TPTP Problem File: SLH0943^1.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_00297_009564__6930318_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1611 ( 653 unt; 339 typ; 0 def)
% Number of atoms : 3371 (1604 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 11201 ( 500 ~; 91 |; 194 &;9143 @)
% ( 0 <=>;1273 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Number of types : 73 ( 72 usr)
% Number of type conns : 638 ( 638 >; 0 *; 0 +; 0 <<)
% Number of symbols : 270 ( 267 usr; 24 con; 0-5 aty)
% Number of variables : 3708 ( 87 ^;3482 !; 139 ?;3708 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:27.005
%------------------------------------------------------------------------------
% Could-be-implicit typings (72)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc1050408459402128056nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
set_Pr6304946757569631943nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc1089560213143673063nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
produc144329077504515498nt_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Small__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc3607442247652080723nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Big__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
produc3997409475329840794nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
set_Pr2560585780119916871nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
produc4909961631098372119st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J_J,type,
set_Pr7878039356161601943nt_nat: $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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc1219242969750017639nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
produc2907033302207676215nt_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_Pr5382606609415531783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
produc1473018763691903991list_a: $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_I_062_It__Int__Oint_Mt__Nat__Onat_J_Mt__Small__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
produc7263902551456197441nt_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Nat__Onat_J_Mt__Big__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
produc2267447564731110970nt_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
set_Pr1394042615296016247nt_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc7709606177366032167list_a: $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__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc5327195440680321047nt_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__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__States__Ostates_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
states2291996930907517375nt_int: $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__Common__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
state_1864181505321353467nt_int: $tType ).
thf(ty_n_t__Small__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
state_7070011053975014053nt_int: $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__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
list_P5707943133018811711nt_int: $tType ).
thf(ty_n_t__Big__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
state_7675739447491938028nt_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
list_list_int_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__List__Olist_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
list_list_int_nat2: $tType ).
thf(ty_n_t__List__Olist_I_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J_J,type,
list_list_a_nat: $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__Current__Ocurrent_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
current_int_nat: $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__States__Ostates_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
states_int_nat: $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__Common__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
state_int_nat: $tType ).
thf(ty_n_t__Stack__Ostack_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
stack_int_nat: $tType ).
thf(ty_n_t__Small__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
state_int_nat2: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
list_int_nat: $tType ).
thf(ty_n_t__Big__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
state_int_nat3: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_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__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 (267)
thf(sy_c_Big_Opop_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
pop_int_nat: state_int_nat3 > produc2267447564731110970nt_nat ).
thf(sy_c_Big_Opop_001t__Int__Oint,type,
pop_int: state_int3 > produc1417387325239031004te_int ).
thf(sy_c_Big_Opop_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
pop_Pr8810604350686413275nt_int: state_7675739447491938028nt_int > produc3997409475329840794nt_int ).
thf(sy_c_Big_Opop_001tf__a,type,
pop_a: state_a3 > produc6972303929186420058tate_a ).
thf(sy_c_Big_Opush_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
push_int_nat: ( int > nat ) > state_int_nat3 > state_int_nat3 ).
thf(sy_c_Big_Opush_001t__Int__Oint,type,
push_int: int > state_int3 > state_int3 ).
thf(sy_c_Big_Opush_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
push_P5578427077328705194nt_int: product_prod_int_int > state_7675739447491938028nt_int > state_7675739447491938028nt_int ).
thf(sy_c_Big_Opush_001tf__a,type,
push_a: a > state_a3 > state_a3 ).
thf(sy_c_Big_Ostate_OCommon_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
common_int_nat: state_int_nat > state_int_nat3 ).
thf(sy_c_Big_Ostate_OCommon_001t__Int__Oint,type,
common_int: state_int > state_int3 ).
thf(sy_c_Big_Ostate_OCommon_001tf__a,type,
common_a: state_a > state_a3 ).
thf(sy_c_Big_Ostate_OReverse_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
reverse_int_nat: current_int_nat > stack_int_nat > list_int_nat > nat > state_int_nat3 ).
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_001tf__a,type,
reverse_a: current_a > stack_a > list_a > nat > state_a3 ).
thf(sy_c_Big__Aux_Olist_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
big_list_int_nat: state_int_nat3 > list_int_nat ).
thf(sy_c_Big__Aux_Olist_001t__Int__Oint,type,
big_list_int: state_int3 > list_int ).
thf(sy_c_Big__Aux_Olist_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
big_li1975423998857797354nt_int: state_7675739447491938028nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_Big__Aux_Olist_001tf__a,type,
big_list_a: state_a3 > list_a ).
thf(sy_c_Big__Aux_Olist__current_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
big_li1428258522215584551nt_nat: state_int_nat3 > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
big_li5865000700487598893nt_int: state_7675739447491938028nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_Big__Aux_Olist__current_001tf__a,type,
big_list_current_a: state_a3 > list_a ).
thf(sy_c_Big__Aux_Osize__new__state__rel_001tf__a,type,
big_si5937185285519891526_rel_a: state_a3 > state_a3 > $o ).
thf(sy_c_Big__Aux_Osize__state__rel_001tf__a,type,
big_size_state_rel_a: state_a3 > state_a3 > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > 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_Olenlex_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
lenlex_int_nat: set_Pr1394042615296016247nt_nat > set_Pr7878039356161601943nt_nat ).
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_001t__List__Olist_Itf__a_J,type,
lenlex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olenlex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lenlex6370358691973319492nt_int: set_Pr2560585780119916871nt_int > set_Pr6304946757569631943nt_int ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
lex_int_nat: set_Pr1394042615296016247nt_nat > set_Pr7878039356161601943nt_nat ).
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_001t__List__Olist_Itf__a_J,type,
lex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
lex_Pr5393148144989827363nt_int: set_Pr2560585780119916871nt_int > set_Pr6304946757569631943nt_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_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
cons_int_nat: ( int > nat ) > list_int_nat > list_int_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_It__Int__Oint_J_Mt__Nat__Onat_J,type,
cons_list_int_nat: ( list_int > nat ) > list_list_int_nat2 > list_list_int_nat2 ).
thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
cons_list_a_nat: ( list_a > nat ) > list_list_a_nat > list_list_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
cons_list_int_nat2: list_int_nat > list_list_int_nat > list_list_int_nat ).
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_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
cons_P3334398858971670639nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
nil_int_nat: list_int_nat ).
thf(sy_c_List_Olist_ONil_001_062_It__List__Olist_It__Int__Oint_J_Mt__Nat__Onat_J,type,
nil_list_int_nat: list_list_int_nat2 ).
thf(sy_c_List_Olist_ONil_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
nil_list_a_nat: list_list_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
nil_list_int_nat2: list_list_int_nat ).
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_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
nil_Pr2300489316682597567nt_int: list_P5707943133018811711nt_int ).
