TPTP Problem File: SLH0945^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0025_States_Proof/prob_00309_009992__6932308_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1532 ( 606 unt; 267 typ; 0 def)
% Number of atoms : 3199 (1605 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 11188 ( 541 ~; 64 |; 185 &;9016 @)
% ( 0 <=>;1382 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 559 ( 559 >; 0 *; 0 +; 0 <<)
% Number of symbols : 218 ( 215 usr; 21 con; 0-5 aty)
% Number of variables : 3884 ( 84 ^;3620 !; 180 ?;3884 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:29.940
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__Int__Oint_J_M_062_It__List__Olist_It__Int__Oint_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
produc4162785005915746057st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J_J,type,
set_Pr8140719670968651609st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
produc8452586669324383225st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__Small__Ostate_It__List__Olist_It__Int__Oint_J_J_J,type,
produc8476525206126224643st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__Big__Ostate_It__List__Olist_It__Int__Oint_J_J_J,type,
produc8132867767721421180st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5834231552977413017st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr765067013931698361st_int: $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__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc1186641810826059865st_int: $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__States__Ostates_It__Int__Oint_J_Mt__Nat__Onat_J,type,
produc4606158825986000493nt_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Small__Ostate_It__Int__Oint_J_J,type,
produc957127447925188579te_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Big__Ostate_It__Int__Oint_J_J,type,
produc1417387325239031004te_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
produc1571854377283420419_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
list_list_list_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
produc7589950997499123219tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
produc6972303929186420058tate_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__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__Current__Ocurrent_It__List__Olist_It__Int__Oint_J_J,type,
current_list_int: $tType ).
thf(ty_n_t__States__Ostates_It__List__Olist_It__Int__Oint_J_J,type,
states_list_int: $tType ).
thf(ty_n_t__Common__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
state_list_int: $tType ).
thf(ty_n_t__Stack__Ostack_It__List__Olist_It__Int__Oint_J_J,type,
stack_list_int: $tType ).
thf(ty_n_t__Small__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
state_list_int2: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Big__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
state_list_int3: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $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__Current__Ocurrent_It__Int__Oint_J,type,
current_int: $tType ).
thf(ty_n_t__States__Ostates_It__Int__Oint_J,type,
states_int: $tType ).
thf(ty_n_t__Common__Ostate_It__Int__Oint_J,type,
state_int: $tType ).
thf(ty_n_t__Stack__Ostack_It__Int__Oint_J,type,
stack_int: $tType ).
thf(ty_n_t__Small__Ostate_It__Int__Oint_J,type,
state_int2: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Big__Ostate_It__Int__Oint_J,type,
state_int3: $tType ).
thf(ty_n_t__Current__Ocurrent_Itf__a_J,type,
current_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__States__Ostates_Itf__a_J,type,
states_a: $tType ).
thf(ty_n_t__Common__Ostate_Itf__a_J,type,
state_a: $tType ).
thf(ty_n_t__Stack__Ostack_Itf__a_J,type,
stack_a: $tType ).
thf(ty_n_t__Small__Ostate_Itf__a_J,type,
state_a2: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Big__Ostate_Itf__a_J,type,
state_a3: $tType ).
thf(ty_n_t__States__Odirection,type,
direction: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $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 (215)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Int__Oint,type,
bNF_Gr1870224194279859149ft_int: set_list_int > int > set_list_int ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
bNF_Gr6350390219475566417cc_int: set_list_int > list_int > set_int ).
thf(sy_c_Big_Opop_001t__Int__Oint,type,
pop_int: state_int3 > produc1417387325239031004te_int ).
thf(sy_c_Big_Opop_001t__List__Olist_It__Int__Oint_J,type,
pop_list_int: state_list_int3 > produc8132867767721421180st_int ).
thf(sy_c_Big_Opop_001tf__a,type,
pop_a: state_a3 > produc6972303929186420058tate_a ).
thf(sy_c_Big_Opush_001t__Int__Oint,type,
push_int: int > state_int3 > state_int3 ).
thf(sy_c_Big_Opush_001t__List__Olist_It__Int__Oint_J,type,
push_list_int: list_int > state_list_int3 > state_list_int3 ).
thf(sy_c_Big_Opush_001tf__a,type,
push_a: a > state_a3 > state_a3 ).
thf(sy_c_Big_Ostate_OCommon_001t__Int__Oint,type,
common_int: state_int > state_int3 ).
thf(sy_c_Big_Ostate_OCommon_001t__List__Olist_It__Int__Oint_J,type,
common_list_int: state_list_int > state_list_int3 ).
thf(sy_c_Big_Ostate_OCommon_001tf__a,type,
common_a: state_a > state_a3 ).
thf(sy_c_Big_Ostate_OReverse_001t__Int__Oint,type,
reverse_int: current_int > stack_int > list_int > nat > state_int3 ).
thf(sy_c_Big_Ostate_OReverse_001t__List__Olist_It__Int__Oint_J,type,
reverse_list_int: current_list_int > stack_list_int > list_list_int > nat > state_list_int3 ).
thf(sy_c_Big_Ostate_OReverse_001tf__a,type,
reverse_a: current_a > stack_a > list_a > nat > state_a3 ).
thf(sy_c_Big__Aux_Olist_001t__Int__Oint,type,
big_list_int: state_int3 > list_int ).
thf(sy_c_Big__Aux_Olist_001t__List__Olist_It__Int__Oint_J,type,
big_list_list_int: state_list_int3 > list_list_int ).
thf(sy_c_Big__Aux_Olist_001tf__a,type,
big_list_a: state_a3 > list_a ).
thf(sy_c_Big__Aux_Olist__current_001t__Int__Oint,type,
big_list_current_int: state_int3 > list_int ).
thf(sy_c_Big__Aux_Olist__current_001t__List__Olist_It__Int__Oint_J,type,
big_li2122662369155428040st_int: state_list_int3 > list_list_int ).
thf(sy_c_Big__Aux_Olist__current_001tf__a,type,
big_list_current_a: state_a3 > list_a ).
thf(sy_c_Common__Aux_Otake__rev_001t__Int__Oint,type,
common_take_rev_int: nat > list_int > list_int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
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_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Odrop_001t__Int__Oint,type,
drop_int: nat > list_int > list_int ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Olenlex_001t__Int__Oint,type,
lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olenlex_001t__List__Olist_It__Int__Oint_J,type,
lenlex_list_int: set_Pr765067013931698361st_int > set_Pr8140719670968651609st_int ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__Int__Oint,type,
lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olex_001t__List__Olist_It__Int__Oint_J,type,
lex_list_int: set_Pr765067013931698361st_int > set_Pr8140719670968651609st_int ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexord_001t__Int__Oint,type,
lexord_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
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_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
cons_list_list_int: list_list_int > list_list_list_int > list_list_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_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
nil_list_list_int: list_list_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_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Int__Oint,type,
hd_int: list_int > int ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
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_It__Int__Oint_J,type,
tl_list_int: list_list_int > list_list_int ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olistrel_001t__Int__Oint_001t__Int__Oint,type,
listrel_int_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onths_001t__Int__Oint,type,
nths_int: list_int > set_nat > list_int ).
thf(sy_c_List_Oremdups__adj_001t__Int__Oint,type,
remdups_adj_int: list_int > list_int ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Otake_001t__Int__Oint,type,
take_int: nat > list_int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_It__Int__Oint_J,type,
size_size_state_int: state_int3 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
size_s4372587533717917555st_int: state_list_int3 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_Itf__a_J,type,
size_size_state_a: state_a3 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_It__Int__Oint_J,type,
size_size_state_int2: state_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_Itf__a_J,type,
size_size_state_a2: state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_Itf__a_J,type,
size_size_current_a: current_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
size_s533118279054570080st_int: list_list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_It__Int__Oint_J,type,
size_size_state_int3: state_int2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
size_s7145721959145893242st_int: state_list_int2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_Itf__a_J,type,
size_size_state_a3: state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__States__Ostates_It__Int__Oint_J,type,
size_size_states_int: states_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__States__Ostates_Itf__a_J,type,
size_size_states_a: states_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
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__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__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $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_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Int__Oint_J_M_062_It__List__Olist_It__Int__Oint_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
produc1538682255923323267st_int: ( list_int > list_int > $o ) > list_list_int > produc4162785005915746057st_int ).
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__Int__Oint_001t__Big__Ostate_It__Int__Oint_J,type,
produc923477050739105812te_int: int > state_int3 > produc1417387325239031004te_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Small__Ostate_It__Int__Oint_J,type,
produc5352538583180469531te_int: int > state_int2 > produc957127447925188579te_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__Big__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
produc1010881625139315636st_int: list_int > state_list_int3 > produc8132867767721421180st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__Small__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
produc2109666439049285947st_int: list_int > state_list_int2 > produc8476525206126224643st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
produc4355665770860423473st_int: list_list_int > list_list_int > produc8452586669324383225st_int ).
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__States__Ostates_It__Int__Oint_J_001t__Nat__Onat,type,
produc5702035587166012709nt_nat: states_int > nat > produc4606158825986000493nt_nat ).
thf(sy_c_Product__Type_OPair_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
produc1877401315875745917_a_nat: states_a > nat > produc1571854377283420419_a_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Big__Ostate_Itf__a_J,type,
produc8641956578966763338tate_a: a > state_a3 > produc6972303929186420058tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Small__Ostate_Itf__a_J,type,
produc1224139502141355779tate_a: a > state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_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_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
collec928425520773505860st_int: ( produc1186641810826059865st_int > $o ) > set_Pr765067013931698361st_int ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Small_Opop_001t__Int__Oint,type,
pop_int2: state_int2 > produc957127447925188579te_int ).
thf(sy_c_Small_Opop_001t__List__Olist_It__Int__Oint_J,type,
pop_list_int2: state_list_int2 > produc8476525206126224643st_int ).
thf(sy_c_Small_Opop_001tf__a,type,
pop_a2: state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opush_001t__Int__Oint,type,
push_int2: int > state_int2 > state_int2 ).
thf(sy_c_Small_Opush_001t__List__Olist_It__Int__Oint_J,type,
push_list_int2: list_int > state_list_int2 > state_list_int2 ).
thf(sy_c_Small_Opush_001tf__a,type,
push_a2: a > state_a2 > state_a2 ).
thf(sy_c_Small_Ostate_OCommon_001t__Int__Oint,type,
common_int2: state_int > state_int2 ).
thf(sy_c_Small_Ostate_OCommon_001t__List__Olist_It__Int__Oint_J,type,
common_list_int2: state_list_int > state_list_int2 ).
thf(sy_c_Small_Ostate_OCommon_001tf__a,type,
common_a2: state_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse1_001t__Int__Oint,type,
reverse1_int: current_int > stack_int > list_int > state_int2 ).
thf(sy_c_Small_Ostate_OReverse1_001t__List__Olist_It__Int__Oint_J,type,
reverse1_list_int: current_list_int > stack_list_int > list_list_int > state_list_int2 ).
thf(sy_c_Small_Ostate_OReverse1_001tf__a,type,
reverse1_a: current_a > stack_a > list_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse2_001t__Int__Oint,type,
reverse2_int: current_int > list_int > stack_int > list_int > nat > state_int2 ).
thf(sy_c_Small_Ostate_OReverse2_001t__List__Olist_It__Int__Oint_J,type,
reverse2_list_int: current_list_int > list_list_int > stack_list_int > list_list_int > nat > state_list_int2 ).
thf(sy_c_Small_Ostate_OReverse2_001tf__a,type,
reverse2_a: current_a > list_a > stack_a > list_a > nat > state_a2 ).
thf(sy_c_Small__Aux_Olist_001t__Int__Oint,type,
small_list_int: state_int2 > list_int ).
thf(sy_c_Small__Aux_Olist_001t__List__Olist_It__Int__Oint_J,type,
small_list_list_int: state_list_int2 > list_list_int ).
thf(sy_c_Small__Aux_Olist_001tf__a,type,
small_list_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Olist__current_001t__Int__Oint,type,
small_1709254455575119551nt_int: state_int2 > list_int ).
thf(sy_c_Small__Aux_Olist__current_001t__List__Olist_It__Int__Oint_J,type,
small_6416132737660784207st_int: state_list_int2 > list_list_int ).
thf(sy_c_Small__Aux_Olist__current_001tf__a,type,
small_list_current_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Osize__new__state__rel_001tf__a,type,
small_1734874108512276223_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_Small__Aux_Osize__state__rel_001tf__a,type,
small_6459853017724497003_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_States_Ostates_OStates_001t__Int__Oint,type,
states_int2: direction > state_int3 > state_int2 > states_int ).
thf(sy_c_States_Ostates_OStates_001t__List__Olist_It__Int__Oint_J,type,
states_list_int2: direction > state_list_int3 > state_list_int2 > states_list_int ).
thf(sy_c_States_Ostates_OStates_001tf__a,type,
states_a2: direction > state_a3 > state_a2 > states_a ).
thf(sy_c_States_Ostates_Osize__states_001t__Int__Oint,type,
size_states_int: ( int > nat ) > states_int > nat ).
thf(sy_c_States_Ostates_Osize__states_001tf__a,type,
size_states_a: ( a > nat ) > states_a > nat ).
thf(sy_c_States_Ostep__states__rel_001tf__a,type,
step_states_rel_a: states_a > states_a > $o ).
thf(sy_c_States__Aux_Olist__big__first_001t__Int__Oint,type,
states9070087747003631121st_int: states_int > list_int ).
thf(sy_c_States__Aux_Olist__big__first_001t__List__Olist_It__Int__Oint_J,type,
states3294962311195589793st_int: states_list_int > list_list_int ).
thf(sy_c_States__Aux_Olist__big__first_001tf__a,type,
states1888450819780863577irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olist__current__big__first_001t__Int__Oint,type,
states8405765007952574210st_int: states_int > list_int ).
thf(sy_c_States__Aux_Olist__current__big__first_001t__List__Olist_It__Int__Oint_J,type,
states2742979599107686674st_int: states_list_int > list_list_int ).
thf(sy_c_States__Aux_Olist__current__big__first_001tf__a,type,
states7295096810965389224irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olist__current__small__first_001t__Int__Oint,type,
states2953871243264257211st_int: states_int > list_int ).
thf(sy_c_States__Aux_Olist__current__small__first_001t__List__Olist_It__Int__Oint_J,type,
states3081394979590915659st_int: states_list_int > list_list_int ).
thf(sy_c_States__Aux_Olist__current__small__first_001tf__a,type,
states7886008410469471791irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olist__small__first_001t__Int__Oint,type,
states4273554455855080522st_int: states_int > list_int ).
thf(sy_c_States__Aux_Olist__small__first_001t__List__Olist_It__Int__Oint_J,type,
states3243460551303850586st_int: states_list_int > list_list_int ).
thf(sy_c_States__Aux_Olist__small__first_001tf__a,type,
states1596304293096088672irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olists_001t__Int__Oint,type,
states_lists_int: states_int > produc1186641810826059865st_int ).
thf(sy_c_States__Aux_Olists_001t__List__Olist_It__Int__Oint_J,type,
states656764333025170780st_int: states_list_int > produc8452586669324383225st_int ).
thf(sy_c_States__Aux_Olists_001tf__a,type,
states_lists_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_States__Aux_Olists__current_001t__Int__Oint,type,
states4446210166810079719nt_int: states_int > produc1186641810826059865st_int ).
thf(sy_c_States__Aux_Olists__current_001tf__a,type,
states7719277857994474499rent_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_States__Aux_Olists__current__rel_001tf__a,type,
states5251248496104418302_rel_a: states_a > states_a > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_It__Int__Oint_J,type,
type_i293623501042546956te_int: state_int3 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
type_i6241818230346301340st_int: state_list_int3 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_Itf__a_J,type,
type_i6304938058965754292tate_a: state_a3 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Small__Ostate_It__Int__Oint_J,type,
type_i606337318274449939te_int: state_int2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Small__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
type_i6069260309285853475st_int: state_list_int2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Small__Ostate_Itf__a_J,type,
type_i464410347872898157tate_a: state_a2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__States__Ostates_It__Int__Oint_J,type,
type_i6677890655034438905es_int: states_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__States__Ostates_It__List__Olist_It__Int__Oint_J_J,type,
type_i3791846131947626761st_int: states_list_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__States__Ostates_Itf__a_J,type,
type_i8221491762852169479ates_a: states_a > $o ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__Big__Ostate_It__Int__Oint_J,type,
type_r7114992325145952472te_int: state_int3 > nat ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__Big__Ostate_Itf__a_J,type,
type_r2494999336194962664tate_a: state_a3 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Big__Ostate_It__Int__Oint_J,type,
type_s92858266581276998te_int: state_int3 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Big__Ostate_Itf__a_J,type,
type_s6530235180886170618tate_a: state_a3 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Common__Ostate_Itf__a_J,type,
type_s8424385952999958455tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Current__Ocurrent_Itf__a_J,type,
type_s933026853152659577rent_a: current_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Small__Ostate_It__Int__Oint_J,type,
type_s2537683556491027789te_int: state_int2 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Small__Ostate_Itf__a_J,type,
type_s6404775287138157491tate_a: state_a2 > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_It__Int__Oint_J,type,
type_s5405555405925207448te_int: state_int3 > state_int3 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
type_s2155344414434909736st_int: state_list_int3 > state_list_int3 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_Itf__a_J,type,
type_s3593206172722485288tate_a: state_a3 > state_a3 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Small__Ostate_It__Int__Oint_J,type,
type_s3231261300627247263te_int: state_int2 > state_int2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Small__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
type_s4628147265906176431st_int: state_list_int2 > state_list_int2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Small__Ostate_Itf__a_J,type,
type_s3703408523585882337tate_a: state_a2 > state_a2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__States__Ostates_It__Int__Oint_J,type,
type_s271243990432398725es_int: states_int > states_int ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__States__Ostates_It__List__Olist_It__Int__Oint_J_J,type,
type_s976745419985777557st_int: states_list_int > states_list_int ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__States__Ostates_Itf__a_J,type,
type_s4923920245906622843ates_a: states_a > states_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Small__Ostate_Itf__a_J,type,
accp_state_a: ( state_a2 > state_a2 > $o ) > state_a2 > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__States__Ostates_Itf__a_J,type,
accp_states_a: ( states_a > states_a > $o ) > states_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
member3583055517200259234st_int: produc8452586669324383225st_int > set_Pr8140719670968651609st_int > $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_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_v_auxS____,type,
auxS: list_a ).
thf(sy_v_big,type,
big: state_a3 ).
thf(sy_v_big_H____,type,
big2: stack_a ).
thf(sy_v_biga____,type,
biga: state_a3 ).
thf(sy_v_count____,type,
count: nat ).
thf(sy_v_current____,type,
current: current_a ).
thf(sy_v_dir,type,
dir: direction ).
thf(sy_v_dira____,type,
dira: direction ).
thf(sy_v_newS____,type,
newS: list_a ).
thf(sy_v_small,type,
small: state_a2 ).
