeque_It__Nat__Onat_J,type,
deque_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Idle__Oidle_Itf__a_J_J,type,
set_idle_a: $tType ).
thf(ty_n_t__RealTimeDeque__Odeque_Itf__a_J,type,
deque_a: $tType ).
thf(ty_n_t__Stack__Ostack_It__Nat__Onat_J,type,
stack_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Idle__Oidle_It__Nat__Onat_J,type,
idle_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Stack__Ostack_Itf__a_J,type,
stack_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Idle__Oidle_Itf__a_J,type,
idle_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 (304)
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_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_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_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_Euclidean__Division_Odivmod__nat,type,
euclidean_divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
euclid4777050414544973029ze_nat: nat > nat ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Idle_Oidle_OIdle_001t__Nat__Onat,type,
idle_nat2: stack_nat > nat > idle_nat ).
thf(sy_c_Idle_Oidle_OIdle_001tf__a,type,
idle_a2: stack_a > nat > idle_a ).
thf(sy_c_Idle_Opop_001t__Nat__Onat,type,
pop_nat: idle_nat > produc6105442644609652625le_nat ).
thf(sy_c_Idle_Opop_001tf__a,type,
pop_a: idle_a > produc7590564867095724333idle_a ).
thf(sy_c_Idle_Opop__rel_001tf__a,type,
pop_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle_Opush_001t__Nat__Onat,type,
push_nat: nat > idle_nat > idle_nat ).
thf(sy_c_Idle_Opush_001tf__a,type,
push_a: a > idle_a > idle_a ).
thf(sy_c_Idle_Opush__rel_001tf__a,type,
push_rel_a: produc7590564867095724333idle_a > produc7590564867095724333idle_a > $o ).
thf(sy_c_Idle__Aux_Oinvar__idle__rel_001tf__a,type,
idle_i6200314614184386870_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Ois__empty__idle__rel_001tf__a,type,
idle_i5927890838958520983_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Olist_001t__Nat__Onat,type,
idle_list_nat: idle_nat > list_nat ).
thf(sy_c_Idle__Aux_Olist_001tf__a,type,
idle_list_a: idle_a > list_a ).
thf(sy_c_Idle__Aux_Olist__rel_001tf__a,type,
idle_list_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Osize__idle__rel_001tf__a,type,
idle_size_idle_rel_a: idle_a > idle_a > $o ).
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__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_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_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001tf__a,type,
bind_nat_a: list_nat > ( nat > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__Nat__Onat,type,
bind_a_nat: list_a > ( a > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( 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_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
drop_P8868858903918902087at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001tf__a,type,
enumerate_a: nat > list_a > list_P2851791750731487283_nat_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__Idle__Oidle_Itf__a_J,type,
last_idle_a: list_idle_a > idle_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
last_P6484183829340986144at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
last_P5509911954246017860_nat_a: list_P2851791750731487283_nat_a > product_prod_nat_a ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
last_P7618890818322975820idle_a: list_P474335768696175027idle_a > produc7590564867095724333idle_a ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
last_P2271748490522340894_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
last_P5659436501804209045eque_a: list_P9016464757445826172eque_a > produc7037199475535947254eque_a ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
last_P8790725268278465478od_a_a: list_P1396940483166286381od_a_a > product_prod_a_a ).
thf(sy_c_List_Olast_001t__RealTimeDeque__Odeque_Itf__a_J,type,
last_deque_a: list_deque_a > deque_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
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__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_Olexn_001t__Nat__Onat,type,
lexn_nat: set_Pr1261947904930325089at_nat > nat > set_Pr3451248702717554689st_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_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_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__Idle__Oidle_Itf__a_J,type,
cons_idle_a: idle_a > list_idle_a > list_idle_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__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
cons_P7029976571565372387idle_a: produc7590564867095724333idle_a > list_P474335768696175027idle_a > list_P474335768696175027idle_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
cons_P8083696988307304876eque_a: produc7037199475535947254eque_a > list_P9016464757445826172eque_a > list_P9016464757445826172eque_a ).
thf(sy_c_List_Olist_OCons_001t__RealTimeDeque__Odeque_Itf__a_J,type,
cons_deque_a: deque_a > list_deque_a > list_deque_a ).
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__Nat__Onat_Mt__Nat__Onat_J,type,
nil_nat_nat: list_nat_nat ).
thf(sy_c_List_Olist_ONil_001t__Idle__Oidle_Itf__a_J,type,
nil_idle_a: list_idle_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__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
nil_Pr1417316670369895453_nat_a: list_P2851791750731487283_nat_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
nil_Pr7402525243500994295_a_nat: list_P3592885314253461005_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
nil_Product_prod_a_a: list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_ONil_001t__RealTimeDeque__Odeque_Itf__a_J,type,
nil_deque_a: list_deque_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Idle__Oidle_Itf__a_J,type,
hd_idle_a: list_idle_a > idle_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
hd_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
hd_Pro2949996684582079736_nat_a: list_P2851791750731487283_nat_a > product_prod_nat_a ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
hd_Pro9081346849183306136idle_a: list_P474335768696175027idle_a > produc7590564867095724333idle_a ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
hd_Pro8935205257713178578_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
hd_Pro5574702693030206177eque_a: list_P9016464757445826172eque_a > produc7037199475535947254eque_a ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
hd_Product_prod_a_a: list_P1396940483166286381od_a_a > product_prod_a_a ).
thf(sy_c_List_Olist_Ohd_001t__RealTimeDeque__Odeque_Itf__a_J,type,
hd_deque_a: list_deque_a > deque_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__Idle__Oidle_Itf__a_J,type,
set_idle_a2: list_idle_a > set_idle_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
set_Pr1944501890106611522idle_a: list_P474335768696175027idle_a > set_Pr7225968907286947725idle_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
set_Pr655576496610006923eque_a: list_P9016464757445826172eque_a > set_Pr3011860922491142998eque_a ).
thf(sy_c_List_Olist_Oset_001t__RealTimeDeque__Odeque_Itf__a_J,type,
set_deque_a2: list_deque_a > set_deque_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
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__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Idle__Oidle_Itf__a_J,type,
list_update_idle_a: list_idle_a > nat > idle_a > list_idle_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
list_u3350830261621523099idle_a: list_P474335768696175027idle_a > nat > produc7590564867095724333idle_a > list_P474335768696175027idle_a ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
list_u2964271099763749988eque_a: list_P9016464757445826172eque_a > nat > produc7037199475535947254eque_a > list_P9016464757445826172eque_a ).
thf(sy_c_List_Olist__update_001t__RealTimeDeque__Odeque_Itf__a_J,type,
list_update_deque_a: list_deque_a > nat > deque_a > list_deque_a ).
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_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
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__Idle__Oidle_Itf__a_J,type,
listrel_a_idle_a: set_Pr7225968907286947725idle_a > set_Pr8671745777326831373idle_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__RealTimeDeque__Odeque_Itf__a_J,type,
listrel_a_deque_a: set_Pr3011860922491142998eque_a > set_Pr775331825050515926eque_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_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__Nat__Onat,type,
map_ta8710832428924958105_a_nat: ( a > nat ) > list_a > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev__rel_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta8615873517111064934at_nat: produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ).
thf(sy_c_List_Omap__tailrec__rev__rel_001tf__a_001t__Nat__Onat,type,
map_ta7397863945511617930_a_nat: produc564279554677168641st_nat > produc564279554677168641st_nat > $o ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
maps_nat_a: ( nat > list_a ) > list_nat > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
maps_a_nat: ( a > list_nat ) > list_a > list_nat ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
measures_nat: list_nat_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
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__Idle__Oidle_Itf__a_J,type,
nth_idle_a: list_idle_a > nat > idle_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
nth_Pr3434503280043392308idle_a: list_P474335768696175027idle_a > nat > produc7590564867095724333idle_a ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
nth_Pr3409035872437518461eque_a: list_P9016464757445826172eque_a > nat > produc7037199475535947254eque_a ).
thf(sy_c_List_Onth_001t__RealTimeDeque__Odeque_Itf__a_J,type,
nth_deque_a: list_deque_a > nat > deque_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oproduct_001tf__a_001t__Idle__Oidle_Itf__a_J,type,
product_a_idle_a: list_a > list_idle_a > list_P474335768696175027idle_a ).
thf(sy_c_List_Oproduct_001tf__a_001t__RealTimeDeque__Odeque_Itf__a_J,type,
product_a_deque_a: list_a > list_deque_a > list_P9016464757445826172eque_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_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_Oreplicate_001t__Idle__Oidle_Itf__a_J,type,
replicate_idle_a: nat > idle_a > list_idle_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_Itf__a_J,type,
replicate_list_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
replic4511300930391394541idle_a: nat > produc7590564867095724333idle_a > list_P474335768696175027idle_a ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J,type,
replic3687150286780796342eque_a: nat > produc7037199475535947254eque_a > list_P9016464757445826172eque_a ).
thf(sy_c_List_Oreplicate_001t__RealTimeDeque__Odeque_Itf__a_J,type,
replicate_deque_a: nat > deque_a > list_deque_a ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
rev_Pr6102188148953555047at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
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_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > 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_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $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__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
take_P2173866234530122223at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001tf__a,type,
zip_nat_a: list_nat > list_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Idle__Oidle_Itf__a_J,type,
zip_a_idle_a: list_a > list_idle_a > list_P474335768696175027idle_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Nat__Onat,type,
zip_a_nat: list_a > list_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Ozip_001tf__a_001t__RealTimeDeque__Odeque_Itf__a_J,type,
zip_a_deque_a: list_a > list_deque_a > list_P9016464757445826172eque_a ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Idle__Oidle_It__Nat__Onat_J,type,
size_size_idle_nat: idle_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Idle__Oidle_Itf__a_J,type,
size_size_idle_a: idle_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Idle__Oidle_Itf__a_J_J,type,
size_s8477453896275132790idle_a: list_idle_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__RealTimeDeque__Odeque_Itf__a_J_J,type,
size_s6425414874530267071eque_a: list_deque_a > 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__RealTimeDeque__Odeque_Itf__a_J,type,
size_size_deque_a: deque_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_It__Nat__Onat_J,type,
size_size_stack_nat: stack_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_Itf__a_J,type,
size_size_stack_a: stack_a > 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__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__Idle__Oidle_Itf__a_J_J_J,type,
ord_le1072348969082359597idle_a: set_Pr7225968907286947725idle_a > set_Pr7225968907286947725idle_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__RealTimeDeque__Odeque_Itf__a_J_J_J,type,
ord_le1319221647696452342eque_a: set_Pr3011860922491142998eque_a > set_Pr3011860922491142998eque_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
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_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_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_001_062_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc6785043063399293179st_nat: ( a > nat ) > produc432399132543013523st_nat > produc564279554677168641st_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__Idle__Oidle_Itf__a_J_J,type,
produc4587228114721326877idle_a: list_a > list_idle_a > produc2967520842975235501idle_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__RealTimeDeque__Odeque_Itf__a_J_J,type,
produc568430809755685862eque_a: list_a > list_deque_a > produc4537397550861375862eque_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__Idle__Oidle_It__Nat__Onat_J,type,
produc9134156765868440393le_nat: nat > idle_nat > produc6105442644609652625le_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_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__Idle__Oidle_Itf__a_J,type,
produc1265230069547855005idle_a: a > idle_a > produc7590564867095724333idle_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__RealTimeDeque__Odeque_Itf__a_J,type,
produc3093615717782868582eque_a: a > deque_a > produc7037199475535947254eque_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_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
thf(sy_c_Product__Type_Ocurry_001tf__a_001t__Idle__Oidle_Itf__a_J_001_Eo,type,
produc7330610896797225991le_a_o: ( produc7590564867095724333idle_a > $o ) > a > idle_a > $o ).
thf(sy_c_Product__Type_Ocurry_001tf__a_001t__RealTimeDeque__Odeque_Itf__a_J_001_Eo,type,
produc2509474620477234558ue_a_o: ( produc7037199475535947254eque_a > $o ) > a > deque_a > $o ).
thf(sy_c_RealTimeDeque_OdeqL_H_001tf__a,type,
deqL_a: deque_a > produc7037199475535947254eque_a ).
thf(sy_c_RealTimeDeque_Odeque_OIdle_001t__Nat__Onat,type,
idle_nat3: idle_nat > idle_nat > deque_nat ).
thf(sy_c_RealTimeDeque_Odeque_OIdle_001tf__a,type,
idle_a3: idle_a > idle_a > deque_a ).
thf(sy_c_RealTimeDeque_Odeque_Osize__deque_001tf__a,type,
size_deque_a: ( a > nat ) > deque_a > nat ).
thf(sy_c_RealTimeDeque__Aux_OlistL_001t__Nat__Onat,type,
realTi2392226096638470052tL_nat: deque_nat > list_nat ).
thf(sy_c_RealTimeDeque__Aux_OlistL_001tf__a,type,
realTi2141985871725878314istL_a: deque_a > list_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Stack_Ofirst_001t__Nat__Onat,type,
first_nat: stack_nat > nat ).
thf(sy_c_Stack_Ofirst_001tf__a,type,
first_a: stack_a > a ).
thf(sy_c_Stack_Opop_001t__Nat__Onat,type,
pop_nat2: stack_nat > stack_nat ).
thf(sy_c_Stack_Opop_001tf__a,type,
pop_a2: stack_a > stack_a ).
thf(sy_c_Stack_Opush_001t__Nat__Onat,type,
push_nat2: nat > stack_nat > stack_nat ).
thf(sy_c_Stack_Opush_001tf__a,type,
push_a2: a > stack_a > stack_a ).
thf(sy_c_Stack__Aux_Olist_001t__Nat__Onat,type,
stack_list_nat: stack_nat > list_nat ).
thf(sy_c_Stack__Aux_Olist_001tf__a,type,
stack_list_a: stack_a > list_a ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Idle__Oidle_It__Nat__Onat_J,type,
type_i5289740830312680605le_nat: idle_nat > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Idle__Oidle_Itf__a_J,type,
type_i8151583586401621767idle_a: idle_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__RealTimeDeque__Odeque_Itf__a_J,type,
type_i7033253005186307024eque_a: deque_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Idle__Oidle_It__Nat__Onat_J,type,
type_i1745286555725844959le_nat: idle_nat > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Idle__Oidle_Itf__a_J,type,
type_i7304311975391125061idle_a: idle_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__Nat__Onat_J,type,
type_i6581939242220632785ck_nat: stack_nat > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_Itf__a_J,type,
type_i3216275384938974675tack_a: stack_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Idle__Oidle_Itf__a_J,type,
accp_idle_a: ( idle_a > idle_a > $o ) > idle_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__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,
accp_P3924000266242761678st_nat: ( produc1616951275169580055st_nat > produc1616951275169580055st_nat > $o ) > produc1616951275169580055st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P7537707418771614264st_nat: ( produc564279554677168641st_nat > produc564279554677168641st_nat > $o ) > produc564279554677168641st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
accp_P2118310398061403062idle_a: ( produc7590564867095724333idle_a > produc7590564867095724333idle_a > $o ) > produc7590564867095724333idle_a > $o ).
thf(sy_c_member_001t__Idle__Oidle_Itf__a_J,type,
member_idle_a: idle_a > set_idle_a > $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_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_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__Idle__Oidle_Itf__a_J_J_J,type,
member9073500730917845590idle_a: produc2967520842975235501idle_a > set_Pr8671745777326831373idle_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__RealTimeDeque__Odeque_Itf__a_J_J_J,type,
member3240663459192582047eque_a: produc4537397550861375862eque_a > set_Pr775331825050515926eque_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__Idle__Oidle_Itf__a_J_J,type,
member3794253416837758678idle_a: produc7590564867095724333idle_a > set_Pr7225968907286947725idle_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__RealTimeDeque__Odeque_Itf__a_J_J,type,
member8785101876683738399eque_a: produc7037199475535947254eque_a > set_Pr3011860922491142998eque_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__RealTimeDeque__Odeque_Itf__a_J,type,
member_deque_a: deque_a > set_deque_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_deque_H,type,
deque: deque_a ).
thf(sy_v_left_H____,type,
left: idle_a ).
thf(sy_v_left____,type,
left2: idle_a ).
thf(sy_v_length__left_H____,type,
length_left: nat ).
thf(sy_v_length__right____,type,
length_right: nat ).
thf(sy_v_right____,type,
right: stack_a ).
thf(sy_v_stack__left_H____,type,
stack_left: stack_a ).
thf(sy_v_xa____,type,
xa: a ).