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_Ohd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
hd_Pro282112905867057956nt_int: list_P5707943133018811711nt_int > product_prod_int_int ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_List_Olist_Otl_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
tl_int_nat: list_int_nat > list_int_nat ).
thf(sy_c_List_Olist_Otl_001t__Int__Oint,type,
tl_int: list_int > 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_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_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_Omeasures_001t__Int__Oint,type,
measures_int: list_int_nat > set_Pr958786334691620121nt_int ).
thf(sy_c_List_Omeasures_001t__List__Olist_It__Int__Oint_J,type,
measures_list_int: list_list_int_nat2 > set_Pr765067013931698361st_int ).
thf(sy_c_List_Omeasures_001t__List__Olist_Itf__a_J,type,
measures_list_a: list_list_a_nat > set_Pr4048851178543822343list_a ).
thf(sy_c_List_On__lists_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
n_lists_int_nat: nat > list_int_nat > list_list_int_nat ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).
thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
replicate_int: nat > int > list_int ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
replic1057375728873637753nt_int: nat > product_prod_int_int > list_P5707943133018811711nt_int ).
thf(sy_c_List_Otake_001t__Int__Oint,type,
take_int: nat > list_int > list_int ).
thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
zip_int_int: list_int > list_int > list_P5707943133018811711nt_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
size_s5894652653385020498nt_nat: state_int_nat3 > 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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
size_s7125611714489952088nt_int: state_7675739447491938028nt_int > 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_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
size_s1172059598526859839nt_nat: list_int_nat > 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_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
size_s5157815400016825771nt_int: list_P5707943133018811711nt_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
size_s7563699036481858393nt_nat: state_int_nat2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_It__Int__Oint_J,type,
size_size_state_int2: state_int2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
size_s2112932827293414673nt_int: state_7070011053975014053nt_int > 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_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_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc4965991708622915748nt_nat: ( ( int > nat ) > ( int > nat ) > $o ) > list_int_nat > produc144329077504515498nt_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__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc740395959522281929st_int: ( int > int ) > produc1186641810826059865st_int > produc4909961631098372119st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
produc7977806053278589903nt_nat: ( int > nat ) > ( int > nat ) > produc5327195440680321047nt_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Nat__Onat_J_001t__Big__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc306892107061677554nt_nat: ( int > nat ) > state_int_nat3 > produc2267447564731110970nt_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Nat__Onat_J_001t__Small__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc3404780059017917689nt_nat: ( int > nat ) > state_int_nat2 > produc7263902551456197441nt_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc3328129369365053992nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > list_P5707943133018811711nt_int > produc1050408459402128056nt_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_001_062_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc8643929849434629545list_a: ( a > a ) > produc9164743771328383783list_a > produc1473018763691903991list_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_001t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
produc5249521469742256623nt_nat: list_int_nat > list_int_nat > produc2907033302207676215nt_nat ).
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__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_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc1932183703851549015nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > produc1089560213143673063nt_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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Big__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc1421668873469546762nt_int: product_prod_int_int > state_7675739447491938028nt_int > produc3997409475329840794nt_int ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Small__Ostate_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc6824091031757186499nt_int: product_prod_int_int > state_7070011053975014053nt_int > produc3607442247652080723nt_int ).
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_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
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_Small_Opop_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
pop_int_nat2: state_int_nat2 > produc7263902551456197441nt_nat ).
thf(sy_c_Small_Opop_001t__Int__Oint,type,
pop_int2: state_int2 > produc957127447925188579te_int ).
thf(sy_c_Small_Opop_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
pop_Pr2845992955501674466nt_int: state_7070011053975014053nt_int > produc3607442247652080723nt_int ).
thf(sy_c_Small_Opop_001tf__a,type,
pop_a2: state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opush_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
push_int_nat2: ( int > nat ) > state_int_nat2 > state_int_nat2 ).
thf(sy_c_Small_Opush_001t__Int__Oint,type,
push_int2: int > state_int2 > state_int2 ).
thf(sy_c_Small_Opush_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
push_P6924061806946900579nt_int: product_prod_int_int > state_7070011053975014053nt_int > state_7070011053975014053nt_int ).
thf(sy_c_Small_Opush_001tf__a,type,
push_a2: a > state_a2 > state_a2 ).
thf(sy_c_Small_Ostate_OCommon_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
common_int_nat2: state_int_nat > state_int_nat2 ).
thf(sy_c_Small_Ostate_OCommon_001t__Int__Oint,type,
common_int2: state_int > state_int2 ).
thf(sy_c_Small_Ostate_OCommon_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
common8420081547569957991nt_int: state_1864181505321353467nt_int > state_7070011053975014053nt_int ).
thf(sy_c_Small_Ostate_OCommon_001tf__a,type,
common_a2: state_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse1_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
reverse1_int_nat: current_int_nat > stack_int_nat > list_int_nat > state_int_nat2 ).
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_001tf__a,type,
reverse1_a: current_a > stack_a > list_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse2_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
reverse2_int_nat: current_int_nat > list_int_nat > stack_int_nat > list_int_nat > nat > state_int_nat2 ).
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_001tf__a,type,
reverse2_a: current_a > list_a > stack_a > list_a > nat > state_a2 ).
thf(sy_c_Small__Aux_Olist_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
small_list_int_nat: state_int_nat2 > list_int_nat ).
thf(sy_c_Small__Aux_Olist_001t__Int__Oint,type,
small_list_int: state_int2 > list_int ).
thf(sy_c_Small__Aux_Olist_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
small_818879167795241585nt_int: state_7070011053975014053nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_Small__Aux_Olist_001tf__a,type,
small_list_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Olist__current_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
small_4712429614954843054nt_nat: state_int_nat2 > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
small_1783718464699920230nt_int: state_7070011053975014053nt_int > list_P5707943133018811711nt_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__state__rel_001tf__a,type,
small_6459853017724497003_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_States_Ostates_OStates_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states_int_nat2: direction > state_int_nat3 > state_int_nat2 > states_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states7113599851329041750nt_int: direction > state_7675739447491938028nt_int > state_7070011053975014053nt_int > states2291996930907517375nt_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_001tf__a,type,
size_states_a: ( a > nat ) > states_a > nat ).
thf(sy_c_States_Ostep__states__rel_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
step_s6801834171724358809nt_nat: states_int_nat > states_int_nat > $o ).
thf(sy_c_States_Ostep__states__rel_001t__Int__Oint,type,
step_states_rel_int: states_int > states_int > $o ).
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_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states7642676829979469824nt_nat: states_int_nat > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states6533906423715856788nt_int: states2291996930907517375nt_int > list_P5707943133018811711nt_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_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states5976382135168326513nt_nat: states_int_nat > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states1026446725800263523nt_int: states2291996930907517375nt_int > list_P5707943133018811711nt_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_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states1525641699331295146nt_nat: states_int_nat > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states6947458959548194410nt_int: states2291996930907517375nt_int > list_P5707943133018811711nt_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_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states5069133381756058809nt_nat: states_int_nat > list_int_nat ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states1431159827715037211nt_int: states2291996930907517375nt_int > list_P5707943133018811711nt_int ).
thf(sy_c_States__Aux_Olist__small__first_001tf__a,type,
states1596304293096088672irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olists_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
states_lists_int_nat: states_int_nat > produc2907033302207676215nt_nat ).
thf(sy_c_States__Aux_Olists_001t__Int__Oint,type,
states_lists_int: states_int > produc1186641810826059865st_int ).
thf(sy_c_States__Aux_Olists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
states4349910033867585945nt_int: states2291996930907517375nt_int > produc1089560213143673063nt_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_001t__Int__Oint,type,
states6576289658430312748el_int: states_int > states_int > $o ).
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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_i719020461660937083nt_nat: state_int_nat3 > $o ).