% Relevant facts (1257)
thf(fact_0__092_060open_062invar_Abig_A_092_060Longrightarrow_062_Ainvar_A_Istep_Abig_J_092_060close_062,axiom,
( ( type_i6304938058965754292tate_a @ biga )
=> ( type_i6304938058965754292tate_a @ ( type_s3593206172722485288tate_a @ biga ) ) ) ).
% \<open>invar big \<Longrightarrow> invar (step big)\<close>
thf(fact_1__092_060open_062invar_A_IReverse2_Acurrent_AauxS_Abig_H_AnewS_Acount_J_A_092_060Longrightarrow_062_Ainvar_A_Istep_A_IReverse2_Acurrent_AauxS_Abig_H_AnewS_Acount_J_J_092_060close_062,axiom,
( ( type_i464410347872898157tate_a @ ( reverse2_a @ current @ auxS @ big2 @ newS @ count ) )
=> ( type_i464410347872898157tate_a @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ current @ auxS @ big2 @ newS @ count ) ) ) ) ).
% \<open>invar (Reverse2 current auxS big' newS count) \<Longrightarrow> invar (step (Reverse2 current auxS big' newS count))\<close>
thf(fact_2__C2__3_Oprems_C_I2_J,axiom,
( ( type_s4923920245906622843ates_a @ ( states_a2 @ dira @ biga @ ( reverse2_a @ current @ auxS @ big2 @ newS @ count ) ) )
= ( states_a2 @ dir @ big @ small ) ) ).
% "2_3.prems"(2)
thf(fact_3__C2__3_Oprems_C_I1_J,axiom,
type_i8221491762852169479ates_a @ ( states_a2 @ dira @ biga @ ( reverse2_a @ current @ auxS @ big2 @ newS @ count ) ) ).
% "2_3.prems"(1)
thf(fact_4_Small__Proof_Oinvar__step,axiom,
! [Small: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( type_i606337318274449939te_int @ ( type_s3231261300627247263te_int @ Small ) ) ) ).
% Small_Proof.invar_step
thf(fact_5_Small__Proof_Oinvar__step,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( type_i464410347872898157tate_a @ ( type_s3703408523585882337tate_a @ Small ) ) ) ).
% Small_Proof.invar_step
thf(fact_6_Big__Proof_Oinvar__step,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( type_i293623501042546956te_int @ ( type_s5405555405925207448te_int @ Big ) ) ) ).
% Big_Proof.invar_step
thf(fact_7_Big__Proof_Oinvar__step,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( type_s3593206172722485288tate_a @ Big ) ) ) ).
% Big_Proof.invar_step
thf(fact_8_Small__Proof_Oinvar__push,axiom,
! [Small: state_int2,X: int] :
( ( type_i606337318274449939te_int @ Small )
=> ( type_i606337318274449939te_int @ ( push_int2 @ X @ Small ) ) ) ).
% Small_Proof.invar_push
thf(fact_9_Small__Proof_Oinvar__push,axiom,
! [Small: state_a2,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( type_i464410347872898157tate_a @ ( push_a2 @ X @ Small ) ) ) ).
% Small_Proof.invar_push
thf(fact_10_Big__Proof_Oinvar__push,axiom,
! [Big: state_int3,X: int] :
( ( type_i293623501042546956te_int @ Big )
=> ( type_i293623501042546956te_int @ ( push_int @ X @ Big ) ) ) ).
% Big_Proof.invar_push
thf(fact_11_Big__Proof_Oinvar__push,axiom,
! [Big: state_a3,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( push_a @ X @ Big ) ) ) ).
% Big_Proof.invar_push
thf(fact_12_Small__Proof_Ostep__list__current,axiom,
! [Small: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( small_1709254455575119551nt_int @ ( type_s3231261300627247263te_int @ Small ) )
= ( small_1709254455575119551nt_int @ Small ) ) ) ).
% Small_Proof.step_list_current
thf(fact_13_Small__Proof_Ostep__list__current,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_current_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_current_a @ Small ) ) ) ).
% Small_Proof.step_list_current
thf(fact_14_Big__Proof_Ostep__list__current,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( big_list_current_int @ ( type_s5405555405925207448te_int @ Big ) )
= ( big_list_current_int @ Big ) ) ) ).
% Big_Proof.step_list_current
thf(fact_15_Big__Proof_Ostep__list__current,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_current_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_current_a @ Big ) ) ) ).
% Big_Proof.step_list_current
thf(fact_16_Big__Proof_Ostep__list,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( big_list_int @ ( type_s5405555405925207448te_int @ Big ) )
= ( big_list_int @ Big ) ) ) ).
% Big_Proof.step_list
thf(fact_17_Big__Proof_Ostep__list,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_a @ Big ) ) ) ).
% Big_Proof.step_list
thf(fact_18_Small__Proof_Ostep__size,axiom,
! [Small: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( size_size_state_int3 @ ( type_s3231261300627247263te_int @ Small ) )
= ( size_size_state_int3 @ Small ) ) ) ).
% Small_Proof.step_size
thf(fact_19_Small__Proof_Ostep__size,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( size_size_state_a3 @ ( type_s3703408523585882337tate_a @ Small ) )
= ( size_size_state_a3 @ Small ) ) ) ).
% Small_Proof.step_size
thf(fact_20_step__list__reverse2,axiom,
! [Small: state_int2,Current: current_int,Aux: list_int,Big: stack_int,New: list_int,Count: nat] :
( ( Small
= ( reverse2_int @ Current @ Aux @ Big @ New @ Count ) )
=> ( ( type_i606337318274449939te_int @ Small )
=> ( ( small_list_int @ ( type_s3231261300627247263te_int @ Small ) )
= ( small_list_int @ Small ) ) ) ) ).
% step_list_reverse2
thf(fact_21_step__list__reverse2,axiom,
! [Small: state_a2,Current: current_a,Aux: list_a,Big: stack_a,New: list_a,Count: nat] :
( ( Small
= ( reverse2_a @ Current @ Aux @ Big @ New @ Count ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_reverse2
thf(fact_22_Big__Proof_Ostep__size,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( size_size_state_int @ Big )
= ( size_size_state_int @ ( type_s5405555405925207448te_int @ Big ) ) ) ) ).
% Big_Proof.step_size
thf(fact_23_Big__Proof_Ostep__size,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( size_size_state_a @ Big )
= ( size_size_state_a @ ( type_s3593206172722485288tate_a @ Big ) ) ) ) ).
% Big_Proof.step_size
thf(fact_24_invar__push__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ ( push_int @ X @ Big ) @ Small ) ) ) ).
% invar_push_big
thf(fact_25_invar__push__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) ) ) ).
% invar_push_big
thf(fact_26_invar__push__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ ( push_int2 @ X @ Small ) ) ) ) ).
% invar_push_small
thf(fact_27_invar__push__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) ) ) ).
% invar_push_small
thf(fact_28_step__states_Osimps_I4_J,axiom,
! [Dir: direction,Left: state_int3,V: current_int,Va: list_int,Vb: stack_int,Vc: list_int,Vd: nat] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ Left @ ( reverse2_int @ V @ Va @ Vb @ Vc @ Vd ) ) )
= ( states_int2 @ Dir @ ( type_s5405555405925207448te_int @ Left ) @ ( type_s3231261300627247263te_int @ ( reverse2_int @ V @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% step_states.simps(4)
thf(fact_29_step__states_Osimps_I4_J,axiom,
! [Dir: direction,Left: state_a3,V: current_a,Va: list_a,Vb: stack_a,Vc: list_a,Vd: nat] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Left @ ( reverse2_a @ V @ Va @ Vb @ Vc @ Vd ) ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ Left ) @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ V @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% step_states.simps(4)
thf(fact_30_Small_Ostate_Oinject_I2_J,axiom,
! [X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat,Y21: current_a,Y22: list_a,Y23: stack_a,Y24: list_a,Y25: nat] :
( ( ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 )
= ( reverse2_a @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% Small.state.inject(2)
thf(fact_31_states_Oinject,axiom,
! [X1: direction,X2: state_int3,X3: state_int2,Y1: direction,Y2: state_int3,Y3: state_int2] :
( ( ( states_int2 @ X1 @ X2 @ X3 )
= ( states_int2 @ Y1 @ Y2 @ Y3 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X3 = Y3 ) ) ) ).
% states.inject
thf(fact_32_states_Oinject,axiom,
! [X1: direction,X2: state_a3,X3: state_a2,Y1: direction,Y2: state_a3,Y3: state_a2] :
( ( ( states_a2 @ X1 @ X2 @ X3 )
= ( states_a2 @ Y1 @ Y2 @ Y3 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X3 = Y3 ) ) ) ).
% states.inject
thf(fact_33_step__lists__current,axiom,
! [States: states_int] :
( ( type_i6677890655034438905es_int @ States )
=> ( ( states4446210166810079719nt_int @ ( type_s271243990432398725es_int @ States ) )
= ( states4446210166810079719nt_int @ States ) ) ) ).
% step_lists_current
thf(fact_34_step__lists__current,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states7719277857994474499rent_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states7719277857994474499rent_a @ States ) ) ) ).
% step_lists_current
thf(fact_35_step__lists,axiom,
! [States: states_int] :
( ( type_i6677890655034438905es_int @ States )
=> ( ( states_lists_int @ ( type_s271243990432398725es_int @ States ) )
= ( states_lists_int @ States ) ) ) ).
% step_lists
thf(fact_36_step__lists,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_lists_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states_lists_a @ States ) ) ) ).
% step_lists
thf(fact_37_Small__Proof_Osize__push,axiom,
! [Small: state_int2,X: int] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( size_size_state_int3 @ ( push_int2 @ X @ Small ) )
= ( suc @ ( size_size_state_int3 @ Small ) ) ) ) ).
% Small_Proof.size_push
thf(fact_38_Small__Proof_Osize__push,axiom,
! [Small: state_a2,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( size_size_state_a3 @ ( push_a2 @ X @ Small ) )
= ( suc @ ( size_size_state_a3 @ Small ) ) ) ) ).
% Small_Proof.size_push
thf(fact_39_Big__Proof_Osize__push,axiom,
! [Big: state_int3,X: int] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( size_size_state_int @ ( push_int @ X @ Big ) )
= ( suc @ ( size_size_state_int @ Big ) ) ) ) ).
% Big_Proof.size_push
thf(fact_40_Big__Proof_Osize__push,axiom,
! [Big: state_a3,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( size_size_state_a @ ( push_a @ X @ Big ) )
= ( suc @ ( size_size_state_a @ Big ) ) ) ) ).
% Big_Proof.size_push
thf(fact_41_step__list__common,axiom,
! [Small: state_int2,Common: state_int] :
( ( Small
= ( common_int2 @ Common ) )
=> ( ( type_i606337318274449939te_int @ Small )
=> ( ( small_list_int @ ( type_s3231261300627247263te_int @ Small ) )
= ( small_list_int @ Small ) ) ) ) ).
% step_list_common
thf(fact_42_step__list__common,axiom,
! [Small: state_a2,Common: state_a] :
( ( Small
= ( common_a2 @ Common ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_common
thf(fact_43_Big__Proof_Oremaining__steps__push,axiom,
! [Big: state_int3,X: int] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( type_r7114992325145952472te_int @ ( push_int @ X @ Big ) )
= ( type_r7114992325145952472te_int @ Big ) ) ) ).
% Big_Proof.remaining_steps_push
thf(fact_44_Big__Proof_Oremaining__steps__push,axiom,
! [Big: state_a3,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( type_r2494999336194962664tate_a @ ( push_a @ X @ Big ) )
= ( type_r2494999336194962664tate_a @ Big ) ) ) ).
% Big_Proof.remaining_steps_push
thf(fact_45_states_Oexhaust,axiom,
! [Y: states_int] :
~ ! [X12: direction,X26: state_int3,X32: state_int2] :
( Y
!= ( states_int2 @ X12 @ X26 @ X32 ) ) ).
% states.exhaust
thf(fact_46_states_Oexhaust,axiom,
! [Y: states_a] :
~ ! [X12: direction,X26: state_a3,X32: state_a2] :
( Y
!= ( states_a2 @ X12 @ X26 @ X32 ) ) ).
% states.exhaust
thf(fact_47_Small_Ostate_Oinject_I3_J,axiom,
! [X3: state_a,Y3: state_a] :
( ( ( common_a2 @ X3 )
= ( common_a2 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% Small.state.inject(3)
thf(fact_48_Small_Ostate_Oinject_I3_J,axiom,
! [X3: state_int,Y3: state_int] :
( ( ( common_int2 @ X3 )
= ( common_int2 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% Small.state.inject(3)
thf(fact_49_Small_Ostate_Odistinct_I5_J,axiom,
! [X21: current_int,X22: list_int,X23: stack_int,X24: list_int,X25: nat,X3: state_int] :
( ( reverse2_int @ X21 @ X22 @ X23 @ X24 @ X25 )
!= ( common_int2 @ X3 ) ) ).
% Small.state.distinct(5)
thf(fact_50_Small_Ostate_Odistinct_I5_J,axiom,
! [X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat,X3: state_a] :
( ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(5)
thf(fact_51_step__states_Osimps_I5_J,axiom,
! [Dir: direction,Left: state_int3,V: state_int] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ Left @ ( common_int2 @ V ) ) )
= ( states_int2 @ Dir @ ( type_s5405555405925207448te_int @ Left ) @ ( type_s3231261300627247263te_int @ ( common_int2 @ V ) ) ) ) ).
% step_states.simps(5)
thf(fact_52_step__states_Osimps_I5_J,axiom,
! [Dir: direction,Left: state_a3,V: state_a] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Left @ ( common_a2 @ V ) ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ Left ) @ ( type_s3703408523585882337tate_a @ ( common_a2 @ V ) ) ) ) ).
% step_states.simps(5)
thf(fact_53_remaining__steps__states_Ocases,axiom,
! [X: states_int] :
~ ! [Uu: direction,Big2: state_int3,Small2: state_int2] :
( X
!= ( states_int2 @ Uu @ Big2 @ Small2 ) ) ).
% remaining_steps_states.cases
thf(fact_54_remaining__steps__states_Ocases,axiom,
! [X: states_a] :
~ ! [Uu: direction,Big2: state_a3,Small2: state_a2] :
( X
!= ( states_a2 @ Uu @ Big2 @ Small2 ) ) ).
% remaining_steps_states.cases
thf(fact_55_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_56_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_57_Big__Proof_Osize__new__push,axiom,
! [Big: state_int3,X: int] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( type_s92858266581276998te_int @ ( push_int @ X @ Big ) )
= ( suc @ ( type_s92858266581276998te_int @ Big ) ) ) ) ).
