TPTP Problem File: SLH0183^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0019_Stack_Proof/prob_00026_000816__6587332_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1586 ( 556 unt; 316 typ; 0 def)
% Number of atoms : 3681 (1644 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 11966 ( 686 ~; 118 |; 248 &;9275 @)
% ( 0 <=>;1639 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 878 ( 878 >; 0 *; 0 +; 0 <<)
% Number of symbols : 279 ( 276 usr; 18 con; 0-4 aty)
% Number of variables : 3670 ( 113 ^;3344 !; 213 ?;3670 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:48:29.825
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $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__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Stack__Ostack_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
stack_8114207454185536789at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_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__Stack__Ostack_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
stack_3651646392800388539od_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Stack__Ostack_Itf__a_J_J,type,
produc5950890884604230459tack_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__Stack__Ostack_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
stack_nat_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Stack__Ostack_It__List__Olist_Itf__a_J_J,type,
stack_list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Stack__Ostack_It__Num__Onum_J,type,
stack_num: $tType ).
thf(ty_n_t__Stack__Ostack_It__Nat__Onat_J,type,
stack_nat: $tType ).
thf(ty_n_t__Stack__Ostack_It__Int__Oint_J,type,
stack_int: $tType ).
thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
list_num: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $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__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (276)
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_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Euclidean__Division_Odivmod__nat,type,
euclidean_divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
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__Int__Oint,type,
times_times_int: int > int > int ).
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__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
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_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__Num__Onum,type,
bind_nat_num: list_nat > ( nat > list_num ) > list_num ).
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_001t__Num__Onum_001t__Int__Oint,type,
bind_num_int: list_num > ( num > list_int ) > list_int ).
thf(sy_c_List_Obind_001t__Num__Onum_001t__Nat__Onat,type,
bind_num_nat: list_num > ( num > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Num__Onum_001t__Num__Onum,type,
bind_num_num: list_num > ( num > list_num ) > list_num ).
thf(sy_c_List_Obind_001t__Num__Onum_001tf__a,type,
bind_num_a: list_num > ( num > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__Int__Oint,type,
bind_a_int: list_a > ( a > list_int ) > list_int ).
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_001t__Num__Onum,type,
bind_a_num: list_a > ( a > list_num ) > list_num ).
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_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_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
gen_length_nat_nat: nat > list_nat_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_Itf__a_J,type,
gen_length_list_a: nat > list_list_a > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Num__Onum,type,
gen_length_num: nat > list_num > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_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__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_Mtf__a_J,type,
last_P8790725268278465478od_a_a: list_P1396940483166286381od_a_a > product_prod_a_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_Olexord_001t__Nat__Onat,type,
lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001_062_It__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__Int__Oint,type,
cons_int: int > list_int > list_int ).
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__Num__Onum,type,
cons_num: num > list_num > list_num ).
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_Mtf__a_J,type,
cons_P7316939126706565853od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_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__Int__Oint,type,
nil_int: list_int ).
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__Num__Onum,type,
nil_num: list_num ).
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_Mtf__a_J,type,
nil_Product_prod_a_a: list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_nat_nat: list_nat_nat > nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Int__Oint,type,
hd_int: list_int > int ).
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__Num__Onum,type,
hd_num: list_num > num ).
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_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_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Nat__Onat,type,
list_all2_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001tf__a,type,
list_all2_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
tl_nat_nat: list_nat_nat > list_nat_nat ).
thf(sy_c_List_Olist_Otl_001t__Int__Oint,type,
tl_int: list_int > list_int ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Num__Onum,type,
tl_num: list_num > list_num ).
thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
tl_Pro4228036916689694320at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
tl_Product_prod_a_a: list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_ex1_nat_nat: ( ( nat > nat ) > $o ) > list_nat_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
list_ex1_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Num__Onum,type,
list_ex1_num: ( num > $o ) > list_num > $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__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_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_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrelp_001tf__a_001tf__a,type,
listrelp_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Num__Onum,type,
maps_nat_num: ( nat > list_num ) > list_nat > list_num ).
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_001t__Num__Onum_001t__Int__Oint,type,
maps_num_int: ( num > list_int ) > list_num > list_int ).
thf(sy_c_List_Omaps_001t__Num__Onum_001t__Nat__Onat,type,
maps_num_nat: ( num > list_nat ) > list_num > list_nat ).
thf(sy_c_List_Omaps_001t__Num__Onum_001t__Num__Onum,type,
maps_num_num: ( num > list_num ) > list_num > list_num ).
thf(sy_c_List_Omaps_001t__Num__Onum_001tf__a,type,
maps_num_a: ( num > list_a ) > list_num > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Int__Oint,type,
maps_a_int: ( a > list_int ) > list_a > list_int ).
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_001t__Num__Onum,type,
maps_a_num: ( a > list_num ) > list_a > list_num ).
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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: list_nat_nat > ( nat > nat ) > $o ).
thf(sy_c_List_Omember_001t__Int__Oint,type,
member_int: list_int > int > $o ).
thf(sy_c_List_Omember_001t__List__Olist_Itf__a_J,type,
member_list_a: list_list_a > list_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Num__Onum,type,
member_num: list_num > num > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_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_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onull_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
null_nat_nat: list_nat_nat > $o ).
thf(sy_c_List_Onull_001t__Int__Oint,type,
null_int: list_int > $o ).
thf(sy_c_List_Onull_001t__List__Olist_Itf__a_J,type,
null_list_a: list_list_a > $o ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001t__Num__Onum,type,
null_num: list_num > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001tf__a,type,
lexordp_eq_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Int__Oint,type,
ord_lexordp_eq_int: list_int > list_int > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat,type,
ord_lexordp_eq_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Num__Onum,type,
ord_lexordp_eq_num: list_num > list_num > $o ).
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_Oremdups__adj__rel_001tf__a,type,
remdups_adj_rel_a: list_a > list_a > $o ).
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_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_Oshuffles_001t__Nat__Onat,type,
shuffles_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Oshuffles__rel_001tf__a,type,
shuffles_rel_a: produc9164743771328383783list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
sorted_wrt_num: ( num > num > $o ) > list_num > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Osplice_001t__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osplice_001tf__a,type,
splice_a: list_a > list_a > list_a ).
thf(sy_c_List_Osplice__rel_001tf__a,type,
splice_rel_a: produc9164743771328383783list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Osuccessively_001t__Nat__Onat,type,
successively_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osuccessively_001tf__a,type,
successively_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Osuccessively__rel_001tf__a,type,
successively_rel_a: produc5032551385658279741list_a > produc5032551385658279741list_a > $o ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Ozip_001t__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__Nat__Onat,type,
zip_a_nat: list_a > list_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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_Itf__a_J,type,
size_size_list_a: list_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_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
ord_min_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Num__Onum,type,
ord_min_num: num > num > num ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_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_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__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_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__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__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Stack__Ostack_Itf__a_J,type,
produc6444587299782406187tack_a: a > stack_a > produc5950890884604230459tack_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
thf(sy_c_Stack_Ofirst_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
first_nat_nat: stack_nat_nat > nat > nat ).
thf(sy_c_Stack_Ofirst_001t__Int__Oint,type,
first_int: stack_int > int ).
thf(sy_c_Stack_Ofirst_001t__List__Olist_Itf__a_J,type,
first_list_a: stack_list_a > list_a ).
thf(sy_c_Stack_Ofirst_001t__Nat__Onat,type,
first_nat: stack_nat > nat ).
thf(sy_c_Stack_Ofirst_001t__Num__Onum,type,
first_num: stack_num > num ).
thf(sy_c_Stack_Ofirst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
first_904984104297451180at_nat: stack_8114207454185536789at_nat > product_prod_nat_nat ).
thf(sy_c_Stack_Ofirst_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
first_1410928715923173714od_a_a: stack_3651646392800388539od_a_a > product_prod_a_a ).
thf(sy_c_Stack_Ofirst_001tf__a,type,
first_a: stack_a > a ).
thf(sy_c_Stack_Ofirst__rel_001tf__a,type,
first_rel_a: stack_a > stack_a > $o ).
thf(sy_c_Stack_Ois__empty__stack__rel_001tf__a,type,
is_empty_stack_rel_a: stack_a > stack_a > $o ).
thf(sy_c_Stack_Opop_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
pop_nat_nat: stack_nat_nat > stack_nat_nat ).
thf(sy_c_Stack_Opop_001t__Int__Oint,type,
pop_int: stack_int > stack_int ).
thf(sy_c_Stack_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a: stack_list_a > stack_list_a ).
thf(sy_c_Stack_Opop_001t__Nat__Onat,type,
pop_nat: stack_nat > stack_nat ).
thf(sy_c_Stack_Opop_001t__Num__Onum,type,
pop_num: stack_num > stack_num ).
thf(sy_c_Stack_Opop_001tf__a,type,
pop_a: stack_a > stack_a ).
thf(sy_c_Stack_Opop__rel_001tf__a,type,
pop_rel_a: stack_a > stack_a > $o ).
thf(sy_c_Stack_Opush_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
push_nat_nat: ( nat > nat ) > stack_nat_nat > stack_nat_nat ).
thf(sy_c_Stack_Opush_001t__Int__Oint,type,
push_int: int > stack_int > stack_int ).
thf(sy_c_Stack_Opush_001t__Nat__Onat,type,
push_nat: nat > stack_nat > stack_nat ).
thf(sy_c_Stack_Opush_001t__Num__Onum,type,
push_num: num > stack_num > stack_num ).
thf(sy_c_Stack_Opush_001tf__a,type,
push_a: a > stack_a > stack_a ).
thf(sy_c_Stack_Opush__rel_001tf__a,type,
push_rel_a: produc5950890884604230459tack_a > produc5950890884604230459tack_a > $o ).
thf(sy_c_Stack_Ostack_OStack_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
stack_nat_nat2: list_nat_nat > list_nat_nat > stack_nat_nat ).
thf(sy_c_Stack_Ostack_OStack_001t__Int__Oint,type,
stack_int2: list_int > list_int > stack_int ).
thf(sy_c_Stack_Ostack_OStack_001t__List__Olist_Itf__a_J,type,
stack_list_a2: list_list_a > list_list_a > stack_list_a ).
thf(sy_c_Stack_Ostack_OStack_001t__Nat__Onat,type,
stack_nat2: list_nat > list_nat > stack_nat ).
thf(sy_c_Stack_Ostack_OStack_001t__Num__Onum,type,
stack_num2: list_num > list_num > stack_num ).
thf(sy_c_Stack_Ostack_OStack_001tf__a,type,
stack_a2: list_a > list_a > stack_a ).
thf(sy_c_Stack__Aux_Olist_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
stack_list_nat_nat: stack_nat_nat > list_nat_nat ).
thf(sy_c_Stack__Aux_Olist_001t__Int__Oint,type,
stack_list_int: stack_int > list_int ).
thf(sy_c_Stack__Aux_Olist_001t__List__Olist_Itf__a_J,type,
stack_list_list_a: stack_list_a > list_list_a ).
thf(sy_c_Stack__Aux_Olist_001t__Nat__Onat,type,
stack_list_nat: stack_nat > list_nat ).
thf(sy_c_Stack__Aux_Olist_001t__Num__Onum,type,
stack_list_num: stack_num > list_num ).
thf(sy_c_Stack__Aux_Olist_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
stack_5270352112346662202at_nat: stack_8114207454185536789at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_Stack__Aux_Olist_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
stack_1749223580616833248od_a_a: stack_3651646392800388539od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Stack__Aux_Olist_001tf__a,type,
stack_list_a3: stack_a > list_a ).
thf(sy_c_Stack__Aux_Olist__rel_001tf__a,type,
stack_list_rel_a: stack_a > stack_a > $o ).
thf(sy_c_Stack__Aux_Osize__stack__rel_001t__Nat__Onat,type,
stack_4774468673732873419el_nat: stack_nat > stack_nat > $o ).
thf(sy_c_Stack__Aux_Osize__stack__rel_001tf__a,type,
stack_9138219648075182787_rel_a: stack_a > stack_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
type_i1695466888102578112at_nat: stack_nat_nat > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__Int__Oint_J,type,
type_i2404088222711436077ck_int: stack_int > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__List__Olist_Itf__a_J_J,type,
type_i4148797458121419353list_a: stack_list_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_It__Num__Onum_J,type,
type_i7290183179169470491ck_num: stack_num > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
type_i299188854013607254at_nat: stack_8114207454185536789at_nat > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
type_i2782212211984832892od_a_a: stack_3651646392800388539od_a_a > $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__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__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,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
accp_P5335913909695611590list_a: ( produc5032551385658279741list_a > produc5032551385658279741list_a > $o ) > produc5032551385658279741list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
accp_P7377042638478740784list_a: ( produc9164743771328383783list_a > produc9164743771328383783list_a > $o ) > produc9164743771328383783list_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Stack__Ostack_Itf__a_J_J,type,
accp_P2456643874756862276tack_a: ( produc5950890884604230459tack_a > produc5950890884604230459tack_a > $o ) > produc5950890884604230459tack_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Stack__Ostack_It__Nat__Onat_J,type,
accp_stack_nat: ( stack_nat > stack_nat > $o ) > stack_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Stack__Ostack_Itf__a_J,type,
accp_stack_a: ( stack_a > stack_a > $o ) > stack_a > $o ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Nat__Onat,type,
mlex_prod_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
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_a2: list_a > set_list_a > $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_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_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_stack,type,
stack: stack_a ).
% Relevant facts (1262)
thf(fact_0_list__empty,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
= nil_list_a )
= ( type_i4148797458121419353list_a @ Stack ) ) ).
% list_empty
thf(fact_1_list__empty,axiom,
! [Stack: stack_num] :
( ( ( stack_list_num @ Stack )
= nil_num )
= ( type_i7290183179169470491ck_num @ Stack ) ) ).
% list_empty
thf(fact_2_list__empty,axiom,
! [Stack: stack_nat] :
( ( ( stack_list_nat @ Stack )
= nil_nat )
= ( type_i6581939242220632785ck_nat @ Stack ) ) ).
% list_empty
thf(fact_3_list__empty,axiom,
! [Stack: stack_int] :
( ( ( stack_list_int @ Stack )
= nil_int )
= ( type_i2404088222711436077ck_int @ Stack ) ) ).
% list_empty
thf(fact_4_list__empty,axiom,
! [Stack: stack_nat_nat] :
( ( ( stack_list_nat_nat @ Stack )
= nil_nat_nat )
= ( type_i1695466888102578112at_nat @ Stack ) ) ).
% list_empty
thf(fact_5_list__empty,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a3 @ Stack )
= nil_a )
= ( type_i3216275384938974675tack_a @ Stack ) ) ).
% list_empty
thf(fact_6_list__empty__2,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
=> ~ ( type_i4148797458121419353list_a @ Stack ) ) ).
% list_empty_2
thf(fact_7_list__empty__2,axiom,
! [Stack: stack_num] :
( ( ( stack_list_num @ Stack )
!= nil_num )
=> ~ ( type_i7290183179169470491ck_num @ Stack ) ) ).
% list_empty_2
thf(fact_8_list__empty__2,axiom,
! [Stack: stack_nat] :
( ( ( stack_list_nat @ Stack )
!= nil_nat )
=> ~ ( type_i6581939242220632785ck_nat @ Stack ) ) ).