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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
type_i400838367537526639nt_int: state_7675739447491938028nt_int > $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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_i6582299730934091266nt_nat: state_int_nat2 > $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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
type_i8101691888268224936nt_int: state_7070011053975014053nt_int > $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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_i7955358610786168808nt_nat: states_int_nat > $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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
type_i7149897499329822530nt_int: states2291996930907517375nt_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_Itf__a_J,type,
type_r2494999336194962664tate_a: state_a3 > 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_Itf__a_J,type,
type_s6404775287138157491tate_a: state_a2 > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_s7127291786553863687nt_nat: state_int_nat3 > state_int_nat3 ).
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_Itf__a_J,type,
type_s3593206172722485288tate_a: state_a3 > state_a3 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Small__Ostate_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_s6378993889865489038nt_nat: state_int_nat2 > state_int_nat2 ).
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_Itf__a_J,type,
type_s3703408523585882337tate_a: state_a2 > state_a2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__States__Ostates_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
type_s2502071056353965684nt_nat: states_int_nat > states_int_nat ).
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_Itf__a_J,type,
type_s4923920245906622843ates_a: states_a > states_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Big__Ostate_Itf__a_J,type,
accp_state_a: ( state_a3 > state_a3 > $o ) > state_a3 > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Small__Ostate_Itf__a_J,type,
accp_state_a2: ( state_a2 > state_a2 > $o ) > state_a2 > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__States__Ostates_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
accp_states_int_nat: ( states_int_nat > states_int_nat > $o ) > states_int_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__States__Ostates_It__Int__Oint_J,type,
accp_states_int: ( states_int > states_int > $o ) > states_int > $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__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
member4491287985726366656nt_nat: produc5327195440680321047nt_nat > set_Pr1394042615296016247nt_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_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_Mt__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J_J,type,
member7939632110732153312nt_nat: produc2907033302207676215nt_nat > set_Pr7878039356161601943nt_nat > $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_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member1390679175989562640nt_int: produc1089560213143673063nt_int > set_Pr6304946757569631943nt_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_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $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_aux____,type,
aux: list_a ).
thf(sy_v_big,type,
big: state_a3 ).
thf(sy_v_biga____,type,
biga: stack_a ).
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_small,type,
small: state_a2 ).
thf(sy_v_smalla____,type,
smalla: state_a2 ).
% Relevant facts (1266)
thf(fact_0__C2__1_Oprems_C_I1_J,axiom,
type_i8221491762852169479ates_a @ ( states_a2 @ dira @ ( reverse_a @ current @ biga @ aux @ ( suc @ count ) ) @ smalla ) ).
% "2_1.prems"(1)
thf(fact_1__C2__1_Oprems_C_I2_J,axiom,
( ( type_s4923920245906622843ates_a @ ( states_a2 @ dira @ ( reverse_a @ current @ biga @ aux @ ( suc @ count ) ) @ smalla ) )
= ( states_a2 @ dir @ big @ small ) ) ).
% "2_1.prems"(2)
thf(fact_2_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_3_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_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_Big_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X14: nat,Y11: current_a,Y12: stack_a,Y13: list_a,Y14: nat] :
( ( ( reverse_a @ X11 @ X12 @ X13 @ X14 )
= ( reverse_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% Big.state.inject(1)
thf(fact_18_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_19_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_20_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_21_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_22_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_23_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_24_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_25_states_Oexhaust,axiom,
! [Y: states_a] :
~ ! [X15: direction,X22: state_a3,X32: state_a2] :
( Y
!= ( states_a2 @ X15 @ X22 @ X32 ) ) ).
% states.exhaust
thf(fact_26_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_27_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_28_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_29_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_30_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_31_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_32_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_33_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_34_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_35_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_36_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_37_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_38_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_39_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_40_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_41_mem__Collect__eq,axiom,
! [A: produc1186641810826059865st_int,P: produc1186641810826059865st_int > $o] :
( ( member6698963635872716290st_int @ A @ ( collec928425520773505860st_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_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_43_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_Pr765067013931698361st_int] :
( ( collec928425520773505860st_int
@ ^ [X5: produc1186641810826059865st_int] : ( member6698963635872716290st_int @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X5: product_prod_int_int] : ( member5262025264175285858nt_int @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X5: int] : ( member_int @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_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_48_size__neq__size__imp__neq,axiom,
! [X: states_a,Y: states_a] :
( ( ( size_size_states_a @ X )
!= ( size_size_states_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_49_size__neq__size__imp__neq,axiom,
! [X: state_a,Y: state_a] :
( ( ( size_size_state_a2 @ X )
!= ( size_size_state_a2 @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_50_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_51_size__neq__size__imp__neq,axiom,
! [X: current_a,Y: current_a] :
( ( ( size_size_current_a @ X )
!= ( size_size_current_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_lists__current_Osimps,axiom,
! [Uu: direction,Big: state_int3,Small: state_int2] :
( ( states4446210166810079719nt_int @ ( states_int2 @ Uu @ Big @ Small ) )
= ( produc364263696895485585st_int @ ( big_list_current_int @ Big ) @ ( small_1709254455575119551nt_int @ Small ) ) ) ).
% lists_current.simps
thf(fact_63_lists__current_Osimps,axiom,
! [Uu: direction,Big: state_a3,Small: state_a2] :
( ( states7719277857994474499rent_a @ ( states_a2 @ Uu @ Big @ Small ) )
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big ) @ ( small_list_current_a @ Small ) ) ) ).
% lists_current.simps
thf(fact_64_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_65_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_66_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_67_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_68_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_69_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_70_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_71_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_72_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_73_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_74_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_75_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_76_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_77_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_78_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_79_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_80_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_81_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_82_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_83_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_84_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_85_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_86_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_87_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_88_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_89_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_90_Big_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X14: nat,X2: state_a] :
( ( reverse_a @ X11 @ X12 @ X13 @ X14 )
!= ( common_a @ X2 ) ) ).
% Big.state.distinct(1)
thf(fact_91_Big_Ostate_Oexhaust,axiom,
! [Y: state_a3] :
( ! [X112: current_a,X122: stack_a,X132: list_a,X142: nat] :
( Y
!= ( reverse_a @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X22: state_a] :
( Y
!= ( common_a @ X22 ) ) ) ).
% Big.state.exhaust
thf(fact_92_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_93_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_94_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_95_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_96_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_97_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_98_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_99_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_100_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_101_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_102_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_103_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_104_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_105_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_106_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_107_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_108_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_109_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_110_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_111_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_112_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_113_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I3 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_114_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_115_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_116_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_117_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_118_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_119_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_120_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_121_Big_Ostep__state_Ocases,axiom,
! [X: state_a3] :
( ! [State: state_a] :
( X
!= ( common_a @ State ) )
=> ( ! [Current: current_a,Uu2: stack_a,Aux: list_a] :
( X
!= ( reverse_a @ Current @ Uu2 @ Aux @ zero_zero_nat ) )
=> ~ ! [Current: current_a,Big2: stack_a,Aux: list_a,V2: nat] :
( X
!= ( reverse_a @ Current @ Big2 @ Aux @ ( suc @ V2 ) ) ) ) ) ).