% Relevant facts (1265)
thf(fact_0__C4_Oprems_C_I1_J,axiom,
type_i7033253005186307024eque_a @ ( idle_a3 @ left2 @ ( idle_a2 @ right @ length_right ) ) ).
% "4.prems"(1)
thf(fact_1__C4_Oprems_C_I2_J,axiom,
( ( realTi2141985871725878314istL_a @ ( idle_a3 @ left2 @ ( idle_a2 @ right @ length_right ) ) )
!= nil_a ) ).
% "4.prems"(2)
thf(fact_2__092_060open_062_092_060And_062x_Aidle_H_O_A_092_060lbrakk_062_092_060not_062_Ais__empty_Aleft_059_Ainvar_Aleft_059_AIdle_Opop_Aleft_A_061_A_Ix_M_Aidle_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ainvar_Aidle_H_092_060close_062,axiom,
! [X: a,Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ left2 )
=> ( ( type_i8151583586401621767idle_a @ left2 )
=> ( ( ( pop_a @ left2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( type_i8151583586401621767idle_a @ Idle ) ) ) ) ).
% \<open>\<And>x idle'. \<lbrakk>\<not> is_empty left; invar left; Idle.pop left = (x, idle')\<rbrakk> \<Longrightarrow> invar idle'\<close>
thf(fact_3__C4_Oprems_C_I3_J,axiom,
( ( deqL_a @ ( idle_a3 @ left2 @ ( idle_a2 @ right @ length_right ) ) )
= ( produc3093615717782868582eque_a @ xa @ deque ) ) ).
% "4.prems"(3)
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062stack__left_H_Alength__left_H_O_Aleft_H_A_061_Aidle_OIdle_Astack__left_H_Alength__left_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Stack_left: stack_a,Length_left: nat] :
( left
!= ( idle_a2 @ Stack_left @ Length_left ) ) ).
% \<open>\<And>thesis. (\<And>stack_left' length_left'. left' = idle.Idle stack_left' length_left' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_pop__left,axiom,
( ( pop_a @ left2 )
= ( produc1265230069547855005idle_a @ xa @ left ) ) ).
% pop_left
thf(fact_6_left_H,axiom,
( left
= ( idle_a2 @ stack_left @ length_left ) ) ).
% left'
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062left_H_O_AIdle_Opop_Aleft_A_061_A_Ix_M_Aleft_H_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Left: idle_a] :
( ( pop_a @ left2 )
!= ( produc1265230069547855005idle_a @ xa @ Left ) ) ).
% \<open>\<And>thesis. (\<And>left'. Idle.pop left = (x, left') \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_Idle__Proof_Oinvar__push,axiom,
! [Idle2: idle_nat,X: nat] :
( ( type_i5289740830312680605le_nat @ Idle2 )
=> ( type_i5289740830312680605le_nat @ ( push_nat @ X @ Idle2 ) ) ) ).
% Idle_Proof.invar_push
thf(fact_9_Idle__Proof_Oinvar__push,axiom,
! [Idle2: idle_a,X: a] :
( ( type_i8151583586401621767idle_a @ Idle2 )
=> ( type_i8151583586401621767idle_a @ ( push_a @ X @ Idle2 ) ) ) ).
% Idle_Proof.invar_push
thf(fact_10_Idle__Proof_Oinvar__pop,axiom,
! [Idle2: idle_nat,X: nat,Idle: idle_nat] :
( ~ ( type_i1745286555725844959le_nat @ Idle2 )
=> ( ( type_i5289740830312680605le_nat @ Idle2 )
=> ( ( ( pop_nat @ Idle2 )
= ( produc9134156765868440393le_nat @ X @ Idle ) )
=> ( type_i5289740830312680605le_nat @ Idle ) ) ) ) ).
% Idle_Proof.invar_pop
thf(fact_11_Idle__Proof_Oinvar__pop,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle2 )
=> ( ( type_i8151583586401621767idle_a @ Idle2 )
=> ( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( type_i8151583586401621767idle_a @ Idle ) ) ) ) ).
% Idle_Proof.invar_pop
thf(fact_12_invar__idle_Oelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
= StackSize ) ) ) ).
% invar_idle.elims(3)
thf(fact_13_invar__idle_Oelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ).
% invar_idle.elims(2)
thf(fact_14_invar__idle_Oelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( Y
= ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ) ).
% invar_idle.elims(1)
thf(fact_15_invar__idle_Osimps,axiom,
! [Stack2: stack_a,StackSize2: nat] :
( ( type_i8151583586401621767idle_a @ ( idle_a2 @ Stack2 @ StackSize2 ) )
= ( ( size_size_stack_a @ Stack2 )
= StackSize2 ) ) ).
% invar_idle.simps
thf(fact_16_is__empty__idle_Ocases,axiom,
! [X: idle_a] :
~ ! [Stack: stack_a,Uu: nat] :
( X
!= ( idle_a2 @ Stack @ Uu ) ) ).
% is_empty_idle.cases
thf(fact_17_deque_Oinject_I4_J,axiom,
! [X51: idle_nat,X52: idle_nat,Y51: idle_nat,Y52: idle_nat] :
( ( ( idle_nat3 @ X51 @ X52 )
= ( idle_nat3 @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% deque.inject(4)
thf(fact_18_deque_Oinject_I4_J,axiom,
! [X51: idle_a,X52: idle_a,Y51: idle_a,Y52: idle_a] :
( ( ( idle_a3 @ X51 @ X52 )
= ( idle_a3 @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% deque.inject(4)
thf(fact_19_idle_Oinject,axiom,
! [X1: stack_a,X2: nat,Y1: stack_a,Y2: nat] :
( ( ( idle_a2 @ X1 @ X2 )
= ( idle_a2 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% idle.inject
thf(fact_20_prod_Oinject,axiom,
! [X1: nat > nat > nat,X2: produc7248412053542808358at_nat,Y1: nat > nat > nat,Y2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ X1 @ X2 )
= ( produc3209952032786966637at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_21_prod_Oinject,axiom,
! [X1: a > nat,X2: produc432399132543013523st_nat,Y1: a > nat,Y2: produc432399132543013523st_nat] :
( ( ( produc6785043063399293179st_nat @ X1 @ X2 )
= ( produc6785043063399293179st_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_22_prod_Oinject,axiom,
! [X1: a > a > $o,X2: list_a,Y1: a > a > $o,Y2: list_a] :
( ( ( produc8111569692950616493list_a @ X1 @ X2 )
= ( produc8111569692950616493list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_23_prod_Oinject,axiom,
! [X1: a,X2: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X2 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_24_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_25_prod_Oinject,axiom,
! [X1: a,X2: idle_a,Y1: a,Y2: idle_a] :
( ( ( produc1265230069547855005idle_a @ X1 @ X2 )
= ( produc1265230069547855005idle_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_26_prod_Oinject,axiom,
! [X1: a,X2: deque_a,Y1: a,Y2: deque_a] :
( ( ( produc3093615717782868582eque_a @ X1 @ X2 )
= ( produc3093615717782868582eque_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_27_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_28_old_Oprod_Oinject,axiom,
! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ A @ B )
= ( produc3209952032786966637at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_29_old_Oprod_Oinject,axiom,
! [A: a > nat,B: produc432399132543013523st_nat,A2: a > nat,B2: produc432399132543013523st_nat] :
( ( ( produc6785043063399293179st_nat @ A @ B )
= ( produc6785043063399293179st_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
! [A: a > a > $o,B: list_a,A2: a > a > $o,B2: list_a] :
( ( ( produc8111569692950616493list_a @ A @ B )
= ( produc8111569692950616493list_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_31_old_Oprod_Oinject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_32_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
! [A: a,B: idle_a,A2: a,B2: idle_a] :
( ( ( produc1265230069547855005idle_a @ A @ B )
= ( produc1265230069547855005idle_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
! [A: a,B: deque_a,A2: a,B2: deque_a] :
( ( ( produc3093615717782868582eque_a @ A @ B )
= ( produc3093615717782868582eque_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_36_Idle_Opush_Ocases,axiom,
! [X: produc7590564867095724333idle_a] :
~ ! [X3: a,Stack: stack_a,StackSize: nat] :
( X
!= ( produc1265230069547855005idle_a @ X3 @ ( idle_a2 @ Stack @ StackSize ) ) ) ).
% Idle.push.cases
thf(fact_37_enqR_Ocases,axiom,
! [X: produc7037199475535947254eque_a] :
~ ! [X3: a,Deque: deque_a] :
( X
!= ( produc3093615717782868582eque_a @ X3 @ Deque ) ) ).
% enqR.cases
thf(fact_38_idle_Oexhaust,axiom,
! [Y: idle_a] :
~ ! [X12: stack_a,X22: nat] :
( Y
!= ( idle_a2 @ X12 @ X22 ) ) ).
% idle.exhaust
thf(fact_39_size__neq__size__imp__neq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_40_size__neq__size__imp__neq,axiom,
! [X: list_idle_a,Y: list_idle_a] :
( ( ( size_s8477453896275132790idle_a @ X )
!= ( size_s8477453896275132790idle_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_41_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_42_size__neq__size__imp__neq,axiom,
! [X: idle_nat,Y: idle_nat] :
( ( ( size_size_idle_nat @ X )
!= ( size_size_idle_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_43_size__neq__size__imp__neq,axiom,
! [X: idle_a,Y: idle_a] :
( ( ( size_size_idle_a @ X )
!= ( size_size_idle_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_44_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_45_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_46_old_Oprod_Oexhaust,axiom,
! [Y: produc4471711990508489141at_nat] :
~ ! [A3: nat > nat > nat,B3: produc7248412053542808358at_nat] :
( Y
!= ( produc3209952032786966637at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_47_old_Oprod_Oexhaust,axiom,
! [Y: produc564279554677168641st_nat] :
~ ! [A3: a > nat,B3: produc432399132543013523st_nat] :
( Y
!= ( produc6785043063399293179st_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_48_old_Oprod_Oexhaust,axiom,
! [Y: produc5032551385658279741list_a] :
~ ! [A3: a > a > $o,B3: list_a] :
( Y
!= ( produc8111569692950616493list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_49_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A3: a,B3: nat] :
( Y
!= ( product_Pair_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_50_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
! [Y: produc7590564867095724333idle_a] :
~ ! [A3: a,B3: idle_a] :
( Y
!= ( produc1265230069547855005idle_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [Y: produc7037199475535947254eque_a] :
~ ! [A3: a,B3: deque_a] :
( Y
!= ( produc3093615717782868582eque_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_54_size__idle_Osimps,axiom,
! [Stack2: stack_nat,Uu2: nat] :
( ( size_size_idle_nat @ ( idle_nat2 @ Stack2 @ Uu2 ) )
= ( size_size_stack_nat @ Stack2 ) ) ).
% size_idle.simps
thf(fact_55_size__idle_Osimps,axiom,
! [Stack2: stack_a,Uu2: nat] :
( ( size_size_idle_a @ ( idle_a2 @ Stack2 @ Uu2 ) )
= ( size_size_stack_a @ Stack2 ) ) ).
% size_idle.simps
thf(fact_56_size__idle_Oelims,axiom,
! [X: idle_nat,Y: nat] :
( ( ( size_size_idle_nat @ X )
= Y )
=> ~ ! [Stack: stack_nat] :
( ? [Uu: nat] :
( X
= ( idle_nat2 @ Stack @ Uu ) )
=> ( Y
!= ( size_size_stack_nat @ Stack ) ) ) ) ).
% size_idle.elims
thf(fact_57_size__idle_Oelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( Y
!= ( size_size_stack_a @ Stack ) ) ) ) ).
% size_idle.elims
thf(fact_58_prod__induct4,axiom,
! [P: produc4471711990508489141at_nat > $o,X: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: nat,C: nat,D: nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ ( product_Pair_nat_nat @ C @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_59_prod__induct3,axiom,
! [P: produc4471711990508489141at_nat > $o,X: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: nat,C: product_prod_nat_nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_60_prod__induct3,axiom,
! [P: produc564279554677168641st_nat > $o,X: produc564279554677168641st_nat] :
( ! [A3: a > nat,B3: list_a,C: list_nat] : ( P @ ( produc6785043063399293179st_nat @ A3 @ ( produc4792949784200893581st_nat @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_61_prod__cases4,axiom,
! [Y: produc4471711990508489141at_nat] :
~ ! [A3: nat > nat > nat,B3: nat,C: nat,D: nat] :
( Y
!= ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ ( product_Pair_nat_nat @ C @ D ) ) ) ) ).