% list_empty_2
thf(fact_9_list__empty__2,axiom,
! [Stack: stack_int] :
( ( ( stack_list_int @ Stack )
!= nil_int )
=> ~ ( type_i2404088222711436077ck_int @ Stack ) ) ).
% list_empty_2
thf(fact_10_list__empty__2,axiom,
! [Stack: stack_nat_nat] :
( ( ( stack_list_nat_nat @ Stack )
!= nil_nat_nat )
=> ~ ( type_i1695466888102578112at_nat @ Stack ) ) ).
% list_empty_2
thf(fact_11_list__empty__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a3 @ Stack )
!= nil_a )
=> ~ ( type_i3216275384938974675tack_a @ Stack ) ) ).
% list_empty_2
thf(fact_12_list__not__empty,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
= ( ~ ( type_i4148797458121419353list_a @ Stack ) ) ) ).
% list_not_empty
thf(fact_13_list__not__empty,axiom,
! [Stack: stack_num] :
( ( ( stack_list_num @ Stack )
!= nil_num )
= ( ~ ( type_i7290183179169470491ck_num @ Stack ) ) ) ).
% list_not_empty
thf(fact_14_list__not__empty,axiom,
! [Stack: stack_nat] :
( ( ( stack_list_nat @ Stack )
!= nil_nat )
= ( ~ ( type_i6581939242220632785ck_nat @ Stack ) ) ) ).
% list_not_empty
thf(fact_15_list__not__empty,axiom,
! [Stack: stack_int] :
( ( ( stack_list_int @ Stack )
!= nil_int )
= ( ~ ( type_i2404088222711436077ck_int @ Stack ) ) ) ).
% list_not_empty
thf(fact_16_list__not__empty,axiom,
! [Stack: stack_nat_nat] :
( ( ( stack_list_nat_nat @ Stack )
!= nil_nat_nat )
= ( ~ ( type_i1695466888102578112at_nat @ Stack ) ) ) ).
% list_not_empty
thf(fact_17_list__not__empty,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a3 @ Stack )
!= nil_a )
= ( ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ).
% list_not_empty
thf(fact_18_first__list,axiom,
! [Stack: stack_list_a] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( first_list_a @ Stack )
= ( hd_list_a @ ( stack_list_list_a @ Stack ) ) ) ) ).
% first_list
thf(fact_19_first__list,axiom,
! [Stack: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack )
=> ( ( first_nat @ Stack )
= ( hd_nat @ ( stack_list_nat @ Stack ) ) ) ) ).
% first_list
thf(fact_20_first__list,axiom,
! [Stack: stack_8114207454185536789at_nat] :
( ~ ( type_i299188854013607254at_nat @ Stack )
=> ( ( first_904984104297451180at_nat @ Stack )
= ( hd_Pro3460610213475200108at_nat @ ( stack_5270352112346662202at_nat @ Stack ) ) ) ) ).
% first_list
thf(fact_21_first__list,axiom,
! [Stack: stack_3651646392800388539od_a_a] :
( ~ ( type_i2782212211984832892od_a_a @ Stack )
=> ( ( first_1410928715923173714od_a_a @ Stack )
= ( hd_Product_prod_a_a @ ( stack_1749223580616833248od_a_a @ Stack ) ) ) ) ).
% first_list
thf(fact_22_first__list,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( first_a @ Stack )
= ( hd_a @ ( stack_list_a3 @ Stack ) ) ) ) ).
% first_list
thf(fact_23_pop__list,axiom,
! [Stack: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ Stack )
=> ( ( stack_list_nat @ ( pop_nat @ Stack ) )
= ( tl_nat @ ( stack_list_nat @ Stack ) ) ) ) ).
% pop_list
thf(fact_24_pop__list,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( stack_list_a3 @ ( pop_a @ Stack ) )
= ( tl_a @ ( stack_list_a3 @ Stack ) ) ) ) ).
% pop_list
thf(fact_25_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_26_list__ex1__simps_I1_J,axiom,
! [P: num > $o] :
~ ( list_ex1_num @ P @ nil_num ) ).
% list_ex1_simps(1)
thf(fact_27_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_28_list__ex1__simps_I1_J,axiom,
! [P: int > $o] :
~ ( list_ex1_int @ P @ nil_int ) ).
% list_ex1_simps(1)
thf(fact_29_list__ex1__simps_I1_J,axiom,
! [P: ( nat > nat ) > $o] :
~ ( list_ex1_nat_nat @ P @ nil_nat_nat ) ).
% list_ex1_simps(1)
thf(fact_30_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_31_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_list_a] :
( ( type_i4148797458121419353list_a @ X )
=> ( X
= ( stack_list_a2 @ nil_list_a @ nil_list_a ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_32_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_num] :
( ( type_i7290183179169470491ck_num @ X )
=> ( X
= ( stack_num2 @ nil_num @ nil_num ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_33_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_int] :
( ( type_i2404088222711436077ck_int @ X )
=> ( X
= ( stack_int2 @ nil_int @ nil_int ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_34_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_nat_nat] :
( ( type_i1695466888102578112at_nat @ X )
=> ( X
= ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_35_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_nat] :
( ( type_i6581939242220632785ck_nat @ X )
=> ( X
= ( stack_nat2 @ nil_nat @ nil_nat ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_36_is__empty__stack_Oelims_I2_J,axiom,
! [X: stack_a] :
( ( type_i3216275384938974675tack_a @ X )
=> ( X
= ( stack_a2 @ nil_a @ nil_a ) ) ) ).
% is_empty_stack.elims(2)
thf(fact_37_is__empty__stack_Osimps_I1_J,axiom,
type_i4148797458121419353list_a @ ( stack_list_a2 @ nil_list_a @ nil_list_a ) ).
% is_empty_stack.simps(1)
thf(fact_38_is__empty__stack_Osimps_I1_J,axiom,
type_i7290183179169470491ck_num @ ( stack_num2 @ nil_num @ nil_num ) ).
% is_empty_stack.simps(1)
thf(fact_39_is__empty__stack_Osimps_I1_J,axiom,
type_i2404088222711436077ck_int @ ( stack_int2 @ nil_int @ nil_int ) ).
% is_empty_stack.simps(1)
thf(fact_40_is__empty__stack_Osimps_I1_J,axiom,
type_i1695466888102578112at_nat @ ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) ).
% is_empty_stack.simps(1)
thf(fact_41_is__empty__stack_Osimps_I1_J,axiom,
type_i6581939242220632785ck_nat @ ( stack_nat2 @ nil_nat @ nil_nat ) ).
% is_empty_stack.simps(1)
thf(fact_42_is__empty__stack_Osimps_I1_J,axiom,
type_i3216275384938974675tack_a @ ( stack_a2 @ nil_a @ nil_a ) ).
% is_empty_stack.simps(1)
thf(fact_43_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_44_bind__simps_I1_J,axiom,
! [F: a > list_num] :
( ( bind_a_num @ nil_a @ F )
= nil_num ) ).
% bind_simps(1)
thf(fact_45_bind__simps_I1_J,axiom,
! [F: a > list_nat] :
( ( bind_a_nat @ nil_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_46_bind__simps_I1_J,axiom,
! [F: a > list_int] :
( ( bind_a_int @ nil_a @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_47_bind__simps_I1_J,axiom,
! [F: num > list_a] :
( ( bind_num_a @ nil_num @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_48_bind__simps_I1_J,axiom,
! [F: num > list_num] :
( ( bind_num_num @ nil_num @ F )
= nil_num ) ).
% bind_simps(1)
thf(fact_49_bind__simps_I1_J,axiom,
! [F: num > list_nat] :
( ( bind_num_nat @ nil_num @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_50_bind__simps_I1_J,axiom,
! [F: num > list_int] :
( ( bind_num_int @ nil_num @ F )
= nil_int ) ).
% bind_simps(1)
thf(fact_51_bind__simps_I1_J,axiom,
! [F: nat > list_a] :
( ( bind_nat_a @ nil_nat @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_52_bind__simps_I1_J,axiom,
! [F: nat > list_num] :
( ( bind_nat_num @ nil_nat @ F )
= nil_num ) ).
% bind_simps(1)
thf(fact_53_member__rec_I2_J,axiom,
! [Y: list_a] :
~ ( member_list_a @ nil_list_a @ Y ) ).
% member_rec(2)
thf(fact_54_member__rec_I2_J,axiom,
! [Y: num] :
~ ( member_num @ nil_num @ Y ) ).
% member_rec(2)
thf(fact_55_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_56_member__rec_I2_J,axiom,
! [Y: int] :
~ ( member_int @ nil_int @ Y ) ).
% member_rec(2)
thf(fact_57_member__rec_I2_J,axiom,
! [Y: nat > nat] :
~ ( member_nat_nat @ nil_nat_nat @ Y ) ).
% member_rec(2)
thf(fact_58_member__rec_I2_J,axiom,
! [Y: a] :
~ ( member_a @ nil_a @ Y ) ).
% member_rec(2)
thf(fact_59_push__list,axiom,
! [X: num,Stack: stack_num] :
( ( stack_list_num @ ( push_num @ X @ Stack ) )
= ( cons_num @ X @ ( stack_list_num @ Stack ) ) ) ).
% push_list
thf(fact_60_push__list,axiom,
! [X: nat,Stack: stack_nat] :
( ( stack_list_nat @ ( push_nat @ X @ Stack ) )
= ( cons_nat @ X @ ( stack_list_nat @ Stack ) ) ) ).
% push_list
thf(fact_61_push__list,axiom,
! [X: int,Stack: stack_int] :
( ( stack_list_int @ ( push_int @ X @ Stack ) )
= ( cons_int @ X @ ( stack_list_int @ Stack ) ) ) ).
% push_list
thf(fact_62_push__list,axiom,
! [X: nat > nat,Stack: stack_nat_nat] :
( ( stack_list_nat_nat @ ( push_nat_nat @ X @ Stack ) )
= ( cons_nat_nat @ X @ ( stack_list_nat_nat @ Stack ) ) ) ).
% push_list
thf(fact_63_push__list,axiom,
! [X: a,Stack: stack_a] :
( ( stack_list_a3 @ ( push_a @ X @ Stack ) )
= ( cons_a @ X @ ( stack_list_a3 @ Stack ) ) ) ).
% push_list
thf(fact_64_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_a @ N @ nil_list_a )
= N ) ).
% gen_length_code(1)
thf(fact_65_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_num @ N @ nil_num )
= N ) ).
% gen_length_code(1)
thf(fact_66_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_int @ N @ nil_int )
= N ) ).
% gen_length_code(1)
thf(fact_67_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat_nat @ N @ nil_nat_nat )
= N ) ).
% gen_length_code(1)
thf(fact_68_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_69_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_70_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_71_maps__simps_I2_J,axiom,
! [F: a > list_num] :
( ( maps_a_num @ F @ nil_a )
= nil_num ) ).
% maps_simps(2)
thf(fact_72_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_73_maps__simps_I2_J,axiom,
! [F: a > list_int] :
( ( maps_a_int @ F @ nil_a )
= nil_int ) ).
% maps_simps(2)
thf(fact_74_maps__simps_I2_J,axiom,
! [F: num > list_a] :
( ( maps_num_a @ F @ nil_num )
= nil_a ) ).
% maps_simps(2)
thf(fact_75_maps__simps_I2_J,axiom,
! [F: num > list_num] :
( ( maps_num_num @ F @ nil_num )
= nil_num ) ).
% maps_simps(2)
thf(fact_76_maps__simps_I2_J,axiom,
! [F: num > list_nat] :
( ( maps_num_nat @ F @ nil_num )
= nil_nat ) ).
% maps_simps(2)
thf(fact_77_maps__simps_I2_J,axiom,
! [F: num > list_int] :
( ( maps_num_int @ F @ nil_num )
= nil_int ) ).
% maps_simps(2)
thf(fact_78_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_79_maps__simps_I2_J,axiom,
! [F: nat > list_num] :
( ( maps_nat_num @ F @ nil_nat )
= nil_num ) ).
% maps_simps(2)
thf(fact_80_null__rec_I2_J,axiom,
null_list_a @ nil_list_a ).
% null_rec(2)
thf(fact_81_null__rec_I2_J,axiom,
null_num @ nil_num ).
% null_rec(2)
thf(fact_82_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
% null_rec(2)
thf(fact_83_null__rec_I2_J,axiom,
null_int @ nil_int ).
% null_rec(2)
thf(fact_84_null__rec_I2_J,axiom,
null_nat_nat @ nil_nat_nat ).
% null_rec(2)
thf(fact_85_null__rec_I2_J,axiom,
null_a @ nil_a ).
% null_rec(2)
thf(fact_86_list_Oinject,axiom,
! [X21: num,X22: list_num,Y21: num,Y22: list_num] :
( ( ( cons_num @ X21 @ X22 )
= ( cons_num @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_87_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_88_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_89_list_Oinject,axiom,
! [X21: nat > nat,X22: list_nat_nat,Y21: nat > nat,Y22: list_nat_nat] :
( ( ( cons_nat_nat @ X21 @ X22 )
= ( cons_nat_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_90_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_91_stack_Oinject,axiom,
! [X1: list_nat,X2: list_nat,Y1: list_nat,Y2: list_nat] :
( ( ( stack_nat2 @ X1 @ X2 )
= ( stack_nat2 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% stack.inject
thf(fact_92_stack_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( stack_a2 @ X1 @ X2 )
= ( stack_a2 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% stack.inject
thf(fact_93_list_Ocollapse,axiom,
! [List: list_P6011104703257516679at_nat] :
( ( List != nil_Pr5478986624290739719at_nat )
=> ( ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ List ) @ ( tl_Pro4228036916689694320at_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_94_list_Ocollapse,axiom,
! [List: list_P1396940483166286381od_a_a] :
( ( List != nil_Product_prod_a_a )
=> ( ( cons_P7316939126706565853od_a_a @ ( hd_Product_prod_a_a @ List ) @ ( tl_Product_prod_a_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_95_list_Ocollapse,axiom,
! [List: list_list_a] :
( ( List != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_96_list_Ocollapse,axiom,
! [List: list_num] :
( ( List != nil_num )
=> ( ( cons_num @ ( hd_num @ List ) @ ( tl_num @ List ) )
= List ) ) ).
% list.collapse
thf(fact_97_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_98_list_Ocollapse,axiom,
! [List: list_int] :
( ( List != nil_int )
=> ( ( cons_int @ ( hd_int @ List ) @ ( tl_int @ List ) )
= List ) ) ).