% Big.step_state.cases
thf(fact_122_states_Osize__neq,axiom,
! [X: states_a] :
( ( size_size_states_a @ X )
!= zero_zero_nat ) ).
% states.size_neq
thf(fact_123_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_124_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_125_remaining__steps__states_Ocases,axiom,
! [X: states_a] :
~ ! [Uu2: direction,Big2: state_a3,Small2: state_a2] :
( X
!= ( states_a2 @ Uu2 @ Big2 @ Small2 ) ) ).
% remaining_steps_states.cases
thf(fact_126_lists__current_Oelims,axiom,
! [X: states_int,Y: produc1186641810826059865st_int] :
( ( ( states4446210166810079719nt_int @ X )
= Y )
=> ~ ! [Uu2: direction,Big2: state_int3,Small2: state_int2] :
( ( X
= ( states_int2 @ Uu2 @ Big2 @ Small2 ) )
=> ( Y
!= ( produc364263696895485585st_int @ ( big_list_current_int @ Big2 ) @ ( small_1709254455575119551nt_int @ Small2 ) ) ) ) ) ).
% lists_current.elims
thf(fact_127_lists__current_Oelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ~ ! [Uu2: direction,Big2: state_a3,Small2: state_a2] :
( ( X
= ( states_a2 @ Uu2 @ Big2 @ Small2 ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big2 ) @ ( small_list_current_a @ Small2 ) ) ) ) ) ).
% lists_current.elims
thf(fact_128_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_129_Small_Ostate_Oinject_I2_J,axiom,
! [X21: current_a,X222: 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 @ X222 @ X23 @ X24 @ X25 )
= ( reverse2_a @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 )
& ( X23 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% Small.state.inject(2)
thf(fact_130_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_131_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X2 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_132_prod_Oinject,axiom,
! [X1: states_a,X2: nat,Y1: states_a,Y2: nat] :
( ( ( produc1877401315875745917_a_nat @ X1 @ X2 )
= ( produc1877401315875745917_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_133_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_134_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_135_prod_Oinject,axiom,
! [X1: list_int,X2: list_int,Y1: list_int,Y2: list_int] :
( ( ( produc364263696895485585st_int @ X1 @ X2 )
= ( produc364263696895485585st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_136_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_137_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_138_old_Oprod_Oinject,axiom,
! [A: states_a,B: nat,A3: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A @ B )
= ( produc1877401315875745917_a_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_139_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_140_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_141_old_Oprod_Oinject,axiom,
! [A: list_int,B: list_int,A3: list_int,B2: list_int] :
( ( ( produc364263696895485585st_int @ A @ B )
= ( produc364263696895485585st_int @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_Small_Ostate_Odistinct_I5_J,axiom,
! [X21: current_a,X222: list_a,X23: stack_a,X24: list_a,X25: nat,X3: state_a] :
( ( reverse2_a @ X21 @ X222 @ X23 @ X24 @ X25 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(5)
thf(fact_152_step__list__reverse2,axiom,
! [Small: state_a2,Current2: current_a,Aux2: list_a,Big: stack_a,New: list_a,Count: nat] :
( ( Small
= ( reverse2_a @ Current2 @ Aux2 @ Big @ New @ Count ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_reverse2
thf(fact_153_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_154_size__ok_H_Ocases,axiom,
! [X: produc1571854377283420419_a_nat] :
~ ! [Uu2: direction,Big2: state_a3,Small2: state_a2,Steps: nat] :
( X
!= ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu2 @ Big2 @ Small2 ) @ Steps ) ) ).
% size_ok'.cases
thf(fact_155_Big_Opush_Ocases,axiom,
! [X: produc6972303929186420058tate_a] :
( ! [X4: a,State: state_a] :
( X
!= ( produc8641956578966763338tate_a @ X4 @ ( common_a @ State ) ) )
=> ~ ! [X4: a,Current: current_a,Big2: stack_a,Aux: list_a,Count2: nat] :
( X
!= ( produc8641956578966763338tate_a @ X4 @ ( reverse_a @ Current @ Big2 @ Aux @ Count2 ) ) ) ) ).
% Big.push.cases
thf(fact_156_Pair__inject,axiom,
! [A: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_157_Pair__inject,axiom,
! [A: states_a,B: nat,A3: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A @ B )
= ( produc1877401315875745917_a_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_158_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_159_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_160_Pair__inject,axiom,
! [A: list_int,B: list_int,A3: list_int,B2: list_int] :
( ( ( produc364263696895485585st_int @ A @ B )
= ( produc364263696895485585st_int @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_161_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_162_prod__cases,axiom,
! [P: produc9164743771328383783list_a > $o,P2: produc9164743771328383783list_a] :
( ! [A4: list_a,B3: list_a] : ( P @ ( produc6837034575241423639list_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_163_prod__cases,axiom,
! [P: produc1571854377283420419_a_nat > $o,P2: produc1571854377283420419_a_nat] :
( ! [A4: states_a,B3: nat] : ( P @ ( produc1877401315875745917_a_nat @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_164_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_165_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_166_prod__cases,axiom,
! [P: produc1186641810826059865st_int > $o,P2: produc1186641810826059865st_int] :
( ! [A4: list_int,B3: list_int] : ( P @ ( produc364263696895485585st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_167_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_168_surj__pair,axiom,
! [P2: produc9164743771328383783list_a] :
? [X4: list_a,Y4: list_a] :
( P2
= ( produc6837034575241423639list_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_169_surj__pair,axiom,
! [P2: produc1571854377283420419_a_nat] :
? [X4: states_a,Y4: nat] :
( P2
= ( produc1877401315875745917_a_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_170_surj__pair,axiom,
! [P2: produc6972303929186420058tate_a] :
? [X4: a,Y4: state_a3] :
( P2
= ( produc8641956578966763338tate_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_171_surj__pair,axiom,
! [P2: produc7589950997499123219tate_a] :
? [X4: a,Y4: state_a2] :
( P2
= ( produc1224139502141355779tate_a @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_172_surj__pair,axiom,
! [P2: produc1186641810826059865st_int] :
? [X4: list_int,Y4: list_int] :
( P2
= ( produc364263696895485585st_int @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_173_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X4: int,Y4: int] :
( P2
= ( product_Pair_int_int @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_174_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A4: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_175_old_Oprod_Oexhaust,axiom,
! [Y: produc1571854377283420419_a_nat] :
~ ! [A4: states_a,B3: nat] :
( Y
!= ( produc1877401315875745917_a_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_176_old_Oprod_Oexhaust,axiom,
! [Y: produc6972303929186420058tate_a] :
~ ! [A4: a,B3: state_a3] :
( Y
!= ( produc8641956578966763338tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_177_old_Oprod_Oexhaust,axiom,
! [Y: produc7589950997499123219tate_a] :
~ ! [A4: a,B3: state_a2] :
( Y
!= ( produc1224139502141355779tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_178_old_Oprod_Oexhaust,axiom,
! [Y: produc1186641810826059865st_int] :
~ ! [A4: list_int,B3: list_int] :
( Y
!= ( produc364263696895485585st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_179_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_180_Big__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a3] :
( ! [State: state_a] :
( X
!= ( common_a @ State ) )
=> ~ ! [Current: current_a,Uu2: stack_a,Uv2: list_a,Uw: nat] :
( X
!= ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) ) ) ).