% prod_cases4
thf(fact_62_prod__cases3,axiom,
! [Y: produc4471711990508489141at_nat] :
~ ! [A3: nat > nat > nat,B3: nat,C: product_prod_nat_nat] :
( Y
!= ( produc3209952032786966637at_nat @ A3 @ ( produc487386426758144856at_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_63_prod__cases3,axiom,
! [Y: produc564279554677168641st_nat] :
~ ! [A3: a > nat,B3: list_a,C: list_nat] :
( Y
!= ( produc6785043063399293179st_nat @ A3 @ ( produc4792949784200893581st_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_64_Pair__inject,axiom,
! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
( ( ( produc3209952032786966637at_nat @ A @ B )
= ( produc3209952032786966637at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_65_Pair__inject,axiom,
! [A: a > nat,B: produc432399132543013523st_nat,A2: a > nat,B2: produc432399132543013523st_nat] :
( ( ( produc6785043063399293179st_nat @ A @ B )
= ( produc6785043063399293179st_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_66_Pair__inject,axiom,
! [A: a > a > $o,B: list_a,A2: a > a > $o,B2: list_a] :
( ( ( produc8111569692950616493list_a @ A @ B )
= ( produc8111569692950616493list_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_67_Pair__inject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_68_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_69_Pair__inject,axiom,
! [A: a,B: idle_a,A2: a,B2: idle_a] :
( ( ( produc1265230069547855005idle_a @ A @ B )
= ( produc1265230069547855005idle_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_70_Pair__inject,axiom,
! [A: a,B: deque_a,A2: a,B2: deque_a] :
( ( ( produc3093615717782868582eque_a @ A @ B )
= ( produc3093615717782868582eque_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_71_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_72_prod__cases,axiom,
! [P: produc4471711990508489141at_nat > $o,P2: produc4471711990508489141at_nat] :
( ! [A3: nat > nat > nat,B3: produc7248412053542808358at_nat] : ( P @ ( produc3209952032786966637at_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_73_prod__cases,axiom,
! [P: produc564279554677168641st_nat > $o,P2: produc564279554677168641st_nat] :
( ! [A3: a > nat,B3: produc432399132543013523st_nat] : ( P @ ( produc6785043063399293179st_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_74_prod__cases,axiom,
! [P: produc5032551385658279741list_a > $o,P2: produc5032551385658279741list_a] :
( ! [A3: a > a > $o,B3: list_a] : ( P @ ( produc8111569692950616493list_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_75_prod__cases,axiom,
! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
( ! [A3: a,B3: nat] : ( P @ ( product_Pair_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_76_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_77_prod__cases,axiom,
! [P: produc7590564867095724333idle_a > $o,P2: produc7590564867095724333idle_a] :
( ! [A3: a,B3: idle_a] : ( P @ ( produc1265230069547855005idle_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_78_prod__cases,axiom,
! [P: produc7037199475535947254eque_a > $o,P2: produc7037199475535947254eque_a] :
( ! [A3: a,B3: deque_a] : ( P @ ( produc3093615717782868582eque_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_79_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_80_surj__pair,axiom,
! [P2: produc7590564867095724333idle_a] :
? [X3: a,Y3: idle_a] :
( P2
= ( produc1265230069547855005idle_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_81_surj__pair,axiom,
! [P2: produc7037199475535947254eque_a] :
? [X3: a,Y3: deque_a] :
( P2
= ( produc3093615717782868582eque_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_82_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P2
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_83_Idle__Proof_Osize__pop,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle2 )
=> ( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( ( suc @ ( size_size_idle_a @ Idle ) )
= ( size_size_idle_a @ Idle2 ) ) ) ) ).
% Idle_Proof.size_pop
thf(fact_84_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_85_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_86_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_87_bind__simps_I1_J,axiom,
! [F: a > list_nat] :
( ( bind_a_nat @ nil_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_88_bind__simps_I1_J,axiom,
! [F: nat > list_a] :
( ( bind_nat_a @ nil_nat @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_89_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_90_member__rec_I2_J,axiom,
! [Y: a] :
~ ( member_a @ nil_a @ Y ) ).
% member_rec(2)
thf(fact_91_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_92_ssubst__Pair__rhs,axiom,
! [R: a,S: idle_a,R2: set_Pr7225968907286947725idle_a,S2: idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_93_ssubst__Pair__rhs,axiom,
! [R: a,S: deque_a,R2: set_Pr3011860922491142998eque_a,S2: deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_94_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_95_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_96_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_97_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_98_curryI,axiom,
! [F: produc7590564867095724333idle_a > $o,A: a,B: idle_a] :
( ( F @ ( produc1265230069547855005idle_a @ A @ B ) )
=> ( produc7330610896797225991le_a_o @ F @ A @ B ) ) ).
% curryI
thf(fact_99_curryI,axiom,
! [F: produc7037199475535947254eque_a > $o,A: a,B: deque_a] :
( ( F @ ( produc3093615717782868582eque_a @ A @ B ) )
=> ( produc2509474620477234558ue_a_o @ F @ A @ B ) ) ).
% curryI
thf(fact_100_curryI,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_101_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_102_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_103_Idle__Proof_Osize__push,axiom,
! [X: a,Idle2: idle_a] :
( ( size_size_idle_a @ ( push_a @ X @ Idle2 ) )
= ( suc @ ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.size_push
thf(fact_104_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_105_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_106_curryD,axiom,
! [F: produc7590564867095724333idle_a > $o,A: a,B: idle_a] :
( ( produc7330610896797225991le_a_o @ F @ A @ B )
=> ( F @ ( produc1265230069547855005idle_a @ A @ B ) ) ) ).
% curryD
thf(fact_107_curryD,axiom,
! [F: produc7037199475535947254eque_a > $o,A: a,B: deque_a] :
( ( produc2509474620477234558ue_a_o @ F @ A @ B )
=> ( F @ ( produc3093615717782868582eque_a @ A @ B ) ) ) ).
% curryD
thf(fact_108_curryD,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryD
thf(fact_109_curryE,axiom,
! [F: produc7590564867095724333idle_a > $o,A: a,B: idle_a] :
( ( produc7330610896797225991le_a_o @ F @ A @ B )
=> ( F @ ( produc1265230069547855005idle_a @ A @ B ) ) ) ).
% curryE
thf(fact_110_curryE,axiom,
! [F: produc7037199475535947254eque_a > $o,A: a,B: deque_a] :
( ( produc2509474620477234558ue_a_o @ F @ A @ B )
=> ( F @ ( produc3093615717782868582eque_a @ A @ B ) ) ) ).
% curryE
thf(fact_111_curryE,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryE
thf(fact_112_Idle_Opush_Oelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Stack @ StackSize ) )
=> ( Y
!= ( idle_a2 @ ( push_a2 @ X @ Stack ) @ ( suc @ StackSize ) ) ) ) ) ).
% Idle.push.elims
thf(fact_113_Idle_Opush_Osimps,axiom,
! [X: a,Stack2: stack_a,StackSize2: nat] :
( ( push_a @ X @ ( idle_a2 @ Stack2 @ StackSize2 ) )
= ( idle_a2 @ ( push_a2 @ X @ Stack2 ) @ ( suc @ StackSize2 ) ) ) ).
% Idle.push.simps
thf(fact_114_is__empty__idle_Oelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ X )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( type_i3216275384938974675tack_a @ Stack ) ) ) ).
% is_empty_idle.elims(3)
thf(fact_115_is__empty__idle_Oelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i7304311975391125061idle_a @ X )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a2 @ Stack @ Uu ) )
=> ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ).
% is_empty_idle.elims(2)
thf(fact_116_is__empty__idle_Oelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i7304311975391125061idle_a @ X )
= Y )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( Y
= ( ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ) ) ).
% is_empty_idle.elims(1)
thf(fact_117_is__empty__idle_Osimps,axiom,
! [Stack2: stack_a,Uu2: nat] :
( ( type_i7304311975391125061idle_a @ ( idle_a2 @ Stack2 @ Uu2 ) )
= ( type_i3216275384938974675tack_a @ Stack2 ) ) ).
% is_empty_idle.simps
thf(fact_118_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_119_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_120_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_121_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_122_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_123_Idle__Proof_Olist__empty,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
= nil_nat )
= ( type_i1745286555725844959le_nat @ Idle2 ) ) ).
% Idle_Proof.list_empty
thf(fact_124_Idle__Proof_Olist__empty,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
= nil_a )
= ( type_i7304311975391125061idle_a @ Idle2 ) ) ).
% Idle_Proof.list_empty
thf(fact_125_Idle__Proof_Olist__empty__2,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
= nil_nat )
=> ( type_i1745286555725844959le_nat @ Idle2 ) ) ).
% Idle_Proof.list_empty_2
thf(fact_126_Idle__Proof_Olist__empty__2,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
= nil_a )
=> ( type_i7304311975391125061idle_a @ Idle2 ) ) ).
% Idle_Proof.list_empty_2
thf(fact_127_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_128_Idle__Proof_Opush__list,axiom,
! [X: a,Idle2: idle_a] :
( ( idle_list_a @ ( push_a @ X @ Idle2 ) )
= ( cons_a @ X @ ( idle_list_a @ Idle2 ) ) ) ).
% Idle_Proof.push_list
thf(fact_129_Idle__Proof_Opush__list,axiom,
! [X: nat,Idle2: idle_nat] :
( ( idle_list_nat @ ( push_nat @ X @ Idle2 ) )
= ( cons_nat @ X @ ( idle_list_nat @ Idle2 ) ) ) ).
% Idle_Proof.push_list
thf(fact_130_Idle__Proof_Opop__list,axiom,
! [Idle2: idle_nat,X: nat,Idle: idle_nat] :
( ~ ( type_i1745286555725844959le_nat @ Idle2 )
=> ( ( ( pop_nat @ Idle2 )
= ( produc9134156765868440393le_nat @ X @ Idle ) )
=> ( ( cons_nat @ X @ ( idle_list_nat @ Idle ) )
= ( idle_list_nat @ Idle2 ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_131_Idle__Proof_Opop__list,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle2 )
=> ( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( ( cons_a @ X @ ( idle_list_a @ Idle ) )
= ( idle_list_a @ Idle2 ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_132_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_133_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_134_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_135_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ~ ! [X3: a,Xs2: list_a,Ys: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys ) ) ) ).
% splice.cases
thf(fact_136_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ~ ! [X3: nat,Xs2: list_nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ Ys ) ) ) ).