% list.collapse
thf(fact_99_list_Ocollapse,axiom,
! [List: list_nat_nat] :
( ( List != nil_nat_nat )
=> ( ( cons_nat_nat @ ( hd_nat_nat @ List ) @ ( tl_nat_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_100_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_101_hd__Cons__tl,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( tl_Pro4228036916689694320at_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_102_hd__Cons__tl,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( ( cons_P7316939126706565853od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( tl_Product_prod_a_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_103_hd__Cons__tl,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs ) @ ( tl_list_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_104_hd__Cons__tl,axiom,
! [Xs: list_num] :
( ( Xs != nil_num )
=> ( ( cons_num @ ( hd_num @ Xs ) @ ( tl_num @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_105_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_106_hd__Cons__tl,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( hd_int @ Xs ) @ ( tl_int @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_107_hd__Cons__tl,axiom,
! [Xs: list_nat_nat] :
( ( Xs != nil_nat_nat )
=> ( ( cons_nat_nat @ ( hd_nat_nat @ Xs ) @ ( tl_nat_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_108_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_109_list_Osel_I3_J,axiom,
! [X21: num,X22: list_num] :
( ( tl_num @ ( cons_num @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_110_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_111_list_Osel_I3_J,axiom,
! [X21: int,X22: list_int] :
( ( tl_int @ ( cons_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_112_list_Osel_I3_J,axiom,
! [X21: nat > nat,X22: list_nat_nat] :
( ( tl_nat_nat @ ( cons_nat_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_113_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_114_list_Osel_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( hd_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_115_list_Osel_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
( ( hd_Pro3460610213475200108at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_116_list_Osel_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] :
( ( hd_Product_prod_a_a @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_117_list_Osel_I1_J,axiom,
! [X21: num,X22: list_num] :
( ( hd_num @ ( cons_num @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_118_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_119_list_Osel_I1_J,axiom,
! [X21: int,X22: list_int] :
( ( hd_int @ ( cons_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_120_list_Osel_I1_J,axiom,
! [X21: nat > nat,X22: list_nat_nat] :
( ( hd_nat_nat @ ( cons_nat_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_121_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_122_null__rec_I1_J,axiom,
! [X: num,Xs: list_num] :
~ ( null_num @ ( cons_num @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_123_null__rec_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_124_null__rec_I1_J,axiom,
! [X: int,Xs: list_int] :
~ ( null_int @ ( cons_int @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_125_null__rec_I1_J,axiom,
! [X: nat > nat,Xs: list_nat_nat] :
~ ( null_nat_nat @ ( cons_nat_nat @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_126_null__rec_I1_J,axiom,
! [X: a,Xs: list_a] :
~ ( null_a @ ( cons_a @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_127_member__rec_I1_J,axiom,
! [X: num,Xs: list_num,Y: num] :
( ( member_num @ ( cons_num @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_num @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_128_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_129_member__rec_I1_J,axiom,
! [X: int,Xs: list_int,Y: int] :
( ( member_int @ ( cons_int @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_int @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_130_member__rec_I1_J,axiom,
! [X: nat > nat,Xs: list_nat_nat,Y: nat > nat] :
( ( member_nat_nat @ ( cons_nat_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_131_member__rec_I1_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( member_a @ ( cons_a @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_132_pop_Osimps_I3_J,axiom,
! [X: list_a,Right: list_list_a] :
( ( pop_list_a @ ( stack_list_a2 @ nil_list_a @ ( cons_list_a @ X @ Right ) ) )
= ( stack_list_a2 @ nil_list_a @ Right ) ) ).
% pop.simps(3)
thf(fact_133_pop_Osimps_I3_J,axiom,
! [X: num,Right: list_num] :
( ( pop_num @ ( stack_num2 @ nil_num @ ( cons_num @ X @ Right ) ) )
= ( stack_num2 @ nil_num @ Right ) ) ).
% pop.simps(3)
thf(fact_134_pop_Osimps_I3_J,axiom,
! [X: int,Right: list_int] :
( ( pop_int @ ( stack_int2 @ nil_int @ ( cons_int @ X @ Right ) ) )
= ( stack_int2 @ nil_int @ Right ) ) ).
% pop.simps(3)
thf(fact_135_pop_Osimps_I3_J,axiom,
! [X: nat > nat,Right: list_nat_nat] :
( ( pop_nat_nat @ ( stack_nat_nat2 @ nil_nat_nat @ ( cons_nat_nat @ X @ Right ) ) )
= ( stack_nat_nat2 @ nil_nat_nat @ Right ) ) ).
% pop.simps(3)
thf(fact_136_pop_Osimps_I3_J,axiom,
! [X: nat,Right: list_nat] :
( ( pop_nat @ ( stack_nat2 @ nil_nat @ ( cons_nat @ X @ Right ) ) )
= ( stack_nat2 @ nil_nat @ Right ) ) ).
% pop.simps(3)
thf(fact_137_pop_Osimps_I3_J,axiom,
! [X: a,Right: list_a] :
( ( pop_a @ ( stack_a2 @ nil_a @ ( cons_a @ X @ Right ) ) )
= ( stack_a2 @ nil_a @ Right ) ) ).
% pop.simps(3)
thf(fact_138_pop_Osimps_I2_J,axiom,
! [X: num,Left: list_num,Right: list_num] :
( ( pop_num @ ( stack_num2 @ ( cons_num @ X @ Left ) @ Right ) )
= ( stack_num2 @ Left @ Right ) ) ).
% pop.simps(2)
thf(fact_139_pop_Osimps_I2_J,axiom,
! [X: int,Left: list_int,Right: list_int] :
( ( pop_int @ ( stack_int2 @ ( cons_int @ X @ Left ) @ Right ) )
= ( stack_int2 @ Left @ Right ) ) ).
% pop.simps(2)
thf(fact_140_pop_Osimps_I2_J,axiom,
! [X: nat > nat,Left: list_nat_nat,Right: list_nat_nat] :
( ( pop_nat_nat @ ( stack_nat_nat2 @ ( cons_nat_nat @ X @ Left ) @ Right ) )
= ( stack_nat_nat2 @ Left @ Right ) ) ).
% pop.simps(2)
thf(fact_141_pop_Osimps_I2_J,axiom,
! [X: nat,Left: list_nat,Right: list_nat] :
( ( pop_nat @ ( stack_nat2 @ ( cons_nat @ X @ Left ) @ Right ) )
= ( stack_nat2 @ Left @ Right ) ) ).
% pop.simps(2)
thf(fact_142_pop_Osimps_I2_J,axiom,
! [X: a,Left: list_a,Right: list_a] :
( ( pop_a @ ( stack_a2 @ ( cons_a @ X @ Left ) @ Right ) )
= ( stack_a2 @ Left @ Right ) ) ).
% pop.simps(2)
thf(fact_143_pop_Osimps_I1_J,axiom,
( ( pop_list_a @ ( stack_list_a2 @ nil_list_a @ nil_list_a ) )
= ( stack_list_a2 @ nil_list_a @ nil_list_a ) ) ).
% pop.simps(1)
thf(fact_144_pop_Osimps_I1_J,axiom,
( ( pop_num @ ( stack_num2 @ nil_num @ nil_num ) )
= ( stack_num2 @ nil_num @ nil_num ) ) ).
% pop.simps(1)
thf(fact_145_pop_Osimps_I1_J,axiom,
( ( pop_int @ ( stack_int2 @ nil_int @ nil_int ) )
= ( stack_int2 @ nil_int @ nil_int ) ) ).
% pop.simps(1)
thf(fact_146_pop_Osimps_I1_J,axiom,
( ( pop_nat_nat @ ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) )
= ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) ) ).
% pop.simps(1)
thf(fact_147_pop_Osimps_I1_J,axiom,
( ( pop_nat @ ( stack_nat2 @ nil_nat @ nil_nat ) )
= ( stack_nat2 @ nil_nat @ nil_nat ) ) ).
% pop.simps(1)
thf(fact_148_pop_Osimps_I1_J,axiom,
( ( pop_a @ ( stack_a2 @ nil_a @ nil_a ) )
= ( stack_a2 @ nil_a @ nil_a ) ) ).
% pop.simps(1)
thf(fact_149_first_Osimps_I2_J,axiom,
! [X: list_a,Right: list_list_a] :
( ( first_list_a @ ( stack_list_a2 @ nil_list_a @ ( cons_list_a @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_150_first_Osimps_I2_J,axiom,
! [X: num,Right: list_num] :
( ( first_num @ ( stack_num2 @ nil_num @ ( cons_num @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_151_first_Osimps_I2_J,axiom,
! [X: int,Right: list_int] :
( ( first_int @ ( stack_int2 @ nil_int @ ( cons_int @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_152_first_Osimps_I2_J,axiom,
! [X: nat > nat,Right: list_nat_nat] :
( ( first_nat_nat @ ( stack_nat_nat2 @ nil_nat_nat @ ( cons_nat_nat @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_153_first_Osimps_I2_J,axiom,
! [X: nat,Right: list_nat] :
( ( first_nat @ ( stack_nat2 @ nil_nat @ ( cons_nat @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_154_first_Osimps_I2_J,axiom,
! [X: a,Right: list_a] :
( ( first_a @ ( stack_a2 @ nil_a @ ( cons_a @ X @ Right ) ) )
= X ) ).
% first.simps(2)
thf(fact_155_first_Osimps_I1_J,axiom,
! [X: num,Left: list_num,Right: list_num] :
( ( first_num @ ( stack_num2 @ ( cons_num @ X @ Left ) @ Right ) )
= X ) ).
% first.simps(1)
thf(fact_156_first_Osimps_I1_J,axiom,
! [X: int,Left: list_int,Right: list_int] :
( ( first_int @ ( stack_int2 @ ( cons_int @ X @ Left ) @ Right ) )
= X ) ).
% first.simps(1)
thf(fact_157_first_Osimps_I1_J,axiom,
! [X: nat > nat,Left: list_nat_nat,Right: list_nat_nat] :
( ( first_nat_nat @ ( stack_nat_nat2 @ ( cons_nat_nat @ X @ Left ) @ Right ) )
= X ) ).
% first.simps(1)
thf(fact_158_first_Osimps_I1_J,axiom,
! [X: nat,Left: list_nat,Right: list_nat] :
( ( first_nat @ ( stack_nat2 @ ( cons_nat @ X @ Left ) @ Right ) )
= X ) ).
% first.simps(1)
thf(fact_159_first_Osimps_I1_J,axiom,
! [X: a,Left: list_a,Right: list_a] :
( ( first_a @ ( stack_a2 @ ( cons_a @ X @ Left ) @ Right ) )
= X ) ).
% first.simps(1)
thf(fact_160_is__empty__stack_Oelims_I3_J,axiom,
! [X: stack_num] :
( ~ ( type_i7290183179169470491ck_num @ X )
=> ( ! [Vb: num,Vc: list_num,Va: list_num] :
( X
!= ( stack_num2 @ ( cons_num @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_num,Vb: num,Vc: list_num] :
( X
!= ( stack_num2 @ V @ ( cons_num @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.elims(3)
thf(fact_161_is__empty__stack_Oelims_I3_J,axiom,
! [X: stack_int] :
( ~ ( type_i2404088222711436077ck_int @ X )
=> ( ! [Vb: int,Vc: list_int,Va: list_int] :
( X
!= ( stack_int2 @ ( cons_int @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_int,Vb: int,Vc: list_int] :
( X
!= ( stack_int2 @ V @ ( cons_int @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.elims(3)
thf(fact_162_is__empty__stack_Oelims_I3_J,axiom,
! [X: stack_nat_nat] :
( ~ ( type_i1695466888102578112at_nat @ X )
=> ( ! [Vb: nat > nat,Vc: list_nat_nat,Va: list_nat_nat] :
( X
!= ( stack_nat_nat2 @ ( cons_nat_nat @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_nat_nat,Vb: nat > nat,Vc: list_nat_nat] :
( X
!= ( stack_nat_nat2 @ V @ ( cons_nat_nat @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.elims(3)
thf(fact_163_is__empty__stack_Oelims_I3_J,axiom,
! [X: stack_nat] :
( ~ ( type_i6581939242220632785ck_nat @ X )
=> ( ! [Vb: nat,Vc: list_nat,Va: list_nat] :
( X
!= ( stack_nat2 @ ( cons_nat @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_nat,Vb: nat,Vc: list_nat] :
( X
!= ( stack_nat2 @ V @ ( cons_nat @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.elims(3)
thf(fact_164_is__empty__stack_Oelims_I3_J,axiom,
! [X: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ X )
=> ( ! [Vb: a,Vc: list_a,Va: list_a] :
( X
!= ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_a,Vb: a,Vc: list_a] :
( X
!= ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.elims(3)
thf(fact_165_is__empty__stack_Osimps_I3_J,axiom,
! [V2: list_num,Vb2: num,Vc2: list_num] :
~ ( type_i7290183179169470491ck_num @ ( stack_num2 @ V2 @ ( cons_num @ Vb2 @ Vc2 ) ) ) ).
% is_empty_stack.simps(3)
thf(fact_166_is__empty__stack_Osimps_I3_J,axiom,
! [V2: list_int,Vb2: int,Vc2: list_int] :
~ ( type_i2404088222711436077ck_int @ ( stack_int2 @ V2 @ ( cons_int @ Vb2 @ Vc2 ) ) ) ).
% is_empty_stack.simps(3)
thf(fact_167_is__empty__stack_Osimps_I3_J,axiom,
! [V2: list_nat_nat,Vb2: nat > nat,Vc2: list_nat_nat] :
~ ( type_i1695466888102578112at_nat @ ( stack_nat_nat2 @ V2 @ ( cons_nat_nat @ Vb2 @ Vc2 ) ) ) ).
% is_empty_stack.simps(3)
thf(fact_168_is__empty__stack_Osimps_I3_J,axiom,
! [V2: list_nat,Vb2: nat,Vc2: list_nat] :
~ ( type_i6581939242220632785ck_nat @ ( stack_nat2 @ V2 @ ( cons_nat @ Vb2 @ Vc2 ) ) ) ).
% is_empty_stack.simps(3)
thf(fact_169_is__empty__stack_Osimps_I3_J,axiom,
! [V2: list_a,Vb2: a,Vc2: list_a] :
~ ( type_i3216275384938974675tack_a @ ( stack_a2 @ V2 @ ( cons_a @ Vb2 @ Vc2 ) ) ) ).