% Big_Aux.size_new_state.cases
thf(fact_181_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_int2 @ 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_193_Small__Proof_Opop__list__current,axiom,
! [Small: state_int_nat2,X: int > nat,Small3: state_int_nat2] :
( ( type_i6582299730934091266nt_nat @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7563699036481858393nt_nat @ Small ) )
=> ( ( ( pop_int_nat2 @ Small )
= ( produc3404780059017917689nt_nat @ X @ Small3 ) )
=> ( ( cons_int_nat @ X @ ( small_4712429614954843054nt_nat @ Small3 ) )
= ( small_4712429614954843054nt_nat @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_194_Small__Proof_Opop__list__current,axiom,
! [Small: state_7070011053975014053nt_int,X: product_prod_int_int,Small3: state_7070011053975014053nt_int] :
( ( type_i8101691888268224936nt_int @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2112932827293414673nt_int @ Small ) )
=> ( ( ( pop_Pr2845992955501674466nt_int @ Small )
= ( produc6824091031757186499nt_int @ X @ Small3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( small_1783718464699920230nt_int @ Small3 ) )
= ( small_1783718464699920230nt_int @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_195_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_196_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_197_Big__Proof_Opop__list__current,axiom,
! [Big: state_int_nat3,X: int > nat,Big3: state_int_nat3] :
( ( type_i719020461660937083nt_nat @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( ( pop_int_nat @ Big )
= ( produc306892107061677554nt_nat @ X @ Big3 ) )
=> ( ( cons_int_nat @ X @ ( big_li1428258522215584551nt_nat @ Big3 ) )
= ( big_li1428258522215584551nt_nat @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_198_Big__Proof_Opop__list__current,axiom,
! [Big: state_7675739447491938028nt_int,X: product_prod_int_int,Big3: state_7675739447491938028nt_int] :
( ( type_i400838367537526639nt_int @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7125611714489952088nt_int @ Big ) )
=> ( ( ( pop_Pr8810604350686413275nt_int @ Big )
= ( produc1421668873469546762nt_int @ X @ Big3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( big_li5865000700487598893nt_int @ Big3 ) )
= ( big_li5865000700487598893nt_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_199_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_200_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_201_Big__Proof_Opop__list,axiom,
! [Big: state_int_nat3,X: int > nat,Big3: state_int_nat3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( type_i719020461660937083nt_nat @ Big )
=> ( ( ( pop_int_nat @ Big )
= ( produc306892107061677554nt_nat @ X @ Big3 ) )
=> ( ( cons_int_nat @ X @ ( big_list_int_nat @ Big3 ) )
= ( big_list_int_nat @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_202_Big__Proof_Opop__list,axiom,
! [Big: state_7675739447491938028nt_int,X: product_prod_int_int,Big3: state_7675739447491938028nt_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7125611714489952088nt_int @ Big ) )
=> ( ( type_i400838367537526639nt_int @ Big )
=> ( ( ( pop_Pr8810604350686413275nt_int @ Big )
= ( produc1421668873469546762nt_int @ X @ Big3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( big_li1975423998857797354nt_int @ Big3 ) )
= ( big_li1975423998857797354nt_int @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_203_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_204_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_205_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_206_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_207_step__states_Ocases,axiom,
! [X: states_a] :
( ! [Dir2: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,CurrentS: current_a,Uu2: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu2 @ 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_208_Small_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,Y11: current_a,Y12: stack_a,Y13: list_a] :
( ( ( reverse1_a @ X11 @ X12 @ X13 )
= ( reverse1_a @ Y11 @ Y12 @ Y13 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 ) ) ) ).
% Small.state.inject(1)
thf(fact_209_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_210_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_211_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_212_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_213_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_214_Big__Proof_Opush__list,axiom,
! [X: int > nat,Big: state_int_nat3] :
( ( big_list_int_nat @ ( push_int_nat @ X @ Big ) )
= ( cons_int_nat @ X @ ( big_list_int_nat @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_215_Big__Proof_Opush__list,axiom,
! [X: product_prod_int_int,Big: state_7675739447491938028nt_int] :
( ( big_li1975423998857797354nt_int @ ( push_P5578427077328705194nt_int @ X @ Big ) )
= ( cons_P3334398858971670639nt_int @ X @ ( big_li1975423998857797354nt_int @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_216_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_217_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_218_Small__Proof_Opush__list__current,axiom,
! [X: int > nat,Small: state_int_nat2] :
( ( small_4712429614954843054nt_nat @ ( push_int_nat2 @ X @ Small ) )
= ( cons_int_nat @ X @ ( small_4712429614954843054nt_nat @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_219_Small__Proof_Opush__list__current,axiom,
! [X: product_prod_int_int,Small: state_7070011053975014053nt_int] :
( ( small_1783718464699920230nt_int @ ( push_P6924061806946900579nt_int @ X @ Small ) )
= ( cons_P3334398858971670639nt_int @ X @ ( small_1783718464699920230nt_int @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_220_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_221_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_222_Big__Proof_Opush__list__current,axiom,
! [X: int > nat,Big: state_int_nat3] :
( ( big_li1428258522215584551nt_nat @ ( push_int_nat @ X @ Big ) )
= ( cons_int_nat @ X @ ( big_li1428258522215584551nt_nat @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_223_Big__Proof_Opush__list__current,axiom,
! [X: product_prod_int_int,Big: state_7675739447491938028nt_int] :
( ( big_li5865000700487598893nt_int @ ( push_P5578427077328705194nt_int @ X @ Big ) )
= ( cons_P3334398858971670639nt_int @ X @ ( big_li5865000700487598893nt_int @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_224_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_225_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_226_push__list__common,axiom,
! [Small: state_int_nat2,Common: state_int_nat,X: int > nat] :
( ( Small
= ( common_int_nat2 @ Common ) )
=> ( ( small_list_int_nat @ ( push_int_nat2 @ X @ Small ) )
= ( cons_int_nat @ X @ ( small_list_int_nat @ Small ) ) ) ) ).
% push_list_common
thf(fact_227_push__list__common,axiom,
! [Small: state_7070011053975014053nt_int,Common: state_1864181505321353467nt_int,X: product_prod_int_int] :
( ( Small
= ( common8420081547569957991nt_int @ Common ) )
=> ( ( small_818879167795241585nt_int @ ( push_P6924061806946900579nt_int @ X @ Small ) )
= ( cons_P3334398858971670639nt_int @ X @ ( small_818879167795241585nt_int @ Small ) ) ) ) ).