% splice.cases
thf(fact_137_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_138_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ( ! [Xs2: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat,Y3: nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_139_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X3: a,Ys: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_140_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ~ ! [P3: nat > nat > $o,X3: nat,Ys: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_141_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X3: a,Y3: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_142_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X3: nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [P3: nat > nat > $o,X3: nat,Y3: nat,Xs2: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_143_small__deque_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ( X
!= ( produc6837034575241423639list_a @ nil_a @ nil_a ) )
=> ( ! [X3: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ nil_a ) @ nil_a ) )
=> ( ! [X3: a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ X3 @ nil_a ) ) )
=> ( ! [X3: a,Y3: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ nil_a ) @ ( cons_a @ Y3 @ nil_a ) ) )
=> ( ! [X3: a,Y3: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ nil_a ) ) @ nil_a ) )
=> ( ! [X3: a,Y3: a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ nil_a ) ) ) )
=> ( ! [X3: a,Y3: a,Z: a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ ( cons_a @ Z @ nil_a ) ) ) ) )
=> ( ! [X3: a,Y3: a,Z: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ ( cons_a @ Z @ nil_a ) ) ) @ nil_a ) )
=> ( ! [X3: a,Y3: a,Z: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ nil_a ) ) @ ( cons_a @ Z @ nil_a ) ) )
=> ( ! [X3: a,Y3: a,Z: a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ nil_a ) @ ( cons_a @ Y3 @ ( cons_a @ Z @ nil_a ) ) ) )
=> ( ! [V: a,Vb: a,Va: a,Vc: a,Ve: list_a,B3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Va @ ( cons_a @ Vc @ Ve ) ) ) ) @ B3 ) )
=> ( ! [V: a,Vb: a,Va: a,Vd: list_a,Vc: a,Ve: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Va @ Vd ) ) ) @ ( cons_a @ Vc @ Ve ) ) )
=> ( ! [V: a,Vb: a,Ve: a,Vf: list_a,Va: a,Vd: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Ve @ Vf ) ) ) @ ( cons_a @ Va @ Vd ) ) )
=> ( ! [V: a,Vb: a,Vc: list_a,Va: a,Ve: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vb @ Vc ) ) @ ( cons_a @ Va @ ( cons_a @ Ve @ Vf ) ) ) )
=> ( ! [V: a,Vd: a,Va: a,Vf: list_a,Vb: a,Vc: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vd @ ( cons_a @ Va @ Vf ) ) ) @ ( cons_a @ Vb @ Vc ) ) )
=> ( ! [V: a,Vd: a,Ve: list_a,Vb: a,Va: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vd @ Ve ) ) @ ( cons_a @ Vb @ ( cons_a @ Va @ Vf ) ) ) )
=> ( ! [V: a,Vc: a,Vf: list_a,Vb: a,Vd: a,Ve: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ ( cons_a @ Vc @ Vf ) ) @ ( cons_a @ Vb @ ( cons_a @ Vd @ Ve ) ) ) )
=> ( ! [V: a,Va: list_a,Vb: a,Vd: a,Vc: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ ( cons_a @ Vb @ ( cons_a @ Vd @ ( cons_a @ Vc @ Vf ) ) ) ) )
=> ( ! [Vb: a,Vd: a,Vc: a,Vf: list_a,V: a,Va: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Vb @ ( cons_a @ Vd @ ( cons_a @ Vc @ Vf ) ) ) @ ( cons_a @ V @ Va ) ) )
=> ( ! [Vb: a,Vd: a,Ve: list_a,V: a,Vc: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Vb @ ( cons_a @ Vd @ Ve ) ) @ ( cons_a @ V @ ( cons_a @ Vc @ Vf ) ) ) )
=> ( ! [Vb: a,Va: a,Vf: list_a,V: a,Vd: a,Ve: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Vb @ ( cons_a @ Va @ Vf ) ) @ ( cons_a @ V @ ( cons_a @ Vd @ Ve ) ) ) )
=> ( ! [Vb: a,Vc: list_a,V: a,Vd: a,Va: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Vb @ Vc ) @ ( cons_a @ V @ ( cons_a @ Vd @ ( cons_a @ Va @ Vf ) ) ) ) )
=> ( ! [Va: a,Ve: a,Vf: list_a,V: a,Vb: a,Vc: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Va @ ( cons_a @ Ve @ Vf ) ) @ ( cons_a @ V @ ( cons_a @ Vb @ Vc ) ) ) )
=> ( ! [Va: a,Vd: list_a,V: a,Vb: a,Ve: a,Vf: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Va @ Vd ) @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Ve @ Vf ) ) ) ) )
=> ( ! [Vc: a,Ve: list_a,V: a,Vb: a,Va: a,Vd: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ Vc @ Ve ) @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Va @ Vd ) ) ) ) )
=> ~ ! [A3: list_a,V: a,Vb: a,Va: a,Vc: a,Ve: list_a] :
( X
!= ( produc6837034575241423639list_a @ A3 @ ( cons_a @ V @ ( cons_a @ Vb @ ( cons_a @ Va @ ( cons_a @ Vc @ Ve ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% small_deque.cases
thf(fact_144_small__deque_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ( X
!= ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
=> ( ! [X3: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ nil_nat ) @ nil_nat ) )
=> ( ! [X3: nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ X3 @ nil_nat ) ) )
=> ( ! [X3: nat,Y3: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ nil_nat ) @ ( cons_nat @ Y3 @ nil_nat ) ) )
=> ( ! [X3: nat,Y3: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) @ nil_nat ) )
=> ( ! [X3: nat,Y3: nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ ( cons_nat @ Z @ nil_nat ) ) ) ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ ( cons_nat @ Z @ nil_nat ) ) ) @ nil_nat ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) @ ( cons_nat @ Z @ nil_nat ) ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ nil_nat ) @ ( cons_nat @ Y3 @ ( cons_nat @ Z @ nil_nat ) ) ) )
=> ( ! [V: nat,Vb: nat,Va: nat,Vc: nat,Ve: list_nat,B3: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Va @ ( cons_nat @ Vc @ Ve ) ) ) ) @ B3 ) )
=> ( ! [V: nat,Vb: nat,Va: nat,Vd: list_nat,Vc: nat,Ve: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Va @ Vd ) ) ) @ ( cons_nat @ Vc @ Ve ) ) )
=> ( ! [V: nat,Vb: nat,Ve: nat,Vf: list_nat,Va: nat,Vd: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Ve @ Vf ) ) ) @ ( cons_nat @ Va @ Vd ) ) )
=> ( ! [V: nat,Vb: nat,Vc: list_nat,Va: nat,Ve: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vb @ Vc ) ) @ ( cons_nat @ Va @ ( cons_nat @ Ve @ Vf ) ) ) )
=> ( ! [V: nat,Vd: nat,Va: nat,Vf: list_nat,Vb: nat,Vc: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vd @ ( cons_nat @ Va @ Vf ) ) ) @ ( cons_nat @ Vb @ Vc ) ) )
=> ( ! [V: nat,Vd: nat,Ve: list_nat,Vb: nat,Va: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vd @ Ve ) ) @ ( cons_nat @ Vb @ ( cons_nat @ Va @ Vf ) ) ) )
=> ( ! [V: nat,Vc: nat,Vf: list_nat,Vb: nat,Vd: nat,Ve: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ ( cons_nat @ Vc @ Vf ) ) @ ( cons_nat @ Vb @ ( cons_nat @ Vd @ Ve ) ) ) )
=> ( ! [V: nat,Va: list_nat,Vb: nat,Vd: nat,Vc: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ ( cons_nat @ Vb @ ( cons_nat @ Vd @ ( cons_nat @ Vc @ Vf ) ) ) ) )
=> ( ! [Vb: nat,Vd: nat,Vc: nat,Vf: list_nat,V: nat,Va: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Vb @ ( cons_nat @ Vd @ ( cons_nat @ Vc @ Vf ) ) ) @ ( cons_nat @ V @ Va ) ) )
=> ( ! [Vb: nat,Vd: nat,Ve: list_nat,V: nat,Vc: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Vb @ ( cons_nat @ Vd @ Ve ) ) @ ( cons_nat @ V @ ( cons_nat @ Vc @ Vf ) ) ) )
=> ( ! [Vb: nat,Va: nat,Vf: list_nat,V: nat,Vd: nat,Ve: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Vb @ ( cons_nat @ Va @ Vf ) ) @ ( cons_nat @ V @ ( cons_nat @ Vd @ Ve ) ) ) )
=> ( ! [Vb: nat,Vc: list_nat,V: nat,Vd: nat,Va: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Vb @ Vc ) @ ( cons_nat @ V @ ( cons_nat @ Vd @ ( cons_nat @ Va @ Vf ) ) ) ) )
=> ( ! [Va: nat,Ve: nat,Vf: list_nat,V: nat,Vb: nat,Vc: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Va @ ( cons_nat @ Ve @ Vf ) ) @ ( cons_nat @ V @ ( cons_nat @ Vb @ Vc ) ) ) )
=> ( ! [Va: nat,Vd: list_nat,V: nat,Vb: nat,Ve: nat,Vf: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Va @ Vd ) @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Ve @ Vf ) ) ) ) )
=> ( ! [Vc: nat,Ve: list_nat,V: nat,Vb: nat,Va: nat,Vd: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ Vc @ Ve ) @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Va @ Vd ) ) ) ) )
=> ~ ! [A3: list_nat,V: nat,Vb: nat,Va: nat,Vc: nat,Ve: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ A3 @ ( cons_nat @ V @ ( cons_nat @ Vb @ ( cons_nat @ Va @ ( cons_nat @ Vc @ Ve ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% small_deque.cases
thf(fact_145_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_146_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_147_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_148_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_149_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_150_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_151_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_152_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_153_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_154_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_155_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_156_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_157_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_158_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_159_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_160_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_161_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_162_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_163_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_164_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 )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_165_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 )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_166_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $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_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_167_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_168_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_169_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys: list_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_170_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys2: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_a @ ( cons_nat @ Y3 @ Ys ) )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys: list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_171_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys2: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys: list_a] : ( P @ nil_nat @ ( cons_a @ Y3 @ Ys ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys: list_a] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_172_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys: list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_173_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_174_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_175_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_176_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_177_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_178_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_179_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_180_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_181_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_182_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_183_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_184_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_185_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_186_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_187_Idle__Proof_Olist__not__empty__2,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
!= nil_nat )
=> ~ ( type_i1745286555725844959le_nat @ Idle2 ) ) ).
% Idle_Proof.list_not_empty_2
thf(fact_188_Idle__Proof_Olist__not__empty__2,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
!= nil_a )
=> ~ ( type_i7304311975391125061idle_a @ Idle2 ) ) ).
% Idle_Proof.list_not_empty_2
thf(fact_189_Idle__Proof_Olist__not__empty,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
!= nil_nat )
= ( ~ ( type_i1745286555725844959le_nat @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty
thf(fact_190_Idle__Proof_Olist__not__empty,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
!= nil_a )
= ( ~ ( type_i7304311975391125061idle_a @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty
thf(fact_191_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_192_Idle__Proof_Opop__list__tl,axiom,
! [Idle2: idle_nat,X: nat,Idle: idle_nat] :
( ~ ( type_i1745286555725844959le_nat @ Idle2 )
=> ( ( ( pop_nat @ Idle2 )
= ( produc9134156765868440393le_nat @ X @ Idle ) )
=> ( ( cons_nat @ X @ ( tl_nat @ ( idle_list_nat @ Idle2 ) ) )
= ( idle_list_nat @ Idle2 ) ) ) ) ).
% Idle_Proof.pop_list_tl
thf(fact_193_Idle__Proof_Opop__list__tl,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle2 )
=> ( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( ( cons_a @ X @ ( tl_a @ ( idle_list_a @ Idle2 ) ) )
= ( idle_list_a @ Idle2 ) ) ) ) ).
% Idle_Proof.pop_list_tl
thf(fact_194_Idle_Opush_Opelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ( accp_P2118310398061403062idle_a @ push_rel_a @ ( produc1265230069547855005idle_a @ X @ Xa ) )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( Y
= ( idle_a2 @ ( push_a2 @ X @ Stack ) @ ( suc @ StackSize ) ) )
=> ~ ( accp_P2118310398061403062idle_a @ push_rel_a @ ( produc1265230069547855005idle_a @ X @ ( idle_a2 @ Stack @ StackSize ) ) ) ) ) ) ) ).
% Idle.push.pelims
thf(fact_195_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_196_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_197_pop__list__tl_H,axiom,
! [Idle2: idle_nat,X: nat,Idle: idle_nat] :
( ( ( pop_nat @ Idle2 )
= ( produc9134156765868440393le_nat @ X @ Idle ) )
=> ( ( idle_list_nat @ Idle )
= ( tl_nat @ ( idle_list_nat @ Idle2 ) ) ) ) ).
% pop_list_tl'
thf(fact_198_pop__list__tl_H,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( ( idle_list_a @ Idle )
= ( tl_a @ ( idle_list_a @ Idle2 ) ) ) ) ).
% pop_list_tl'
thf(fact_199_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_200_map__tailrec__rev_Oelims,axiom,
! [X: a > nat,Xa: list_a,Xb: list_nat,Y: list_nat] :
( ( ( map_ta8710832428924958105_a_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y != Xb ) )
=> ~ ! [A3: a,As: list_a] :
( ( Xa
= ( cons_a @ A3 @ As ) )
=> ( Y
!= ( map_ta8710832428924958105_a_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_201_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_202_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_a @ N @ nil_a )
= nil_Pr1417316670369895453_nat_a ) ).
% enumerate_simps(1)
thf(fact_203_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_204_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_205_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_206_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_207_list_Osel_I3_J,axiom,
! [X21: nat,X222: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_208_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_209_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_210_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_211_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_212_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_213_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_214_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A: nat,As2: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As2 ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_215_cons__tl,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= Ys2 )
=> ( Xs
= ( tl_nat @ Ys2 ) ) ) ).
% cons_tl
thf(fact_216_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_217_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_218_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_219_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_220_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_221_map__tailrec__rev_Opelims,axiom,
! [X: a > nat,Xa: list_a,Xb: list_nat,Y: list_nat] :
( ( ( map_ta8710832428924958105_a_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P7537707418771614264st_nat @ map_ta7397863945511617930_a_nat @ ( produc6785043063399293179st_nat @ X @ ( produc4792949784200893581st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_a )
=> ( ( Y = Xb )
=> ~ ( accp_P7537707418771614264st_nat @ map_ta7397863945511617930_a_nat @ ( produc6785043063399293179st_nat @ X @ ( produc4792949784200893581st_nat @ nil_a @ Xb ) ) ) ) )
=> ~ ! [A3: a,As: list_a] :
( ( Xa
= ( cons_a @ A3 @ As ) )
=> ( ( Y
= ( map_ta8710832428924958105_a_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P7537707418771614264st_nat @ map_ta7397863945511617930_a_nat @ ( produc6785043063399293179st_nat @ X @ ( produc4792949784200893581st_nat @ ( cons_a @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_222_map__tailrec__rev_Opelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
=> ~ ! [A3: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A3 @ As ) )
=> ( ( Y
= ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_223_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 ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_224_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 ) )
=> ~ ! [X3: a,Y3: nat,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys ) @ ( listrel_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_225_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 ) )
=> ~ ! [X3: nat,Y3: a,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys ) )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y3 ) @ R )
=> ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs2 @ Ys ) @ ( listrel_nat_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_226_listrel_Ocases,axiom,
! [A1: list_a,A22: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ A1 @ A22 ) @ ( listrel_a_idle_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_idle_a ) )
=> ~ ! [X3: a,Y3: idle_a,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys: list_idle_a] :
( ( A22
= ( cons_idle_a @ Y3 @ Ys ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X3 @ Y3 ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs2 @ Ys ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_227_listrel_Ocases,axiom,
! [A1: list_a,A22: list_deque_a,R: set_Pr3011860922491142998eque_a] :
( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ A1 @ A22 ) @ ( listrel_a_deque_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_deque_a ) )
=> ~ ! [X3: a,Y3: deque_a,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys: list_deque_a] :
( ( A22
= ( cons_deque_a @ Y3 @ Ys ) )
=> ( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X3 @ Y3 ) @ R )
=> ~ ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs2 @ Ys ) @ ( listrel_a_deque_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_228_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 ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_229_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 ) )
| ? [X4: a,Y4: a,Xs3: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_a @ Y4 @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_230_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 ) )
| ? [X4: a,Y4: nat,Xs3: list_a,Ys3: list_nat] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y4 ) @ R )
& ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys3 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_231_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 ) )
| ? [X4: nat,Y4: a,Xs3: list_nat,Ys3: list_a] :
( ( A1
= ( cons_nat @ X4 @ Xs3 ) )
& ( A22
= ( cons_a @ Y4 @ Ys3 ) )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y4 ) @ R )
& ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs3 @ Ys3 ) @ ( listrel_nat_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_232_listrel_Osimps,axiom,
! [A1: list_a,A22: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ A1 @ A22 ) @ ( listrel_a_idle_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_idle_a ) )
| ? [X4: a,Y4: idle_a,Xs3: list_a,Ys3: list_idle_a] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_idle_a @ Y4 @ Ys3 ) )
& ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X4 @ Y4 ) @ R )
& ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs3 @ Ys3 ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_233_listrel_Osimps,axiom,
! [A1: list_a,A22: list_deque_a,R: set_Pr3011860922491142998eque_a] :
( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ A1 @ A22 ) @ ( listrel_a_deque_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_deque_a ) )
| ? [X4: a,Y4: deque_a,Xs3: list_a,Ys3: list_deque_a] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_deque_a @ Y4 @ Ys3 ) )
& ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X4 @ Y4 ) @ R )
& ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs3 @ Ys3 ) @ ( listrel_a_deque_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_234_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 ) )
| ? [X4: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_235_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_236_listrel_ONil,axiom,
! [R: set_Pr4934435412358123699_a_nat] : ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ nil_a @ nil_nat ) @ ( listrel_a_nat @ R ) ) ).
% listrel.Nil
thf(fact_237_listrel_ONil,axiom,
! [R: set_Pr4193341848836149977_nat_a] : ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ nil_nat @ nil_a ) @ ( listrel_nat_a @ R ) ) ).
% listrel.Nil
thf(fact_238_listrel_ONil,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).