% is_empty_stack.simps(3)
thf(fact_170_is__empty__stack_Osimps_I2_J,axiom,
! [Vb2: num,Vc2: list_num,Va2: list_num] :
~ ( type_i7290183179169470491ck_num @ ( stack_num2 @ ( cons_num @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% is_empty_stack.simps(2)
thf(fact_171_is__empty__stack_Osimps_I2_J,axiom,
! [Vb2: int,Vc2: list_int,Va2: list_int] :
~ ( type_i2404088222711436077ck_int @ ( stack_int2 @ ( cons_int @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% is_empty_stack.simps(2)
thf(fact_172_is__empty__stack_Osimps_I2_J,axiom,
! [Vb2: nat > nat,Vc2: list_nat_nat,Va2: list_nat_nat] :
~ ( type_i1695466888102578112at_nat @ ( stack_nat_nat2 @ ( cons_nat_nat @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% is_empty_stack.simps(2)
thf(fact_173_is__empty__stack_Osimps_I2_J,axiom,
! [Vb2: nat,Vc2: list_nat,Va2: list_nat] :
~ ( type_i6581939242220632785ck_nat @ ( stack_nat2 @ ( cons_nat @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% is_empty_stack.simps(2)
thf(fact_174_is__empty__stack_Osimps_I2_J,axiom,
! [Vb2: a,Vc2: list_a,Va2: list_a] :
~ ( type_i3216275384938974675tack_a @ ( stack_a2 @ ( cons_a @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% is_empty_stack.simps(2)
thf(fact_175_pop_Ocases,axiom,
! [X: stack_list_a] :
( ( X
!= ( stack_list_a2 @ nil_list_a @ nil_list_a ) )
=> ( ! [X3: list_a,Left2: list_list_a,Right2: list_list_a] :
( X
!= ( stack_list_a2 @ ( cons_list_a @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: list_a,Right2: list_list_a] :
( X
!= ( stack_list_a2 @ nil_list_a @ ( cons_list_a @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_176_pop_Ocases,axiom,
! [X: stack_num] :
( ( X
!= ( stack_num2 @ nil_num @ nil_num ) )
=> ( ! [X3: num,Left2: list_num,Right2: list_num] :
( X
!= ( stack_num2 @ ( cons_num @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: num,Right2: list_num] :
( X
!= ( stack_num2 @ nil_num @ ( cons_num @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_177_pop_Ocases,axiom,
! [X: stack_int] :
( ( X
!= ( stack_int2 @ nil_int @ nil_int ) )
=> ( ! [X3: int,Left2: list_int,Right2: list_int] :
( X
!= ( stack_int2 @ ( cons_int @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: int,Right2: list_int] :
( X
!= ( stack_int2 @ nil_int @ ( cons_int @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_178_pop_Ocases,axiom,
! [X: stack_nat_nat] :
( ( X
!= ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) )
=> ( ! [X3: nat > nat,Left2: list_nat_nat,Right2: list_nat_nat] :
( X
!= ( stack_nat_nat2 @ ( cons_nat_nat @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: nat > nat,Right2: list_nat_nat] :
( X
!= ( stack_nat_nat2 @ nil_nat_nat @ ( cons_nat_nat @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_179_pop_Ocases,axiom,
! [X: stack_nat] :
( ( X
!= ( stack_nat2 @ nil_nat @ nil_nat ) )
=> ( ! [X3: nat,Left2: list_nat,Right2: list_nat] :
( X
!= ( stack_nat2 @ ( cons_nat @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: nat,Right2: list_nat] :
( X
!= ( stack_nat2 @ nil_nat @ ( cons_nat @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_180_pop_Ocases,axiom,
! [X: stack_a] :
( ( X
!= ( stack_a2 @ nil_a @ nil_a ) )
=> ( ! [X3: a,Left2: list_a,Right2: list_a] :
( X
!= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ~ ! [X3: a,Right2: list_a] :
( X
!= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) ) ) ) ).
% pop.cases
thf(fact_181_pop_Oelims,axiom,
! [X: stack_list_a,Y: stack_list_a] :
( ( ( pop_list_a @ X )
= Y )
=> ( ( ( X
= ( stack_list_a2 @ nil_list_a @ nil_list_a ) )
=> ( Y
!= ( stack_list_a2 @ nil_list_a @ nil_list_a ) ) )
=> ( ! [X3: list_a,Left2: list_list_a,Right2: list_list_a] :
( ( X
= ( stack_list_a2 @ ( cons_list_a @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_list_a2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: list_a,Right2: list_list_a] :
( ( X
= ( stack_list_a2 @ nil_list_a @ ( cons_list_a @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_list_a2 @ nil_list_a @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_182_pop_Oelims,axiom,
! [X: stack_num,Y: stack_num] :
( ( ( pop_num @ X )
= Y )
=> ( ( ( X
= ( stack_num2 @ nil_num @ nil_num ) )
=> ( Y
!= ( stack_num2 @ nil_num @ nil_num ) ) )
=> ( ! [X3: num,Left2: list_num,Right2: list_num] :
( ( X
= ( stack_num2 @ ( cons_num @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_num2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: num,Right2: list_num] :
( ( X
= ( stack_num2 @ nil_num @ ( cons_num @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_num2 @ nil_num @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_183_pop_Oelims,axiom,
! [X: stack_int,Y: stack_int] :
( ( ( pop_int @ X )
= Y )
=> ( ( ( X
= ( stack_int2 @ nil_int @ nil_int ) )
=> ( Y
!= ( stack_int2 @ nil_int @ nil_int ) ) )
=> ( ! [X3: int,Left2: list_int,Right2: list_int] :
( ( X
= ( stack_int2 @ ( cons_int @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_int2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: int,Right2: list_int] :
( ( X
= ( stack_int2 @ nil_int @ ( cons_int @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_int2 @ nil_int @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_184_pop_Oelims,axiom,
! [X: stack_nat_nat,Y: stack_nat_nat] :
( ( ( pop_nat_nat @ X )
= Y )
=> ( ( ( X
= ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) )
=> ( Y
!= ( stack_nat_nat2 @ nil_nat_nat @ nil_nat_nat ) ) )
=> ( ! [X3: nat > nat,Left2: list_nat_nat,Right2: list_nat_nat] :
( ( X
= ( stack_nat_nat2 @ ( cons_nat_nat @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_nat_nat2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: nat > nat,Right2: list_nat_nat] :
( ( X
= ( stack_nat_nat2 @ nil_nat_nat @ ( cons_nat_nat @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_nat_nat2 @ nil_nat_nat @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_185_pop_Oelims,axiom,
! [X: stack_nat,Y: stack_nat] :
( ( ( pop_nat @ X )
= Y )
=> ( ( ( X
= ( stack_nat2 @ nil_nat @ nil_nat ) )
=> ( Y
!= ( stack_nat2 @ nil_nat @ nil_nat ) ) )
=> ( ! [X3: nat,Left2: list_nat,Right2: list_nat] :
( ( X
= ( stack_nat2 @ ( cons_nat @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_nat2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: nat,Right2: list_nat] :
( ( X
= ( stack_nat2 @ nil_nat @ ( cons_nat @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_nat2 @ nil_nat @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_186_pop_Oelims,axiom,
! [X: stack_a,Y: stack_a] :
( ( ( pop_a @ X )
= Y )
=> ( ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ( Y
!= ( stack_a2 @ nil_a @ nil_a ) ) )
=> ( ! [X3: a,Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ( Y
!= ( stack_a2 @ Left2 @ Right2 ) ) )
=> ~ ! [X3: a,Right2: list_a] :
( ( X
= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) )
=> ( Y
!= ( stack_a2 @ nil_a @ Right2 ) ) ) ) ) ) ).
% pop.elims
thf(fact_187_list_Oexpand,axiom,
! [List: list_list_a,List2: list_list_a] :
( ( ( List = nil_list_a )
= ( List2 = nil_list_a ) )
=> ( ( ( List != nil_list_a )
=> ( ( List2 != nil_list_a )
=> ( ( ( hd_list_a @ List )
= ( hd_list_a @ List2 ) )
& ( ( tl_list_a @ List )
= ( tl_list_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_188_list_Oexpand,axiom,
! [List: list_num,List2: list_num] :
( ( ( List = nil_num )
= ( List2 = nil_num ) )
=> ( ( ( List != nil_num )
=> ( ( List2 != nil_num )
=> ( ( ( hd_num @ List )
= ( hd_num @ List2 ) )
& ( ( tl_num @ List )
= ( tl_num @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_189_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_190_list_Oexpand,axiom,
! [List: list_int,List2: list_int] :
( ( ( List = nil_int )
= ( List2 = nil_int ) )
=> ( ( ( List != nil_int )
=> ( ( List2 != nil_int )
=> ( ( ( hd_int @ List )
= ( hd_int @ List2 ) )
& ( ( tl_int @ List )
= ( tl_int @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_191_list_Oexpand,axiom,
! [List: list_nat_nat,List2: list_nat_nat] :
( ( ( List = nil_nat_nat )
= ( List2 = nil_nat_nat ) )
=> ( ( ( List != nil_nat_nat )
=> ( ( List2 != nil_nat_nat )
=> ( ( ( hd_nat_nat @ List )
= ( hd_nat_nat @ List2 ) )
& ( ( tl_nat_nat @ List )
= ( tl_nat_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_192_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_193_push_Oelims,axiom,
! [X: a,Xa: stack_a,Y: stack_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( Xa
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( Y
!= ( stack_a2 @ ( cons_a @ X @ Left2 ) @ Right2 ) ) ) ) ).
% push.elims
thf(fact_194_push_Osimps,axiom,
! [X: a,Left: list_a,Right: list_a] :
( ( push_a @ X @ ( stack_a2 @ Left @ Right ) )
= ( stack_a2 @ ( cons_a @ X @ Left ) @ Right ) ) ).
% push.simps
thf(fact_195_first_Ocases,axiom,
! [X: stack_a] :
( ! [X3: a,Left2: list_a,Right2: list_a] :
( X
!= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ( ! [X3: a,Right2: list_a] :
( X
!= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) )
=> ( X
= ( stack_a2 @ nil_a @ nil_a ) ) ) ) ).
% first.cases
thf(fact_196_stack_Oexhaust,axiom,
! [Y: stack_a] :
~ ! [X12: list_a,X23: list_a] :
( Y
!= ( stack_a2 @ X12 @ X23 ) ) ).
% stack.exhaust
thf(fact_197_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_198_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_199_is__empty__stack_Ocases,axiom,
! [X: stack_a] :
( ( X
!= ( stack_a2 @ nil_a @ nil_a ) )
=> ( ! [Vb: a,Vc: list_a,Va: list_a] :
( X
!= ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) )
=> ~ ! [V: list_a,Vb: a,Vc: list_a] :
( X
!= ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) ) ) ) ).
% is_empty_stack.cases
thf(fact_200_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_201_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_202_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_203_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_204_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_205_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_206_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_207_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_208_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_209_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_210_is__empty__stack_Oelims_I1_J,axiom,
! [X: stack_a,Y: $o] :
( ( ( type_i3216275384938974675tack_a @ X )
= Y )
=> ( ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ~ Y )
=> ( ( ? [Vb: a,Vc: list_a,Va: list_a] :
( X
= ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) )
=> Y )
=> ~ ( ? [V: list_a,Vb: a,Vc: list_a] :
( X
= ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) )
=> Y ) ) ) ) ).
% is_empty_stack.elims(1)
thf(fact_211_eq__Nil__null,axiom,
! [Xs: list_a] :
( ( Xs = nil_a )
= ( null_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_212_cons__tl,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_213_cons__hd,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( X
= ( hd_a @ Ys ) ) ) ).
% cons_hd
thf(fact_214_first_Oelims,axiom,
! [X: stack_a,Y: a] :
( ( ( first_a @ X )
= Y )
=> ( ! [X3: a] :
( ? [Left2: list_a,Right2: list_a] :
( X
= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ( Y != X3 ) )
=> ( ! [X3: a] :
( ? [Right2: list_a] :
( X
= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) )
=> ( Y != X3 ) )
=> ~ ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ( Y != undefined_a ) ) ) ) ) ).
% first.elims
thf(fact_215_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_216_Cons__in__shuffles__iff,axiom,
! [Z: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a2 @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z )
& ( member_list_a2 @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z )
& ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_217_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_218_size__stack_Ocases,axiom,
! [X: stack_a] :
~ ! [Left2: list_a,Right2: list_a] :
( X
!= ( stack_a2 @ Left2 @ Right2 ) ) ).
% size_stack.cases
thf(fact_219_Nil__in__shuffles,axiom,
! [Xs: list_a,Ys: list_a] :
( ( member_list_a2 @ nil_a @ ( shuffles_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_in_shuffles
thf(fact_220_Nil__in__shufflesI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = nil_a )
=> ( ( Ys = nil_a )
=> ( member_list_a2 @ nil_a @ ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_221_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_222_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a2 @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs3 ) )
=> ! [Z2: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z2 @ Zs2 ) )
=> ( ( X3 = Z2 )
=> ~ ( member_list_a2 @ Zs2 @ ( shuffles_a @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: a,Ys4: list_a] :
( ( Ys
= ( cons_a @ Y3 @ Ys4 ) )
=> ! [Z2: a,Zs2: list_a] :
( ( Zs
= ( cons_a @ Z2 @ Zs2 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member_list_a2 @ Zs2 @ ( shuffles_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_223_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_224_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_225_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_226_first_Opelims,axiom,
! [X: stack_a,Y: a] :
( ( ( first_a @ X )
= Y )
=> ( ( accp_stack_a @ first_rel_a @ X )
=> ( ! [X3: a,Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ( ( Y = X3 )
=> ~ ( accp_stack_a @ first_rel_a @ ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) ) ) )
=> ( ! [X3: a,Right2: list_a] :
( ( X
= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) )
=> ( ( Y = X3 )
=> ~ ( accp_stack_a @ first_rel_a @ ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) ) ) )
=> ~ ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ( ( Y = undefined_a )
=> ~ ( accp_stack_a @ first_rel_a @ ( stack_a2 @ nil_a @ nil_a ) ) ) ) ) ) ) ) ).
% first.pelims
thf(fact_227_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_228_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: a > a > $o,X: a,Xs: list_a] :
~ ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ nil_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_229_listrelp_Osimps,axiom,
( listrelp_a_a
= ( ^ [R: a > a > $o,A1: list_a,A2: list_a] :
( ( ( A1 = nil_a )
& ( A2 = nil_a ) )
| ? [X4: a,Y4: a,Xs4: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs4 ) )
& ( A2
= ( cons_a @ Y4 @ Ys3 ) )
& ( R @ X4 @ Y4 )
& ( listrelp_a_a @ R @ Xs4 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_230_listrelp_Ocases,axiom,
! [R2: a > a > $o,A12: list_a,A22: list_a] :
( ( listrelp_a_a @ R2 @ A12 @ A22 )
=> ( ( ( A12 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys2 ) )
=> ( ( R2 @ X3 @ Y3 )
=> ~ ( listrelp_a_a @ R2 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_231_successively__Cons,axiom,
! [P: a > a > $o,X: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X @ Xs ) )
= ( ( Xs = nil_a )
| ( ( P @ X @ ( hd_a @ Xs ) )
& ( successively_a @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_232_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_233_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ( lexordp_eq_a @ Less @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_234_successively_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( successively_a @ P @ nil_a ) ).
% successively.simps(1)
thf(fact_235_ord_Olexordp__eq_ONil,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_236_listrelp_ONil,axiom,
! [R2: a > a > $o] : ( listrelp_a_a @ R2 @ nil_a @ nil_a ) ).
% listrelp.Nil
thf(fact_237_successively_Oelims_I2_J,axiom,
! [X: a > a > $o,Xa: list_a] :
( ( successively_a @ X @ Xa )
=> ( ( Xa != nil_a )
=> ( ! [X3: a] :
( Xa
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ~ ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_238_successively_Oelims_I1_J,axiom,
! [X: a > a > $o,Xa: list_a,Y: $o] :
( ( ( successively_a @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_a )
=> ~ Y )
=> ( ( ? [X3: a] :
( Xa
= ( cons_a @ X3 @ nil_a ) )
=> ~ Y )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_239_successively_Osimps_I2_J,axiom,
! [P: a > a > $o,X: a] : ( successively_a @ P @ ( cons_a @ X @ nil_a ) ) ).