% Big_Proof.size_new_push
thf(fact_58_Big__Proof_Osize__new__push,axiom,
! [Big: state_a3,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( type_s6530235180886170618tate_a @ ( push_a @ X @ Big ) )
= ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% Big_Proof.size_new_push
thf(fact_59_Small__Proof_Osize__new__push,axiom,
! [Small: state_int2,X: int] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( type_s2537683556491027789te_int @ ( push_int2 @ X @ Small ) )
= ( suc @ ( type_s2537683556491027789te_int @ Small ) ) ) ) ).
% Small_Proof.size_new_push
thf(fact_60_Small__Proof_Osize__new__push,axiom,
! [Small: state_a2,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( type_s6404775287138157491tate_a @ ( push_a2 @ X @ Small ) )
= ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_new_push
thf(fact_61_States__Aux_Olists_Osimps_I4_J,axiom,
! [Uv: direction,Big: state_int3,V: state_int] :
( ( states_lists_int @ ( states_int2 @ Uv @ Big @ ( common_int2 @ V ) ) )
= ( produc364263696895485585st_int @ ( big_list_int @ Big ) @ ( small_list_int @ ( common_int2 @ V ) ) ) ) ).
% States_Aux.lists.simps(4)
thf(fact_62_States__Aux_Olists_Osimps_I4_J,axiom,
! [Uv: direction,Big: state_a3,V: state_a] :
( ( states_lists_a @ ( states_a2 @ Uv @ Big @ ( common_a2 @ V ) ) )
= ( produc6837034575241423639list_a @ ( big_list_a @ Big ) @ ( small_list_a @ ( common_a2 @ V ) ) ) ) ).
% States_Aux.lists.simps(4)
thf(fact_63_States__Aux_Olists_Osimps_I3_J,axiom,
! [Uv: direction,Big: state_int3,V: current_int,Va: list_int,Vb: stack_int,Vc: list_int,Vd: nat] :
( ( states_lists_int @ ( states_int2 @ Uv @ Big @ ( reverse2_int @ V @ Va @ Vb @ Vc @ Vd ) ) )
= ( produc364263696895485585st_int @ ( big_list_int @ Big ) @ ( small_list_int @ ( reverse2_int @ V @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% States_Aux.lists.simps(3)
thf(fact_64_States__Aux_Olists_Osimps_I3_J,axiom,
! [Uv: direction,Big: state_a3,V: current_a,Va: list_a,Vb: stack_a,Vc: list_a,Vd: nat] :
( ( states_lists_a @ ( states_a2 @ Uv @ Big @ ( reverse2_a @ V @ Va @ Vb @ Vc @ Vd ) ) )
= ( produc6837034575241423639list_a @ ( big_list_a @ Big ) @ ( small_list_a @ ( reverse2_a @ V @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% States_Aux.lists.simps(3)
thf(fact_65_step__states_Osimps_I2_J,axiom,
! [Dir: direction,V: current_int,Va: stack_int,Vb: list_int,Vd: nat,Right: state_int2] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ ( reverse_int @ V @ Va @ Vb @ ( suc @ Vd ) ) @ Right ) )
= ( states_int2 @ Dir @ ( type_s5405555405925207448te_int @ ( reverse_int @ V @ Va @ Vb @ ( suc @ Vd ) ) ) @ ( type_s3231261300627247263te_int @ Right ) ) ) ).
% step_states.simps(2)
thf(fact_66_step__states_Osimps_I2_J,axiom,
! [Dir: direction,V: current_a,Va: stack_a,Vb: list_a,Vd: nat,Right: state_a2] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( reverse_a @ V @ Va @ Vb @ ( suc @ Vd ) ) @ Right ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ ( reverse_a @ V @ Va @ Vb @ ( suc @ Vd ) ) ) @ ( type_s3703408523585882337tate_a @ Right ) ) ) ).
% step_states.simps(2)
thf(fact_67_lists__current_Osimps,axiom,
! [Uu2: direction,Big: state_int3,Small: state_int2] :
( ( states4446210166810079719nt_int @ ( states_int2 @ Uu2 @ Big @ Small ) )
= ( produc364263696895485585st_int @ ( big_list_current_int @ Big ) @ ( small_1709254455575119551nt_int @ Small ) ) ) ).
% lists_current.simps
thf(fact_68_lists__current_Osimps,axiom,
! [Uu2: direction,Big: state_a3,Small: state_a2] :
( ( states7719277857994474499rent_a @ ( states_a2 @ Uu2 @ Big @ Small ) )
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big ) @ ( small_list_current_a @ Small ) ) ) ).
% lists_current.simps
thf(fact_69_lists__current_Oelims,axiom,
! [X: states_int,Y: produc1186641810826059865st_int] :
( ( ( states4446210166810079719nt_int @ X )
= Y )
=> ~ ! [Uu: direction,Big2: state_int3,Small2: state_int2] :
( ( X
= ( states_int2 @ Uu @ Big2 @ Small2 ) )
=> ( Y
!= ( produc364263696895485585st_int @ ( big_list_current_int @ Big2 ) @ ( small_1709254455575119551nt_int @ Small2 ) ) ) ) ) ).
% lists_current.elims
thf(fact_70_lists__current_Oelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ~ ! [Uu: direction,Big2: state_a3,Small2: state_a2] :
( ( X
= ( states_a2 @ Uu @ Big2 @ Small2 ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big2 ) @ ( small_list_current_a @ Small2 ) ) ) ) ) ).
% lists_current.elims
thf(fact_71_Big__Proof_Oremaining__steps__step__0,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ( type_r7114992325145952472te_int @ Big )
= zero_zero_nat )
=> ( ( type_r7114992325145952472te_int @ ( type_s5405555405925207448te_int @ Big ) )
= zero_zero_nat ) ) ) ).
% Big_Proof.remaining_steps_step_0
thf(fact_72_Big__Proof_Oremaining__steps__step__0,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( type_r2494999336194962664tate_a @ Big )
= zero_zero_nat )
=> ( ( type_r2494999336194962664tate_a @ ( type_s3593206172722485288tate_a @ Big ) )
= zero_zero_nat ) ) ) ).
% Big_Proof.remaining_steps_step_0
thf(fact_73_mem__Collect__eq,axiom,
! [A: produc1186641810826059865st_int,P: produc1186641810826059865st_int > $o] :
( ( member6698963635872716290st_int @ A @ ( collec928425520773505860st_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: list_int,P: list_int > $o] :
( ( member_list_int @ A @ ( collect_list_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_Pr765067013931698361st_int] :
( ( collec928425520773505860st_int
@ ^ [X4: produc1186641810826059865st_int] : ( member6698963635872716290st_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_list_int] :
( ( collect_list_int
@ ^ [X4: list_int] : ( member_list_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_84_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_85_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_86_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_87_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_88_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_89_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ X5 @ Y4 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_90_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_91_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_92_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_93_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_94_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_95_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_96_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_97_size__neq__size__imp__neq,axiom,
! [X: state_int2,Y: state_int2] :
( ( ( size_size_state_int3 @ X )
!= ( size_size_state_int3 @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_98_size__neq__size__imp__neq,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( size_size_state_a3 @ X )
!= ( size_size_state_a3 @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_99_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_100_size__neq__size__imp__neq,axiom,
! [X: state_int3,Y: state_int3] :
( ( ( size_size_state_int @ X )
!= ( size_size_state_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_101_size__neq__size__imp__neq,axiom,
! [X: state_a3,Y: state_a3] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_102_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_103_Big_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X14: nat,Y11: current_a,Y12: stack_a,Y13: list_a,Y14: nat] :
( ( ( reverse_a @ X11 @ X122 @ X13 @ X14 )
= ( reverse_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% Big.state.inject(1)
thf(fact_104_states_Osize__gen,axiom,
! [X: int > nat,X1: direction,X2: state_int3,X3: state_int2] :
( ( size_states_int @ X @ ( states_int2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size_gen
thf(fact_105_states_Osize__gen,axiom,
! [X: a > nat,X1: direction,X2: state_a3,X3: state_a2] :
( ( size_states_a @ X @ ( states_a2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size_gen
thf(fact_106_prod_Oinject,axiom,
! [X1: int,X2: state_int3,Y1: int,Y2: state_int3] :
( ( ( produc923477050739105812te_int @ X1 @ X2 )
= ( produc923477050739105812te_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_107_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_108_prod_Oinject,axiom,
! [X1: int > int > $o,X2: list_int,Y1: int > int > $o,Y2: list_int] :
( ( ( produc8618682346314911123st_int @ X1 @ X2 )
= ( produc8618682346314911123st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_109_prod_Oinject,axiom,
! [X1: a,X2: state_a2,Y1: a,Y2: state_a2] :
( ( ( produc1224139502141355779tate_a @ X1 @ X2 )
= ( produc1224139502141355779tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_110_prod_Oinject,axiom,
! [X1: a,X2: state_a3,Y1: a,Y2: state_a3] :
( ( ( produc8641956578966763338tate_a @ X1 @ X2 )
= ( produc8641956578966763338tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_111_old_Oprod_Oinject,axiom,
! [A: int,B: state_int3,A3: int,B2: state_int3] :
( ( ( produc923477050739105812te_int @ A @ B )
= ( produc923477050739105812te_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_112_old_Oprod_Oinject,axiom,
! [A: int,B: int,A3: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_113_old_Oprod_Oinject,axiom,
! [A: int > int > $o,B: list_int,A3: int > int > $o,B2: list_int] :
( ( ( produc8618682346314911123st_int @ A @ B )
= ( produc8618682346314911123st_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_114_old_Oprod_Oinject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_115_old_Oprod_Oinject,axiom,
! [A: a,B: state_a3,A3: a,B2: state_a3] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_116_states_Osize_I2_J,axiom,
! [X1: direction,X2: state_int3,X3: state_int2] :
( ( size_size_states_int @ ( states_int2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size(2)
thf(fact_117_states_Osize_I2_J,axiom,
! [X1: direction,X2: state_a3,X3: state_a2] :
( ( size_size_states_a @ ( states_a2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size(2)
thf(fact_118_Big__Proof_Oremaining__steps__step,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r7114992325145952472te_int @ Big ) )
=> ( ( suc @ ( type_r7114992325145952472te_int @ ( type_s5405555405925207448te_int @ Big ) ) )
= ( type_r7114992325145952472te_int @ Big ) ) ) ) ).
% Big_Proof.remaining_steps_step
thf(fact_119_Big__Proof_Oremaining__steps__step,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r2494999336194962664tate_a @ Big ) )
=> ( ( suc @ ( type_r2494999336194962664tate_a @ ( type_s3593206172722485288tate_a @ Big ) ) )
= ( type_r2494999336194962664tate_a @ Big ) ) ) ) ).
% Big_Proof.remaining_steps_step
thf(fact_120_push__small__lists,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,Big3: list_list_int,Small3: list_list_int,X: list_int] :
( ( type_i3791846131947626761st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
=> ( ( ( states656764333025170780st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
= ( produc4355665770860423473st_int @ Big3 @ Small3 ) )
=> ( ( states656764333025170780st_int @ ( states_list_int2 @ Dir @ Big @ ( push_list_int2 @ X @ Small ) ) )
= ( produc4355665770860423473st_int @ Big3 @ ( cons_list_int @ X @ Small3 ) ) ) ) ) ).
% push_small_lists
thf(fact_121_push__small__lists,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,Big3: list_int,Small3: list_int,X: int] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ( states_lists_int @ ( states_int2 @ Dir @ Big @ Small ) )
= ( produc364263696895485585st_int @ Big3 @ Small3 ) )
=> ( ( states_lists_int @ ( states_int2 @ Dir @ Big @ ( push_int2 @ X @ Small ) ) )
= ( produc364263696895485585st_int @ Big3 @ ( cons_int @ X @ Small3 ) ) ) ) ) ).
% push_small_lists
thf(fact_122_push__small__lists,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Big3: list_a,Small3: list_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big3 @ Small3 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( produc6837034575241423639list_a @ Big3 @ ( cons_a @ X @ Small3 ) ) ) ) ) ).
% push_small_lists
thf(fact_123_States__Aux_Olists_Osimps_I2_J,axiom,
! [Uv: direction,V: state_int,Small: state_int2] :
( ( states_lists_int @ ( states_int2 @ Uv @ ( common_int @ V ) @ Small ) )
= ( produc364263696895485585st_int @ ( big_list_int @ ( common_int @ V ) ) @ ( small_list_int @ Small ) ) ) ).
% States_Aux.lists.simps(2)
thf(fact_124_States__Aux_Olists_Osimps_I2_J,axiom,
! [Uv: direction,V: state_a,Small: state_a2] :
( ( states_lists_a @ ( states_a2 @ Uv @ ( common_a @ V ) @ Small ) )
= ( produc6837034575241423639list_a @ ( big_list_a @ ( common_a @ V ) ) @ ( small_list_a @ Small ) ) ) ).
% States_Aux.lists.simps(2)
thf(fact_125_push__big,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,Big3: list_list_int,Small3: list_list_int,X: list_int] :
( ( ( states656764333025170780st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
= ( produc4355665770860423473st_int @ Big3 @ Small3 ) )
=> ( ( states656764333025170780st_int @ ( states_list_int2 @ Dir @ ( push_list_int @ X @ Big ) @ Small ) )
= ( produc4355665770860423473st_int @ ( cons_list_int @ X @ Big3 ) @ Small3 ) ) ) ).
% push_big
thf(fact_126_push__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,Big3: list_int,Small3: list_int,X: int] :
( ( ( states_lists_int @ ( states_int2 @ Dir @ Big @ Small ) )
= ( produc364263696895485585st_int @ Big3 @ Small3 ) )
=> ( ( states_lists_int @ ( states_int2 @ Dir @ ( push_int @ X @ Big ) @ Small ) )
= ( produc364263696895485585st_int @ ( cons_int @ X @ Big3 ) @ Small3 ) ) ) ).
% push_big
thf(fact_127_push__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Big3: list_a,Small3: list_a,X: a] :
( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big3 @ Small3 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( produc6837034575241423639list_a @ ( cons_a @ X @ Big3 ) @ Small3 ) ) ) ).
% push_big
thf(fact_128_Big_Ostate_Oinject_I2_J,axiom,
! [X2: state_a,Y2: state_a] :
( ( ( common_a @ X2 )
= ( common_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% Big.state.inject(2)
thf(fact_129_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_130_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_131_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_132_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_133_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_134_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_135_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_136_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_137_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_138_Big__Proof_Opush__list,axiom,
! [X: list_int,Big: state_list_int3] :
( ( big_list_list_int @ ( push_list_int @ X @ Big ) )
= ( cons_list_int @ X @ ( big_list_list_int @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_139_Big__Proof_Opush__list,axiom,
! [X: a,Big: state_a3] :
( ( big_list_a @ ( push_a @ X @ Big ) )
= ( cons_a @ X @ ( big_list_a @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_140_Big__Proof_Opush__list,axiom,
! [X: int,Big: state_int3] :
( ( big_list_int @ ( push_int @ X @ Big ) )
= ( cons_int @ X @ ( big_list_int @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_141_Small__Proof_Opush__list__current,axiom,
! [X: list_int,Small: state_list_int2] :
( ( small_6416132737660784207st_int @ ( push_list_int2 @ X @ Small ) )
= ( cons_list_int @ X @ ( small_6416132737660784207st_int @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_142_Small__Proof_Opush__list__current,axiom,
! [X: int,Small: state_int2] :
( ( small_1709254455575119551nt_int @ ( push_int2 @ X @ Small ) )
= ( cons_int @ X @ ( small_1709254455575119551nt_int @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_143_Small__Proof_Opush__list__current,axiom,
! [X: a,Small: state_a2] :
( ( small_list_current_a @ ( push_a2 @ X @ Small ) )
= ( cons_a @ X @ ( small_list_current_a @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_144_Big__Proof_Opush__list__current,axiom,
! [X: list_int,Big: state_list_int3] :
( ( big_li2122662369155428040st_int @ ( push_list_int @ X @ Big ) )
= ( cons_list_int @ X @ ( big_li2122662369155428040st_int @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_145_Big__Proof_Opush__list__current,axiom,
! [X: int,Big: state_int3] :
( ( big_list_current_int @ ( push_int @ X @ Big ) )
= ( cons_int @ X @ ( big_list_current_int @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_146_Big__Proof_Opush__list__current,axiom,
! [X: a,Big: state_a3] :
( ( big_list_current_a @ ( push_a @ X @ Big ) )
= ( cons_a @ X @ ( big_list_current_a @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_147_push__list__common,axiom,
! [Small: state_list_int2,Common: state_list_int,X: list_int] :
( ( Small
= ( common_list_int2 @ Common ) )
=> ( ( small_list_list_int @ ( push_list_int2 @ X @ Small ) )
= ( cons_list_int @ X @ ( small_list_list_int @ Small ) ) ) ) ).
% push_list_common
thf(fact_148_push__list__common,axiom,
! [Small: state_a2,Common: state_a,X: a] :
( ( Small
= ( common_a2 @ Common ) )
=> ( ( small_list_a @ ( push_a2 @ X @ Small ) )
= ( cons_a @ X @ ( small_list_a @ Small ) ) ) ) ).