% push_list_common
thf(fact_228_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_229_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_230_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,X: int > nat,Big3: state_int_nat3] :
( ( type_i7955358610786168808nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( ( pop_int_nat @ Big )
= ( produc306892107061677554nt_nat @ X @ Big3 ) )
=> ( ( cons_int_nat @ X @ ( states7642676829979469824nt_nat @ ( states_int_nat2 @ Dir @ Big3 @ Small ) ) )
= ( states7642676829979469824nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_231_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,X: product_prod_int_int,Big3: state_7675739447491938028nt_int] :
( ( type_i7149897499329822530nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7125611714489952088nt_int @ Big ) )
=> ( ( ( pop_Pr8810604350686413275nt_int @ Big )
= ( produc1421668873469546762nt_int @ X @ Big3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( states6533906423715856788nt_int @ ( states7113599851329041750nt_int @ Dir @ Big3 @ Small ) ) )
= ( states6533906423715856788nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_232_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_233_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_234_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,X: int > nat,Big3: state_int_nat3] :
( ( type_i7955358610786168808nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( ( pop_int_nat @ Big )
= ( produc306892107061677554nt_nat @ X @ Big3 ) )
=> ( ( cons_int_nat @ X @ ( states5976382135168326513nt_nat @ ( states_int_nat2 @ Dir @ Big3 @ Small ) ) )
= ( states5976382135168326513nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_235_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,X: product_prod_int_int,Big3: state_7675739447491938028nt_int] :
( ( type_i7149897499329822530nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7125611714489952088nt_int @ Big ) )
=> ( ( ( pop_Pr8810604350686413275nt_int @ Big )
= ( produc1421668873469546762nt_int @ X @ Big3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( states1026446725800263523nt_int @ ( states7113599851329041750nt_int @ Dir @ Big3 @ Small ) ) )
= ( states1026446725800263523nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_236_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_237_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_238_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_239_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_240_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_241_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_242_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_243_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_244_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_245_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_246_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_247_Small_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X21: current_a,X222: list_a,X23: stack_a,X24: list_a,X25: nat] :
( ( reverse1_a @ X11 @ X12 @ X13 )
!= ( reverse2_a @ X21 @ X222 @ X23 @ X24 @ X25 ) ) ).
% Small.state.distinct(1)
thf(fact_248_Small_Ostate_Odistinct_I3_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X3: state_a] :
( ( reverse1_a @ X11 @ X12 @ X13 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(3)
thf(fact_249_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_250_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_251_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_252_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_253_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_254_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X4 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_255_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_256_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_257_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_258_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_259_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_260_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M: nat] :
( M6
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_261_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_262_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_263_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_264_invar__list__big__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1888450819780863577irst_a @ States )
= ( states7295096810965389224irst_a @ States ) ) ) ).
% invar_list_big_first
thf(fact_265_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_266_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_267_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_268_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_269_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_270_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_271_Small__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a2] :
( ! [State: state_a] :
( X
!= ( common_a2 @ State ) )
=> ( ! [Current: current_a,Uu2: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
!= ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw @ Ux ) )
=> ~ ! [Current: current_a,Uy: stack_a,Uz: list_a] :
( X
!= ( reverse1_a @ Current @ Uy @ Uz ) ) ) ) ).
% Small_Aux.size_new_state.cases
thf(fact_272_Small_Ostep__state_Ocases,axiom,
! [X: state_a2] :
( ! [State: state_a] :
( X
!= ( common_a2 @ State ) )
=> ( ! [Current: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( reverse1_a @ Current @ Small2 @ AuxS ) )
=> ~ ! [Current: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( reverse2_a @ Current @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ).
% Small.step_state.cases
thf(fact_273_Small_Ostate_Oexhaust,axiom,
! [Y: state_a2] :
( ! [X112: current_a,X122: stack_a,X132: list_a] :
( Y
!= ( reverse1_a @ X112 @ X122 @ X132 ) )
=> ( ! [X212: current_a,X223: list_a,X232: stack_a,X242: list_a,X252: nat] :
( Y
!= ( reverse2_a @ X212 @ X223 @ X232 @ X242 @ X252 ) )
=> ~ ! [X32: state_a] :
( Y
!= ( common_a2 @ X32 ) ) ) ) ).
% Small.state.exhaust
thf(fact_274_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_275_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_276_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_277_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_278_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_279_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_280_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_281_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_282_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_283_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_284_Small_Opush_Ocases,axiom,
! [X: produc7589950997499123219tate_a] :
( ! [X4: a,State: state_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( common_a2 @ State ) ) )
=> ( ! [X4: a,Current: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse1_a @ Current @ Small2 @ AuxS ) ) )
=> ~ ! [X4: a,Current: current_a,AuxS: list_a,Big2: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse2_a @ Current @ AuxS @ Big2 @ NewS @ Count2 ) ) ) ) ) ).
% Small.push.cases
thf(fact_285_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_286_push__big,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,Big3: list_int_nat,Small3: list_int_nat,X: int > nat] :
( ( ( states_lists_int_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
= ( produc5249521469742256623nt_nat @ Big3 @ Small3 ) )
=> ( ( states_lists_int_nat @ ( states_int_nat2 @ Dir @ ( push_int_nat @ X @ Big ) @ Small ) )
= ( produc5249521469742256623nt_nat @ ( cons_int_nat @ X @ Big3 ) @ Small3 ) ) ) ).
% push_big
thf(fact_287_push__big,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,Big3: list_P5707943133018811711nt_int,Small3: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( states4349910033867585945nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
= ( produc1932183703851549015nt_int @ Big3 @ Small3 ) )
=> ( ( states4349910033867585945nt_int @ ( states7113599851329041750nt_int @ Dir @ ( push_P5578427077328705194nt_int @ X @ Big ) @ Small ) )
= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Big3 ) @ Small3 ) ) ) ).
% push_big
thf(fact_288_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_289_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_290_States__Aux_Olists_Ocases,axiom,
! [X: states_a] :
( ! [Uu2: direction,CurrentB: current_a,Big2: stack_a,AuxB: list_a,Count2: nat,CurrentS: current_a,Small2: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Uu2 @ ( 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_291_push__small__lists,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,Big3: list_int_nat,Small3: list_int_nat,X: int > nat] :
( ( type_i7955358610786168808nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
=> ( ( ( states_lists_int_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
= ( produc5249521469742256623nt_nat @ Big3 @ Small3 ) )
=> ( ( states_lists_int_nat @ ( states_int_nat2 @ Dir @ Big @ ( push_int_nat2 @ X @ Small ) ) )
= ( produc5249521469742256623nt_nat @ Big3 @ ( cons_int_nat @ X @ Small3 ) ) ) ) ) ).
% push_small_lists
thf(fact_292_push__small__lists,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,Big3: list_P5707943133018811711nt_int,Small3: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( type_i7149897499329822530nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
=> ( ( ( states4349910033867585945nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
= ( produc1932183703851549015nt_int @ Big3 @ Small3 ) )
=> ( ( states4349910033867585945nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ ( push_P6924061806946900579nt_int @ X @ Small ) ) )
= ( produc1932183703851549015nt_int @ Big3 @ ( cons_P3334398858971670639nt_int @ X @ Small3 ) ) ) ) ) ).