% successively.simps(2)
thf(fact_240_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_241_ord_Olexordp__eq_Ocases,axiom,
! [Less: a > a > $o,A12: list_a,A22: list_a] :
( ( lexordp_eq_a @ Less @ A12 @ A22 )
=> ( ( A12 != nil_a )
=> ( ! [X3: a] :
( ? [Xs2: list_a] :
( A12
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Y3: a] :
( ? [Ys2: list_a] :
( A22
= ( cons_a @ Y3 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y3 ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y3 )
=> ( ~ ( Less @ Y3 @ X3 )
=> ~ ( lexordp_eq_a @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_242_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_a
= ( ^ [Less2: a > a > $o,A1: list_a,A2: list_a] :
( ? [Ys3: list_a] :
( ( A1 = nil_a )
& ( A2 = Ys3 ) )
| ? [X4: a,Y4: a,Xs4: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs4 ) )
& ( A2
= ( cons_a @ Y4 @ Ys3 ) )
& ( Less2 @ X4 @ Y4 ) )
| ? [X4: a,Y4: a,Xs4: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs4 ) )
& ( A2
= ( cons_a @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y4 )
& ~ ( Less2 @ Y4 @ X4 )
& ( lexordp_eq_a @ Less2 @ Xs4 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_243_pop_Opelims,axiom,
! [X: stack_a,Y: stack_a] :
( ( ( pop_a @ X )
= Y )
=> ( ( accp_stack_a @ pop_rel_a @ X )
=> ( ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ( ( Y
= ( stack_a2 @ nil_a @ nil_a ) )
=> ~ ( accp_stack_a @ pop_rel_a @ ( stack_a2 @ nil_a @ nil_a ) ) ) )
=> ( ! [X3: a,Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) )
=> ( ( Y
= ( stack_a2 @ Left2 @ Right2 ) )
=> ~ ( accp_stack_a @ pop_rel_a @ ( stack_a2 @ ( cons_a @ X3 @ Left2 ) @ Right2 ) ) ) )
=> ~ ! [X3: a,Right2: list_a] :
( ( X
= ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) )
=> ( ( Y
= ( stack_a2 @ nil_a @ Right2 ) )
=> ~ ( accp_stack_a @ pop_rel_a @ ( stack_a2 @ nil_a @ ( cons_a @ X3 @ Right2 ) ) ) ) ) ) ) ) ) ).
% pop.pelims
thf(fact_244_is__empty__stack_Opelims_I1_J,axiom,
! [X: stack_a,Y: $o] :
( ( ( type_i3216275384938974675tack_a @ X )
= Y )
=> ( ( accp_stack_a @ is_empty_stack_rel_a @ X )
=> ( ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ( Y
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ nil_a @ nil_a ) ) ) )
=> ( ! [Vb: a,Vc: list_a,Va: list_a] :
( ( X
= ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) )
=> ( ~ Y
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) ) ) )
=> ~ ! [V: list_a,Vb: a,Vc: list_a] :
( ( X
= ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) )
=> ( ~ Y
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) ) ) ) ) ) ) ) ).
% is_empty_stack.pelims(1)
thf(fact_245_is__empty__stack_Opelims_I2_J,axiom,
! [X: stack_a] :
( ( type_i3216275384938974675tack_a @ X )
=> ( ( accp_stack_a @ is_empty_stack_rel_a @ X )
=> ~ ( ( X
= ( stack_a2 @ nil_a @ nil_a ) )
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ nil_a @ nil_a ) ) ) ) ) ).
% is_empty_stack.pelims(2)
thf(fact_246_is__empty__stack_Opelims_I3_J,axiom,
! [X: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ X )
=> ( ( accp_stack_a @ is_empty_stack_rel_a @ X )
=> ( ! [Vb: a,Vc: list_a,Va: list_a] :
( ( X
= ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) )
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ ( cons_a @ Vb @ Vc ) @ Va ) ) )
=> ~ ! [V: list_a,Vb: a,Vc: list_a] :
( ( X
= ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) )
=> ~ ( accp_stack_a @ is_empty_stack_rel_a @ ( stack_a2 @ V @ ( cons_a @ Vb @ Vc ) ) ) ) ) ) ) ).
% is_empty_stack.pelims(3)
thf(fact_247_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_nat
= ( ^ [A1: list_nat,A2: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A2 = Ys3 ) )
| ? [X4: nat,Y4: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs4 ) )
& ( A2
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ord_less_nat @ X4 @ Y4 ) )
| ? [X4: nat,Y4: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs4 ) )
& ( A2
= ( cons_nat @ Y4 @ Ys3 ) )
& ~ ( ord_less_nat @ X4 @ Y4 )
& ~ ( ord_less_nat @ Y4 @ X4 )
& ( ord_lexordp_eq_nat @ Xs4 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_248_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_num
= ( ^ [A1: list_num,A2: list_num] :
( ? [Ys3: list_num] :
( ( A1 = nil_num )
& ( A2 = Ys3 ) )
| ? [X4: num,Y4: num,Xs4: list_num,Ys3: list_num] :
( ( A1
= ( cons_num @ X4 @ Xs4 ) )
& ( A2
= ( cons_num @ Y4 @ Ys3 ) )
& ( ord_less_num @ X4 @ Y4 ) )
| ? [X4: num,Y4: num,Xs4: list_num,Ys3: list_num] :
( ( A1
= ( cons_num @ X4 @ Xs4 ) )
& ( A2
= ( cons_num @ Y4 @ Ys3 ) )
& ~ ( ord_less_num @ X4 @ Y4 )
& ~ ( ord_less_num @ Y4 @ X4 )
& ( ord_lexordp_eq_num @ Xs4 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_249_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_int
= ( ^ [A1: list_int,A2: list_int] :
( ? [Ys3: list_int] :
( ( A1 = nil_int )
& ( A2 = Ys3 ) )
| ? [X4: int,Y4: int,Xs4: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X4 @ Xs4 ) )
& ( A2
= ( cons_int @ Y4 @ Ys3 ) )
& ( ord_less_int @ X4 @ Y4 ) )
| ? [X4: int,Y4: int,Xs4: list_int,Ys3: list_int] :
( ( A1
= ( cons_int @ X4 @ Xs4 ) )
& ( A2
= ( cons_int @ Y4 @ Ys3 ) )
& ~ ( ord_less_int @ X4 @ Y4 )
& ~ ( ord_less_int @ Y4 @ X4 )
& ( ord_lexordp_eq_int @ Xs4 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_250_lexordp__eq_Ocases,axiom,
! [A12: list_nat,A22: list_nat] :
( ( ord_lexordp_eq_nat @ A12 @ A22 )
=> ( ( A12 != nil_nat )
=> ( ! [X3: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y3: nat] :
( ? [Ys2: list_nat] :
( A22
= ( cons_nat @ Y3 @ Ys2 ) )
=> ~ ( ord_less_nat @ X3 @ Y3 ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ~ ( ord_less_nat @ Y3 @ X3 )
=> ~ ( ord_lexordp_eq_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_251_lexordp__eq_Ocases,axiom,
! [A12: list_num,A22: list_num] :
( ( ord_lexordp_eq_num @ A12 @ A22 )
=> ( ( A12 != nil_num )
=> ( ! [X3: num] :
( ? [Xs2: list_num] :
( A12
= ( cons_num @ X3 @ Xs2 ) )
=> ! [Y3: num] :
( ? [Ys2: list_num] :
( A22
= ( cons_num @ Y3 @ Ys2 ) )
=> ~ ( ord_less_num @ X3 @ Y3 ) ) )
=> ~ ! [X3: num,Y3: num,Xs2: list_num] :
( ( A12
= ( cons_num @ X3 @ Xs2 ) )
=> ! [Ys2: list_num] :
( ( A22
= ( cons_num @ Y3 @ Ys2 ) )
=> ( ~ ( ord_less_num @ X3 @ Y3 )
=> ( ~ ( ord_less_num @ Y3 @ X3 )
=> ~ ( ord_lexordp_eq_num @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_252_lexordp__eq_Ocases,axiom,
! [A12: list_int,A22: list_int] :
( ( ord_lexordp_eq_int @ A12 @ A22 )
=> ( ( A12 != nil_int )
=> ( ! [X3: int] :
( ? [Xs2: list_int] :
( A12
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Y3: int] :
( ? [Ys2: list_int] :
( A22
= ( cons_int @ Y3 @ Ys2 ) )
=> ~ ( ord_less_int @ X3 @ Y3 ) ) )
=> ~ ! [X3: int,Y3: int,Xs2: list_int] :
( ( A12
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A22
= ( cons_int @ Y3 @ Ys2 ) )
=> ( ~ ( ord_less_int @ X3 @ Y3 )
=> ( ~ ( ord_less_int @ Y3 @ X3 )
=> ~ ( ord_lexordp_eq_int @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_253_splice_Oelims,axiom,
! [X: list_a,Xa: list_a,Y: list_a] :
( ( ( splice_a @ X @ Xa )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != Xa ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( X
= ( cons_a @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_a @ X3 @ ( splice_a @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_254_push_Opelims,axiom,
! [X: a,Xa: stack_a,Y: stack_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ( accp_P2456643874756862276tack_a @ push_rel_a @ ( produc6444587299782406187tack_a @ X @ Xa ) )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( Xa
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( ( Y
= ( stack_a2 @ ( cons_a @ X @ Left2 ) @ Right2 ) )
=> ~ ( accp_P2456643874756862276tack_a @ push_rel_a @ ( produc6444587299782406187tack_a @ X @ ( stack_a2 @ Left2 @ Right2 ) ) ) ) ) ) ) ).
% push.pelims
thf(fact_255_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_256_list_Orel__sel,axiom,
( list_all2_a_a
= ( ^ [R3: a > a > $o,A: list_a,B: list_a] :
( ( ( A = nil_a )
= ( B = nil_a ) )
& ( ( A != nil_a )
=> ( ( B != nil_a )
=> ( ( R3 @ ( hd_a @ A ) @ ( hd_a @ B ) )
& ( list_all2_a_a @ R3 @ ( tl_a @ A ) @ ( tl_a @ B ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_257_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_258_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_259_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_260_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_261_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_262_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_263_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_264_append_Oright__neutral,axiom,
! [A3: list_a] :
( ( append_a @ A3 @ nil_a )
= A3 ) ).
% append.right_neutral
thf(fact_265_list__all2__Nil2,axiom,
! [P: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% list_all2_Nil2
thf(fact_266_list__all2__Nil,axiom,
! [P: a > a > $o,Ys: list_a] :
( ( list_all2_a_a @ P @ nil_a @ Ys )
= ( Ys = nil_a ) ) ).
% list_all2_Nil
thf(fact_267_split__Nil__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( splice_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% split_Nil_iff
thf(fact_268_splice__Nil2,axiom,
! [Xs: list_a] :
( ( splice_a @ Xs @ nil_a )
= Xs ) ).
% splice_Nil2
thf(fact_269_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_270_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_271_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_272_lexordp__eq__simps_I4_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ~ ( ord_less_nat @ Y @ X )
& ( ord_lexordp_eq_nat @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_273_lexordp__eq__simps_I4_J,axiom,
! [X: num,Xs: list_num,Y: num,Ys: list_num] :
( ( ord_lexordp_eq_num @ ( cons_num @ X @ Xs ) @ ( cons_num @ Y @ Ys ) )
= ( ( ord_less_num @ X @ Y )
| ( ~ ( ord_less_num @ Y @ X )
& ( ord_lexordp_eq_num @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_274_lexordp__eq__simps_I4_J,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int] :
( ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
= ( ( ord_less_int @ X @ Y )
| ( ~ ( ord_less_int @ Y @ X )
& ( ord_lexordp_eq_int @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_275_list_Octr__transfer_I1_J,axiom,
! [R4: a > a > $o] : ( list_all2_a_a @ R4 @ nil_a @ nil_a ) ).
% list.ctr_transfer(1)
thf(fact_276_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_277_append_Oleft__neutral,axiom,
! [A3: list_a] :
( ( append_a @ nil_a @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_278_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_279_push_Ocases,axiom,
! [X: produc5950890884604230459tack_a] :
~ ! [X3: a,Left2: list_a,Right2: list_a] :
( X
!= ( produc6444587299782406187tack_a @ X3 @ ( stack_a2 @ Left2 @ Right2 ) ) ) ).
% push.cases
thf(fact_280_splice_Osimps_I1_J,axiom,
! [Ys: list_a] :
( ( splice_a @ nil_a @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_281_list_Orel__cases,axiom,
! [R4: a > a > $o,A3: list_a,B2: list_a] :
( ( list_all2_a_a @ R4 @ A3 @ B2 )
=> ( ( ( A3 = nil_a )
=> ( B2 != nil_a ) )
=> ~ ! [X12: a,X23: list_a] :
( ( A3
= ( cons_a @ X12 @ X23 ) )
=> ! [Y12: a,Y23: list_a] :
( ( B2
= ( cons_a @ Y12 @ Y23 ) )
=> ( ( R4 @ X12 @ Y12 )
=> ~ ( list_all2_a_a @ R4 @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_282_list_Orel__induct,axiom,
! [R4: a > a > $o,X: list_a,Y: list_a,Q: list_a > list_a > $o] :
( ( list_all2_a_a @ R4 @ X @ Y )
=> ( ( Q @ nil_a @ nil_a )
=> ( ! [A21: a,A222: list_a,B21: a,B22: list_a] :
( ( R4 @ A21 @ B21 )
=> ( ( Q @ A222 @ B22 )
=> ( Q @ ( cons_a @ A21 @ A222 ) @ ( cons_a @ B21 @ B22 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_283_list__all2__induct,axiom,
! [P: a > a > $o,Xs: list_a,Ys: list_a,R4: list_a > list_a > $o] :
( ( list_all2_a_a @ P @ Xs @ Ys )
=> ( ( R4 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P @ X3 @ Y3 )
=> ( ( list_all2_a_a @ P @ Xs2 @ Ys2 )
=> ( ( R4 @ Xs2 @ Ys2 )
=> ( R4 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) )
=> ( R4 @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_284_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_285_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_286_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_287_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_288_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_289_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs4: list_a,Xs5: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_290_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_291_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_292_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_293_lexordp__eq_OCons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_294_lexordp__eq_OCons,axiom,
! [X: num,Y: num,Xs: list_num,Ys: list_num] :
( ( ord_less_num @ X @ Y )
=> ( ord_lexordp_eq_num @ ( cons_num @ X @ Xs ) @ ( cons_num @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_295_lexordp__eq_OCons,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ( ord_less_int @ X @ Y )
=> ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_296_lexordp__eq_OCons__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ~ ( ord_less_nat @ Y @ X )
=> ( ( ord_lexordp_eq_nat @ Xs @ Ys )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_297_lexordp__eq_OCons__eq,axiom,
! [X: num,Y: num,Xs: list_num,Ys: list_num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ~ ( ord_less_num @ Y @ X )
=> ( ( ord_lexordp_eq_num @ Xs @ Ys )
=> ( ord_lexordp_eq_num @ ( cons_num @ X @ Xs ) @ ( cons_num @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_298_lexordp__eq_OCons__eq,axiom,
! [X: int,Y: int,Xs: list_int,Ys: list_int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ~ ( ord_less_int @ Y @ X )
=> ( ( ord_lexordp_eq_int @ Xs @ Ys )
=> ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_299_Stack__Aux_Olist_Osimps,axiom,
! [Left: list_a,Right: list_a] :
( ( stack_list_a3 @ ( stack_a2 @ Left @ Right ) )
= ( append_a @ Left @ Right ) ) ).
% Stack_Aux.list.simps
thf(fact_300_list_Oelims,axiom,
! [X: stack_a,Y: list_a] :
( ( ( stack_list_a3 @ X )
= Y )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( Y
!= ( append_a @ Left2 @ Right2 ) ) ) ) ).
% list.elims
thf(fact_301_list_Opelims,axiom,
! [X: stack_a,Y: list_a] :
( ( ( stack_list_a3 @ X )
= Y )
=> ( ( accp_stack_a @ stack_list_rel_a @ X )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( ( Y
= ( append_a @ Left2 @ Right2 ) )
=> ~ ( accp_stack_a @ stack_list_rel_a @ ( stack_a2 @ Left2 @ Right2 ) ) ) ) ) ) ).