% listrel.Nil
thf(fact_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_listrel_OCons,axiom,
! [X: a,Y: idle_a,R: set_Pr7225968907286947725idle_a,Xs: list_a,Ys2: list_idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ R )
=> ( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs @ Ys2 ) @ ( listrel_a_idle_a @ R ) )
=> ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ ( cons_a @ X @ Xs ) @ ( cons_idle_a @ Y @ Ys2 ) ) @ ( listrel_a_idle_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_253_listrel_OCons,axiom,
! [X: a,Y: deque_a,R: set_Pr3011860922491142998eque_a,Xs: list_a,Ys2: list_deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X @ Y ) @ R )
=> ( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs @ Ys2 ) @ ( listrel_a_deque_a @ R ) )
=> ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ ( cons_a @ X @ Xs ) @ ( cons_deque_a @ Y @ Ys2 ) ) @ ( listrel_a_deque_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_254_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_255_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_idle_a @ R ) )
=> ~ ! [Y3: idle_a,Ys: list_idle_a] :
( ( Xs
= ( cons_idle_a @ Y3 @ Ys ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ Y @ Y3 ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Ys2 @ Ys ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_256_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_deque_a,R: set_Pr3011860922491142998eque_a] :
( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_deque_a @ R ) )
=> ~ ! [Y3: deque_a,Ys: list_deque_a] :
( ( Xs
= ( cons_deque_a @ Y3 @ Ys ) )
=> ( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ Y @ Y3 ) @ R )
=> ~ ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Ys2 @ Ys ) @ ( listrel_a_deque_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_257_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 ) )
=> ~ ! [Y3: nat,Ys: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_258_listrel__Cons2,axiom,
! [Xs: list_a,Y: idle_a,Ys2: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs @ ( cons_idle_a @ Y @ Ys2 ) ) @ ( listrel_a_idle_a @ R ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs2 ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X3 @ Y ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs2 @ Ys2 ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_259_listrel__Cons2,axiom,
! [Xs: list_a,Y: deque_a,Ys2: list_deque_a,R: set_Pr3011860922491142998eque_a] :
( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs @ ( cons_deque_a @ Y @ Ys2 ) ) @ ( listrel_a_deque_a @ R ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs2 ) )
=> ( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X3 @ Y ) @ R )
=> ~ ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs2 @ Ys2 ) @ ( listrel_a_deque_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_260_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 ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_261_lexord__Nil__left,axiom,
! [Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R ) )
= ( ? [A4: a,X4: list_a] :
( Y
= ( cons_a @ A4 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_262_lexord__Nil__left,axiom,
! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
= ( ? [A4: nat,X4: list_nat] :
( Y
= ( cons_nat @ A4 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_263_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_264_lexn__length,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,N: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lexn_nat @ R @ N ) )
=> ( ( ( size_size_list_nat @ Xs )
= N )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ).
% lexn_length
thf(fact_265_Cons__listrel1__Cons,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 ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_266_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_267_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_268_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y4: a,Ys3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ Y4 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_269_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_278_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_279_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_280_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_281_append__self__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_a ) ) ).
% append_self_conv
thf(fact_282_append__self__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_283_self__append__conv,axiom,
! [Y: list_a,Ys2: list_a] :
( ( Y
= ( append_a @ Y @ Ys2 ) )
= ( Ys2 = nil_a ) ) ).
% self_append_conv
thf(fact_284_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_285_append__self__conv2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_286_append__self__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_287_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_288_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_299_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_300_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_301_append__listrel1I,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).
% append_listrel1I
thf(fact_302_lexord__append__leftI,axiom,
! [U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_leftI
thf(fact_303_lexord__append__leftD,axiom,
! [X: list_nat,U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) )
=> ( ! [A3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_304_lexord__append__rightI,axiom,
! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
( ? [B4: nat,Z2: list_nat] :
( Y
= ( cons_nat @ B4 @ Z2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_rightI
thf(fact_305_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_306_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_307_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_308_append__Nil,axiom,
! [Ys2: list_a] :
( ( append_a @ nil_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_309_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_310_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_311_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_312_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_a @ nil_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_313_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_314_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_315_listrel1I,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_316_listrel1E,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ! [Us3: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append_nat @ Us3 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_317_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_318_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_319_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_320_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_321_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_322_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 ) ) )
| ? [Ys4: list_a] :
( ( Ys2
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_323_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 ) ) )
| ? [Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_324_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 ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_325_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 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_326_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_327_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_328_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_329_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_330_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_331_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_332_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_333_listrel1I2,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys2 ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I2
thf(fact_334_not__listrel1__Nil,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_335_not__listrel1__Nil,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_336_not__Nil__listrel1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_337_not__Nil__listrel1,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_338_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_339_lexord__linear,axiom,
! [R: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
( ! [A3: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R )
| ( A3 = B3 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A3 ) @ 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_340_lexord__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_341_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_342_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_343_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_344_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,X3: a,Xs4: list_a,Y3: a,Ys5: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_345_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,X3: nat,Xs4: list_nat,Y3: nat,Ys5: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_346_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_347_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_348_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_349_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ 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_350_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ 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_351_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_352_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_353_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( ! [Y3: nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_354_listrel1I1,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_355_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_356_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_357_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_358_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_359_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_360_length__product,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys2 ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_product
thf(fact_361_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys2: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys2 = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs3: list_a,Xs5: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys2
= ( append_a @ ( concat_a @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_362_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys2 = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_363_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_364_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_365_lexord__sufI,axiom,
! [U: list_nat,W2: list_nat,R: set_Pr1261947904930325089at_nat,V2: list_nat,Z3: 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 @ V2 ) @ ( append_nat @ W2 @ Z3 ) ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_sufI
thf(fact_366_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_367_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_368_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_369_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_370_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_371_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_372_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_373_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_374_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_375_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_376_concat__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_377_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_378_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_379_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_380_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_381_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_382_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_383_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_384_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_385_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_386_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_387_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_388_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_389_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_390_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_391_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_392_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_393_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_394_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_395_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_396_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_397_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_398_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_399_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_400_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_401_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_402_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_403_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_404_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_405_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_406_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_407_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_408_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_409_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_410_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_411_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_412_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_413_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_414_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_415_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_416_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_417_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_418_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_419_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_420_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_421_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_422_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_423_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_424_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_425_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_426_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_427_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_428_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_429_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_430_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_431_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_432_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_433_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_434_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_435_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_436_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_437_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_438_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_439_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_440_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_441_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_442_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_443_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_444_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_445_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_446_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_447_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_448_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_449_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_450_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_451_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_452_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_453_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_454_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_455_remdups__adj_Oelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ( ! [X3: a] :
( ( X
= ( cons_a @ X3 @ nil_a ) )
=> ( Y
!= ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( X
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y
= ( remdups_adj_a @ ( cons_a @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y
= ( cons_a @ X3 @ ( remdups_adj_a @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_456_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X3: nat] :
( ( X
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_457_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_458_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_459_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_460_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_461_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_462_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_463_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_464_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_465_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_466_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_467_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_468_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_469_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_470_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_471_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs2: list_nat,Xs4: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs4 ) @ Xss23 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs4 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_472_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_473_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl2 )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_474_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_475_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_476_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_477_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_478_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_479_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs @ Ys2 ) @ ( listrel_a_idle_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s8477453896275132790idle_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ ( nth_a @ Xs @ N4 ) @ ( nth_idle_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_480_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_deque_a,R: set_Pr3011860922491142998eque_a] :
( ( member3240663459192582047eque_a @ ( produc568430809755685862eque_a @ Xs @ Ys2 ) @ ( listrel_a_deque_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s6425414874530267071eque_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ ( nth_a @ Xs @ N4 ) @ ( nth_deque_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_481_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_482_Idle__Proof_Olist__not__empty__size__2,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty_size_2
thf(fact_483_Idle__Proof_Olist__not__empty__size__2,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
= nil_nat )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_nat @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty_size_2
thf(fact_484_Idle__Proof_Olist__not__empty__size,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
!= nil_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty_size
thf(fact_485_Idle__Proof_Olist__not__empty__size,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
!= nil_nat )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_nat @ Idle2 ) ) ) ).
% Idle_Proof.list_not_empty_size
thf(fact_486_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_487_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_488_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_489_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_490_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_491_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_492_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_493_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_494_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_495_last__remdups__adj,axiom,
! [Xs: list_nat] :
( ( last_nat @ ( remdups_adj_nat @ Xs ) )
= ( last_nat @ Xs ) ) ).
% last_remdups_adj
thf(fact_496_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_497_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_498_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_499_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_500_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_501_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_502_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_503_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_504_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_505_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_506_last__appendR,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ).
% last_appendR
thf(fact_507_last__appendR,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_508_last__appendL,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_509_last__appendL,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_510_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_511_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_512_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_513_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_514_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_515_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_516_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_517_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_518_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_519_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_520_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_521_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_522_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_523_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_524_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_525_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_526_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_527_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_528_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_529_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_530_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_531_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_532_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_533_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_534_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_535_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_536_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_537_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_538_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_539_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_540_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_541_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_542_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_543_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_544_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_545_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_546_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_547_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_548_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_549_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_550_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_551_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_552_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_553_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_554_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_555_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_556_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_557_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_558_longest__common__suffix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ss: list_a,Xs4: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys2
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_559_longest__common__suffix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs4 @ Ss ) )
& ( Ys2
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs4 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs4 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_560_last__append,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) )
& ( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ) ).
% last_append
thf(fact_561_last__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_562_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_563_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_564_deque_Osize_I11_J,axiom,
! [X51: idle_a,X52: idle_a] :
( ( size_size_deque_a @ ( idle_a3 @ X51 @ X52 ) )
= ( suc @ zero_zero_nat ) ) ).
% deque.size(11)
thf(fact_565_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_566_last__tl,axiom,
! [Xs: list_nat] :
( ( ( Xs = nil_nat )
| ( ( tl_nat @ Xs )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_tl
thf(fact_567_Idle__Proof_Osize__empty,axiom,
! [Idle2: idle_a] :
( ( ( size_size_idle_a @ Idle2 )
= zero_zero_nat )
= ( type_i7304311975391125061idle_a @ Idle2 ) ) ).
% Idle_Proof.size_empty
thf(fact_568_Idle__Proof_Osize__empty__2,axiom,
! [Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle2 )
=> ( zero_zero_nat
!= ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.size_empty_2
thf(fact_569_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_570_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_571_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_572_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_573_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_574_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_575_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_576_Idle__Proof_Olist__empty__size__2,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
!= nil_a )
=> ( zero_zero_nat
!= ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.list_empty_size_2
thf(fact_577_Idle__Proof_Olist__empty__size__2,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
!= nil_nat )
=> ( zero_zero_nat
!= ( size_size_idle_nat @ Idle2 ) ) ) ).
% Idle_Proof.list_empty_size_2
thf(fact_578_Idle__Proof_Olist__empty__size,axiom,
! [Idle2: idle_a] :
( ( ( idle_list_a @ Idle2 )
= nil_a )
= ( zero_zero_nat
= ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.list_empty_size
thf(fact_579_Idle__Proof_Olist__empty__size,axiom,
! [Idle2: idle_nat] :
( ( ( idle_list_nat @ Idle2 )
= nil_nat )
= ( zero_zero_nat
= ( size_size_idle_nat @ Idle2 ) ) ) ).
% Idle_Proof.list_empty_size
thf(fact_580_Idle__Proof_Osize__not__empty__2,axiom,
! [Idle2: idle_a] :
( ( type_i7304311975391125061idle_a @ Idle2 )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle2 ) ) ) ).
% Idle_Proof.size_not_empty_2
thf(fact_581_Idle__Proof_Osize__not__empty,axiom,
! [Idle2: idle_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle2 ) )
= ( ~ ( type_i7304311975391125061idle_a @ Idle2 ) ) ) ).
% Idle_Proof.size_not_empty
thf(fact_582_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_583_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_584_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys2: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys2 )
= ( ( Ys2 != nil_a )
& ( ( butlast_a @ Ys2 )
= Xs )
& ( ( last_a @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_585_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs )
& ( ( last_nat @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_586_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_587_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_588_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_589_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_590_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_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_600_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_601_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_602_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_603_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_604_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_605_take__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ! [I2: nat] :
( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ).
% take_equalityI
thf(fact_606_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_607_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_608_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_609_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_610_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_611_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_612_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 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_613_lex__take__index,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_614_take__last__length,axiom,
! [Xs: list_a] :
( ( ( take_a @ ( suc @ zero_zero_nat ) @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ Xs ) ) ) ).
% take_last_length
thf(fact_615_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_616_subrelI,axiom,
! [R: set_Pr7225968907286947725idle_a,S: set_Pr7225968907286947725idle_a] :
( ! [X3: a,Y3: idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X3 @ Y3 ) @ R )
=> ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le1072348969082359597idle_a @ R @ S ) ) ).
% subrelI
thf(fact_617_subrelI,axiom,
! [R: set_Pr3011860922491142998eque_a,S: set_Pr3011860922491142998eque_a] :
( ! [X3: a,Y3: deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X3 @ Y3 ) @ R )
=> ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X3 @ Y3 ) @ S ) )
=> ( ord_le1319221647696452342eque_a @ R @ S ) ) ).
% subrelI
thf(fact_618_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_619_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 ) )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) ) )
& ( ( take_nat @ I3 @ X )
= ( take_nat @ I3 @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X @ I3 ) @ ( nth_nat @ Y @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_620_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_621_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R ) )
= ( ? [Y4: nat,N4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ Y4 ) @ R )
& ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
& ( Ys2
= ( list_update_nat @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_622_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_623_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_624_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_625_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_626_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_627_rev__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( rev_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( rev_nat @ Ys2 ) @ ( rev_nat @ Xs ) ) ) ).
% rev_append
thf(fact_628_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X4: list_nat] : X4 ) ) ).
% drop0
thf(fact_629_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_630_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_631_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_632_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_633_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_634_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_635_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_636_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_637_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_638_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_639_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_640_singleton__rev__conv,axiom,
! [X: nat,Xs: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_641_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_642_rev__singleton__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_643_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_644_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_645_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_646_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_647_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_648_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_649_butlast__rev,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( tl_nat @ Xs ) ) ) ).
% butlast_rev
thf(fact_650_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys2 ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys2 ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_651_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
= ( Xs
= ( append_nat @ ( rev_nat @ Ys2 ) @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_663_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_664_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_665_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_666_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_667_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q )
= ( ord_min_nat @ ( times_times_nat @ M @ Q ) @ ( times_times_nat @ N @ Q ) ) ) ).
% nat_mult_min_left
thf(fact_668_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q ) ) ) ).