% push_list_common
thf(fact_149_push__list__common,axiom,
! [Small: state_int2,Common: state_int,X: int] :
( ( Small
= ( common_int2 @ Common ) )
=> ( ( small_list_int @ ( push_int2 @ X @ Small ) )
= ( cons_int @ X @ ( small_list_int @ Small ) ) ) ) ).
% push_list_common
thf(fact_150_Big_Opush_Ocases,axiom,
! [X: produc1417387325239031004te_int] :
( ! [X5: int,State: state_int] :
( X
!= ( produc923477050739105812te_int @ X5 @ ( common_int @ State ) ) )
=> ~ ! [X5: int,Current2: current_int,Big2: stack_int,Aux2: list_int,Count2: nat] :
( X
!= ( produc923477050739105812te_int @ X5 @ ( reverse_int @ Current2 @ Big2 @ Aux2 @ Count2 ) ) ) ) ).
% Big.push.cases
thf(fact_151_Big_Opush_Ocases,axiom,
! [X: produc6972303929186420058tate_a] :
( ! [X5: a,State: state_a] :
( X
!= ( produc8641956578966763338tate_a @ X5 @ ( common_a @ State ) ) )
=> ~ ! [X5: a,Current2: current_a,Big2: stack_a,Aux2: list_a,Count2: nat] :
( X
!= ( produc8641956578966763338tate_a @ X5 @ ( reverse_a @ Current2 @ Big2 @ Aux2 @ Count2 ) ) ) ) ).
% Big.push.cases
thf(fact_152_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_153_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_154_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_155_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_156_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_157_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_158_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_159_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_160_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_161_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_162_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_163_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_164_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_165_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_166_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_167_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_168_Big_Ostate_Oexhaust,axiom,
! [Y: state_a3] :
( ! [X112: current_a,X123: stack_a,X132: list_a,X142: nat] :
( Y
!= ( reverse_a @ X112 @ X123 @ X132 @ X142 ) )
=> ~ ! [X26: state_a] :
( Y
!= ( common_a @ X26 ) ) ) ).
% Big.state.exhaust
thf(fact_169_Big_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X14: nat,X2: state_a] :
( ( reverse_a @ X11 @ X122 @ X13 @ X14 )
!= ( common_a @ X2 ) ) ).
% Big.state.distinct(1)
thf(fact_170_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_171_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_172_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_173_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_174_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_175_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_176_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_177_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_178_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_179_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_180_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_181_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_182_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_183_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_184_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_185_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_186_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_187_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_188_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_189_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_190_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_191_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_192_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_193_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_194_size__ok_H_Ocases,axiom,
! [X: produc4606158825986000493nt_nat] :
~ ! [Uu: direction,Big2: state_int3,Small2: state_int2,Steps: nat] :
( X
!= ( produc5702035587166012709nt_nat @ ( states_int2 @ Uu @ Big2 @ Small2 ) @ Steps ) ) ).
% size_ok'.cases
thf(fact_195_size__ok_H_Ocases,axiom,
! [X: produc1571854377283420419_a_nat] :
~ ! [Uu: direction,Big2: state_a3,Small2: state_a2,Steps: nat] :
( X
!= ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big2 @ Small2 ) @ Steps ) ) ).
% size_ok'.cases
thf(fact_196_Big_Ostep__state_Ocases,axiom,
! [X: state_a3] :
( ! [State: state_a] :
( X
!= ( common_a @ State ) )
=> ( ! [Current2: current_a,Uu: stack_a,Aux2: list_a] :
( X
!= ( reverse_a @ Current2 @ Uu @ Aux2 @ zero_zero_nat ) )
=> ~ ! [Current2: current_a,Big2: stack_a,Aux2: list_a,V2: nat] :
( X
!= ( reverse_a @ Current2 @ Big2 @ Aux2 @ ( suc @ V2 ) ) ) ) ) ).
% Big.step_state.cases
thf(fact_197_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_198_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_199_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_200_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_201_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_202_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_203_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_204_Pair__inject,axiom,
! [A: int,B: state_int3,A3: int,B2: state_int3] :
( ( ( produc923477050739105812te_int @ A @ B )
= ( produc923477050739105812te_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_205_Pair__inject,axiom,
! [A: int,B: int,A3: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_206_Pair__inject,axiom,
! [A: int > int > $o,B: list_int,A3: int > int > $o,B2: list_int] :
( ( ( produc8618682346314911123st_int @ A @ B )
= ( produc8618682346314911123st_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_207_Pair__inject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_208_Pair__inject,axiom,
! [A: a,B: state_a3,A3: a,B2: state_a3] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_209_prod__cases,axiom,
! [P: produc1417387325239031004te_int > $o,P2: produc1417387325239031004te_int] :
( ! [A4: int,B3: state_int3] : ( P @ ( produc923477050739105812te_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_210_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A4: int,B3: int] : ( P @ ( product_Pair_int_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_211_prod__cases,axiom,
! [P: produc5834231552977413017st_int > $o,P2: produc5834231552977413017st_int] :
( ! [A4: int > int > $o,B3: list_int] : ( P @ ( produc8618682346314911123st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_212_prod__cases,axiom,
! [P: produc7589950997499123219tate_a > $o,P2: produc7589950997499123219tate_a] :
( ! [A4: a,B3: state_a2] : ( P @ ( produc1224139502141355779tate_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_213_prod__cases,axiom,
! [P: produc6972303929186420058tate_a > $o,P2: produc6972303929186420058tate_a] :
( ! [A4: a,B3: state_a3] : ( P @ ( produc8641956578966763338tate_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_214_surj__pair,axiom,
! [P2: produc1417387325239031004te_int] :
? [X5: int,Y4: state_int3] :
( P2
= ( produc923477050739105812te_int @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_215_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X5: int,Y4: int] :
( P2
= ( product_Pair_int_int @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_216_surj__pair,axiom,
! [P2: produc5834231552977413017st_int] :
? [X5: int > int > $o,Y4: list_int] :
( P2
= ( produc8618682346314911123st_int @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_217_surj__pair,axiom,
! [P2: produc7589950997499123219tate_a] :
? [X5: a,Y4: state_a2] :
( P2
= ( produc1224139502141355779tate_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_218_surj__pair,axiom,
! [P2: produc6972303929186420058tate_a] :
? [X5: a,Y4: state_a3] :
( P2
= ( produc8641956578966763338tate_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_219_old_Oprod_Oexhaust,axiom,
! [Y: produc1417387325239031004te_int] :
~ ! [A4: int,B3: state_int3] :
( Y
!= ( produc923477050739105812te_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_220_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_221_old_Oprod_Oexhaust,axiom,
! [Y: produc5834231552977413017st_int] :
~ ! [A4: int > int > $o,B3: list_int] :
( Y
!= ( produc8618682346314911123st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_222_old_Oprod_Oexhaust,axiom,
! [Y: produc7589950997499123219tate_a] :
~ ! [A4: a,B3: state_a2] :
( Y
!= ( produc1224139502141355779tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_223_old_Oprod_Oexhaust,axiom,
! [Y: produc6972303929186420058tate_a] :
~ ! [A4: a,B3: state_a3] :
( Y
!= ( produc8641956578966763338tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_224_step__states_Osimps_I3_J,axiom,
! [Dir: direction,V: state_int,Right: state_int2] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ ( common_int @ V ) @ Right ) )
= ( states_int2 @ Dir @ ( type_s5405555405925207448te_int @ ( common_int @ V ) ) @ ( type_s3231261300627247263te_int @ Right ) ) ) ).
% step_states.simps(3)
thf(fact_225_step__states_Osimps_I3_J,axiom,
! [Dir: direction,V: state_a,Right: state_a2] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( common_a @ V ) @ Right ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ ( common_a @ V ) ) @ ( type_s3703408523585882337tate_a @ Right ) ) ) ).
% step_states.simps(3)
thf(fact_226_Big__Proof_Osize__size__new,axiom,
! [Big: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s92858266581276998te_int @ Big ) ) ) ) ).
% Big_Proof.size_size_new
thf(fact_227_Big__Proof_Osize__size__new,axiom,
! [Big: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% Big_Proof.size_size_new
thf(fact_228_Small__Proof_Osize__size__new,axiom,
! [Small: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s2537683556491027789te_int @ Small ) ) ) ) ).
% Small_Proof.size_size_new
thf(fact_229_Small__Proof_Osize__size__new,axiom,
! [Small: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_size_new
thf(fact_230_Small__Proof_Opop__list__current,axiom,
! [Small: state_list_int2,X: list_int,Small3: state_list_int2] :
( ( type_i6069260309285853475st_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7145721959145893242st_int @ Small ) )
=> ( ( ( pop_list_int2 @ Small )
= ( produc2109666439049285947st_int @ X @ Small3 ) )
=> ( ( cons_list_int @ X @ ( small_6416132737660784207st_int @ Small3 ) )
= ( small_6416132737660784207st_int @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_231_Small__Proof_Opop__list__current,axiom,
! [Small: state_int2,X: int,Small3: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( cons_int @ X @ ( small_1709254455575119551nt_int @ Small3 ) )
= ( small_1709254455575119551nt_int @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_232_Small__Proof_Opop__list__current,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( cons_a @ X @ ( small_list_current_a @ Small3 ) )
= ( small_list_current_a @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_233_Big__Proof_Opop__list__current,axiom,
! [Big: state_list_int3,X: list_int,Big3: state_list_int3] :
( ( type_i6241818230346301340st_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s4372587533717917555st_int @ Big ) )
=> ( ( ( pop_list_int @ Big )
= ( produc1010881625139315636st_int @ X @ Big3 ) )
=> ( ( cons_list_int @ X @ ( big_li2122662369155428040st_int @ Big3 ) )
= ( big_li2122662369155428040st_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_234_Big__Proof_Opop__list__current,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( cons_int @ X @ ( big_list_current_int @ Big3 ) )
= ( big_list_current_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_235_Big__Proof_Opop__list__current,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( cons_a @ X @ ( big_list_current_a @ Big3 ) )
= ( big_list_current_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_236_Big__Proof_Opop__list,axiom,
! [Big: state_list_int3,X: list_int,Big3: state_list_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s4372587533717917555st_int @ Big ) )
=> ( ( type_i6241818230346301340st_int @ Big )
=> ( ( ( pop_list_int @ Big )
= ( produc1010881625139315636st_int @ X @ Big3 ) )
=> ( ( cons_list_int @ X @ ( big_list_list_int @ Big3 ) )
= ( big_list_list_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_237_Big__Proof_Opop__list,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( type_i293623501042546956te_int @ Big )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( cons_int @ X @ ( big_list_int @ Big3 ) )
= ( big_list_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_238_Big__Proof_Opop__list,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( cons_a @ X @ ( big_list_a @ Big3 ) )
= ( big_list_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_239_invars__pop__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Small3: state_int2] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( type_i293623501042546956te_int @ Big )
& ( type_i606337318274449939te_int @ Small3 ) ) ) ) ) ).
% invars_pop_small
thf(fact_240_invars__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small3: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
& ( type_i464410347872898157tate_a @ Small3 ) ) ) ) ) ).
% invars_pop_small
thf(fact_241_invars__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( type_i293623501042546956te_int @ Big3 )
& ( type_i606337318274449939te_int @ Small ) ) ) ) ) ).
% invars_pop_big
thf(fact_242_invars__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( type_i6304938058965754292tate_a @ Big3 )
& ( type_i464410347872898157tate_a @ Small ) ) ) ) ) ).
% invars_pop_big
thf(fact_243_Small__Proof_Osize__new__pop,axiom,
! [Small: state_int2,X: int,Small3: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s2537683556491027789te_int @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( suc @ ( type_s2537683556491027789te_int @ Small3 ) )
= ( type_s2537683556491027789te_int @ Small ) ) ) ) ) ).
% Small_Proof.size_new_pop
thf(fact_244_Small__Proof_Osize__new__pop,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( suc @ ( type_s6404775287138157491tate_a @ Small3 ) )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% Small_Proof.size_new_pop
thf(fact_245_Small__Proof_Osize__pop,axiom,
! [Small: state_int2,X: int,Small3: state_int2] :
( ( type_i606337318274449939te_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( suc @ ( size_size_state_int3 @ Small3 ) )
= ( size_size_state_int3 @ Small ) ) ) ) ) ).
% Small_Proof.size_pop
thf(fact_246_Small__Proof_Osize__pop,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( suc @ ( size_size_state_a3 @ Small3 ) )
= ( size_size_state_a3 @ Small ) ) ) ) ) ).
% Small_Proof.size_pop
thf(fact_247_Big__Proof_Osize__new__pop,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s92858266581276998te_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( suc @ ( type_s92858266581276998te_int @ Big3 ) )
= ( type_s92858266581276998te_int @ Big ) ) ) ) ) ).
% Big_Proof.size_new_pop
thf(fact_248_Big__Proof_Osize__new__pop,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( suc @ ( type_s6530235180886170618tate_a @ Big3 ) )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% Big_Proof.size_new_pop
thf(fact_249_Big__Proof_Osize__pop,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( suc @ ( size_size_state_int @ Big3 ) )
= ( size_size_state_int @ Big ) ) ) ) ) ).
% Big_Proof.size_pop
thf(fact_250_Big__Proof_Osize__pop,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( suc @ ( size_size_state_a @ Big3 ) )
= ( size_size_state_a @ Big ) ) ) ) ) ).
% Big_Proof.size_pop
thf(fact_251_Big__Proof_Oinvar__pop,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( type_i293623501042546956te_int @ Big )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( type_i293623501042546956te_int @ Big3 ) ) ) ) ).
% Big_Proof.invar_pop
thf(fact_252_Big__Proof_Oinvar__pop,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( type_i6304938058965754292tate_a @ Big3 ) ) ) ) ).
% Big_Proof.invar_pop
thf(fact_253_Small__Proof_Oinvar__pop,axiom,
! [Small: state_int2,X: int,Small3: state_int2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( type_i606337318274449939te_int @ Small )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( type_i606337318274449939te_int @ Small3 ) ) ) ) ).
% Small_Proof.invar_pop
thf(fact_254_Small__Proof_Oinvar__pop,axiom,
! [Small: state_a2,X: a,Small3: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( type_i464410347872898157tate_a @ Small3 ) ) ) ) ).
% Small_Proof.invar_pop
thf(fact_255_invar__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big3 @ Small ) ) ) ) ) ).
% invar_pop_big
thf(fact_256_invar__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big3 @ Small ) ) ) ) ) ).
% invar_pop_big
thf(fact_257_invar__pop__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Small3: state_int2] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small3 ) ) ) ) ) ).
% invar_pop_small
thf(fact_258_invar__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small3: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small3 ) ) ) ) ) ).
% invar_pop_small
thf(fact_259_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,X: list_int,Big3: state_list_int3] :
( ( type_i3791846131947626761st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s4372587533717917555st_int @ Big ) )
=> ( ( ( pop_list_int @ Big )
= ( produc1010881625139315636st_int @ X @ Big3 ) )
=> ( ( cons_list_int @ X @ ( states2742979599107686674st_int @ ( states_list_int2 @ Dir @ Big3 @ Small ) ) )
= ( states2742979599107686674st_int @ ( states_list_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_260_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( cons_int @ X @ ( states8405765007952574210st_int @ ( states_int2 @ Dir @ Big3 @ Small ) ) )
= ( states8405765007952574210st_int @ ( states_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_261_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( cons_a @ X @ ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_262_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,X: list_int,Big3: state_list_int3] :
( ( type_i3791846131947626761st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s4372587533717917555st_int @ Big ) )
=> ( ( ( pop_list_int @ Big )
= ( produc1010881625139315636st_int @ X @ Big3 ) )
=> ( ( cons_list_int @ X @ ( states3294962311195589793st_int @ ( states_list_int2 @ Dir @ Big3 @ Small ) ) )
= ( states3294962311195589793st_int @ ( states_list_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_263_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( cons_int @ X @ ( states9070087747003631121st_int @ ( states_int2 @ Dir @ Big3 @ Small ) ) )
= ( states9070087747003631121st_int @ ( states_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_264_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( cons_a @ X @ ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) ) )
= ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_265_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,X: list_int,Small3: state_list_int2] :
( ( type_i3791846131947626761st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7145721959145893242st_int @ Small ) )
=> ( ( ( pop_list_int2 @ Small )
= ( produc2109666439049285947st_int @ X @ Small3 ) )
=> ( ( cons_list_int @ X @ ( states3081394979590915659st_int @ ( states_list_int2 @ Dir @ Big @ Small3 ) ) )
= ( states3081394979590915659st_int @ ( states_list_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_266_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Small3: state_int2] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( cons_int @ X @ ( states2953871243264257211st_int @ ( states_int2 @ Dir @ Big @ Small3 ) ) )
= ( states2953871243264257211st_int @ ( states_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_267_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small3: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( cons_a @ X @ ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small3 ) ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_268_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_list_int3,Small: state_list_int2,X: list_int,Small3: state_list_int2] :
( ( type_i3791846131947626761st_int @ ( states_list_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7145721959145893242st_int @ Small ) )
=> ( ( ( pop_list_int2 @ Small )
= ( produc2109666439049285947st_int @ X @ Small3 ) )
=> ( ( cons_list_int @ X @ ( states3243460551303850586st_int @ ( states_list_int2 @ Dir @ Big @ Small3 ) ) )
= ( states3243460551303850586st_int @ ( states_list_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_269_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Small3: state_int2] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( cons_int @ X @ ( states4273554455855080522st_int @ ( states_int2 @ Dir @ Big @ Small3 ) ) )
= ( states4273554455855080522st_int @ ( states_int2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_270_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small3: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( cons_a @ X @ ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small3 ) ) )
= ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_271_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_272_list_Oinject,axiom,
! [X21: list_int,X22: list_list_int,Y21: list_int,Y22: list_list_int] :
( ( ( cons_list_int @ X21 @ X22 )
= ( cons_list_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_273_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_274_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_275_Big__Proof_Oremaining__steps__pop,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( type_i293623501042546956te_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ord_less_eq_nat @ ( type_r7114992325145952472te_int @ Big3 ) @ ( type_r7114992325145952472te_int @ Big ) ) ) ) ) ).