% list.pelims
thf(fact_302_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_303_old_Oprod_Oinject,axiom,
! [A3: nat,B2: nat,A4: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B2 )
= ( product_Pair_nat_nat @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_304_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_305_in__measure,axiom,
! [X: nat,Y: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_306_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_307_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_308_successively__append__iff,axiom,
! [P: a > a > $o,Xs: list_a,Ys: list_a] :
( ( successively_a @ P @ ( append_a @ Xs @ Ys ) )
= ( ( successively_a @ P @ Xs )
& ( successively_a @ P @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( P @ ( last_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_309_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_310_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_311_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_312_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_313_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_314_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_315_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_316_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_317_butlast__rev,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( rev_a @ Xs ) )
= ( rev_a @ ( tl_a @ Xs ) ) ) ).
% butlast_rev
thf(fact_318_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_319_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_320_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_321_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_322_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_323_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_324_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_325_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_326_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_327_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_328_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P2: a > a > $o,X3: a,Y3: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_329_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ~ ! [P2: a > a > $o,X3: a,Ys2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_330_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_331_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_332_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_333_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_334_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_335_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_336_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys4 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys4 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_337_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_338_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_339_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_340_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_341_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_342_Pair__inject,axiom,
! [A3: nat,B2: nat,A4: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B2 )
= ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_343_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
( ! [A5: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B4 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_344_surj__pair,axiom,
! [P3: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P3
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_345_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A5: nat,B4: nat] :
( Y
!= ( product_Pair_nat_nat @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_346_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_347_splice_Opelims,axiom,
! [X: list_a,Xa: list_a,Y: list_a] :
( ( ( splice_a @ X @ Xa )
= Y )
=> ( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ X @ Xa ) )
=> ( ( ( X = nil_a )
=> ( ( Y = Xa )
=> ~ ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Xa ) ) ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( X
= ( cons_a @ X3 @ Xs2 ) )
=> ( ( Y
= ( cons_a @ X3 @ ( splice_a @ Xa @ Xs2 ) ) )
=> ~ ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_348_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_349_successively_Opelims_I2_J,axiom,
! [X: a > a > $o,Xa: list_a] :
( ( successively_a @ X @ Xa )
=> ( ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ Xa ) )
=> ( ( ( Xa = nil_a )
=> ~ ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ nil_a ) ) )
=> ( ! [X3: a] :
( ( Xa
= ( cons_a @ X3 @ nil_a ) )
=> ~ ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) )
=> ~ ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_350_successively_Opelims_I1_J,axiom,
! [X: a > a > $o,Xa: list_a,Y: $o] :
( ( ( successively_a @ X @ Xa )
= Y )
=> ( ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ Xa ) )
=> ( ( ( Xa = nil_a )
=> ( Y
=> ~ ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ nil_a ) ) ) )
=> ( ! [X3: a] :
( ( Xa
= ( cons_a @ X3 @ nil_a ) )
=> ( Y
=> ~ ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( Y
= ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) )
=> ~ ( accp_P5335913909695611590list_a @ successively_rel_a @ ( produc8111569692950616493list_a @ X @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_351_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_352_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_353_splice_Opinduct,axiom,
! [A0: list_a,A12: list_a,P: list_a > list_a > $o] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ A0 @ A12 ) )
=> ( ! [Ys2: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( P @ nil_a @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Ys2: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) )
=> ( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ) ).
% splice.pinduct
thf(fact_354_splice_Opsimps_I1_J,axiom,
! [Ys: list_a] :
( ( accp_P7377042638478740784list_a @ splice_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ( ( splice_a @ nil_a @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_355_shuffles_Opinduct,axiom,
! [A0: list_a,A12: list_a,P: list_a > list_a > $o] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ A0 @ A12 ) )
=> ( ! [Ys2: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( P @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ( P @ Xs2 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( accp_P7377042638478740784list_a @ shuffles_rel_a @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( ( P @ Xs2 @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ( P @ ( cons_a @ X3 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_356_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_357_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_358_lexord__append__left__rightI,axiom,
! [A3: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B2 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A3 @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B2 @ Y ) ) ) @ ( lexord_nat @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_359_listrel_Ocases,axiom,
! [A12: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A12 @ A22 ) @ ( listrel_a_a @ R2 ) )
=> ( ( ( A12 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y3 @ Ys2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_360_listrel_Ocases,axiom,
! [A12: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
=> ( ( ( A12 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_361_listrel_Osimps,axiom,
! [A12: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A12 @ A22 ) @ ( listrel_a_a @ R2 ) )
= ( ( ( A12 = nil_a )
& ( A22 = nil_a ) )
| ? [X4: a,Y4: a,Xs4: list_a,Ys3: list_a] :
( ( A12
= ( cons_a @ X4 @ Xs4 ) )
& ( A22
= ( cons_a @ Y4 @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs4 @ Ys3 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_362_listrel_Osimps,axiom,
! [A12: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( A12 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X4: nat,Y4: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X4 @ Xs4 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs4 @ Ys3 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_363_Cons__listrel1__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_364_lexord__cons__cons,axiom,
! [A3: nat,X: list_nat,B2: nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ X ) @ ( cons_nat @ B2 @ Y ) ) @ ( lexord_nat @ R2 ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B2 ) @ R2 )
| ( ( A3 = B2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_365_lexord__Nil__left,axiom,
! [Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R2 ) )
= ( ? [A: a,X4: list_a] :
( Y
= ( cons_a @ A @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_366_not__listrel1__Nil,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_367_not__Nil__listrel1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_368_lexord__linear,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
( ! [A5: nat,B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R2 )
| ( A5 = B4 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A5 ) @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) )
| ( X = Y )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( lexord_nat @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_369_lexord__irreflexive,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_370_listrel__Nil2,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R2 ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_371_listrel__Nil1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R2 ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_372_listrel_ONil,axiom,
! [R2: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R2 ) ) ).
% listrel.Nil
thf(fact_373_lexord__Nil__right,axiom,
! [X: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ nil_a ) @ ( lexord_a @ R2 ) ) ).
% lexord_Nil_right
thf(fact_374_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_375_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( ! [Y3: nat] :
( ( Ys
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs3: list_nat] :
( ( Ys
= ( cons_nat @ X @ Zs3 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_376_listrel1I1,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_377_listrel__Cons2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_378_listrel__Cons1,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys2 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_379_listrel_OCons,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_380_lexord__append__leftD,axiom,
! [X: list_nat,U: list_nat,V2: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R2 ) )
=> ( ! [A5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A5 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_381_listrel1I,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_382_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ! [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us2 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_383_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_384_mlex__iff,axiom,
! [X: nat,Y: nat,F: nat > nat,R4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) ) ) ) ).
% mlex_iff
thf(fact_385_mlex__less,axiom,
! [F: nat > nat,X: nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) ) ) ).
% mlex_less
thf(fact_386_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_387_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_388_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_389_hd__remdups__adj,axiom,
! [Xs: list_a] :
( ( hd_a @ ( remdups_adj_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% hd_remdups_adj
thf(fact_390_remdups__adj__Cons__alt,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) )
= ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_391_mlex__leq,axiom,
! [F: nat > nat,X: nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) ) ) ) ).
% mlex_leq
thf(fact_392_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_393_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_394_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_395_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_396_remdups__adj_Opelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X )
= Y )
=> ( ( accp_list_a @ remdups_adj_rel_a @ X )
=> ( ( ( X = nil_a )
=> ( ( Y = nil_a )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ nil_a ) ) )
=> ( ! [X3: a] :
( ( X
= ( cons_a @ X3 @ nil_a ) )
=> ( ( Y
= ( cons_a @ X3 @ nil_a ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( 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 ) ) ) ) ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_397_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_398_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_399_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_400_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_401_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_402_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_403_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_404_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_405_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_406_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_407_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_408_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_409_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_410_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_411_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less_nat @ N @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K @ I3 )
=> ( P @ I3 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_412_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_413_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_414_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_415_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_416_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_417_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_418_order__less__le__subst2,axiom,
! [A3: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_419_order__less__le__subst2,axiom,
! [A3: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_420_order__less__le__subst2,axiom,
! [A3: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_421_order__less__le__subst2,axiom,
! [A3: num,B2: num,F: num > int,C: int] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_422_order__less__le__subst2,axiom,
! [A3: int,B2: int,F: int > num,C: num] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_423_order__less__le__subst2,axiom,
! [A3: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_424_order__less__le__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_425_order__less__le__subst2,axiom,
! [A3: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_426_order__less__le__subst2,axiom,
! [A3: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_427_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_428_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_429_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_430_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_431_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_432_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_433_order_Oasym,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A3 ) ) ).
% order.asym
thf(fact_434_order_Oasym,axiom,
! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ~ ( ord_less_num @ B2 @ A3 ) ) ).
% order.asym
thf(fact_435_order_Oasym,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ~ ( ord_less_int @ B2 @ A3 ) ) ).
% order.asym
thf(fact_436_ord__eq__less__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( A3 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_437_ord__eq__less__trans,axiom,
! [A3: num,B2: num,C: num] :
( ( A3 = B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_438_ord__eq__less__trans,axiom,
! [A3: int,B2: int,C: int] :
( ( A3 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_439_ord__less__eq__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_440_ord__less__eq__trans,axiom,
! [A3: num,B2: num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_441_ord__less__eq__trans,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_442_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_443_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_444_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_445_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_446_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_447_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_448_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_449_dual__order_Oasym,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B2 ) ) ).
% dual_order.asym
thf(fact_450_dual__order_Oasym,axiom,
! [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
=> ~ ( ord_less_num @ A3 @ B2 ) ) ).
% dual_order.asym
thf(fact_451_dual__order_Oasym,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
=> ~ ( ord_less_int @ A3 @ B2 ) ) ).
% dual_order.asym
thf(fact_452_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_453_dual__order_Oirrefl,axiom,
! [A3: num] :
~ ( ord_less_num @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_454_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_455_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P5 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_456_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_457_linorder__less__wlog,axiom,
! [P: num > num > $o,A3: num,B2: num] :
( ! [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: num] : ( P @ A5 @ A5 )
=> ( ! [A5: num,B4: num] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_458_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B2: int] :
( ! [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_459_order_Ostrict__trans,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_460_order_Ostrict__trans,axiom,
! [A3: num,B2: num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_461_order_Ostrict__trans,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_462_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_463_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_464_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_465_dual__order_Ostrict__trans,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_466_dual__order_Ostrict__trans,axiom,
! [B2: num,A3: num,C: num] :
( ( ord_less_num @ B2 @ A3 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_467_dual__order_Ostrict__trans,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_468_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( A3 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_469_order_Ostrict__implies__not__eq,axiom,
! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( A3 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_470_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( A3 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_471_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( A3 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_472_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
=> ( A3 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_473_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( A3 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_474_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_475_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_476_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_477_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_478_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_479_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_480_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_481_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_482_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_483_order__less__asym_H,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A3 ) ) ).
% order_less_asym'
thf(fact_484_order__less__asym_H,axiom,
! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ~ ( ord_less_num @ B2 @ A3 ) ) ).
% order_less_asym'
thf(fact_485_order__less__asym_H,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ~ ( ord_less_int @ B2 @ A3 ) ) ).
% order_less_asym'
thf(fact_486_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_487_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_488_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_489_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_490_ord__eq__less__subst,axiom,
! [A3: num,F: nat > num,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_491_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B2: nat,C: nat] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_492_ord__eq__less__subst,axiom,
! [A3: nat,F: num > nat,B2: num,C: num] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_493_ord__eq__less__subst,axiom,
! [A3: num,F: num > num,B2: num,C: num] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_494_ord__eq__less__subst,axiom,
! [A3: int,F: num > int,B2: num,C: num] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_495_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B2: int,C: int] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_496_ord__eq__less__subst,axiom,
! [A3: num,F: int > num,B2: int,C: int] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_497_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B2: int,C: int] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_498_ord__less__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_499_ord__less__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_500_ord__less__eq__subst,axiom,
! [A3: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_501_ord__less__eq__subst,axiom,
! [A3: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_502_ord__less__eq__subst,axiom,
! [A3: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_503_ord__less__eq__subst,axiom,
! [A3: num,B2: num,F: num > int,C: int] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_504_ord__less__eq__subst,axiom,
! [A3: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_505_ord__less__eq__subst,axiom,
! [A3: int,B2: int,F: int > num,C: num] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_506_ord__less__eq__subst,axiom,
! [A3: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_507_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_508_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_509_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_510_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_511_order__less__subst1,axiom,
! [A3: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_512_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_513_order__less__subst1,axiom,
! [A3: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_514_order__less__subst1,axiom,
! [A3: num,F: num > num,B2: num,C: num] :
( ( ord_less_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_515_order__less__subst1,axiom,
! [A3: num,F: int > num,B2: int,C: int] :
( ( ord_less_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_516_order__less__subst1,axiom,
! [A3: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_517_order__less__subst1,axiom,
! [A3: int,F: num > int,B2: num,C: num] :
( ( ord_less_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_518_order__less__subst1,axiom,
! [A3: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_519_order__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_520_order__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_521_order__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_522_order__less__subst2,axiom,
! [A3: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_523_order__less__subst2,axiom,
! [A3: num,B2: num,F: num > num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_524_order__less__subst2,axiom,
! [A3: num,B2: num,F: num > int,C: int] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_525_order__less__subst2,axiom,
! [A3: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_526_order__less__subst2,axiom,
! [A3: int,B2: int,F: int > num,C: num] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_527_order__less__subst2,axiom,
! [A3: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_528_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_529_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_530_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_531_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_532_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_533_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_534_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_535_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_536_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_537_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_538_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_539_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_540_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_541_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_542_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_543_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_544_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_545_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_546_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_547_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_548_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_549_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_550_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_551_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_552_nless__le,axiom,
! [A3: num,B2: num] :
( ( ~ ( ord_less_num @ A3 @ B2 ) )
= ( ~ ( ord_less_eq_num @ A3 @ B2 )
| ( A3 = B2 ) ) ) ).
% nless_le
thf(fact_553_nless__le,axiom,
! [A3: int,B2: int] :
( ( ~ ( ord_less_int @ A3 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B2 )
| ( A3 = B2 ) ) ) ).
% nless_le
thf(fact_554_nless__le,axiom,
! [A3: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A3 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ).