% nat_mult_min_right
thf(fact_669_list__update_Osimps_I1_J,axiom,
! [I: nat,V2: a] :
( ( list_update_a @ nil_a @ I @ V2 )
= nil_a ) ).
% list_update.simps(1)
thf(fact_670_list__update_Osimps_I1_J,axiom,
! [I: nat,V2: nat] :
( ( list_update_nat @ nil_nat @ I @ V2 )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_671_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_672_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_673_tl__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% tl_drop
thf(fact_674_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_675_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_676_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_677_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_678_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_679_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_680_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_681_rev__app__single,axiom,
! [Xs: list_a,X: a] :
( ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) )
= ( rev_a @ ( cons_a @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_682_rev__app__single,axiom,
! [Xs: list_nat,X: nat] :
( ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X @ nil_nat ) )
= ( rev_nat @ ( cons_nat @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_683_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_684_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_685_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_686_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_687_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_688_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_689_RealTimeDeque__Aux_OlistL_Osimps_I5_J,axiom,
! [Left2: idle_nat,Right: idle_nat] :
( ( realTi2392226096638470052tL_nat @ ( idle_nat3 @ Left2 @ Right ) )
= ( append_nat @ ( idle_list_nat @ Left2 ) @ ( rev_nat @ ( idle_list_nat @ Right ) ) ) ) ).
% RealTimeDeque_Aux.listL.simps(5)
thf(fact_690_RealTimeDeque__Aux_OlistL_Osimps_I5_J,axiom,
! [Left2: idle_a,Right: idle_a] :
( ( realTi2141985871725878314istL_a @ ( idle_a3 @ Left2 @ Right ) )
= ( append_a @ ( idle_list_a @ Left2 ) @ ( rev_a @ ( idle_list_a @ Right ) ) ) ) ).
% RealTimeDeque_Aux.listL.simps(5)
thf(fact_691_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_692_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_693_deque_Osize__gen_I5_J,axiom,
! [X: a > nat,X51: idle_a,X52: idle_a] :
( ( size_deque_a @ X @ ( idle_a3 @ X51 @ X52 ) )
= ( suc @ zero_zero_nat ) ) ).
% deque.size_gen(5)
thf(fact_694_hd__remdups__adj,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( remdups_adj_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% hd_remdups_adj
thf(fact_695_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_696_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_697_hd__take,axiom,
! [J: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_nat @ ( take_nat @ J @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_698_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_699_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_700_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_701_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_702_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_703_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_704_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_705_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_706_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_707_list_Osel_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_708_cons__hd,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= Ys2 )
=> ( X
= ( hd_nat @ Ys2 ) ) ) ).
% cons_hd
thf(fact_709_hd__concat,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( ( hd_list_a @ Xs )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs ) )
= ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_710_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_711_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_712_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_713_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_714_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_715_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_716_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_717_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_718_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_719_last__rev,axiom,
! [Xs: list_nat] :
( ( last_nat @ ( rev_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% last_rev
thf(fact_720_hd__rev,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( rev_nat @ Xs ) )
= ( last_nat @ Xs ) ) ).
% hd_rev
thf(fact_721_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_722_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_723_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_724_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_725_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_726_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
| ( Ys2 = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys2 ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys2 ) ) ) ) ).
% remdups_adj_append'
thf(fact_727_remdups__adj__append_H,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
| ( Ys2 = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys2 ) ) )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( remdups_adj_nat @ Xs ) @ ( remdups_adj_nat @ Ys2 ) ) ) ) ).
% remdups_adj_append'
thf(fact_728_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_729_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_730_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_731_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_732_rev__tl__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( rev_a @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) )
= ( rev_a @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_733_rev__tl__hd,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( rev_nat @ ( tl_nat @ Xs ) ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) )
= ( rev_nat @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_734_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_735_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_736_remdups__adj__singleton__iff,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ ( remdups_adj_a @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ ( hd_a @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_737_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_738_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_a @ ( replicate_list_a @ I @ nil_a ) )
= nil_a ) ).
% concat_replicate_trivial
thf(fact_739_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_740_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_741_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_742_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_743_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_744_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_745_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_746_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_747_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_748_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_749_hd__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_750_last__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_751_take__replicate,axiom,
! [I: nat,K: nat,X: nat] :
( ( take_nat @ I @ ( replicate_nat @ K @ X ) )
= ( replicate_nat @ ( ord_min_nat @ I @ K ) @ X ) ) ).
% take_replicate
thf(fact_752_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_753_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_754_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_755_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_756_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_757_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_758_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_759_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Ys2 @ Xs ) )
=> ? [M6: nat,N2: nat,Zs2: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M6 @ Zs2 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= Ys2 ) ) ) ).
% comm_append_are_replicate
thf(fact_760_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_761_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_762_remdups__adj__replicate,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= ( cons_a @ X @ nil_a ) ) ) ) ).
% remdups_adj_replicate
thf(fact_763_remdups__adj__replicate,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_764_remdups__adj__singleton,axiom,
! [Xs: list_a,X: a] :
( ( ( remdups_adj_a @ Xs )
= ( cons_a @ X @ nil_a ) )
=> ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_765_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_766_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_767_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_768_last__drop__rev,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( last_a @ Xs ) @ ( drop_a @ one_one_nat @ ( rev_a @ Xs ) ) )
= ( rev_a @ Xs ) ) ) ).
% last_drop_rev
thf(fact_769_last__drop__rev,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( last_nat @ Xs ) @ ( drop_nat @ one_one_nat @ ( rev_nat @ Xs ) ) )
= ( rev_nat @ Xs ) ) ) ).
% last_drop_rev
thf(fact_770_take__last,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ one_one_nat @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) ).
% take_last
thf(fact_771_take__last,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ one_one_nat @ ( rev_nat @ Xs ) )
= ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) ) ) ).
% take_last
thf(fact_772_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_773_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys2 ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys2 )
& ( ( Xs = nil_a )
| ( Ys2 = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_774_distinct__adj__append__iff,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( ( distinct_adj_nat @ Xs )
& ( distinct_adj_nat @ Ys2 )
& ( ( Xs = nil_nat )
| ( Ys2 = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_775_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_776_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_777_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_778_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_779_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_780_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_781_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_782_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_783_distinct__adj__Cons__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_784_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_785_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_786_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_787_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_788_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_789_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_790_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_791_distinct__adj__appendD2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys2 ) )
=> ( distinct_adj_nat @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_792_distinct__adj__appendD1,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys2 ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_793_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_794_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_795_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_796_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_797_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_798_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_799_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_800_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_801_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_802_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_803_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_804_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_805_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_806_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_807_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_808_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_809_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_810_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_811_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_812_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_813_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_814_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_815_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_816_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_817_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_818_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_819_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_820_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_821_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_822_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_823_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_824_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_825_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_826_distinct__adj__ConsD,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_827_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_828_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_829_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_830_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_831_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_832_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_833_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_834_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_835_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_836_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_837_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_838_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_839_replicate__add,axiom,
! [N: nat,M: nat,X: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M ) @ X )
= ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ M @ X ) ) ) ).
% replicate_add
thf(fact_840_take__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_841_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_842_distinct__adj__singleton,axiom,
! [X: nat] : ( distinct_adj_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_843_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N4: nat,Xs3: list_nat] : ( plus_plus_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_844_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_845_take__add,axiom,
! [I: nat,J: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_846_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_847_distinct__adj__Cons,axiom,
! [X: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_848_distinct__adj__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_849_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_850_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_851_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_852_comm__append__is__replicate,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Ys2 @ Xs ) )
=> ? [N2: nat,Zs2: list_a] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_a @ ( replicate_list_a @ N2 @ Zs2 ) )
= ( append_a @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_853_comm__append__is__replicate,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_nat )
=> ( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Ys2 @ Xs ) )
=> ? [N2: nat,Zs2: list_nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= ( append_nat @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_854_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F2: a > nat,Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_a @ Xs3 ) ) @ ( size_list_a @ F2 @ ( tl_a @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_855_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs3: list_nat] : ( if_nat @ ( Xs3 = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_nat @ Xs3 ) ) @ ( size_list_nat @ F2 @ ( tl_nat @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_856_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys2: list_idle_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8477453896275132790idle_a @ Ys2 ) )
=> ( ( nth_Pr3434503280043392308idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) @ I )
= ( produc1265230069547855005idle_a @ ( nth_a @ Xs @ I ) @ ( nth_idle_a @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_857_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys2: list_deque_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s6425414874530267071eque_a @ Ys2 ) )
=> ( ( nth_Pr3409035872437518461eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) @ I )
= ( produc3093615717782868582eque_a @ ( nth_a @ Xs @ I ) @ ( nth_deque_a @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_858_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_859_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_860_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_861_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_862_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_863_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_864_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_865_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_866_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_867_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_868_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_869_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_870_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_871_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_872_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_873_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_874_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_875_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_876_drop__replicate,axiom,
! [I: nat,K: nat,X: nat] :
( ( drop_nat @ I @ ( replicate_nat @ K @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_877_size__list__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
( ( size_list_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ F @ Ys2 ) ) ) ).
% size_list_append
thf(fact_878_Nil__eq__zip__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( nil_Product_prod_a_a
= ( zip_a_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
| ( Ys2 = nil_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_879_Nil__eq__zip__iff,axiom,
! [Xs: list_a,Ys2: list_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( zip_a_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
| ( Ys2 = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_880_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys2: list_a] :
( ( nil_Pr1417316670369895453_nat_a
= ( zip_nat_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
| ( Ys2 = nil_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_881_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
| ( Ys2 = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_882_zip__eq__Nil__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( zip_a_a @ Xs @ Ys2 )
= nil_Product_prod_a_a )
= ( ( Xs = nil_a )
| ( Ys2 = nil_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_883_zip__eq__Nil__iff,axiom,
! [Xs: list_a,Ys2: list_nat] :
( ( ( zip_a_nat @ Xs @ Ys2 )
= nil_Pr7402525243500994295_a_nat )
= ( ( Xs = nil_a )
| ( Ys2 = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_884_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys2: list_a] :
( ( ( zip_nat_a @ Xs @ Ys2 )
= nil_Pr1417316670369895453_nat_a )
= ( ( Xs = nil_nat )
| ( Ys2 = nil_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_885_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( zip_nat_nat @ Xs @ Ys2 )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_nat )
| ( Ys2 = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_886_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_887_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_888_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_889_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_890_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_891_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_892_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_893_tl__replicate,axiom,
! [N: nat,X: nat] :
( ( tl_nat @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_894_zip__replicate,axiom,
! [I: nat,X: a,J: nat,Y: idle_a] :
( ( zip_a_idle_a @ ( replicate_a @ I @ X ) @ ( replicate_idle_a @ J @ Y ) )
= ( replic4511300930391394541idle_a @ ( ord_min_nat @ I @ J ) @ ( produc1265230069547855005idle_a @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_895_zip__replicate,axiom,
! [I: nat,X: a,J: nat,Y: deque_a] :
( ( zip_a_deque_a @ ( replicate_a @ I @ X ) @ ( replicate_deque_a @ J @ Y ) )
= ( replic3687150286780796342eque_a @ ( ord_min_nat @ I @ J ) @ ( produc3093615717782868582eque_a @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_896_zip__replicate,axiom,
! [I: nat,X: nat,J: nat,Y: nat] :
( ( zip_nat_nat @ ( replicate_nat @ I @ X ) @ ( replicate_nat @ J @ Y ) )
= ( replic4235873036481779905at_nat @ ( ord_min_nat @ I @ J ) @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_897_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: idle_a,Ys2: list_idle_a] :
( ( zip_a_idle_a @ ( cons_a @ X @ Xs ) @ ( cons_idle_a @ Y @ Ys2 ) )
= ( cons_P7029976571565372387idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ ( zip_a_idle_a @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_898_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: deque_a,Ys2: list_deque_a] :
( ( zip_a_deque_a @ ( cons_a @ X @ Xs ) @ ( cons_deque_a @ Y @ Ys2 ) )
= ( cons_P8083696988307304876eque_a @ ( produc3093615717782868582eque_a @ X @ Y ) @ ( zip_a_deque_a @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_899_zip__Cons__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_900_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys2: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_901_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_902_length__zip,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_zip
thf(fact_903_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_904_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_905_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_906_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_907_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_908_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_909_take__zip,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( take_P2173866234530122223at_nat @ N @ ( zip_nat_nat @ Xs @ Ys2 ) )
= ( zip_nat_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ N @ Ys2 ) ) ) ).
% take_zip
thf(fact_910_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_911_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_912_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_913_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_914_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_915_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_916_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_917_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_918_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_919_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_920_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_921_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_922_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_923_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_924_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_925_drop__zip,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs @ Ys2 ) )
= ( zip_nat_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_926_zip__rev,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys2 ) )
= ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ) ).
% zip_rev
thf(fact_927_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys2: list_idle_a,Y: idle_a] :
( ( zip_a_idle_a @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_idle_a @ Ys2 @ I @ Y ) )
= ( list_u3350830261621523099idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) @ I @ ( produc1265230069547855005idle_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_928_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys2: list_deque_a,Y: deque_a] :
( ( zip_a_deque_a @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_deque_a @ Ys2 @ I @ Y ) )
= ( list_u2964271099763749988eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) @ I @ ( produc3093615717782868582eque_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_929_zip__update,axiom,
! [Xs: list_nat,I: nat,X: nat,Ys2: list_nat,Y: nat] :
( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys2 @ I @ Y ) )
= ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) @ I @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_930_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_931_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_932_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_933_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_934_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_935_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_936_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_937_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_938_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_939_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_940_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_941_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_942_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_943_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_944_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_945_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_946_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_947_hd__zip,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( hd_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys2 ) )
= ( product_Pair_a_a @ ( hd_a @ Xs ) @ ( hd_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_948_hd__zip,axiom,
! [Xs: list_a,Ys2: list_nat] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ ( zip_a_nat @ Xs @ Ys2 ) )
= ( product_Pair_a_nat @ ( hd_a @ Xs ) @ ( hd_nat @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_949_hd__zip,axiom,
! [Xs: list_nat,Ys2: list_a] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_a )
=> ( ( hd_Pro2949996684582079736_nat_a @ ( zip_nat_a @ Xs @ Ys2 ) )
= ( product_Pair_nat_a @ ( hd_nat @ Xs ) @ ( hd_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_950_hd__zip,axiom,
! [Xs: list_a,Ys2: list_idle_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_idle_a )
=> ( ( hd_Pro9081346849183306136idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) )
= ( produc1265230069547855005idle_a @ ( hd_a @ Xs ) @ ( hd_idle_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_951_hd__zip,axiom,
! [Xs: list_a,Ys2: list_deque_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_deque_a )
=> ( ( hd_Pro5574702693030206177eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) )
= ( produc3093615717782868582eque_a @ ( hd_a @ Xs ) @ ( hd_deque_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_952_hd__zip,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) )
= ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_953_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_954_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_955_zip__obtain__same__length,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_P6011104703257516679at_nat > $o] :
( ! [Zs2: list_nat,Ws2: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( N2
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( Zs2
= ( take_nat @ N2 @ Xs ) )
=> ( ( Ws2
= ( take_nat @ N2 @ Ys2 ) )
=> ( P @ ( zip_nat_nat @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ).