% Big_Proof.remaining_steps_pop
thf(fact_276_Big__Proof_Oremaining__steps__pop,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ord_less_eq_nat @ ( type_r2494999336194962664tate_a @ Big3 ) @ ( type_r2494999336194962664tate_a @ Big ) ) ) ) ) ).
% Big_Proof.remaining_steps_pop
thf(fact_277_Small__Aux_Osize__state_Osimps_I1_J,axiom,
! [State2: state_int] :
( ( size_size_state_int3 @ ( common_int2 @ State2 ) )
= ( size_size_state_int2 @ State2 ) ) ).
% Small_Aux.size_state.simps(1)
thf(fact_278_Small__Aux_Osize__state_Osimps_I1_J,axiom,
! [State2: state_a] :
( ( size_size_state_a3 @ ( common_a2 @ State2 ) )
= ( size_size_state_a2 @ State2 ) ) ).
% Small_Aux.size_state.simps(1)
thf(fact_279_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_280_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_281_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_282_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_283_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_284_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_285_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_286_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_287_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_288_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_289_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_290_impossible__Cons,axiom,
! [Xs: list_list_int,Ys: list_list_int,X: list_int] :
( ( ord_less_eq_nat @ ( size_s533118279054570080st_int @ Xs ) @ ( size_s533118279054570080st_int @ Ys ) )
=> ( Xs
!= ( cons_list_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_291_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_292_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_293_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_294_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_295_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_296_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_297_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_298_list__small__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2] :
( ( ( states4273554455855080522st_int @ ( states_int2 @ Dir @ Big @ Small ) )
= ( states2953871243264257211st_int @ ( states_int2 @ Dir @ Big @ Small ) ) )
= ( ( states9070087747003631121st_int @ ( states_int2 @ Dir @ Big @ Small ) )
= ( states8405765007952574210st_int @ ( states_int2 @ Dir @ Big @ Small ) ) ) ) ).
% list_small_big
thf(fact_299_list__small__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2] :
( ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
= ( ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ).
% list_small_big
thf(fact_300_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_301_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_302_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_303_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_304_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_305_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z: nat] :
( ( R @ X5 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X5 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_306_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_307_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_308_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_309_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_310_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_311_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_312_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_313_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_314_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_315_invar__list__big__first,axiom,
! [States: states_int] :
( ( type_i6677890655034438905es_int @ States )
=> ( ( states9070087747003631121st_int @ States )
= ( states8405765007952574210st_int @ States ) ) ) ).
% invar_list_big_first
thf(fact_316_invar__list__big__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1888450819780863577irst_a @ States )
= ( states7295096810965389224irst_a @ States ) ) ) ).
% invar_list_big_first
thf(fact_317_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_318_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_319_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_320_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_321_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_322_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_323_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_324_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s533118279054570080st_int @ Xs ) )
= ( ? [X4: list_int,Ys2: list_list_int] :
( ( Xs
= ( cons_list_int @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s533118279054570080st_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_325_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_326_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_327_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_328_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_329_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_330_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_331_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_332_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_333_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_334_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_335_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_336_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_337_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_338_not__Cons__self2,axiom,
! [X: list_int,Xs: list_list_int] :
( ( cons_list_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_339_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_340_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys3: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys3 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_341_lists__small__first__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( states4273554455855080522st_int @ ( states_int2 @ Dir @ Big3 @ Small ) )
= ( states2953871243264257211st_int @ ( states_int2 @ Dir @ Big3 @ Small ) ) ) ) ) ) ).
% lists_small_first_pop_big
thf(fact_342_lists__small__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) ) ) ) ) ) ).
% lists_small_first_pop_big
thf(fact_343_lists__small__first__pop__small,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Small3: state_int2] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int3 @ Small ) )
=> ( ( ( pop_int2 @ Small )
= ( produc5352538583180469531te_int @ X @ Small3 ) )
=> ( ( states4273554455855080522st_int @ ( states_int2 @ Dir @ Big @ Small3 ) )
= ( states2953871243264257211st_int @ ( states_int2 @ Dir @ Big @ Small3 ) ) ) ) ) ) ).
% lists_small_first_pop_small
thf(fact_344_lists__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small3: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small3 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small3 ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small3 ) ) ) ) ) ) ).
% lists_small_first_pop_small
thf(fact_345_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_346_length__Suc__conv,axiom,
! [Xs: list_list_int,N: nat] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: list_int,Ys2: list_list_int] :
( ( Xs
= ( cons_list_int @ Y6 @ Ys2 ) )
& ( ( size_s533118279054570080st_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_347_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_348_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y6: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_349_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_int] :
( ( ( suc @ N )
= ( size_s533118279054570080st_int @ Xs ) )
= ( ? [Y6: list_int,Ys2: list_list_int] :
( ( Xs
= ( cons_list_int @ Y6 @ Ys2 ) )
& ( ( size_s533118279054570080st_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_350_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y6: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_351_lists__big__first__pop__big,axiom,
! [Dir: direction,Big: state_int3,Small: state_int2,X: int,Big3: state_int3] :
( ( type_i6677890655034438905es_int @ ( states_int2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( states9070087747003631121st_int @ ( states_int2 @ Dir @ Big3 @ Small ) )
= ( states8405765007952574210st_int @ ( states_int2 @ Dir @ Big3 @ Small ) ) ) ) ) ) ).
% lists_big_first_pop_big
thf(fact_352_lists__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big3: state_a3] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big3 @ Small ) ) ) ) ) ) ).
% lists_big_first_pop_big
thf(fact_353_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_354_Big__Aux_Osize__state_Osimps_I1_J,axiom,
! [State2: state_int] :
( ( size_size_state_int @ ( common_int @ State2 ) )
= ( size_size_state_int2 @ State2 ) ) ).
% Big_Aux.size_state.simps(1)
thf(fact_355_Big__Aux_Osize__state_Osimps_I1_J,axiom,
! [State2: state_a] :
( ( size_size_state_a @ ( common_a @ State2 ) )
= ( size_size_state_a2 @ State2 ) ) ).
% Big_Aux.size_state.simps(1)
thf(fact_356_Big__Proof_Opop__list__tl,axiom,
! [Big: state_int3,X: int,Big3: state_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( type_i293623501042546956te_int @ Big )
=> ( ( ( pop_int @ Big )
= ( produc923477050739105812te_int @ X @ Big3 ) )
=> ( ( big_list_int @ Big3 )
= ( tl_int @ ( big_list_int @ Big ) ) ) ) ) ) ).
% Big_Proof.pop_list_tl
thf(fact_357_Big__Proof_Opop__list__tl,axiom,
! [Big: state_a3,X: a,Big3: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big3 ) )
=> ( ( big_list_a @ Big3 )
= ( tl_a @ ( big_list_a @ Big ) ) ) ) ) ) ).
% Big_Proof.pop_list_tl
thf(fact_358_Big__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a3] :
( ! [State: state_a] :
( X
!= ( common_a @ State ) )
=> ~ ! [Current2: current_a,Uu: stack_a,Uv2: list_a,Uw: nat] :
( X
!= ( reverse_a @ Current2 @ Uu @ Uv2 @ Uw ) ) ) ).
% Big_Aux.size_new_state.cases
thf(fact_359_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_360_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_361_length__Cons,axiom,
! [X: list_int,Xs: list_list_int] :
( ( size_s533118279054570080st_int @ ( cons_list_int @ X @ Xs ) )
= ( suc @ ( size_s533118279054570080st_int @ Xs ) ) ) ).
% length_Cons
thf(fact_362_length__Cons,axiom,
! [X: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_363_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_364_list_Osel_I3_J,axiom,
! [X21: list_int,X22: list_list_int] :
( ( tl_list_int @ ( cons_list_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_365_list_Osel_I3_J,axiom,
! [X21: int,X22: list_int] :
( ( tl_int @ ( cons_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_366_cons__tl,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_367_cons__tl,axiom,
! [X: list_int,Xs: list_list_int,Ys: list_list_int] :
( ( ( cons_list_int @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_list_int @ Ys ) ) ) ).
% cons_tl
thf(fact_368_cons__tl,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_int @ Ys ) ) ) ).
% cons_tl
thf(fact_369_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_370_Cons__lenlex__iff,axiom,
! [M: list_int,Ms: list_list_int,N: list_int,Ns: list_list_int,R2: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ ( cons_list_int @ M @ Ms ) @ ( cons_list_int @ N @ Ns ) ) @ ( lenlex_list_int @ R2 ) )
= ( ( ord_less_nat @ ( size_s533118279054570080st_int @ Ms ) @ ( size_s533118279054570080st_int @ Ns ) )
| ( ( ( size_s533118279054570080st_int @ Ms )
= ( size_s533118279054570080st_int @ Ns ) )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Ms @ Ns ) @ ( lenlex_list_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_371_Cons__lenlex__iff,axiom,
! [M: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_372_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_373_Cons__in__lex,axiom,
! [X: list_int,Xs: list_list_int,Y: list_int,Ys: list_list_int,R2: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ ( cons_list_int @ X @ Xs ) @ ( cons_list_int @ Y @ Ys ) ) @ ( lex_list_int @ R2 ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ R2 )
& ( ( size_s533118279054570080st_int @ Xs )
= ( size_s533118279054570080st_int @ Ys ) ) )
| ( ( X = Y )
& ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Xs @ Ys ) @ ( lex_list_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_374_Cons__in__lex,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_375_step__states_Ocases,axiom,
! [X: states_int] :
( ! [Dir2: direction,CurrentB: current_int,Big2: stack_int,AuxB: list_int,CurrentS: current_int,Uu: stack_int,AuxS: list_int] :
( X
!= ( states_int2 @ Dir2 @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int @ CurrentS @ Uu @ AuxS ) ) )
=> ( ! [Dir2: direction,V2: current_int,Va2: stack_int,Vb2: list_int,Vd2: nat,Right2: state_int2] :
( X
!= ( states_int2 @ Dir2 @ ( reverse_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ! [Dir2: direction,V2: state_int,Right2: state_int2] :
( X
!= ( states_int2 @ Dir2 @ ( common_int @ V2 ) @ Right2 ) )
=> ( ! [Dir2: direction,Left2: state_int3,V2: current_int,Va2: list_int,Vb2: stack_int,Vc2: list_int,Vd2: nat] :
( X
!= ( states_int2 @ Dir2 @ Left2 @ ( reverse2_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Dir2: direction,Left2: state_int3,V2: state_int] :
( X
!= ( states_int2 @ Dir2 @ Left2 @ ( common_int2 @ V2 ) ) ) ) ) ) ) ).
% step_states.cases
thf(fact_376_step__states_Ocases,axiom,
! [X: states_a] :
( ! [Dir2: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( ! [Dir2: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( X
!= ( states_a2 @ Dir2 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ! [Dir2: direction,V2: state_a,Right2: state_a2] :
( X
!= ( states_a2 @ Dir2 @ ( common_a @ V2 ) @ Right2 ) )
=> ( ! [Dir2: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( X
!= ( states_a2 @ Dir2 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Dir2: direction,Left2: state_a3,V2: state_a] :
( X
!= ( states_a2 @ Dir2 @ Left2 @ ( common_a2 @ V2 ) ) ) ) ) ) ) ).
% step_states.cases
thf(fact_377_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_378_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_379_Small_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,Y11: current_a,Y12: stack_a,Y13: list_a] :
( ( ( reverse1_a @ X11 @ X122 @ X13 )
= ( reverse1_a @ Y11 @ Y12 @ Y13 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 ) ) ) ).
% Small.state.inject(1)
thf(fact_380_Small_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X21: current_a,X22: list_a,X23: stack_a,X24: list_a,X25: nat] :
( ( reverse1_a @ X11 @ X122 @ X13 )
!= ( reverse2_a @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% Small.state.distinct(1)
thf(fact_381_Small_Ostate_Odistinct_I3_J,axiom,
! [X11: current_int,X122: stack_int,X13: list_int,X3: state_int] :
( ( reverse1_int @ X11 @ X122 @ X13 )
!= ( common_int2 @ X3 ) ) ).
% Small.state.distinct(3)
thf(fact_382_Small_Ostate_Odistinct_I3_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X3: state_a] :
( ( reverse1_a @ X11 @ X122 @ X13 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(3)
thf(fact_383_Small__Aux_Osize__new__state_Ocases,axiom,
! [X: state_int2] :
( ! [State: state_int] :
( X
!= ( common_int2 @ State ) )
=> ( ! [Current2: current_int,Uu: list_int,Uv2: stack_int,Uw: list_int,Ux: nat] :
( X
!= ( reverse2_int @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ~ ! [Current2: current_int,Uy: stack_int,Uz: list_int] :
( X
!= ( reverse1_int @ Current2 @ Uy @ Uz ) ) ) ) ).
% Small_Aux.size_new_state.cases
thf(fact_384_Small__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a2] :
( ! [State: state_a] :
( X
!= ( common_a2 @ State ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
!= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( X
!= ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ).
% Small_Aux.size_new_state.cases
thf(fact_385_Small_Ostate_Oexhaust,axiom,
! [Y: state_int2] :
( ! [X112: current_int,X123: stack_int,X132: list_int] :
( Y
!= ( reverse1_int @ X112 @ X123 @ X132 ) )
=> ( ! [X212: current_int,X222: list_int,X232: stack_int,X242: list_int,X252: nat] :
( Y
!= ( reverse2_int @ X212 @ X222 @ X232 @ X242 @ X252 ) )
=> ~ ! [X32: state_int] :
( Y
!= ( common_int2 @ X32 ) ) ) ) ).
% Small.state.exhaust
thf(fact_386_Small_Ostate_Oexhaust,axiom,
! [Y: state_a2] :
( ! [X112: current_a,X123: stack_a,X132: list_a] :
( Y
!= ( reverse1_a @ X112 @ X123 @ X132 ) )
=> ( ! [X212: current_a,X222: list_a,X232: stack_a,X242: list_a,X252: nat] :
( Y
!= ( reverse2_a @ X212 @ X222 @ X232 @ X242 @ X252 ) )
=> ~ ! [X32: state_a] :
( Y
!= ( common_a2 @ X32 ) ) ) ) ).
% Small.state.exhaust
thf(fact_387_Small_Ostep__state_Ocases,axiom,
! [X: state_int2] :
( ! [State: state_int] :
( X
!= ( common_int2 @ State ) )
=> ( ! [Current2: current_int,Small2: stack_int,AuxS: list_int] :
( X
!= ( reverse1_int @ Current2 @ Small2 @ AuxS ) )
=> ~ ! [Current2: current_int,AuxS: list_int,Big2: stack_int,NewS: list_int,Count2: nat] :
( X
!= ( reverse2_int @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ).
% Small.step_state.cases
thf(fact_388_Small_Ostep__state_Ocases,axiom,
! [X: state_a2] :
( ! [State: state_a] :
( X
!= ( common_a2 @ State ) )
=> ( ! [Current2: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( reverse1_a @ Current2 @ Small2 @ AuxS ) )
=> ~ ! [Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ).
% Small.step_state.cases
thf(fact_389_lenlex__irreflexive,axiom,
! [R2: set_Pr765067013931698361st_int,Xs: list_list_int] :
( ! [X5: list_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X5 @ X5 ) @ R2 )
=> ~ ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Xs @ Xs ) @ ( lenlex_list_int @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_390_lenlex__irreflexive,axiom,
! [R2: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X5: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ X5 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_391_Small_Opush_Ocases,axiom,
! [X: produc957127447925188579te_int] :
( ! [X5: int,State: state_int] :
( X
!= ( produc5352538583180469531te_int @ X5 @ ( common_int2 @ State ) ) )
=> ( ! [X5: int,Current2: current_int,Small2: stack_int,AuxS: list_int] :
( X
!= ( produc5352538583180469531te_int @ X5 @ ( reverse1_int @ Current2 @ Small2 @ AuxS ) ) )
=> ~ ! [X5: int,Current2: current_int,AuxS: list_int,Big2: stack_int,NewS: list_int,Count2: nat] :
( X
!= ( produc5352538583180469531te_int @ X5 @ ( reverse2_int @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ).