% nless_le
thf(fact_555_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_556_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_557_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_558_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_559_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_560_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_561_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_562_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_563_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_564_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_565_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_566_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_567_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A: num,B: num] :
( ( ord_less_num @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_568_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_569_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_570_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_571_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_572_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_573_order_Ostrict__trans1,axiom,
! [A3: num,B2: num,C: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_574_order_Ostrict__trans1,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_575_order_Ostrict__trans1,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_576_order_Ostrict__trans2,axiom,
! [A3: num,B2: num,C: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_less_eq_num @ B2 @ C )
=> ( ord_less_num @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_577_order_Ostrict__trans2,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_578_order_Ostrict__trans2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_579_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
& ~ ( ord_less_eq_num @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_580_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ~ ( ord_less_eq_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_581_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_582_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B: num,A: num] :
( ( ord_less_num @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_583_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_584_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_585_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_586_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_587_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_588_dual__order_Ostrict__trans1,axiom,
! [B2: num,A3: num,C: num] :
( ( ord_less_eq_num @ B2 @ A3 )
=> ( ( ord_less_num @ C @ B2 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_589_dual__order_Ostrict__trans1,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_eq_int @ B2 @ A3 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_590_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_591_dual__order_Ostrict__trans2,axiom,
! [B2: num,A3: num,C: num] :
( ( ord_less_num @ B2 @ A3 )
=> ( ( ord_less_eq_num @ C @ B2 )
=> ( ord_less_num @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_592_dual__order_Ostrict__trans2,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_593_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_594_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
& ~ ( ord_less_eq_num @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_595_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ~ ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_596_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_597_order_Ostrict__implies__order,axiom,
! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ord_less_eq_num @ A3 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_598_order_Ostrict__implies__order,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_599_order_Ostrict__implies__order,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_600_dual__order_Ostrict__implies__order,axiom,
! [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
=> ( ord_less_eq_num @ B2 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_601_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ord_less_eq_int @ B2 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_602_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_603_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_num @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_604_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_605_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_606_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_607_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_608_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_609_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_610_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_611_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_612_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_613_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_614_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_615_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_616_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_617_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_618_order__le__neq__trans,axiom,
! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_num @ A3 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_619_order__le__neq__trans,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_620_order__le__neq__trans,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_621_order__neq__le__trans,axiom,
! [A3: num,B2: num] :
( ( A3 != B2 )
=> ( ( ord_less_eq_num @ A3 @ B2 )
=> ( ord_less_num @ A3 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_622_order__neq__le__trans,axiom,
! [A3: int,B2: int] :
( ( A3 != B2 )
=> ( ( ord_less_eq_int @ A3 @ B2 )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_623_order__neq__le__trans,axiom,
! [A3: nat,B2: nat] :
( ( A3 != B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_624_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_625_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_626_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_627_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_628_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_629_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_630_order__le__less__subst1,axiom,
! [A3: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_eq_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_631_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_632_order__le__less__subst1,axiom,
! [A3: num,F: num > num,B2: num,C: num] :
( ( ord_less_eq_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_633_order__le__less__subst1,axiom,
! [A3: int,F: num > int,B2: num,C: num] :
( ( ord_less_eq_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_634_order__le__less__subst1,axiom,
! [A3: num,F: int > num,B2: int,C: int] :
( ( ord_less_eq_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_635_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_636_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_637_order__le__less__subst1,axiom,
! [A3: nat,F: num > nat,B2: num,C: num] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_num @ B2 @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_638_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_639_order__le__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_num @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_640_order__le__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_641_order__le__less__subst2,axiom,
! [A3: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_642_order__less__le__subst1,axiom,
! [A3: num,F: nat > num,B2: nat,C: nat] :
( ( ord_less_num @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_643_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_644_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_645_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_646_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_647_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_648_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_649_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_650_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_651_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_652_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_653_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_654_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_655_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_656_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_657_verit__comp__simplify1_I3_J,axiom,
! [B3: num,A4: num] :
( ( ~ ( ord_less_eq_num @ B3 @ A4 ) )
= ( ord_less_num @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_658_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_659_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_660_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_661_verit__comp__simplify1_I1_J,axiom,
! [A3: num] :
~ ( ord_less_num @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_662_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_663_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_664_pinf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_665_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_666_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_667_pinf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_668_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_669_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_670_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_671_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_672_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_673_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_674_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_675_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_676_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ~ ( ord_less_num @ X6 @ T ) ) ).
% pinf(5)
thf(fact_677_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_678_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_679_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ Z2 @ X6 )
=> ( ord_less_num @ T @ X6 ) ) ).
% pinf(7)
thf(fact_680_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_681_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_682_minf_I1_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_683_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_684_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_685_minf_I2_J,axiom,
! [P: num > $o,P6: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_686_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_687_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_688_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_689_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_690_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_691_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_692_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_693_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_694_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ( ord_less_num @ X6 @ T ) ) ).
% minf(5)
thf(fact_695_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_696_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_697_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z2 )
=> ~ ( ord_less_num @ T @ X6 ) ) ).
% minf(7)
thf(fact_698_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_699_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_700_complete__interval,axiom,
! [A3: int,B2: int,P: int > $o] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A3 @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A3 @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A3 @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_701_complete__interval,axiom,
! [A3: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A3 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A3 @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_702_lexord__sufI,axiom,
! [U: list_nat,W: list_nat,R2: set_Pr1261947904930325089at_nat,V2: list_nat,Z: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W ) @ ( lexord_nat @ R2 ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ W ) @ ( size_size_list_nat @ U ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V2 ) @ ( append_nat @ W @ Z ) ) @ ( lexord_nat @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_703_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_704_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_705_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_706_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_707_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_708_list__all2__lengthD,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_709_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_710_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_711_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_712_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_713_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_714_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_715_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_716_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: nat,Zs3: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_717_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z2: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_718_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z2: nat,Zs3: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_719_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z2: a,Zs3: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_720_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z2: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_721_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z2: a,Zs3: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_722_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z2: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z2 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_723_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_724_list__all2__append2,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ? [Us3: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ Vs3 ) )
& ( ( size_size_list_nat @ Us3 )
= ( size_size_list_nat @ Ys ) )
& ( ( size_size_list_nat @ Vs3 )
= ( size_size_list_nat @ Zs ) )
& ( list_all2_nat_nat @ P @ Us3 @ Ys )
& ( list_all2_nat_nat @ P @ Vs3 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_725_list__all2__append1,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( list_all2_nat_nat @ P @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( ? [Us3: list_nat,Vs3: list_nat] :
( ( Zs
= ( append_nat @ Us3 @ Vs3 ) )
& ( ( size_size_list_nat @ Us3 )
= ( size_size_list_nat @ Xs ) )
& ( ( size_size_list_nat @ Vs3 )
= ( size_size_list_nat @ Ys ) )
& ( list_all2_nat_nat @ P @ Xs @ Us3 )
& ( list_all2_nat_nat @ P @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_726_list__all2__append,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > nat > $o,Us: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( list_all2_nat_nat @ P @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) )
= ( ( list_all2_nat_nat @ P @ Xs @ Ys )
& ( list_all2_nat_nat @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_727_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_728_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_729_listrel__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_730_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs3: list_a,Y3: a,Ys4: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_731_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs3 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_732_lexord__sufE,axiom,
! [Xs: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Qs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_733_Cons__lenlex__iff,axiom,
! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
= ( ( 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 @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_734_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_735_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_736_lenlex__append1,axiom,
! [Us: list_nat,Xs: list_nat,R4: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs ) @ ( lenlex_nat @ R4 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs @ Ys ) ) @ ( lenlex_nat @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_737_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R2 ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_738_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_739_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_740_Skolem__list__nth,axiom,
! [K2: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X7: nat] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_741_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_742_Nil2__notin__lex,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_743_Nil__notin__lex,axiom,
! [Ys: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R2 ) ) ).
% Nil_notin_lex
thf(fact_744_list__all2__nthD,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,P3: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( ord_less_nat @ P3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ P3 ) @ ( nth_nat @ Ys @ P3 ) ) ) ) ).
% list_all2_nthD
thf(fact_745_list__all2__nthD2,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,P3: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( ord_less_nat @ P3 @ ( size_size_list_nat @ Ys ) )
=> ( P @ ( nth_nat @ Xs @ P3 ) @ ( nth_nat @ Ys @ P3 ) ) ) ) ).
% list_all2_nthD2
thf(fact_746_list__all2__all__nthI,axiom,
! [A3: list_nat,B2: list_nat,P: nat > nat > $o] :
( ( ( size_size_list_nat @ A3 )
= ( size_size_list_nat @ B2 ) )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ A3 ) )
=> ( P @ ( nth_nat @ A3 @ N3 ) @ ( nth_nat @ B2 @ N3 ) ) )
=> ( list_all2_nat_nat @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_747_list__all2__conv__all__nth,axiom,
( list_all2_nat_nat
= ( ^ [P5: nat > nat > $o,Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( P5 @ ( nth_nat @ Xs4 @ I4 ) @ ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_748_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_749_lenlex__irreflexive,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_750_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_751_lex__append__leftD,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_752_lex__append__left__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_753_lex__append__rightI,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_754_lexord__lex,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R2 ) )
= ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) )
& ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ) ).
% lexord_lex
thf(fact_755_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_756_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
= ( ? [Y4: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y4 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_757_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_758_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_759_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_760_list__update__nonempty,axiom,
! [Xs: list_a,K2: nat,X: a] :
( ( ( list_update_a @ Xs @ K2 @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_761_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_762_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_763_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_764_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_765_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_766_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_767_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_768_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_769_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_770_successively__if__sorted__wrt,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( successively_nat @ P @ Xs ) ) ).
% successively_if_sorted_wrt
thf(fact_771_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_772_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_773_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_774_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_775_sorted__wrt_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( sorted_wrt_a @ P @ nil_a ) ).
% sorted_wrt.simps(1)
thf(fact_776_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_777_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_778_strict__sorted__imp__sorted,axiom,
! [Xs: list_num] :
( ( sorted_wrt_num @ ord_less_num @ Xs )
=> ( sorted_wrt_num @ ord_less_eq_num @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_779_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_780_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_781_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_782_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_783_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_784_strict__sorted__simps_I1_J,axiom,
sorted_wrt_num @ ord_less_num @ nil_num ).
% strict_sorted_simps(1)
thf(fact_785_strict__sorted__simps_I1_J,axiom,
sorted_wrt_int @ ord_less_int @ nil_int ).
% strict_sorted_simps(1)
thf(fact_786_sorted__wrt1,axiom,
! [P: a > a > $o,X: a] : ( sorted_wrt_a @ P @ ( cons_a @ X @ nil_a ) ) ).
% sorted_wrt1
thf(fact_787_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_788_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_789_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_790_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_791_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_792_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_793_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P5: nat > nat > $o,Xs4: list_nat] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs4 ) )
=> ( P5 @ ( nth_nat @ Xs4 @ I4 ) @ ( nth_nat @ Xs4 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_794_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_795_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_796_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_797_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_798_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_799_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_800_nth__take__lemma,axiom,
! [K2: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K2 @ Xs )
= ( take_nat @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_801_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_802_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_803_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_804_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_805_lexord__take__index__conv,axiom,
! [X: list_nat,Y: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R2 ) )
= ( ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
& ( ( take_nat @ ( size_size_list_nat @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) ) )
& ( ( take_nat @ I4 @ X )
= ( take_nat @ I4 @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X @ I4 ) @ ( nth_nat @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_806_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I4 ) ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_807_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_808_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_809_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_810_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_811_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_812_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_813_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_814_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_815_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_816_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,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I2 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_817_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_818_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_819_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_820_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_821_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_822_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_823_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_824_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_825_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_826_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_827_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_828_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_829_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_830_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_831_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_832_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_833_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_834_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_835_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_836_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_837_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_838_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_839_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_840_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_841_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_842_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_843_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_844_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_845_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_846_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_847_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_848_take__tl,axiom,
! [N: nat,Xs: list_a] :
( ( take_a @ N @ ( tl_a @ Xs ) )
= ( tl_a @ ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_849_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_850_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_851_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_852_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_853_nth__tl,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( tl_a @ Xs ) ) )
=> ( ( nth_a @ ( tl_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_854_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_855_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_856_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_857_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_858_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_859_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_860_take__hd_H,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys ) )
=> ( ( hd_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_861_take__hd_H,axiom,
! [Ys: list_nat,X: nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys ) @ ( cons_nat @ X @ Xs ) )
= ( take_nat @ ( suc @ ( size_size_list_nat @ Xs ) ) @ Ys ) )
=> ( ( hd_nat @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_862_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_863_min_Oabsorb3,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_min_nat @ A3 @ B2 )
= A3 ) ) ).
% min.absorb3
thf(fact_864_min_Oabsorb3,axiom,
! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( ( ord_min_num @ A3 @ B2 )
= A3 ) ) ).
% min.absorb3
thf(fact_865_min_Oabsorb3,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_min_int @ A3 @ B2 )
= A3 ) ) ).
% min.absorb3
thf(fact_866_min_Oabsorb4,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
=> ( ( ord_min_nat @ A3 @ B2 )
= B2 ) ) ).
% min.absorb4
thf(fact_867_min_Oabsorb4,axiom,
! [B2: num,A3: num] :
( ( ord_less_num @ B2 @ A3 )
=> ( ( ord_min_num @ A3 @ B2 )
= B2 ) ) ).
% min.absorb4
thf(fact_868_min_Oabsorb4,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ( ord_min_int @ A3 @ B2 )
= B2 ) ) ).
% min.absorb4
thf(fact_869_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_870_min__less__iff__conj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_min_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
& ( ord_less_num @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_871_min__less__iff__conj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
& ( ord_less_int @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_872_min_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A3: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_873_min_Ostrict__coboundedI2,axiom,
! [B2: num,C: num,A3: num] :
( ( ord_less_num @ B2 @ C )
=> ( ord_less_num @ ( ord_min_num @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_874_min_Ostrict__coboundedI2,axiom,
! [B2: int,C: int,A3: int] :
( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ ( ord_min_int @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_875_min_Ostrict__coboundedI1,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A3 @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_876_min_Ostrict__coboundedI1,axiom,
! [A3: num,C: num,B2: num] :
( ( ord_less_num @ A3 @ C )
=> ( ord_less_num @ ( ord_min_num @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_877_min_Ostrict__coboundedI1,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ A3 @ C )
=> ( ord_less_int @ ( ord_min_int @ A3 @ B2 ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_878_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
& ( A != B ) ) ) ) ).
% min.strict_order_iff
thf(fact_879_min_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [A: num,B: num] :
( ( A
= ( ord_min_num @ A @ B ) )
& ( A != B ) ) ) ) ).
% min.strict_order_iff
thf(fact_880_min_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( A
= ( ord_min_int @ A @ B ) )
& ( A != B ) ) ) ) ).
% min.strict_order_iff
thf(fact_881_min_Ostrict__boundedE,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( ord_min_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A3 @ B2 )
=> ~ ( ord_less_nat @ A3 @ C ) ) ) ).
% min.strict_boundedE
thf(fact_882_min_Ostrict__boundedE,axiom,
! [A3: num,B2: num,C: num] :
( ( ord_less_num @ A3 @ ( ord_min_num @ B2 @ C ) )
=> ~ ( ( ord_less_num @ A3 @ B2 )
=> ~ ( ord_less_num @ A3 @ C ) ) ) ).
% min.strict_boundedE
thf(fact_883_min_Ostrict__boundedE,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ ( ord_min_int @ B2 @ C ) )
=> ~ ( ( ord_less_int @ A3 @ B2 )
=> ~ ( ord_less_int @ A3 @ C ) ) ) ).
% min.strict_boundedE
thf(fact_884_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_nat @ X @ Z )
| ( ord_less_nat @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_885_min__less__iff__disj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ ( ord_min_num @ X @ Y ) @ Z )
= ( ( ord_less_num @ X @ Z )
| ( ord_less_num @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_886_min__less__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_int @ X @ Z )
| ( ord_less_int @ Y @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_887_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_888_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_889_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_890_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_891_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_892_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_893_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_894_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_895_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_896_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_897_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_898_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_899_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_900_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_901_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_902_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_903_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs4: list_a] : nil_a ) ) ).