% zip_obtain_same_length
thf(fact_956_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_957_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys2: list_idle_a,Xy: produc7590564867095724333idle_a,Xys: list_P474335768696175027idle_a] :
( ( ( zip_a_idle_a @ Xs @ Ys2 )
= ( cons_P7029976571565372387idle_a @ Xy @ Xys ) )
=> ~ ! [X3: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs4 ) )
=> ! [Y3: idle_a,Ys5: list_idle_a] :
( ( Ys2
= ( cons_idle_a @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc1265230069547855005idle_a @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_a_idle_a @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_958_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys2: list_deque_a,Xy: produc7037199475535947254eque_a,Xys: list_P9016464757445826172eque_a] :
( ( ( zip_a_deque_a @ Xs @ Ys2 )
= ( cons_P8083696988307304876eque_a @ Xy @ Xys ) )
=> ~ ! [X3: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs4 ) )
=> ! [Y3: deque_a,Ys5: list_deque_a] :
( ( Ys2
= ( cons_deque_a @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( produc3093615717782868582eque_a @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_a_deque_a @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_959_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys2 )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs4 ) )
=> ! [Y3: nat,Ys5: list_nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_960_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_961_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_962_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_963_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_964_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_965_last__zip,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( last_P8790725268278465478od_a_a @ ( zip_a_a @ Xs @ Ys2 ) )
= ( product_Pair_a_a @ ( last_a @ Xs ) @ ( last_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_966_last__zip,axiom,
! [Xs: list_a,Ys2: list_idle_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_idle_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s8477453896275132790idle_a @ Ys2 ) )
=> ( ( last_P7618890818322975820idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) )
= ( produc1265230069547855005idle_a @ ( last_a @ Xs ) @ ( last_idle_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_967_last__zip,axiom,
! [Xs: list_a,Ys2: list_deque_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_deque_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s6425414874530267071eque_a @ Ys2 ) )
=> ( ( last_P5659436501804209045eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) )
= ( produc3093615717782868582eque_a @ ( last_a @ Xs ) @ ( last_deque_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_968_last__zip,axiom,
! [Xs: list_a,Ys2: list_nat] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_nat )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( last_P2271748490522340894_a_nat @ ( zip_a_nat @ Xs @ Ys2 ) )
= ( product_Pair_a_nat @ ( last_a @ Xs ) @ ( last_nat @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_969_last__zip,axiom,
! [Xs: list_nat,Ys2: list_a] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_a )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( last_P5509911954246017860_nat_a @ ( zip_nat_a @ Xs @ Ys2 ) )
= ( product_Pair_nat_a @ ( last_nat @ Xs ) @ ( last_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_970_last__zip,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys2 != nil_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_971_zip__append2,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( zip_nat_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ ( take_nat @ ( size_size_list_nat @ Ys2 ) @ Xs ) @ Ys2 ) @ ( zip_nat_nat @ ( drop_nat @ ( size_size_list_nat @ Ys2 ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_972_zip__append1,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( zip_nat_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) @ ( zip_nat_nat @ Ys2 @ ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_973_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_974_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_975_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_976_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_Idle__Proof_Osize__pop__sub,axiom,
! [Idle2: idle_a,X: a,Idle: idle_a] :
( ( ( pop_a @ Idle2 )
= ( produc1265230069547855005idle_a @ X @ Idle ) )
=> ( ( size_size_idle_a @ Idle )
= ( minus_minus_nat @ ( size_size_idle_a @ Idle2 ) @ one_one_nat ) ) ) ).
% Idle_Proof.size_pop_sub
thf(fact_988_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_989_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_990_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_991_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_992_Cons__replicate__eq,axiom,
! [X: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_993_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_994_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_995_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_996_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_997_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_998_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_999_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1000_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_a @ N @ nil_a )
= nil_a ) ).
% take_rev_empty
thf(fact_1001_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_nat @ N @ nil_nat )
= nil_nat ) ).
% take_rev_empty
thf(fact_1002_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_1003_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_1004_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1005_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1006_sorted__wrt1,axiom,
! [P: a > a > $o,X: a] : ( sorted_wrt_a @ P @ ( cons_a @ X @ nil_a ) ) ).
% sorted_wrt1
thf(fact_1007_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1008_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_1009_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_1010_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_1011_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_1012_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1013_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_1014_sorted__wrt_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( sorted_wrt_a @ P @ nil_a ) ).
% sorted_wrt.simps(1)
thf(fact_1015_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1016_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_1017_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_1018_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1019_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_1020_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1021_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P4: nat > nat > $o,Xs3: list_nat] :
! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P4 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1022_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_1023_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_1024_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1025_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_1026_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ ( suc @ I3 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1027_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1028_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I3: nat,J3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1029_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_1030_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_1031_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_1032_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I3 ) ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1033_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1034_Idle_Opop_Oelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a @ X )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( Y
!= ( produc1265230069547855005idle_a @ ( first_a @ Stack ) @ ( idle_a2 @ ( pop_a2 @ Stack ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) ) ) ) ).
% Idle.pop.elims
thf(fact_1035_Idle_Opop_Osimps,axiom,
! [Stack2: stack_a,StackSize2: nat] :
( ( pop_a @ ( idle_a2 @ Stack2 @ StackSize2 ) )
= ( produc1265230069547855005idle_a @ ( first_a @ Stack2 ) @ ( idle_a2 @ ( pop_a2 @ Stack2 ) @ ( minus_minus_nat @ StackSize2 @ one_one_nat ) ) ) ) ).
% Idle.pop.simps
thf(fact_1036_first__take__pop,axiom,
! [Stack2: stack_nat,X: nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( cons_nat @ ( first_nat @ Stack2 ) @ ( take_nat @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( stack_list_nat @ ( pop_nat2 @ Stack2 ) ) ) )
= ( take_nat @ X @ ( stack_list_nat @ Stack2 ) ) ) ) ) ).
% first_take_pop
thf(fact_1037_first__take__tl,axiom,
! [Big: stack_nat,Count: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_nat @ Big ) )
=> ( ( cons_nat @ ( first_nat @ Big ) @ ( take_nat @ Count @ ( tl_nat @ ( stack_list_nat @ Big ) ) ) )
= ( take_nat @ ( suc @ Count ) @ ( stack_list_nat @ Big ) ) ) ) ).
% first_take_tl
thf(fact_1038_size__list__length,axiom,
! [Stack2: stack_nat] :
( ( size_size_list_nat @ ( stack_list_nat @ Stack2 ) )
= ( size_size_stack_nat @ Stack2 ) ) ).
% size_list_length
thf(fact_1039_pop__tl,axiom,
! [Stack2: stack_nat] :
( ( stack_list_nat @ ( pop_nat2 @ Stack2 ) )
= ( tl_nat @ ( stack_list_nat @ Stack2 ) ) ) ).
% pop_tl
thf(fact_1040_Stack__Proof_Opush__list,axiom,
! [X: nat,Stack2: stack_nat] :
( ( stack_list_nat @ ( push_nat2 @ X @ Stack2 ) )
= ( cons_nat @ X @ ( stack_list_nat @ Stack2 ) ) ) ).
% Stack_Proof.push_list
thf(fact_1041_Stack__Proof_Opop__list,axiom,
! [Stack2: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( stack_list_nat @ ( pop_nat2 @ Stack2 ) )
= ( tl_nat @ ( stack_list_nat @ Stack2 ) ) ) ) ).
% Stack_Proof.pop_list
thf(fact_1042_first__list,axiom,
! [Stack2: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( first_nat @ Stack2 )
= ( hd_nat @ ( stack_list_nat @ Stack2 ) ) ) ) ).
% first_list
thf(fact_1043_pop__list__length,axiom,
! [Stack2: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( suc @ ( size_size_list_nat @ ( stack_list_nat @ ( pop_nat2 @ Stack2 ) ) ) )
= ( size_size_list_nat @ ( stack_list_nat @ Stack2 ) ) ) ) ).
% pop_list_length
thf(fact_1044_first__pop,axiom,
! [Stack2: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( cons_nat @ ( first_nat @ Stack2 ) @ ( stack_list_nat @ ( pop_nat2 @ Stack2 ) ) )
= ( stack_list_nat @ Stack2 ) ) ) ).
% first_pop
thf(fact_1045_first__hd,axiom,
( first_nat
= ( ^ [Stack3: stack_nat] : ( hd_nat @ ( stack_list_nat @ Stack3 ) ) ) ) ).
% first_hd
thf(fact_1046_Idle__Aux_Olist_Oelims,axiom,
! [X: idle_a,Y: list_a] :
( ( ( idle_list_a @ X )
= Y )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( Y
!= ( stack_list_a @ Stack ) ) ) ) ).
% Idle_Aux.list.elims
thf(fact_1047_Idle__Aux_Olist_Osimps,axiom,
! [Stack2: stack_a,Uu2: nat] :
( ( idle_list_a @ ( idle_a2 @ Stack2 @ Uu2 ) )
= ( stack_list_a @ Stack2 ) ) ).
% Idle_Aux.list.simps
thf(fact_1048_Stack__Proof_Olist__not__empty__2,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
= nil_a )
=> ( type_i3216275384938974675tack_a @ Stack2 ) ) ).
% Stack_Proof.list_not_empty_2
thf(fact_1049_Stack__Proof_Olist__not__empty__2,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
= nil_nat )
=> ( type_i6581939242220632785ck_nat @ Stack2 ) ) ).
% Stack_Proof.list_not_empty_2
thf(fact_1050_Stack__Proof_Olist__not__empty,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
!= nil_a )
= ( ~ ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty
thf(fact_1051_Stack__Proof_Olist__not__empty,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
!= nil_nat )
= ( ~ ( type_i6581939242220632785ck_nat @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty
thf(fact_1052_Stack__Proof_Olist__empty__2,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
!= nil_a )
=> ~ ( type_i3216275384938974675tack_a @ Stack2 ) ) ).
% Stack_Proof.list_empty_2
thf(fact_1053_Stack__Proof_Olist__empty__2,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
!= nil_nat )
=> ~ ( type_i6581939242220632785ck_nat @ Stack2 ) ) ).
% Stack_Proof.list_empty_2
thf(fact_1054_Stack__Proof_Olist__empty,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
= nil_a )
= ( type_i3216275384938974675tack_a @ Stack2 ) ) ).
% Stack_Proof.list_empty
thf(fact_1055_Stack__Proof_Olist__empty,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
= nil_nat )
= ( type_i6581939242220632785ck_nat @ Stack2 ) ) ).
% Stack_Proof.list_empty
thf(fact_1056_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
!= nil_a )
=> ( ( size_size_stack_a @ Stack2 )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_1057_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
!= nil_nat )
=> ( ( size_size_stack_nat @ Stack2 )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_1058_Stack__Proof_Olist__empty__size,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
= nil_a )
= ( ( size_size_stack_a @ Stack2 )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_1059_Stack__Proof_Olist__empty__size,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
= nil_nat )
= ( ( size_size_stack_nat @ Stack2 )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_1060_pop__drop,axiom,
! [Stack2: stack_nat] :
( ( stack_list_nat @ ( pop_nat2 @ Stack2 ) )
= ( drop_nat @ one_one_nat @ ( stack_list_nat @ Stack2 ) ) ) ).
% pop_drop
thf(fact_1061_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_1062_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
= nil_nat )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_nat @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_1063_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack2: stack_a] :
( ( ( stack_list_a @ Stack2 )
!= nil_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_1064_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack2: stack_nat] :
( ( ( stack_list_nat @ Stack2 )
!= nil_nat )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_nat @ Stack2 ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_1065_take__first,axiom,
! [S1: stack_nat,S22: stack_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_nat @ S1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_nat @ S22 ) )
=> ( ( ( take_nat @ ( size_size_stack_nat @ S1 ) @ ( stack_list_nat @ S22 ) )
= ( take_nat @ ( size_size_stack_nat @ S22 ) @ ( stack_list_nat @ S1 ) ) )
=> ( ( first_nat @ S1 )
= ( first_nat @ S22 ) ) ) ) ) ).
% take_first
thf(fact_1066_first__take,axiom,
! [Stack2: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack2 )
=> ( ( cons_a @ ( first_a @ Stack2 ) @ nil_a )
= ( take_a @ one_one_nat @ ( stack_list_a @ Stack2 ) ) ) ) ).
% first_take
thf(fact_1067_first__take,axiom,
! [Stack2: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack2 )
=> ( ( cons_nat @ ( first_nat @ Stack2 ) @ nil_nat )
= ( take_nat @ one_one_nat @ ( stack_list_nat @ Stack2 ) ) ) ) ).
% first_take
thf(fact_1068_Idle_Opop_Opelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a @ X )
= Y )
=> ( ( accp_idle_a @ pop_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( Y
= ( produc1265230069547855005idle_a @ ( first_a @ Stack ) @ ( idle_a2 @ ( pop_a2 @ Stack ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) )
=> ~ ( accp_idle_a @ pop_rel_a @ ( idle_a2 @ Stack @ StackSize ) ) ) ) ) ) ).
% Idle.pop.pelims
thf(fact_1069_drop__Cons__numeral,axiom,
! [V2: num,X: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V2 ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1070_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V2: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1071_take__Cons__numeral,axiom,
! [V2: num,X: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V2 ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1072_Idle__Aux_Olist_Opelims,axiom,
! [X: idle_a,Y: list_a] :
( ( ( idle_list_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_list_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( ( Y
= ( stack_list_a @ Stack ) )
=> ~ ( accp_idle_a @ idle_list_rel_a @ ( idle_a2 @ Stack @ Uu ) ) ) ) ) ) ).
% Idle_Aux.list.pelims
thf(fact_1073_size__idle_Opelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_size_idle_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( ( Y
= ( size_size_stack_a @ Stack ) )
=> ~ ( accp_idle_a @ idle_size_idle_rel_a @ ( idle_a2 @ Stack @ Uu ) ) ) ) ) ) ).