% push_small_lists
thf(fact_293_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_294_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_295_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_int2 @ 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_296_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,X: int > nat,Small3: state_int_nat2] :
( ( type_i7955358610786168808nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7563699036481858393nt_nat @ Small ) )
=> ( ( ( pop_int_nat2 @ Small )
= ( produc3404780059017917689nt_nat @ X @ Small3 ) )
=> ( ( cons_int_nat @ X @ ( states1525641699331295146nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small3 ) ) )
= ( states1525641699331295146nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_297_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,X: product_prod_int_int,Small3: state_7070011053975014053nt_int] :
( ( type_i7149897499329822530nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2112932827293414673nt_int @ Small ) )
=> ( ( ( pop_Pr2845992955501674466nt_int @ Small )
= ( produc6824091031757186499nt_int @ X @ Small3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( states6947458959548194410nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small3 ) ) )
= ( states6947458959548194410nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_298_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_299_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_int2 @ 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_300_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_int_nat3,Small: state_int_nat2,X: int > nat,Small3: state_int_nat2] :
( ( type_i7955358610786168808nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7563699036481858393nt_nat @ Small ) )
=> ( ( ( pop_int_nat2 @ Small )
= ( produc3404780059017917689nt_nat @ X @ Small3 ) )
=> ( ( cons_int_nat @ X @ ( states5069133381756058809nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small3 ) ) )
= ( states5069133381756058809nt_nat @ ( states_int_nat2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_301_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_7675739447491938028nt_int,Small: state_7070011053975014053nt_int,X: product_prod_int_int,Small3: state_7070011053975014053nt_int] :
( ( type_i7149897499329822530nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2112932827293414673nt_int @ Small ) )
=> ( ( ( pop_Pr2845992955501674466nt_int @ Small )
= ( produc6824091031757186499nt_int @ X @ Small3 ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ ( states1431159827715037211nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small3 ) ) )
= ( states1431159827715037211nt_int @ ( states7113599851329041750nt_int @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_302_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_303_step__states_Oelims,axiom,
! [X: states_int_nat,Y: states_int_nat] :
( ( ( type_s2502071056353965684nt_nat @ X )
= Y )
=> ( ! [Dir2: direction,CurrentB: current_int_nat,Big2: stack_int_nat,AuxB: list_int_nat,CurrentS: current_int_nat,Uu2: stack_int_nat,AuxS: list_int_nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int_nat @ CurrentS @ Uu2 @ AuxS ) ) )
=> ( Y
!= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( reverse_int_nat @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_int_nat @ CurrentS @ AuxS @ Big2 @ nil_int_nat @ zero_zero_nat ) ) ) )
=> ( ! [Dir2: direction,V2: current_int_nat,Va2: stack_int_nat,Vb2: list_int_nat,Vd2: nat,Right2: state_int_nat2] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( Y
!= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( reverse_int_nat @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s6378993889865489038nt_nat @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_int_nat,Right2: state_int_nat2] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( common_int_nat @ V2 ) @ Right2 ) )
=> ( Y
!= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( common_int_nat @ V2 ) ) @ ( type_s6378993889865489038nt_nat @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_int_nat3,V2: current_int_nat,Va2: list_int_nat,Vb2: stack_int_nat,Vc2: list_int_nat,Vd2: nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ Left2 @ ( reverse2_int_nat @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ Left2 ) @ ( type_s6378993889865489038nt_nat @ ( reverse2_int_nat @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_int_nat3,V2: state_int_nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ Left2 @ ( common_int_nat2 @ V2 ) ) )
=> ( Y
!= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ Left2 ) @ ( type_s6378993889865489038nt_nat @ ( common_int_nat2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% step_states.elims
thf(fact_304_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,Uu2: stack_int,AuxS: list_int] :
( ( X
= ( states_int2 @ Dir2 @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int @ CurrentS @ Uu2 @ 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_305_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,Uu2: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu2 @ 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_306_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X5: a,Ys: list_a] :
( ( Xs
= ( cons_a @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_307_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s1172059598526859839nt_nat @ Xs ) )
= ( ? [X5: int > nat,Ys: list_int_nat] :
( ( Xs
= ( cons_int_nat @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_s1172059598526859839nt_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_308_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5157815400016825771nt_int @ Xs ) )
= ( ? [X5: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_309_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X5: int,Ys: list_int] :
( ( Xs
= ( cons_int @ X5 @ Ys ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_310_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_int_nat,Big: stack_int_nat,AuxB2: list_int_nat,CurrentS2: current_int_nat,Uu: stack_int_nat,AuxS2: list_int_nat] :
( ( type_s2502071056353965684nt_nat @ ( states_int_nat2 @ Dir @ ( reverse_int_nat @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_int_nat @ CurrentS2 @ Uu @ AuxS2 ) ) )
= ( states_int_nat2 @ Dir @ ( type_s7127291786553863687nt_nat @ ( reverse_int_nat @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) ) @ ( reverse2_int_nat @ CurrentS2 @ AuxS2 @ Big @ nil_int_nat @ zero_zero_nat ) ) ) ).
% step_states.simps(1)
thf(fact_311_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_int,Big: stack_int,AuxB2: list_int,CurrentS2: current_int,Uu: stack_int,AuxS2: list_int] :
( ( type_s271243990432398725es_int @ ( states_int2 @ Dir @ ( reverse_int @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_int @ CurrentS2 @ Uu @ 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_312_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_a,Big: stack_a,AuxB2: list_a,CurrentS2: current_a,Uu: stack_a,AuxS2: list_a] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_a @ CurrentS2 @ Uu @ 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_313_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: 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 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_314_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y6: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_315_Suc__length__conv,axiom,
! [N: nat,Xs: list_int_nat] :
( ( ( suc @ N )
= ( size_s1172059598526859839nt_nat @ Xs ) )
= ( ? [Y6: int > nat,Ys: list_int_nat] :
( ( Xs
= ( cons_int_nat @ Y6 @ Ys ) )
& ( ( size_s1172059598526859839nt_nat @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_316_Suc__length__conv,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( suc @ N )
= ( size_s5157815400016825771nt_int @ Xs ) )
= ( ? [Y6: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys ) )
& ( ( size_s5157815400016825771nt_int @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_317_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y6: int,Ys: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys ) )
& ( ( size_size_list_int @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_318_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_319_length__0__conv,axiom,
! [Xs: list_int_nat] :
( ( ( size_s1172059598526859839nt_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_int_nat ) ) ).
% length_0_conv
thf(fact_320_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_321_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_322_length__greater__0__conv,axiom,
! [Xs: list_int_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1172059598526859839nt_nat @ Xs ) )
= ( Xs != nil_int_nat ) ) ).