% take0
thf(fact_904_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_905_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_906_hd__take,axiom,
! [J: nat,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_a @ ( take_a @ J @ Xs ) )
= ( hd_a @ Xs ) ) ) ).
% hd_take
thf(fact_907_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_908_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_909_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_910_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_911_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_912_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_913_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_914_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_915_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_916_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_917_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_918_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_919_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_920_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_921_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_922_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_923_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_924_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_925_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_926_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_927_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_928_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_929_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_930_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_931_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_932_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs4: list_nat] : ( if_nat @ ( Xs4 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_933_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_934_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_935_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_936_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_937_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_938_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_939_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_940_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_941_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_942_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_943_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_944_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_945_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_946_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_947_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_948_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_949_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_950_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_951_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_952_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_953_SuccI,axiom,
! [Kl: list_a,K2: a,Kl2: set_list_a] :
( ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K2 @ nil_a ) ) @ Kl2 )
=> ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ ( drop_a @ ( suc @ N ) @ Xs ) )
= ( drop_a @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_963_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_964_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_965_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_966_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_967_tl__drop,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( drop_a @ N @ Xs ) )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% tl_drop
thf(fact_968_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_969_tl__drop__2,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( drop_a @ N @ Xs ) )
= ( drop_a @ ( suc @ N ) @ Xs ) ) ).
% tl_drop_2
thf(fact_970_drop__Suc,axiom,
! [N: nat,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ Xs )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% drop_Suc
thf(fact_971_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_972_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_973_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N @ Xs ) )
= ( nth_a @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_974_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_975_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_976_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_977_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_978_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A3: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A3 )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A3 @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_979_SuccD,axiom,
! [K2: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a2 @ ( append_a @ Kl @ ( cons_a @ K2 @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_980_empty__Shift,axiom,
! [Kl2: set_list_a,K2: a] :
( ( member_list_a2 @ nil_a @ Kl2 )
=> ( ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a2 @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_981_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_982_diff__gt__0__iff__gt,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B2 ) )
= ( ord_less_int @ B2 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_983_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_984_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_985_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_986_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_987_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_988_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_989_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_990_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_991_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_992_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_993_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A: int,B: int] : ( ord_less_int @ ( minus_minus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_994_diff__strict__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_995_diff__strict__left__mono,axiom,
! [B2: int,A3: int,C: int] :
( ( ord_less_int @ B2 @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_996_diff__eq__diff__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A3 @ B2 )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_997_diff__strict__mono,axiom,
! [A3: int,B2: int,D2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_998_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_999_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1000_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1001_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1002_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1003_diff__less__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1004_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1005_Suc__sub,axiom,
! [N: nat,M2: nat] :
( ( ( suc @ N )
= M2 )
=> ( N
= ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_1006_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_1007_tl__take,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( take_a @ N @ Xs ) )
= ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_a @ Xs ) ) ) ).
% tl_take
thf(fact_1008_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_1009_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1010_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_1011_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1012_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs4: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_1018_butlast__list__update,axiom,
! [K2: nat,Xs: list_nat,X: nat] :
( ( ( K2
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K2 @ X ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K2 @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K2 @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1019_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_1020_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_1021_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_1022_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_1023_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_1024_rev__update,axiom,
! [K2: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K2 @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K2 ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1025_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_1026_last__list__update,axiom,
! [Xs: list_a,K2: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K2
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X ) )
= X ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1027_last__list__update,axiom,
! [Xs: list_nat,K2: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K2
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K2 @ X ) )
= X ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K2 @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1028_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_1029_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_1030_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_1031_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1032_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1033_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1034_add__less__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1035_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1036_add__less__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1037_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1038_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1039_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_a @ ( replicate_list_a @ I @ nil_a ) )
= nil_a ) ).
% concat_replicate_trivial
thf(fact_1040_add__less__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1041_add__less__same__cancel1,axiom,
! [B2: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A3 ) @ B2 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1042_add__less__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1043_add__less__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ B2 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1044_less__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1045_less__add__same__cancel1,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1046_less__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1047_less__add__same__cancel2,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B2 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1048_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1049_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1050_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1051_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_1052_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1053_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1054_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_1055_hd__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( hd_a @ ( replicate_a @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_1056_length__splice,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_splice
thf(fact_1057_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1058_tl__replicate,axiom,
! [N: nat,X: a] :
( ( tl_a @ ( replicate_a @ N @ X ) )
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_1059_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_1060_less__diff__eq,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ A3 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_1061_diff__less__eq,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ C )
= ( ord_less_int @ A3 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_1062_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ~ ( ord_less_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1063_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: int,B2: int] :
( ~ ( ord_less_int @ A3 @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A3 @ B2 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1064_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1065_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_1066_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1067_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_1068_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1069_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_1070_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1071_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1072_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1073_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1074_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1075_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1076_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1077_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1078_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1079_add__less__le__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1080_add__less__le__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1081_add__le__less__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1082_add__le__less__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1083_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1084_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1085_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1086_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1087_pos__add__strict,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1088_pos__add__strict,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1089_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A3 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1090_add__pos__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1091_add__pos__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1092_add__neg__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1093_add__neg__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1094_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1095_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1096_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1097_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1098_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1099_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1100_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1101_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1102_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1103_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1104_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1105_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1106_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1107_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1108_add__strict__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1109_add__strict__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1110_add__strict__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1111_add__strict__left__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1112_add__strict__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1113_add__strict__right__mono,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1114_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1115_add__less__imp__less__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1116_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1117_add__less__imp__less__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1118_add__mono1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1119_add__mono1,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1120_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1121_less__add__one,axiom,
! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).
% less_add_one
thf(fact_1122_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_1123_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1124_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1125_length__shuffles,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( size_size_list_nat @ Zs )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_1126_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs4: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_1127_add__neg__nonpos,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1128_add__neg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1129_add__nonneg__pos,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1130_add__nonneg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1131_add__nonpos__neg,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1132_add__nonpos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1133_add__pos__nonneg,axiom,
! [A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1134_add__pos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1135_add__strict__increasing,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1136_add__strict__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1137_add__strict__increasing2,axiom,
! [A3: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1138_add__strict__increasing2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1139_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1140_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1141_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_1142_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ( ( ord_less_nat @ A3 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1143_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1144_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1145_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_1146_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_1147_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_1148_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1149_comm__append__is__replicate,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( ( append_a @ Xs @ Ys )
= ( append_a @ Ys @ Xs ) )
=> ? [N3: nat,Zs3: list_a] :
( ( ord_less_nat @ one_one_nat @ N3 )
& ( ( concat_a @ ( replicate_list_a @ N3 @ Zs3 ) )
= ( append_a @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1150_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F2: a > nat,Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_a @ Xs4 ) ) @ ( size_list_a @ F2 @ ( tl_a @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1151_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A0: nat > nat > nat,A12: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A12 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
=> ( ! [F3: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) )
=> ( ( ~ ( ord_less_nat @ B4 @ A5 )
=> ( P @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F3 @ A5 @ Acc ) ) )
=> ( P @ F3 @ A5 @ B4 @ Acc ) ) )
=> ( P @ A0 @ A12 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1152_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F3: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1153_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1154_fold__atLeastAtMost__nat_Opsimps,axiom,
! [F: nat > nat > nat,A3: nat,B2: nat,Acc2: nat] :
( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less_nat @ B2 @ A3 )
=> ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less_nat @ B2 @ A3 )
=> ( ( set_fo2584398358068434914at_nat @ F @ A3 @ B2 @ Acc2 )
= ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_1155_fold__atLeastAtMost__nat_Opelims,axiom,
! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less_nat @ Xb @ Xa )
=> ( Y
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_1156_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1157_size__stack_Osimps,axiom,
! [Left: list_a,Right: list_a] :
( ( size_size_stack_a @ ( stack_a2 @ Left @ Right ) )
= ( plus_plus_nat @ ( size_size_list_a @ Left ) @ ( size_size_list_a @ Right ) ) ) ).
% size_stack.simps
thf(fact_1158_size__stack_Osimps,axiom,
! [Left: list_nat,Right: list_nat] :
( ( size_size_stack_nat @ ( stack_nat2 @ Left @ Right ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Left ) @ ( size_size_list_nat @ Right ) ) ) ).
% size_stack.simps
thf(fact_1159_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_a @ N @ nil_a )
= nil_Pr1417316670369895453_nat_a ) ).
% enumerate_simps(1)
thf(fact_1160_length__enumerate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_1161_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_1162_enumerate__append__eq,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1163_size__stack_Oelims,axiom,
! [X: stack_a,Y: nat] :
( ( ( size_size_stack_a @ X )
= Y )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( Y
!= ( plus_plus_nat @ ( size_size_list_a @ Left2 ) @ ( size_size_list_a @ Right2 ) ) ) ) ) ).
% size_stack.elims
thf(fact_1164_size__stack_Oelims,axiom,
! [X: stack_nat,Y: nat] :
( ( ( size_size_stack_nat @ X )
= Y )
=> ~ ! [Left2: list_nat,Right2: list_nat] :
( ( X
= ( stack_nat2 @ Left2 @ Right2 ) )
=> ( Y
!= ( plus_plus_nat @ ( size_size_list_nat @ Left2 ) @ ( size_size_list_nat @ Right2 ) ) ) ) ) ).
% size_stack.elims
thf(fact_1165_size__stack_Opelims,axiom,
! [X: stack_a,Y: nat] :
( ( ( size_size_stack_a @ X )
= Y )
=> ( ( accp_stack_a @ stack_9138219648075182787_rel_a @ X )
=> ~ ! [Left2: list_a,Right2: list_a] :
( ( X
= ( stack_a2 @ Left2 @ Right2 ) )
=> ( ( Y
= ( plus_plus_nat @ ( size_size_list_a @ Left2 ) @ ( size_size_list_a @ Right2 ) ) )
=> ~ ( accp_stack_a @ stack_9138219648075182787_rel_a @ ( stack_a2 @ Left2 @ Right2 ) ) ) ) ) ) ).
% size_stack.pelims
thf(fact_1166_size__stack_Opelims,axiom,
! [X: stack_nat,Y: nat] :
( ( ( size_size_stack_nat @ X )
= Y )
=> ( ( accp_stack_nat @ stack_4774468673732873419el_nat @ X )
=> ~ ! [Left2: list_nat,Right2: list_nat] :
( ( X
= ( stack_nat2 @ Left2 @ Right2 ) )
=> ( ( Y
= ( plus_plus_nat @ ( size_size_list_nat @ Left2 ) @ ( size_size_list_nat @ Right2 ) ) )
=> ~ ( accp_stack_nat @ stack_4774468673732873419el_nat @ ( stack_nat2 @ Left2 @ Right2 ) ) ) ) ) ) ).
% size_stack.pelims
thf(fact_1167_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1168_zip__eq__Nil__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( zip_a_a @ Xs @ Ys )
= nil_Product_prod_a_a )
= ( ( Xs = nil_a )
| ( Ys = nil_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_1169_Nil__eq__zip__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_Product_prod_a_a
= ( zip_a_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
| ( Ys = nil_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_1170_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_1171_zip__Cons__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1172_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1173_length__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_zip
thf(fact_1174_zip__rev,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_1175_hd__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( hd_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( hd_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1176_hd__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_1177_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs3 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1178_zip__update,axiom,
! [Xs: list_nat,I: nat,X: nat,Ys: list_nat,Y: nat] :
( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_1179_zip__obtain__same__length,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_P6011104703257516679at_nat > $o] :
( ! [Zs3: list_nat,Ws: list_nat,N3: nat] :
( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( N3
= ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( Zs3
= ( take_nat @ N3 @ Xs ) )
=> ( ( Ws
= ( take_nat @ N3 @ Ys ) )
=> ( P @ ( zip_nat_nat @ Zs3 @ Ws ) ) ) ) ) )
=> ( P @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_1180_last__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( last_P8790725268278465478od_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( last_a @ Xs ) @ ( last_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1181_last__zip,axiom,
! [Xs: list_a,Ys: list_nat] :
( ( Xs != nil_a )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P2271748490522340894_a_nat @ ( zip_a_nat @ Xs @ Ys ) )
= ( product_Pair_a_nat @ ( last_a @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1182_last__zip,axiom,
! [Xs: list_nat,Ys: list_a] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_a )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( last_P5509911954246017860_nat_a @ ( zip_nat_a @ Xs @ Ys ) )
= ( product_Pair_nat_a @ ( last_nat @ Xs ) @ ( last_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1183_last__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1184_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1185_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1186_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_1187_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less_num @ M2 @ one ) ).
% semiring_norm(75)
thf(fact_1188_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_1189_mod__induct,axiom,
! [P: nat > $o,N: nat,P3: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( ord_less_nat @ M2 @ P3 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P3 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P3 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1190_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1191_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M @ N2 ) @ M @ ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1192_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1193_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1194_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_1195_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_1196_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1197_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1198_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1199_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs4: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) @ ( take_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ) ).
% rotate_drop_take
thf(fact_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_rotate__append,axiom,
! [L: list_nat,Q4: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q4 ) )
= ( append_nat @ Q4 @ L ) ) ).
% rotate_append
thf(fact_1207_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N2: nat,Xs4: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ).
% rotate_conv_mod
thf(fact_1208_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_1209_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1210_verit__le__mono__div,axiom,
! [A6: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A6 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A6 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1211_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_1212_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1213_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1214_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1215_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1216_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1217_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1218_div__less__mono,axiom,
! [A6: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A6 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A6 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A6 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1219_div__if,axiom,
( divide_divide_nat
= ( ^ [M: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1220_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1221_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_1222_Euclidean__Division_Odivmod__nat__def,axiom,
( euclidean_divmod_nat
= ( ^ [M: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M @ N2 ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).
% Euclidean_Division.divmod_nat_def
thf(fact_1223_mod__double__modulus,axiom,
! [M2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_int @ X @ M2 ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1224_mod__double__modulus,axiom,
! [M2: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( modulo_modulo_nat @ X @ M2 ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1225_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K2 @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_1226_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1227_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1228_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1229_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1230_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1231_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1232_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1233_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1234_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1235_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1236_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1237_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1238_less__add__iff2,axiom,
! [A3: int,E: int,C: int,B2: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_1239_less__add__iff1,axiom,
! [A3: int,E: int,C: int,B2: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B2 ) @ E ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_1240_mult__le__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1241_mult__le__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1242_mult__left__less__imp__less,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1243_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_1244_mult__strict__mono,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1245_mult__strict__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1246_mult__less__cancel__left,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1247_mult__right__less__imp__less,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1248_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_1249_mult__strict__mono_H,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1250_mult__strict__mono_H,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1251_mult__less__cancel__right,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A3 @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1252_mult__le__cancel__left__neg,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1253_mult__le__cancel__left__pos,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1254_mult__left__le__imp__le,axiom,
! [C: int,A3: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1255_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_1256_mult__right__le__imp__le,axiom,
! [A3: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1257_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_1258_mult__le__less__imp__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1259_mult__le__less__imp__less,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1260_mult__less__le__imp__less,axiom,
! [A3: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1261_mult__less__le__imp__less,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
( ( stack_list_a3 @ stack )
= nil_a ) ).
thf(conj_1,hypothesis,
~ ( type_i3216275384938974675tack_a @ stack ) ).
thf(conj_2,conjecture,
$false ).
%------------------------------------------------------------------------------