% Small.push.cases
thf(fact_392_Small_Opush_Ocases,axiom,
! [X: produc7589950997499123219tate_a] :
( ! [X5: a,State: state_a] :
( X
!= ( produc1224139502141355779tate_a @ X5 @ ( common_a2 @ State ) ) )
=> ( ! [X5: a,Current2: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( produc1224139502141355779tate_a @ X5 @ ( reverse1_a @ Current2 @ Small2 @ AuxS ) ) )
=> ~ ! [X5: a,Current2: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( produc1224139502141355779tate_a @ X5 @ ( reverse2_a @ Current2 @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ).
% Small.push.cases
thf(fact_393_lenlex__length,axiom,
! [Ms: list_int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) ) ) ).
% lenlex_length
thf(fact_394_States__Aux_Olists_Ocases,axiom,
! [X: states_int] :
( ! [Uu: direction,CurrentB: current_int,Big2: stack_int,AuxB: list_int,Count2: nat,CurrentS: current_int,Small2: stack_int,AuxS: list_int] :
( X
!= ( states_int2 @ Uu @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ Count2 ) @ ( reverse1_int @ CurrentS @ Small2 @ AuxS ) ) )
=> ( ! [Uv2: direction,V2: state_int,Small2: state_int2] :
( X
!= ( states_int2 @ Uv2 @ ( common_int @ V2 ) @ Small2 ) )
=> ( ! [Uv2: direction,Big2: state_int3,V2: current_int,Va2: list_int,Vb2: stack_int,Vc2: list_int,Vd2: nat] :
( X
!= ( states_int2 @ Uv2 @ Big2 @ ( reverse2_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Uv2: direction,Big2: state_int3,V2: state_int] :
( X
!= ( states_int2 @ Uv2 @ Big2 @ ( common_int2 @ V2 ) ) ) ) ) ) ).
% States_Aux.lists.cases
thf(fact_395_States__Aux_Olists_Ocases,axiom,
! [X: states_a] :
( ! [Uu: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,Count2: nat,CurrentS: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Uu @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ Count2 ) @ ( reverse1_a @ CurrentS @ Small2 @ AuxS ) ) )
=> ( ! [Uv2: direction,V2: state_a,Small2: state_a2] :
( X
!= ( states_a2 @ Uv2 @ ( common_a @ V2 ) @ Small2 ) )
=> ( ! [Uv2: direction,Big2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( X
!= ( states_a2 @ Uv2 @ Big2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Uv2: direction,Big2: state_a3,V2: state_a] :
( X
!= ( states_a2 @ Uv2 @ Big2 @ ( common_a2 @ V2 ) ) ) ) ) ) ).
% States_Aux.lists.cases
thf(fact_396_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_397_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_398_step__states_Oelims,axiom,
! [X: states_list_int,Y: states_list_int] :
( ( ( type_s976745419985777557st_int @ X )
= Y )
=> ( ! [Dir2: direction,CurrentB: current_list_int,Big2: stack_list_int,AuxB: list_list_int,CurrentS: current_list_int,Uu: stack_list_int,AuxS: list_list_int] :
( ( X
= ( states_list_int2 @ Dir2 @ ( reverse_list_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_list_int @ CurrentS @ Uu @ AuxS ) ) )
=> ( Y
!= ( states_list_int2 @ Dir2 @ ( type_s2155344414434909736st_int @ ( reverse_list_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_list_int @ CurrentS @ AuxS @ Big2 @ nil_list_int @ zero_zero_nat ) ) ) )
=> ( ! [Dir2: direction,V2: current_list_int,Va2: stack_list_int,Vb2: list_list_int,Vd2: nat,Right2: state_list_int2] :
( ( X
= ( states_list_int2 @ Dir2 @ ( reverse_list_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( Y
!= ( states_list_int2 @ Dir2 @ ( type_s2155344414434909736st_int @ ( reverse_list_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s4628147265906176431st_int @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_list_int,Right2: state_list_int2] :
( ( X
= ( states_list_int2 @ Dir2 @ ( common_list_int @ V2 ) @ Right2 ) )
=> ( Y
!= ( states_list_int2 @ Dir2 @ ( type_s2155344414434909736st_int @ ( common_list_int @ V2 ) ) @ ( type_s4628147265906176431st_int @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_list_int3,V2: current_list_int,Va2: list_list_int,Vb2: stack_list_int,Vc2: list_list_int,Vd2: nat] :
( ( X
= ( states_list_int2 @ Dir2 @ Left2 @ ( reverse2_list_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( states_list_int2 @ Dir2 @ ( type_s2155344414434909736st_int @ Left2 ) @ ( type_s4628147265906176431st_int @ ( reverse2_list_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_list_int3,V2: state_list_int] :
( ( X
= ( states_list_int2 @ Dir2 @ Left2 @ ( common_list_int2 @ V2 ) ) )
=> ( Y
!= ( states_list_int2 @ Dir2 @ ( type_s2155344414434909736st_int @ Left2 ) @ ( type_s4628147265906176431st_int @ ( common_list_int2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% step_states.elims
thf(fact_399_step__states_Oelims,axiom,
! [X: states_int,Y: states_int] :
( ( ( type_s271243990432398725es_int @ X )
= Y )
=> ( ! [Dir2: direction,CurrentB: current_int,Big2: stack_int,AuxB: list_int,CurrentS: current_int,Uu: stack_int,AuxS: list_int] :
( ( X
= ( states_int2 @ Dir2 @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int @ CurrentS @ Uu @ AuxS ) ) )
=> ( Y
!= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_int @ CurrentS @ AuxS @ Big2 @ nil_int @ zero_zero_nat ) ) ) )
=> ( ! [Dir2: direction,V2: current_int,Va2: stack_int,Vb2: list_int,Vd2: nat,Right2: state_int2] :
( ( X
= ( states_int2 @ Dir2 @ ( reverse_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( Y
!= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ ( reverse_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3231261300627247263te_int @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_int,Right2: state_int2] :
( ( X
= ( states_int2 @ Dir2 @ ( common_int @ V2 ) @ Right2 ) )
=> ( Y
!= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ ( common_int @ V2 ) ) @ ( type_s3231261300627247263te_int @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_int3,V2: current_int,Va2: list_int,Vb2: stack_int,Vc2: list_int,Vd2: nat] :
( ( X
= ( states_int2 @ Dir2 @ Left2 @ ( reverse2_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ Left2 ) @ ( type_s3231261300627247263te_int @ ( reverse2_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_int3,V2: state_int] :
( ( X
= ( states_int2 @ Dir2 @ Left2 @ ( common_int2 @ V2 ) ) )
=> ( Y
!= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ Left2 ) @ ( type_s3231261300627247263te_int @ ( common_int2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% step_states.elims
thf(fact_400_step__states_Oelims,axiom,
! [X: states_a,Y: states_a] :
( ( ( type_s4923920245906622843ates_a @ X )
= Y )
=> ( ! [Dir2: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( Y
!= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS @ AuxS @ Big2 @ nil_a @ zero_zero_nat ) ) ) )
=> ( ! [Dir2: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( Y
!= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_a,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir2 @ ( common_a @ V2 ) @ Right2 ) )
=> ( Y
!= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( common_a @ V2 ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Dir2 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Dir2 @ Left2 @ ( common_a2 @ V2 ) ) )
=> ( Y
!= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% step_states.elims
thf(fact_401_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_list_int,Big: stack_list_int,AuxB2: list_list_int,CurrentS2: current_list_int,Uu2: stack_list_int,AuxS2: list_list_int] :
( ( type_s976745419985777557st_int @ ( states_list_int2 @ Dir @ ( reverse_list_int @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_list_int @ CurrentS2 @ Uu2 @ AuxS2 ) ) )
= ( states_list_int2 @ Dir @ ( type_s2155344414434909736st_int @ ( reverse_list_int @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) ) @ ( reverse2_list_int @ CurrentS2 @ AuxS2 @ Big @ nil_list_int @ zero_zero_nat ) ) ) ).
% step_states.simps(1)
thf(fact_402_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_int,Big: stack_int,AuxB2: list_int,CurrentS2: current_int,Uu2: stack_int,AuxS2: list_int] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ ( reverse_int @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_int @ CurrentS2 @ Uu2 @ AuxS2 ) ) )
= ( states_int2 @ Dir @ ( type_s5405555405925207448te_int @ ( reverse_int @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) ) @ ( reverse2_int @ CurrentS2 @ AuxS2 @ Big @ nil_int @ zero_zero_nat ) ) ) ).
% step_states.simps(1)
thf(fact_403_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_a,Big: stack_a,AuxB2: list_a,CurrentS2: current_a,Uu2: stack_a,AuxS2: list_a] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_a @ CurrentS2 @ Uu2 @ AuxS2 ) ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS2 @ AuxS2 @ Big @ nil_a @ zero_zero_nat ) ) ) ).
% step_states.simps(1)
thf(fact_404_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_405_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_406_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_407_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_408_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_409_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_410_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_411_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_412_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A3 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_413_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A3: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A3 ) )
= ( ord_less_int @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_414_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_415_length__0__conv,axiom,
! [Xs: list_list_int] :
( ( ( size_s533118279054570080st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_int ) ) ).
% length_0_conv
thf(fact_416_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_417_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_418_Nil__lenlex__iff1,axiom,
! [Ns: list_list_int,R2: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ nil_list_int @ Ns ) @ ( lenlex_list_int @ R2 ) )
= ( Ns != nil_list_int ) ) ).
% Nil_lenlex_iff1
thf(fact_419_Nil__lenlex__iff1,axiom,
! [Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ns ) @ ( lenlex_int @ R2 ) )
= ( Ns != nil_int ) ) ).
% Nil_lenlex_iff1
thf(fact_420_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_421_length__greater__0__conv,axiom,
! [Xs: list_list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s533118279054570080st_int @ Xs ) )
= ( Xs != nil_list_int ) ) ).
% length_greater_0_conv
thf(fact_422_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_423_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X5: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_424_transpose_Ocases,axiom,
! [X: list_list_list_int] :
( ( X != nil_list_list_int )
=> ( ! [Xss: list_list_list_int] :
( X
!= ( cons_list_list_int @ nil_list_int @ Xss ) )
=> ~ ! [X5: list_int,Xs2: list_list_int,Xss: list_list_list_int] :
( X
!= ( cons_list_list_int @ ( cons_list_int @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_425_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X5: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_426_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X5: a,Ys4: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_427_sorted__wrt_Ocases,axiom,
! [X: produc4162785005915746057st_int] :
( ! [P3: list_int > list_int > $o] :
( X
!= ( produc1538682255923323267st_int @ P3 @ nil_list_int ) )
=> ~ ! [P3: list_int > list_int > $o,X5: list_int,Ys4: list_list_int] :
( X
!= ( produc1538682255923323267st_int @ P3 @ ( cons_list_int @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_428_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X5: int,Ys4: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_429_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X5: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X5 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X5: a,Y4: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X5 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_430_successively_Ocases,axiom,
! [X: produc4162785005915746057st_int] :
( ! [P3: list_int > list_int > $o] :
( X
!= ( produc1538682255923323267st_int @ P3 @ nil_list_int ) )
=> ( ! [P3: list_int > list_int > $o,X5: list_int] :
( X
!= ( produc1538682255923323267st_int @ P3 @ ( cons_list_int @ X5 @ nil_list_int ) ) )
=> ~ ! [P3: list_int > list_int > $o,X5: list_int,Y4: list_int,Xs2: list_list_int] :
( X
!= ( produc1538682255923323267st_int @ P3 @ ( cons_list_int @ X5 @ ( cons_list_int @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_431_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ( ! [P3: int > int > $o,X5: int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X5 @ nil_int ) ) )
=> ~ ! [P3: int > int > $o,X5: int,Y4: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X5 @ ( cons_int @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_432_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X5: a] : ( P @ ( cons_a @ X5 @ nil_a ) )
=> ( ! [X5: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_433_list__nonempty__induct,axiom,
! [Xs: list_list_int,P: list_list_int > $o] :
( ( Xs != nil_list_int )
=> ( ! [X5: list_int] : ( P @ ( cons_list_int @ X5 @ nil_list_int ) )
=> ( ! [X5: list_int,Xs2: list_list_int] :
( ( Xs2 != nil_list_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_434_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X5: int] : ( P @ ( cons_int @ X5 @ nil_int ) )
=> ( ! [X5: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_435_list__induct2_H,axiom,
! [P: list_int > list_a > $o,Xs: list_int,Ys: list_a] :
( ( P @ nil_int @ nil_a )
=> ( ! [X5: int,Xs2: list_int] : ( P @ ( cons_int @ X5 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P @ nil_int @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X5: int,Xs2: list_int,Y4: a,Ys4: list_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_436_list__induct2_H,axiom,
! [P: list_int > list_list_int > $o,Xs: list_int,Ys: list_list_int] :
( ( P @ nil_int @ nil_list_int )
=> ( ! [X5: int,Xs2: list_int] : ( P @ ( cons_int @ X5 @ Xs2 ) @ nil_list_int )
=> ( ! [Y4: list_int,Ys4: list_list_int] : ( P @ nil_int @ ( cons_list_int @ Y4 @ Ys4 ) )
=> ( ! [X5: int,Xs2: list_int,Y4: list_int,Ys4: list_list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_437_list__induct2_H,axiom,
! [P: list_a > list_int > $o,Xs: list_a,Ys: list_int] :
( ( P @ nil_a @ nil_int )
=> ( ! [X5: a,Xs2: list_a] : ( P @ ( cons_a @ X5 @ Xs2 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_a @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X5: a,Xs2: list_a,Y4: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_438_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X5: a,Xs2: list_a] : ( P @ ( cons_a @ X5 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys4: list_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_439_list__induct2_H,axiom,
! [P: list_a > list_list_int > $o,Xs: list_a,Ys: list_list_int] :
( ( P @ nil_a @ nil_list_int )
=> ( ! [X5: a,Xs2: list_a] : ( P @ ( cons_a @ X5 @ Xs2 ) @ nil_list_int )
=> ( ! [Y4: list_int,Ys4: list_list_int] : ( P @ nil_a @ ( cons_list_int @ Y4 @ Ys4 ) )
=> ( ! [X5: a,Xs2: list_a,Y4: list_int,Ys4: list_list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_440_list__induct2_H,axiom,
! [P: list_list_int > list_int > $o,Xs: list_list_int,Ys: list_int] :
( ( P @ nil_list_int @ nil_int )
=> ( ! [X5: list_int,Xs2: list_list_int] : ( P @ ( cons_list_int @ X5 @ Xs2 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_list_int @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_441_list__induct2_H,axiom,
! [P: list_list_int > list_a > $o,Xs: list_list_int,Ys: list_a] :
( ( P @ nil_list_int @ nil_a )
=> ( ! [X5: list_int,Xs2: list_list_int] : ( P @ ( cons_list_int @ X5 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P @ nil_list_int @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: a,Ys4: list_a] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_442_list__induct2_H,axiom,
! [P: list_list_int > list_list_int > $o,Xs: list_list_int,Ys: list_list_int] :
( ( P @ nil_list_int @ nil_list_int )
=> ( ! [X5: list_int,Xs2: list_list_int] : ( P @ ( cons_list_int @ X5 @ Xs2 ) @ nil_list_int )
=> ( ! [Y4: list_int,Ys4: list_list_int] : ( P @ nil_list_int @ ( cons_list_int @ Y4 @ Ys4 ) )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: list_int,Ys4: list_list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_443_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X5: int,Xs2: list_int] : ( P @ ( cons_int @ X5 @ Xs2 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X5: int,Xs2: list_int,Y4: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_444_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y6: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y6 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_445_neq__Nil__conv,axiom,
! [Xs: list_list_int] :
( ( Xs != nil_list_int )
= ( ? [Y6: list_int,Ys2: list_list_int] :
( Xs
= ( cons_list_int @ Y6 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_446_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y6: int,Ys2: list_int] :
( Xs
= ( cons_int @ Y6 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_447_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X5: a] :
( X
!= ( cons_a @ X5 @ nil_a ) )
=> ~ ! [X5: a,Y4: a,Xs2: list_a] :
( X
!= ( cons_a @ X5 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_448_remdups__adj_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [X5: list_int] :
( X
!= ( cons_list_int @ X5 @ nil_list_int ) )
=> ~ ! [X5: list_int,Y4: list_int,Xs2: list_list_int] :
( X
!= ( cons_list_int @ X5 @ ( cons_list_int @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_449_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X5: int] :
( X
!= ( cons_int @ X5 @ nil_int ) )
=> ~ ! [X5: int,Y4: int,Xs2: list_int] :
( X
!= ( cons_int @ X5 @ ( cons_int @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_450_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X5: int,Xs2: list_int] :
( X
!= ( cons_int @ X5 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_451_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_452_list_Oexhaust,axiom,
! [Y: list_list_int] :
( ( Y != nil_list_int )
=> ~ ! [X212: list_int,X222: list_list_int] :
( Y
!= ( cons_list_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_453_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_454_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_455_list_OdiscI,axiom,
! [List: list_list_int,X21: list_int,X22: list_list_int] :
( ( List
= ( cons_list_int @ X21 @ X22 ) )
=> ( List != nil_list_int ) ) ).