% size_idle.pelims
thf(fact_1074_is__empty__idle_Opelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ X )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a2 @ Stack @ Uu ) )
=> ( type_i3216275384938974675tack_a @ Stack ) ) ) ) ) ).
% is_empty_idle.pelims(3)
thf(fact_1075_is__empty__idle_Opelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i7304311975391125061idle_a @ X )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a2 @ Stack @ Uu ) )
=> ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ) ) ).
% is_empty_idle.pelims(2)
thf(fact_1076_is__empty__idle_Opelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i7304311975391125061idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a2 @ Stack @ Uu ) )
=> ( ( Y
= ( type_i3216275384938974675tack_a @ Stack ) )
=> ~ ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a2 @ Stack @ Uu ) ) ) ) ) ) ).
% is_empty_idle.pelims(1)
thf(fact_1077_invar__idle_Opelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( Y
= ( ( size_size_stack_a @ Stack )
= StackSize ) )
=> ~ ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a2 @ Stack @ StackSize ) ) ) ) ) ) ).
% invar_idle.pelims(1)
thf(fact_1078_invar__idle_Opelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a2 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ) ) ).
% invar_idle.pelims(2)
thf(fact_1079_invar__idle_Opelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a2 @ Stack @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a2 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
= StackSize ) ) ) ) ) ).
% invar_idle.pelims(3)
thf(fact_1080_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1081_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1082_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1083_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1084_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1085_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1086_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1087_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1088_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1089_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1090_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( ord_less_nat @ M @ P2 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P2 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1091_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1092_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1093_product__nth,axiom,
! [N: nat,Xs: list_a,Ys2: list_idle_a] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s8477453896275132790idle_a @ Ys2 ) ) )
=> ( ( nth_Pr3434503280043392308idle_a @ ( product_a_idle_a @ Xs @ Ys2 ) @ N )
= ( produc1265230069547855005idle_a @ ( nth_a @ Xs @ ( divide_divide_nat @ N @ ( size_s8477453896275132790idle_a @ Ys2 ) ) ) @ ( nth_idle_a @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s8477453896275132790idle_a @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_1094_product__nth,axiom,
! [N: nat,Xs: list_a,Ys2: list_deque_a] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s6425414874530267071eque_a @ Ys2 ) ) )
=> ( ( nth_Pr3409035872437518461eque_a @ ( product_a_deque_a @ Xs @ Ys2 ) @ N )
= ( produc3093615717782868582eque_a @ ( nth_a @ Xs @ ( divide_divide_nat @ N @ ( size_s6425414874530267071eque_a @ Ys2 ) ) ) @ ( nth_deque_a @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6425414874530267071eque_a @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_1095_product__nth,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys2 ) @ N )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_1096_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N4: nat,Xs3: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) @ ( take_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_1097_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( rotate_a @ N @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate_is_Nil_conv
thf(fact_1098_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( rotate_nat @ N @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_1099_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_1100_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1101_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1102_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1103_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1104_rotate__append,axiom,
! [L: list_nat,Q: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q ) )
= ( append_nat @ Q @ L ) ) ).
% rotate_append
thf(fact_1105_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N4: nat,Xs3: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_1106_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1107_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N4 )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1108_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1109_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1110_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1111_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1112_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1113_hd__rotate__conv__nth,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( rotate_a @ N @ Xs ) )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_1114_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_1115_Euclidean__Division_Odivmod__nat__def,axiom,
( euclidean_divmod_nat
= ( ^ [M2: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N4 ) @ ( modulo_modulo_nat @ M2 @ N4 ) ) ) ) ).
% Euclidean_Division.divmod_nat_def
thf(fact_1116_euclidean__relation__natI,axiom,
! [N: nat,Q: nat,R: nat,M: nat] :
( ( ( N = zero_zero_nat )
=> ( ( Q = zero_zero_nat )
& ( R = M ) ) )
=> ( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( ( R = zero_zero_nat )
& ( M
= ( times_times_nat @ Q @ N ) ) ) ) )
=> ( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( dvd_dvd_nat @ N @ M )
=> ( ( ord_less_nat @ R @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ Q @ N ) @ R ) ) ) ) )
=> ( ( product_Pair_nat_nat @ ( divide_divide_nat @ M @ N ) @ ( modulo_modulo_nat @ M @ N ) )
= ( product_Pair_nat_nat @ Q @ R ) ) ) ) ) ).
% euclidean_relation_natI
thf(fact_1117_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1118_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1119_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1120_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1121_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1122_nat__gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X3: nat,Y3: nat] :
( X
!= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% nat_gcd.cases
thf(fact_1123_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1124_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1125_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1126_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1127_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1128_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1129_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1130_dvd__minus__add,axiom,
! [Q: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq_nat @ Q @ N )
=> ( ( ord_less_eq_nat @ Q @ ( times_times_nat @ R @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1131_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1132_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1133_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1134_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1135_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_a] :
( ( ( concat_a @ Xss2 )
= nil_a )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
=> ( X4 = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1136_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1137_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_a] :
( ( nil_a
= ( concat_a @ Xss2 ) )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
=> ( X4 = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1138_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1139_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1140_in__set__impl__in__set__zip1,axiom,
! [Xs: list_a,Ys2: list_idle_a,X: a] :
( ( ( size_size_list_a @ Xs )
= ( size_s8477453896275132790idle_a @ Ys2 ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ~ ! [Y3: idle_a] :
~ ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y3 ) @ ( set_Pr1944501890106611522idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1141_in__set__impl__in__set__zip1,axiom,
! [Xs: list_a,Ys2: list_deque_a,X: a] :
( ( ( size_size_list_a @ Xs )
= ( size_s6425414874530267071eque_a @ Ys2 ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ~ ! [Y3: deque_a] :
~ ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X @ Y3 ) @ ( set_Pr655576496610006923eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1142_in__set__impl__in__set__zip1,axiom,
! [Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ~ ! [Y3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1143_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys2: list_idle_a,Y: idle_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s8477453896275132790idle_a @ Ys2 ) )
=> ( ( member_idle_a @ Y @ ( set_idle_a2 @ Ys2 ) )
=> ~ ! [X3: a] :
~ ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X3 @ Y ) @ ( set_Pr1944501890106611522idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1144_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys2: list_deque_a,Y: deque_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s6425414874530267071eque_a @ Ys2 ) )
=> ( ( member_deque_a @ Y @ ( set_deque_a2 @ Ys2 ) )
=> ~ ! [X3: a] :
~ ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X3 @ Y ) @ ( set_Pr655576496610006923eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1145_in__set__impl__in__set__zip2,axiom,
! [Xs: list_nat,Ys2: list_nat,Y: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat2 @ Y @ ( set_nat2 @ Ys2 ) )
=> ~ ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1146_zip__same,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
= ( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1147_in__set__zipE,axiom,
! [X: a,Y: idle_a,Xs: list_a,Ys2: list_idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ ( set_Pr1944501890106611522idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) ) )
=> ~ ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ~ ( member_idle_a @ Y @ ( set_idle_a2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1148_in__set__zipE,axiom,
! [X: a,Y: deque_a,Xs: list_a,Ys2: list_deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X @ Y ) @ ( set_Pr655576496610006923eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) ) )
=> ~ ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ~ ( member_deque_a @ Y @ ( set_deque_a2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1149_in__set__zipE,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
=> ~ ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_1150_set__zip__leftD,axiom,
! [X: a,Y: idle_a,Xs: list_a,Ys2: list_idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ ( set_Pr1944501890106611522idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) ) )
=> ( member_a2 @ X @ ( set_a2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1151_set__zip__leftD,axiom,
! [X: a,Y: deque_a,Xs: list_a,Ys2: list_deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X @ Y ) @ ( set_Pr655576496610006923eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) ) )
=> ( member_a2 @ X @ ( set_a2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1152_set__zip__leftD,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1153_set__zip__rightD,axiom,
! [X: a,Y: idle_a,Xs: list_a,Ys2: list_idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ ( set_Pr1944501890106611522idle_a @ ( zip_a_idle_a @ Xs @ Ys2 ) ) )
=> ( member_idle_a @ Y @ ( set_idle_a2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1154_set__zip__rightD,axiom,
! [X: a,Y: deque_a,Xs: list_a,Ys2: list_deque_a] :
( ( member8785101876683738399eque_a @ ( produc3093615717782868582eque_a @ X @ Y ) @ ( set_Pr655576496610006923eque_a @ ( zip_a_deque_a @ Xs @ Ys2 ) ) )
=> ( member_deque_a @ Y @ ( set_deque_a2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1155_set__zip__rightD,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys2 ) ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_1156_in__set__dropD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1157_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1158_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1159_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1160_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1161_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1162_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1163_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1164_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1165_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1166_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_1167_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1168_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Xs6: list_nat,Ys7: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys7 ) ) )
= ( ( Xs = Xs6 )
& ( Ys2 = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1169_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1170_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1171_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1172_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1173_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1174_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1175_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat2 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1176_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1177_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1178_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1179_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a2 @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1180_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1181_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a2 @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1182_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1183_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a2 @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a2 @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1184_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X: nat] :
( ( A != nil_nat )
=> ( ( member_nat2 @ X @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1185_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1186_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a2 @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1187_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1188_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1189_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X4: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X4 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1190_Cons__in__subseqsD,axiom,
! [Y: nat,Ys2: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys2 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1191_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1192_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1193_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q4: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q4 @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q4 @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1194_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys2 )
=> ( ( ( set_nat2 @ Ys2 )
= ( set_nat2 @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_1195_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys2 ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Ys2 ) )
=> ( P @ X4 @ Y4 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1196_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_1197_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_1198_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1199_sorted__simps_I2_J,axiom,
! [X: nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys2 ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys2 ) )
=> ( ord_less_eq_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys2 ) ) ) ).
% sorted_simps(2)
thf(fact_1200_strict__sorted__simps_I2_J,axiom,
! [X: nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys2 ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys2 ) )
=> ( ord_less_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys2 ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1201_sorted__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Ys2 ) )
=> ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).
% sorted_append
thf(fact_1202_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1203_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1204_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1205_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1206_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1207_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1208_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1209_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1210_lexord__partial__trans,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X3: nat,Y3: nat,Z: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R ) ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( lexord_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1211_lexord__same__pref__iff,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R ) )
= ( ? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R ) )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1212_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_1213_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_1214_card__set__1__iff__replicate,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ? [X4: a] :
( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1215_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X4: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1216_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_1217_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_1218_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_1219_rotate1__fixpoint__card,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= Xs )
=> ( ( Xs = nil_a )
| ( ( finite_card_a @ ( set_a2 @ Xs ) )
= one_one_nat ) ) ) ).
% rotate1_fixpoint_card
thf(fact_1220_rotate1__fixpoint__card,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= Xs )
=> ( ( Xs = nil_nat )
| ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= one_one_nat ) ) ) ).
% rotate1_fixpoint_card
thf(fact_1221_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A5 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A5 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1222_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N2: nat] :
! [X6: list_nat] :
( ( member_list_nat @ X6 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1223_card__le__Suc0__iff__eq,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A5 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
=> ! [Y4: nat] :
( ( member_nat2 @ Y4 @ A5 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1224_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A5: set_nat,L: list_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A5 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A5 ) ) )
= ( ( linord2614967742042102400et_nat @ A5 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1225_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A5: set_nat] :
( ~ ( finite_finite_nat @ A5 )
=> ( ( linord2614967742042102400et_nat @ A5 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1226_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A5 ) )
= A5 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1227_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A5: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A5 ) )
= ( finite_card_nat @ A5 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1228_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A5: set_nat,B5: set_nat] :
( ( ( linord2614967742042102400et_nat @ A5 )
= ( linord2614967742042102400et_nat @ B5 ) )
=> ( ( finite_finite_nat @ A5 )
=> ( ( finite_finite_nat @ B5 )
=> ( A5 = B5 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1229_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A5: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A5 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1230_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A5: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A5 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1231_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1232_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1233_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_1234_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_1235_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_1236_length__remove1,axiom,
! [X: nat,Xs: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1237_remove1__append,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( remove1_nat @ X @ Xs ) @ Ys2 ) ) )
& ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( remove1_nat @ X @ Ys2 ) ) ) ) ) ).
% remove1_append
thf(fact_1238_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_1239_remove1_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( remove1_nat @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1240_remove1_Osimps_I1_J,axiom,
! [X: a] :
( ( remove1_a @ X @ nil_a )
= nil_a ) ).
% remove1.simps(1)
thf(fact_1241_remove1_Osimps_I1_J,axiom,
! [X: nat] :
( ( remove1_nat @ X @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_1242_remove1__split,axiom,
! [A: nat,Xs: list_nat,Ys2: list_nat] :
( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A @ Xs )
= Ys2 )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat2 @ A @ ( set_nat2 @ Ls ) )
& ( Ys2
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1243_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F3: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] :
( X
!= ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1244_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1245_euclidean__relationI,axiom,
! [B: nat,Q: nat,R: nat,A: nat] :
( ( ( B = zero_zero_nat )
=> ( ( Q = zero_zero_nat )
& ( R = A ) ) )
=> ( ( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( R = zero_zero_nat )
& ( A
= ( times_times_nat @ Q @ B ) ) ) ) )
=> ( ( ( B != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ B @ A )
=> ( ( ( euclid3398187327856392827nt_nat @ R )
= ( euclid3398187327856392827nt_nat @ B ) )
& ( ord_less_nat @ ( euclid4777050414544973029ze_nat @ R ) @ ( euclid4777050414544973029ze_nat @ B ) )
& ( A
= ( plus_plus_nat @ ( times_times_nat @ Q @ B ) @ R ) ) ) ) )
=> ( ( product_Pair_nat_nat @ ( divide_divide_nat @ A @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= ( product_Pair_nat_nat @ Q @ R ) ) ) ) ) ).
% euclidean_relationI
thf(fact_1246_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_1247_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_1248_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_1249_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_1250_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1251_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1252_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1253_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1254_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1255_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1256_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1257_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1258_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1259_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_1260_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_1261_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1262_enumerate__eq__zip,axiom,
( enumerate_nat
= ( ^ [N4: nat,Xs3: list_nat] : ( zip_nat_nat @ ( upt @ N4 @ ( plus_plus_nat @ N4 @ ( size_size_list_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_1263_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1264_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
% 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_i8151583586401621767idle_a @ left ).
%------------------------------------------------------------------------------