% list.discI
thf(fact_456_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_457_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_458_list_Odistinct_I1_J,axiom,
! [X21: list_int,X22: list_list_int] :
( nil_list_int
!= ( cons_list_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_459_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_460_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_461_list_Osize_I3_J,axiom,
( ( size_s533118279054570080st_int @ nil_list_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_462_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_463_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 )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_464_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_int,P: list_a > list_list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s533118279054570080st_int @ Ys ) )
=> ( ( P @ nil_a @ nil_list_int )
=> ( ! [X5: a,Xs2: list_a,Y4: list_int,Ys4: list_list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s533118279054570080st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_465_list__induct2,axiom,
! [Xs: list_list_int,Ys: list_a,P: list_list_int > list_a > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_int @ nil_a )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: a,Ys4: list_a] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_466_list__induct2,axiom,
! [Xs: list_list_int,Ys: list_list_int,P: list_list_int > list_list_int > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_s533118279054570080st_int @ Ys ) )
=> ( ( P @ nil_list_int @ nil_list_int )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: list_int,Ys4: list_list_int] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_s533118279054570080st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_467_list__induct2,axiom,
! [Xs: list_a,Ys: list_int,P: list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_a @ nil_int )
=> ( ! [X5: a,Xs2: list_a,Y4: int,Ys4: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_468_list__induct2,axiom,
! [Xs: list_list_int,Ys: list_int,P: list_list_int > list_int > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_list_int @ nil_int )
=> ( ! [X5: list_int,Xs2: list_list_int,Y4: int,Ys4: list_int] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_469_list__induct2,axiom,
! [Xs: list_int,Ys: list_a,P: list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_int @ nil_a )
=> ( ! [X5: int,Xs2: list_int,Y4: a,Ys4: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_470_list__induct2,axiom,
! [Xs: list_int,Ys: list_list_int,P: list_int > list_list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s533118279054570080st_int @ Ys ) )
=> ( ( P @ nil_int @ nil_list_int )
=> ( ! [X5: int,Xs2: list_int,Y4: list_int,Ys4: list_list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s533118279054570080st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_list_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_471_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X5: int,Xs2: list_int,Y4: int,Ys4: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_472_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X5: int,Xs2: list_int,Y4: int,Ys4: list_int,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_473_list__induct4,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_int,P: list_int > list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P @ nil_int @ nil_int @ nil_int @ nil_int )
=> ( ! [X5: int,Xs2: list_int,Y4: int,Ys4: list_int,Z: int,Zs2: list_int,W: int,Ws2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_474_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X5: int,Xs2: list_int,Y4: int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_475_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ~ ! [X5: int,Xs2: list_int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_476_Nil__tl,axiom,
! [Xs: list_int] :
( ( nil_int
= ( tl_int @ Xs ) )
= ( ( Xs = nil_int )
| ? [X4: int] :
( Xs
= ( cons_int @ X4 @ nil_int ) ) ) ) ).
% Nil_tl
thf(fact_477_tl__Nil,axiom,
! [Xs: list_int] :
( ( ( tl_int @ Xs )
= nil_int )
= ( ( Xs = nil_int )
| ? [X4: int] :
( Xs
= ( cons_int @ X4 @ nil_int ) ) ) ) ).
% tl_Nil
thf(fact_478_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_479_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_480_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_481_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_482_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_483_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_484_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_485_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_486_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_487_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_488_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_489_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z2 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_490_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_491_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_492_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_493_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_494_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_495_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_496_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_497_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_498_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_499_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_500_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_501_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z2: int] :
! [X5: int] :
( ( ord_less_int @ Z2 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_502_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_503_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_504_size__list,axiom,
! [Big: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_a @ Big )
!= nil_a ) ) ) ).
% size_list
thf(fact_505_Big__Proof_Olist__current__size,axiom,
! [Big: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( big_list_current_a @ Big )
= nil_a )
=> ~ ( type_i6304938058965754292tate_a @ Big ) ) ) ).
% Big_Proof.list_current_size
thf(fact_506_Small__Proof_Olist__current__size,axiom,
! [Small: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( small_list_current_a @ Small )
= nil_a )
=> ~ ( type_i464410347872898157tate_a @ Small ) ) ) ).
% Small_Proof.list_current_size
thf(fact_507_step__states_Opelims,axiom,
! [X: states_a,Y: states_a] :
( ( ( type_s4923920245906622843ates_a @ X )
= Y )
=> ( ( accp_states_a @ step_states_rel_a @ X )
=> ( ! [Dir2: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( ( Y
= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS @ AuxS @ Big2 @ nil_a @ zero_zero_nat ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) ) ) )
=> ( ! [Dir2: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ( Y
= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir2 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_a,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir2 @ ( common_a @ V2 ) @ Right2 ) )
=> ( ( Y
= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ ( common_a @ V2 ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir2 @ ( common_a @ V2 ) @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Dir2 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( ( Y
= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir2 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Dir2 @ Left2 @ ( common_a2 @ V2 ) ) )
=> ( ( Y
= ( states_a2 @ Dir2 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( common_a2 @ V2 ) ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir2 @ Left2 @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% step_states.pelims
thf(fact_508_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_509_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_510_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_511_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_512_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_513_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_514_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_515_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_516_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_517_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_518_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_519_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_520_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_521_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_522_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_523_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_524_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_525_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_526_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_527_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_528_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_529_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_530_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_531_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_532_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_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_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_535_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_536_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_537_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_538_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_539_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_540_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_541_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_542_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_543_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_544_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_545_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_546_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_547_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_548_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_549_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_550_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_551_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_552_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_553_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_554_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_555_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_556_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_557_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_558_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_559_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_560_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_561_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_562_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_563_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_564_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_565_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_566_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_567_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_568_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_569_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_570_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_571_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_572_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_573_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_574_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_575_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_576_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P6: nat > $o] :
? [N4: nat] :
( ( P6 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P6 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_577_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_578_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_579_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_580_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_581_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_582_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_583_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_584_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_585_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_586_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_587_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_588_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_589_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_590_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_591_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_592_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_593_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_594_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_595_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_596_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_597_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_598_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_599_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_600_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_601_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_602_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_603_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_604_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_605_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_606_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_607_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_608_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_609_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_610_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_611_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_612_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_613_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_614_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_615_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_616_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_617_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_618_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_619_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_620_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_621_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_622_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_623_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_624_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_625_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_626_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_627_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_628_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_629_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_630_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_631_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_632_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_633_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_634_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_635_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_636_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_int @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_637_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_638_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_639_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_640_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_641_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_642_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_643_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_644_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_645_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_646_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_647_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_648_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_649_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_650_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_651_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_652_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_653_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_654_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_655_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_656_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X5: int,Y4: int] :
( ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_657_lists__current_Opelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ( ( accp_states_a @ states5251248496104418302_rel_a @ X )
=> ~ ! [Uu: direction,Big2: state_a3,Small2: state_a2] :
( ( X
= ( states_a2 @ Uu @ Big2 @ Small2 ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big2 ) @ ( small_list_current_a @ Small2 ) ) )
=> ~ ( accp_states_a @ states5251248496104418302_rel_a @ ( states_a2 @ Uu @ Big2 @ Small2 ) ) ) ) ) ) ).
% lists_current.pelims
thf(fact_658_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_659_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_660_gen__length__code_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_661_insert__Nil,axiom,
! [X: int] :
( ( insert_int @ X @ nil_int )
= ( cons_int @ X @ nil_int ) ) ).
% insert_Nil
thf(fact_662_remdups__adj__Cons__alt,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ ( tl_int @ ( remdups_adj_int @ ( cons_int @ X @ Xs ) ) ) )
= ( remdups_adj_int @ ( cons_int @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_663_remdups__adj_Osimps_I3_J,axiom,
! [X: int,Y: int,Xs: list_int] :
( ( ( X = Y )
=> ( ( remdups_adj_int @ ( cons_int @ X @ ( cons_int @ Y @ Xs ) ) )
= ( remdups_adj_int @ ( cons_int @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_int @ ( cons_int @ X @ ( cons_int @ Y @ Xs ) ) )
= ( cons_int @ X @ ( remdups_adj_int @ ( cons_int @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_664_remdups__adj_Oelims,axiom,
! [X: list_int,Y: list_int] :
( ( ( remdups_adj_int @ X )
= Y )
=> ( ( ( X = nil_int )
=> ( Y != nil_int ) )
=> ( ! [X5: int] :
( ( X
= ( cons_int @ X5 @ nil_int ) )
=> ( Y
!= ( cons_int @ X5 @ nil_int ) ) )
=> ~ ! [X5: int,Y4: int,Xs2: list_int] :
( ( X
= ( cons_int @ X5 @ ( cons_int @ Y4 @ Xs2 ) ) )
=> ~ ( ( ( X5 = Y4 )
=> ( Y
= ( remdups_adj_int @ ( cons_int @ X5 @ Xs2 ) ) ) )
& ( ( X5 != Y4 )
=> ( Y
= ( cons_int @ X5 @ ( remdups_adj_int @ ( cons_int @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_665_remdups__adj_Osimps_I2_J,axiom,
! [X: int] :
( ( remdups_adj_int @ ( cons_int @ X @ nil_int ) )
= ( cons_int @ X @ nil_int ) ) ).
% remdups_adj.simps(2)
thf(fact_666_nths__singleton,axiom,
! [A2: set_nat,X: int] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= ( cons_int @ X @ nil_int ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= nil_int ) ) ) ).
% nths_singleton
thf(fact_667_listrel_Osimps,axiom,
! [A1: list_int,A22: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R2 ) )
= ( ( ( A1 = nil_int )
& ( A22 = nil_int ) )
| ? [X4: int,Y6: int,Xs3: list_int,Ys2: list_int] :
( ( A1
= ( cons_int @ X4 @ Xs3 ) )
& ( A22
= ( cons_int @ Y6 @ Ys2 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y6 ) @ R2 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys2 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_668_listrel_Ocases,axiom,
! [A1: list_int,A22: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R2 ) )
=> ( ( ( A1 = nil_int )
=> ( A22 != nil_int ) )
=> ~ ! [X5: int,Y4: int,Xs2: list_int] :
( ( A1
= ( cons_int @ X5 @ Xs2 ) )
=> ! [Ys4: list_int] :
( ( A22
= ( cons_int @ Y4 @ Ys4 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys4 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_669_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_670_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_671_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_672_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_673_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_674_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_675_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_676_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_677_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_678_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_679_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_680_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_681_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_682_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_683_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_684_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_685_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_686_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_687_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_688_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_689_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_690_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_691_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_692_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_693_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_694_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_695_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_696_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_697_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_698_listrel_OCons,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_699_listrel__Cons1,axiom,
! [Y: int,Ys: list_int,Xs: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listrel_int_int @ R2 ) )
=> ~ ! [Y4: int,Ys4: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys4 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys4 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_700_listrel__Cons2,axiom,
! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R2 ) )
=> ~ ! [X5: int,Xs2: list_int] :
( ( Xs
= ( cons_int @ X5 @ Xs2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_701_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_702_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_703_lexord__Nil__left,axiom,
! [Y: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Y ) @ ( lexord_int @ R2 ) )
= ( ? [A5: int,X4: list_int] :
( Y
= ( cons_int @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_704_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_705_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_706_lexord__cons__cons,axiom,
! [A: int,X: list_int,B: int,Y: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ A @ X ) @ ( cons_int @ B @ Y ) ) @ ( lexord_int @ R2 ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
| ( ( A = B )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_707_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_708_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_709_nth__equal__first__eq,axiom,
! [X: int,Xs: list_int,N: nat] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_710_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( nth_int @ Xs @ I ) @ ( drop_int @ ( suc @ I ) @ Xs ) )
= ( drop_int @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_711_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_712_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_713_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_714_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_715_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_716_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_717_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_718_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_719_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_720_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_721_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_722_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_723_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_724_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_725_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_726_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_727_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_728_take__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_729_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_730_drop__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_731_not__in__set__insert,axiom,
! [X: int,Xs: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X @ Xs )
= ( cons_int @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_732_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_733_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_734_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_735_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_736_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_737_set__ConsD,axiom,
! [Y: int,X: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_738_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int @ E @ ( set_int2 @ A ) )
=> ( ! [Z22: list_int] :
( A
!= ( cons_int @ E @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_739_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_740_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X22 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_741_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_742_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_743_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_744_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_745_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_746_nth__via__drop,axiom,
! [N: nat,Xs: list_int,Y: int,Ys: list_int] :
( ( ( drop_int @ N @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ( ( nth_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_747_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_748_int__cases,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_749_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_750_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_751_List_Oinsert__def,axiom,
( insert_int
= ( ^ [X4: int,Xs3: list_int] : ( if_list_int @ ( member_int @ X4 @ ( set_int2 @ Xs3 ) ) @ Xs3 @ ( cons_int @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_752_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_753_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_754_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_755_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_756_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_757_the__elem__set,axiom,
! [X: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% the_elem_set
thf(fact_758_id__take__nth__drop,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( Xs
= ( append_int @ ( take_int @ I @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I ) @ ( drop_int @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_759_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( take_int @ ( suc @ I ) @ Xs )
= ( append_int @ ( take_int @ I @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I ) @ nil_int ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_760_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_761_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_762_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_763_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_764_rev__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X5: int] : ( P @ ( cons_int @ X5 @ nil_int ) )
=> ( ! [X5: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X5 @ nil_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_765_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X @ Xs ) ) )
| ? [Ys5: list_int] :
( ( Ys
= ( cons_int @ X @ Ys5 ) )
& ( ( append_int @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_766_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs ) )
| ? [Ys5: list_int] :
( ( ( cons_int @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_int @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_767_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys4: list_int,Y4: int] :
( Xs
!= ( append_int @ Ys4 @ ( cons_int @ Y4 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_768_rev__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ( P @ nil_int )
=> ( ! [X5: int,Xs2: list_int] :
( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X5 @ nil_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_769_split__list__first__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_int,X4: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y6: int] :
( ( member_int @ Y6 @ ( set_int2 @ Ys2 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_770_split__list__last__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_int,X4: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y6: int] :
( ( member_int @ Y6 @ ( set_int2 @ Zs3 ) )
=> ~ ( P @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_771_in__set__conv__decomp__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_772_in__set__conv__decomp__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_773_split__list__first__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_774_split__list__last__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_int,X5: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_775_split__list__first__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_776_split__list__last__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_int,X5: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_777_in__set__conv__decomp,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys2: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_778_append__Cons__eq__iff,axiom,
! [X: int,Xs: list_int,Ys: list_int,Xs4: list_int,Ys6: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X @ Ys ) )
= ( append_int @ Xs4 @ ( cons_int @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_779_split__list__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys4: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_780_split__list__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_781_split__list__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys4: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_782_split__list__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_783_split__list,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys4: list_int,Zs2: list_int] :
( Xs
= ( append_int @ Ys4 @ ( cons_int @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_784_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X5: int,Xs5: list_int,Y4: int,Ys7: list_int] :
( ( X5 != Y4 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X5 @ nil_int ) @ Xs5 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y4 @ nil_int ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_785_remdups__adj__append__two,axiom,
! [Xs: list_int,X: int,Y: int] :
( ( remdups_adj_int @ ( append_int @ Xs @ ( cons_int @ X @ ( cons_int @ Y @ nil_int ) ) ) )
= ( append_int @ ( remdups_adj_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) @ ( if_list_int @ ( X = Y ) @ nil_int @ ( cons_int @ Y @ nil_int ) ) ) ) ).
% remdups_adj_append_two
thf(fact_786_lexord__append__rightI,axiom,
! [Y: list_int,X: list_int,R2: set_Pr958786334691620121nt_int] :
( ? [B5: int,Z2: list_int] :
( Y
= ( cons_int @ B5 @ Z2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ ( append_int @ X @ Y ) ) @ ( lexord_int @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_787_length__Suc__conv__rev,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ Y6 @ nil_int ) ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_788_length__append__singleton,axiom,
! [Xs: list_int,X: int] :
( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_789_lexord__append__left__rightI,axiom,
! [A: int,B: int,R2: set_Pr958786334691620121nt_int,U: list_int,X: list_int,Y: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ U @ ( cons_int @ A @ X ) ) @ ( append_int @ U @ ( cons_int @ B @ Y ) ) ) @ ( lexord_int @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_790_remdups__adj__append,axiom,
! [Xs_1: list_int,X: int,Xs_2: list_int] :
( ( remdups_adj_int @ ( append_int @ Xs_1 @ ( cons_int @ X @ Xs_2 ) ) )
= ( append_int @ ( remdups_adj_int @ ( append_int @ Xs_1 @ ( cons_int @ X @ nil_int ) ) ) @ ( tl_int @ ( remdups_adj_int @ ( cons_int @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_791_take__hd__drop,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( append_int @ ( take_int @ N @ Xs ) @ ( cons_int @ ( hd_int @ ( drop_int @ N @ Xs ) ) @ nil_int ) )
= ( take_int @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_792_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_int,A: int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( list_update_int @ Xs @ I @ A )
= ( append_int @ ( take_int @ I @ Xs ) @ ( cons_int @ A @ ( drop_int @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_793_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_794_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_795_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_796_list__update__length,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( list_update_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) @ Y )
= ( append_int @ Xs @ ( cons_int @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_797_list_Ocollapse,axiom,
! [List: list_int] :
( ( List != nil_int )
=> ( ( cons_int @ ( hd_int @ List ) @ ( tl_int @ List ) )
= List ) ) ).
% list.collapse
thf(fact_798_hd__Cons__tl,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( hd_int @ Xs ) @ ( tl_int @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_799_take__hd,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( take_int @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ).
% take_hd
thf(fact_800_hd__drop__1,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( hd_int @ Xs ) @ ( drop_int @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_801_hd__drop,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( hd_int @ ( drop_int @ N @ Xs ) ) @ ( drop_int @ ( suc @ N ) @ Xs ) )
= ( drop_int @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_802_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_min
thf(fact_803_of__nat__min,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_min_nat @ X @ Y ) )
= ( ord_min_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_min
thf(fact_804_list_Osel_I1_J,axiom,
! [X21: int,X22: list_int] :
( ( hd_int @ ( cons_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_805_cons__hd,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= Ys )
=> ( X
= ( hd_int @ Ys ) ) ) ).
% cons_hd
thf(fact_806_list__update__code_I2_J,axiom,
! [X: int,Xs: list_int,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_int @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_807_list__update__code_I3_J,axiom,
! [X: int,Xs: list_int,I: nat,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_int @ X @ ( list_update_int @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_808_list_Oexhaust__sel,axiom,
! [List: list_int] :
( ( List != nil_int )
=> ( List
= ( cons_int @ ( hd_int @ List ) @ ( tl_int @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_809_take__hd_H,axiom,
! [Ys: list_int,X: int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ( ( take_int @ ( size_size_list_int @ Ys ) @ ( cons_int @ X @ Xs ) )
= ( take_int @ ( suc @ ( size_size_list_int @ Xs ) ) @ Ys ) )
=> ( ( hd_int @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_810_take__Suc,axiom,
! [Xs: list_int,N: nat] :
( ( Xs != nil_int )
=> ( ( take_int @ ( suc @ N ) @ Xs )
= ( cons_int @ ( hd_int @ Xs ) @ ( take_int @ N @ ( tl_int @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_811_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_812_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_813_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_814_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_815_min__less__iff__conj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z3 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z3 @ X )
& ( ord_less_nat @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_816_min__less__iff__conj,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ Z3 @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z3 @ X )
& ( ord_less_int @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_817_min_Ostrict__coboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_818_min_Ostrict__coboundedI2,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_819_min_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_820_min_Ostrict__coboundedI1,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ C2 )
=> ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_821_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_822_min_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( A5
= ( ord_min_int @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_823_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_824_min_Ostrict__boundedE,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ ( ord_min_int @ B @ C2 ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_825_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_nat @ X @ Z3 )
| ( ord_less_nat @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_826_min__less__iff__disj,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ( ord_less_int @ X @ Z3 )
| ( ord_less_int @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_827_Small__Aux_Osize__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a3 @ X )
= Y )
=> ( ! [State: state_a] :
( ( X
= ( common_a2 @ State ) )
=> ( Y
!= ( size_size_state_a2 @ State ) ) )
=> ( ! [Current2: current_a] :
( ? [Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ) ) ).
% Small_Aux.size_state.elims
thf(fact_828_rotate1__hd__tl,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( rotate1_int @ Xs )
= ( append_int @ ( tl_int @ Xs ) @ ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ) ).
% rotate1_hd_tl
thf(fact_829_SuccD,axiom,
! [K: int,Kl: set_list_int,Kl2: list_int] :
( ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) )
=> ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl ) ) ).
% SuccD
thf(fact_830_Small__Aux_Osize__new__state_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv: stack_a,Uw2: list_a,Ux2: nat] :
( ( type_s6404775287138157491tate_a @ ( reverse2_a @ Current @ Uu2 @ Uv @ Uw2 @ Ux2 ) )
= ( type_s933026853152659577rent_a @ Current ) ) ).
% Small_Aux.size_new_state.simps(2)
thf(fact_831_Small__Aux_Osize__state_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv: stack_a,Uw2: list_a,Ux2: nat] :
( ( size_size_state_a3 @ ( reverse2_a @ Current @ Uu2 @ Uv @ Uw2 @ Ux2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ).
% Small_Aux.size_state.simps(2)
thf(fact_832_rotate1_Osimps_I2_J,axiom,
! [X: int,Xs: list_int] :
( ( rotate1_int @ ( cons_int @ X @ Xs ) )
= ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) ).
% rotate1.simps(2)
thf(fact_833_SuccI,axiom,
! [Kl2: list_int,K: int,Kl: set_list_int] :
( ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl )
=> ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_834_Small__Aux_Osize__new__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6404775287138157491tate_a @ X )
= Y )
=> ( ! [State: state_a] :
( ( X
= ( common_a2 @ State ) )
=> ( Y
!= ( type_s8424385952999958455tate_a @ State ) ) )
=> ( ! [Current2: current_a] :
( ? [Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ) ).
% Small_Aux.size_new_state.elims
thf(fact_835_Succ__Shift,axiom,
! [Kl: set_list_int,K: int,Kl2: list_int] :
( ( bNF_Gr6350390219475566417cc_int @ ( bNF_Gr1870224194279859149ft_int @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6350390219475566417cc_int @ Kl @ ( cons_int @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_836_Cons__listrel1__Cons,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_837_listrel1I2,axiom,
! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,X: int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ X @ Ys ) ) @ ( listrel1_int @ R2 ) ) ) ).
% listrel1I2
thf(fact_838_ShiftD,axiom,
! [Kl2: list_int,Kl: set_list_int,K: int] :
( ( member_list_int @ Kl2 @ ( bNF_Gr1870224194279859149ft_int @ Kl @ K ) )
=> ( member_list_int @ ( cons_int @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_839_Cons__listrel1E2,axiom,
! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R2 ) )
=> ( ! [X5: int] :
( ( Xs
= ( cons_int @ X5 @ Ys ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Xs
= ( cons_int @ Y @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_840_Cons__listrel1E1,axiom,
! [X: int,Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) @ ( listrel1_int @ R2 ) )
=> ( ! [Y4: int] :
( ( Ys
= ( cons_int @ Y4 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Ys
= ( cons_int @ X @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_841_listrel1I1,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R2 ) ) ) ).
% listrel1I1
thf(fact_842_listrel1E,axiom,
! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
=> ~ ! [X5: int,Y4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
=> ! [Us: list_int,Vs: list_int] :
( ( Xs
= ( append_int @ Us @ ( cons_int @ X5 @ Vs ) ) )
=> ( Ys
!= ( append_int @ Us @ ( cons_int @ Y4 @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_843_listrel1I,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Us2: list_int,Vs2: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_int @ Us2 @ ( cons_int @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_int @ Us2 @ ( cons_int @ Y @ Vs2 ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_844_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_845_Small__Aux_Osize__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a3 @ X )
= Y )
=> ( ( accp_state_a @ small_6459853017724497003_rel_a @ X )
=> ( ! [State: state_a] :
( ( X
= ( common_a2 @ State ) )
=> ( ( Y
= ( size_size_state_a2 @ State ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( common_a2 @ State ) ) ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) ) ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ( accp_state_a @ small_6459853017724497003_rel_a @ ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.size_state.pelims
thf(fact_846_replicate__Suc,axiom,
! [N: nat,X: int] :
( ( replicate_int @ ( suc @ N ) @ X )
= ( cons_int @ X @ ( replicate_int @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_847_replicate__app__Cons__same,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( append_int @ ( replicate_int @ N @ X ) @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( append_int @ ( replicate_int @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_848_replicate__append__same,axiom,
! [I: nat,X: int] :
( ( append_int @ ( replicate_int @ I @ X ) @ ( cons_int @ X @ nil_int ) )
= ( cons_int @ X @ ( replicate_int @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_849_remdups__adj__replicate,axiom,
! [N: nat,X: int] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_int @ ( replicate_int @ N @ X ) )
= nil_int ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_int @ ( replicate_int @ N @ X ) )
= ( cons_int @ X @ nil_int ) ) ) ) ).
% remdups_adj_replicate
thf(fact_850_remdups__adj__singleton,axiom,
! [Xs: list_int,X: int] :
( ( ( remdups_adj_int @ Xs )
= ( cons_int @ X @ nil_int ) )
=> ( Xs
= ( replicate_int @ ( size_size_list_int @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_851_Small__Aux_Osize__new__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6404775287138157491tate_a @ X )
= Y )
=> ( ( accp_state_a @ small_1734874108512276223_rel_a @ X )
=> ( ! [State: state_a] :
( ( X
= ( common_a2 @ State ) )
=> ( ( Y
= ( type_s8424385952999958455tate_a @ State ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( common_a2 @ State ) ) ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) ) ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current2 @ Uy @ Uz ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ small_1734874108512276223_rel_a @ ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.size_new_state.pelims
thf(fact_852_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_853_take__rev__step,axiom,
! [Xs: list_int,N: nat,Acc: list_int] :
( ( Xs != nil_int )
=> ( ( append_int @ ( common_take_rev_int @ N @ ( tl_int @ Xs ) ) @ ( cons_int @ ( hd_int @ Xs ) @ Acc ) )
= ( append_int @ ( common_take_rev_int @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_854_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_855_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_856_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_857_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_858_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_859_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_860_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_861_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_862_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_863_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_864_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_865_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_866_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_867_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_868_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_869_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_870_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_871_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_872_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_873_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_874_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_875_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_876_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_877_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_878_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_879_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_880_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_881_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_882_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_883_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_884_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( A = B )
= ( C2 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_885_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_886_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_887_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_888_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_889_eq__iff__diff__eq__0,axiom,
( ( ^ [Y7: int,Z4: int] : ( Y7 = Z4 ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_890_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_891_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_892_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_893_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_894_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_895_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_896_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_897_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_898_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_899_diff__mono,axiom,
! [A: int,B: int,D2: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_900_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_901_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_902_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_903_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_904_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_905_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z3: int] :
( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z3 )
= ( ord_min_int @ ( minus_minus_int @ X @ Z3 ) @ ( minus_minus_int @ Y @ Z3 ) ) ) ).
% min_diff_distrib_left
thf(fact_906_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_907_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_908_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_909_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_910_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_911_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_912_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_913_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_914_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_int )
=> ( ( append_int @ ( common_take_rev_int @ N @ Xs ) @ Ys )
= ( append_int @ ( common_take_rev_int @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_int @ Xs ) ) @ ( cons_int @ ( hd_int @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_915_take__rev__nth,axiom,
! [N: nat,Xs: list_int,X: int,Ys: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( X
= ( nth_int @ Xs @ N ) )
=> ( ( cons_int @ X @ ( append_int @ ( common_take_rev_int @ N @ Xs ) @ Ys ) )
= ( append_int @ ( common_take_rev_int @ ( suc @ N ) @ Xs ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_916_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_917_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_918_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_919_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_920_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_921_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_922_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_923_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_924_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_925_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_926_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_927_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_928_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_929_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_930_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_931_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_932_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_933_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_934_mod__induct,axiom,
! [P: nat > $o,N: nat,P2: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P2 )
=> ( ( ord_less_nat @ M @ P2 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P2 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_935_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_936_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N4 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).
% mod_if
thf(fact_937_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_938_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_939_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_940_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_941_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_942_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_943_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_944_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_945_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
=> ( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M2 @ N2 ) )
=> ( P @ M2 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_946_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_947_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_948_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_949_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_950_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_951_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_952_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_953_min__0__1_I1_J,axiom,
( ( ord_min_int @ zero_zero_int @ one_one_int )
= zero_zero_int ) ).
% min_0_1(1)
thf(fact_954_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_955_min__0__1_I2_J,axiom,
( ( ord_min_int @ one_one_int @ zero_zero_int )
= zero_zero_int ) ).
% min_0_1(2)
thf(fact_956_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_957_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_958_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_959_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_960_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_961_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_962_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_963_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_964_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_965_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_966_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_967_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_968_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_969_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_970_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_971_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_972_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_973_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_974_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_975_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_976_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_977_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_978_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_979_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_980_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_981_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_982_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_983_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_984_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_985_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_986_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_987_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_988_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_989_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_990_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_991_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_992_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_993_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_994_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_995_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_996_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_997_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_998_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_999_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1000_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_1001_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1002_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1003_drop__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1004_length__one__hd,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ).
% length_one_hd
thf(fact_1005_take__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X @ Xs ) )
= nil_int ) )
& ( ( N != zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( take_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1006_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1007_Cons__replicate__eq,axiom,
! [X: int,Xs: list_int,N: nat,Y: int] :
( ( ( cons_int @ X @ Xs )
= ( replicate_int @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_int @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1008_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_1009_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1010_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1011_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1012_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1013_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1014_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1015_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1016_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_1017_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1018_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1019_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_1020_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_1021_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_1022_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_1023_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1024_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_1025_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_1026_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_1027_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_1028_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_1029_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1030_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1031_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1032_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1033_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1034_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_1035_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_1036_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_1037_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1038_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_1039_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1040_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1041_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_1042_abs__if,axiom,
( abs_abs_int
= ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% abs_if
thf(fact_1043_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_1044_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% abs_if_raw
thf(fact_1045_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_1046_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_1047_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1048_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1049_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1050_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_1051_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_1052_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1053_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1054_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1055_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1056_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1057_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1058_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1059_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1060_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1061_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1062_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1063_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1064_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1065_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1066_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1067_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1068_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1069_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1070_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1071_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1072_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1073_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1074_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1075_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1076_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1077_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1078_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1079_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1080_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1081_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1082_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1083_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1084_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1085_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1086_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1087_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1088_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_1089_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1090_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1091_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1092_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1093_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_1094_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1095_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1096_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1097_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1098_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1099_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1100_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1101_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1102_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1103_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1104_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1105_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1106_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1107_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1108_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1109_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1110_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1111_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1112_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1113_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1114_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1115_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1116_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1117_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1118_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1119_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1120_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1121_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1122_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_1123_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_1124_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1125_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1126_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1127_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1128_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1129_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_1130_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_1131_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1132_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1133_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_1134_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1135_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1136_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1137_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1138_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1139_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1140_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1141_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1142_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1143_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1144_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1145_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1146_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1147_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1148_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1149_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1150_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1151_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1152_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1153_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1154_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1155_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1156_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1157_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1158_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1159_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1160_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1161_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1162_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1163_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1164_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1165_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1166_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1167_group__cancel_Osub2,axiom,
! [B6: int,K: int,B: int,A: int] :
( ( B6
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B6 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1168_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1169_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1170_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1171_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1172_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1173_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1174_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1175_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1176_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_1177_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z3 ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).
% min_add_distrib_right
thf(fact_1178_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).
% min_add_distrib_left
thf(fact_1179_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1180_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_1181_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1182_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1183_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1184_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1185_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A5 @ C ) ) ) ) ).
% le_iff_add
thf(fact_1186_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1187_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_1188_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1189_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1190_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_1191_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1192_add__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_1193_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1194_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1195_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1196_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1197_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1198_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1199_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1200_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1201_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1202_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1203_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1204_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1205_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1206_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1207_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1208_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1209_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1210_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1211_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1212_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1213_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1214_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1215_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1216_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1217_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1218_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1219_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_1220_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_1221_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_1222_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1223_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_1224_group__cancel_Oadd2,axiom,
! [B6: nat,K: nat,B: nat,A: nat] :
( ( B6
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1225_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1226_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1227_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1228_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1229_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1230_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1231_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1232_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1233_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1234_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1235_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1236_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1237_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1238_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1239_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1240_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1241_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1242_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1243_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1244_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1245_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1246_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1247_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1248_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1249_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1250_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1251_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1252_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1253_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1254_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1255_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1256_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( type_i6304938058965754292tate_a @ big )
& ( type_i464410347872898157tate_a @ small ) ) ).
%------------------------------------------------------------------------------