% length_greater_0_conv
thf(fact_323_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_324_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X4: a,Ys2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_325_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X4: int,Ys2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_326_sorted__wrt_Ocases,axiom,
! [X: produc144329077504515498nt_nat] :
( ! [P3: ( int > nat ) > ( int > nat ) > $o] :
( X
!= ( produc4965991708622915748nt_nat @ P3 @ nil_int_nat ) )
=> ~ ! [P3: ( int > nat ) > ( int > nat ) > $o,X4: int > nat,Ys2: list_int_nat] :
( X
!= ( produc4965991708622915748nt_nat @ P3 @ ( cons_int_nat @ X4 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_327_sorted__wrt_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P3: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P3 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [P3: product_prod_int_int > product_prod_int_int > $o,X4: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P3 @ ( cons_P3334398858971670639nt_int @ X4 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_328_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X4: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X4: a,Y4: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_329_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ( ! [P3: int > int > $o,X4: int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ nil_int ) ) )
=> ~ ! [P3: int > int > $o,X4: int,Y4: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ ( cons_int @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_330_successively_Ocases,axiom,
! [X: produc144329077504515498nt_nat] :
( ! [P3: ( int > nat ) > ( int > nat ) > $o] :
( X
!= ( produc4965991708622915748nt_nat @ P3 @ nil_int_nat ) )
=> ( ! [P3: ( int > nat ) > ( int > nat ) > $o,X4: int > nat] :
( X
!= ( produc4965991708622915748nt_nat @ P3 @ ( cons_int_nat @ X4 @ nil_int_nat ) ) )
=> ~ ! [P3: ( int > nat ) > ( int > nat ) > $o,X4: int > nat,Y4: int > nat,Xs2: list_int_nat] :
( X
!= ( produc4965991708622915748nt_nat @ P3 @ ( cons_int_nat @ X4 @ ( cons_int_nat @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_331_successively_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P3: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P3 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [P3: product_prod_int_int > product_prod_int_int > $o,X4: product_prod_int_int] :
( X
!= ( produc3328129369365053992nt_int @ P3 @ ( cons_P3334398858971670639nt_int @ X4 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [P3: product_prod_int_int > product_prod_int_int > $o,X4: product_prod_int_int,Y4: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P3 @ ( cons_P3334398858971670639nt_int @ X4 @ ( cons_P3334398858971670639nt_int @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_332_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_333_list_Osize_I3_J,axiom,
( ( size_s1172059598526859839nt_nat @ nil_int_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_334_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_335_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F2: a > a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs ) ) )
=> ~ ! [F2: a > a,A4: a,As: list_a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_336_map__tailrec__rev_Ocases,axiom,
! [X: produc4909961631098372119st_int] :
( ! [F2: int > int,Bs: list_int] :
( X
!= ( produc740395959522281929st_int @ F2 @ ( produc364263696895485585st_int @ nil_int @ Bs ) ) )
=> ~ ! [F2: int > int,A4: int,As: list_int,Bs: list_int] :
( X
!= ( produc740395959522281929st_int @ F2 @ ( produc364263696895485585st_int @ ( cons_int @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_337_shuffles_Ocases,axiom,
! [X: produc2907033302207676215nt_nat] :
( ! [Ys2: list_int_nat] :
( X
!= ( produc5249521469742256623nt_nat @ nil_int_nat @ Ys2 ) )
=> ( ! [Xs2: list_int_nat] :
( X
!= ( produc5249521469742256623nt_nat @ Xs2 @ nil_int_nat ) )
=> ~ ! [X4: int > nat,Xs2: list_int_nat,Y4: int > nat,Ys2: list_int_nat] :
( X
!= ( produc5249521469742256623nt_nat @ ( cons_int_nat @ X4 @ Xs2 ) @ ( cons_int_nat @ Y4 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_338_shuffles_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ( ! [Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ Xs2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X4: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y4: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X4 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y4 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_339_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X4: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_340_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X4: int,Xs2: list_int,Y4: int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y4 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_341_splice_Ocases,axiom,
! [X: produc2907033302207676215nt_nat] :
( ! [Ys2: list_int_nat] :
( X
!= ( produc5249521469742256623nt_nat @ nil_int_nat @ Ys2 ) )
=> ~ ! [X4: int > nat,Xs2: list_int_nat,Ys2: list_int_nat] :
( X
!= ( produc5249521469742256623nt_nat @ ( cons_int_nat @ X4 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_342_splice_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ~ ! [X4: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X4 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_343_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X4: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_344_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ~ ! [X4: int,Xs2: list_int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_345_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_346_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_347_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_348_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_349_size__list,axiom,
! [Big: state_int_nat3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( type_i719020461660937083nt_nat @ Big )
=> ( ( big_list_int_nat @ Big )
!= nil_int_nat ) ) ) ).
% size_list
thf(fact_350_size__list,axiom,
! [Big: state_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( type_i293623501042546956te_int @ Big )
=> ( ( big_list_int @ Big )
!= nil_int ) ) ) ).
% size_list
thf(fact_351_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_352_Big__Proof_Olist__current__size,axiom,
! [Big: state_int_nat3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5894652653385020498nt_nat @ Big ) )
=> ( ( ( big_li1428258522215584551nt_nat @ Big )
= nil_int_nat )
=> ~ ( type_i719020461660937083nt_nat @ Big ) ) ) ).
% Big_Proof.list_current_size
thf(fact_353_Big__Proof_Olist__current__size,axiom,
! [Big: state_int3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Big ) )
=> ( ( ( big_list_current_int @ Big )
= nil_int )
=> ~ ( type_i293623501042546956te_int @ Big ) ) ) ).
% Big_Proof.list_current_size
thf(fact_354_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_355_Small__Proof_Olist__current__size,axiom,
! [Small: state_int_nat2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7563699036481858393nt_nat @ Small ) )
=> ( ( ( small_4712429614954843054nt_nat @ Small )
= nil_int_nat )
=> ~ ( type_i6582299730934091266nt_nat @ Small ) ) ) ).
% Small_Proof.list_current_size
thf(fact_356_Small__Proof_Olist__current__size,axiom,
! [Small: state_int2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int2 @ Small ) )
=> ( ( ( small_1709254455575119551nt_int @ Small )
= nil_int )
=> ~ ( type_i606337318274449939te_int @ Small ) ) ) ).
% Small_Proof.list_current_size
thf(fact_357_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_358_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_359_length__Suc__conv,axiom,
! [Xs: list_int_nat,N: nat] :
( ( ( size_s1172059598526859839nt_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int > nat,Ys: list_int_nat] :
( ( Xs
= ( cons_int_nat @ Y6 @ Ys ) )
& ( ( size_s1172059598526859839nt_nat @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_360_length__Suc__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ( size_s5157815400016825771nt_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
( ( Xs
= ( cons_P3334398858971670639nt_int @ Y6 @ Ys ) )
& ( ( size_s5157815400016825771nt_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_361_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys ) )
& ( ( size_size_list_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_362_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_363_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s1172059598526859839nt_nat
= ( ^ [Xs3: list_int_nat] : ( if_nat @ ( Xs3 = nil_int_nat ) @ zero_zero_nat @ ( suc @ ( size_s1172059598526859839nt_nat @ ( tl_int_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_364_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_int
= ( ^ [Xs3: list_int] : ( if_nat @ ( Xs3 = nil_int ) @ zero_zero_nat @ ( suc @ ( size_size_list_int @ ( tl_int @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_365_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_366_length__Cons,axiom,
! [X: int > nat,Xs: list_int_nat] :
( ( size_s1172059598526859839nt_nat @ ( cons_int_nat @ X @ Xs ) )
= ( suc @ ( size_s1172059598526859839nt_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_367_length__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( suc @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ).
% length_Cons
thf(fact_368_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_369_step__states_Opelims,axiom,
! [X: states_int_nat,Y: states_int_nat] :
( ( ( type_s2502071056353965684nt_nat @ X )
= Y )
=> ( ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ X )
=> ( ! [Dir2: direction,CurrentB: current_int_nat,Big2: stack_int_nat,AuxB: list_int_nat,CurrentS: current_int_nat,Uu2: stack_int_nat,AuxS: list_int_nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int_nat @ CurrentS @ Uu2 @ AuxS ) ) )
=> ( ( Y
= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( reverse_int_nat @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_int_nat @ CurrentS @ AuxS @ Big2 @ nil_int_nat @ zero_zero_nat ) ) )
=> ~ ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int_nat @ CurrentS @ Uu2 @ AuxS ) ) ) ) )
=> ( ! [Dir2: direction,V2: current_int_nat,Va2: stack_int_nat,Vb2: list_int_nat,Vd2: nat,Right2: state_int_nat2] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ( Y
= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( reverse_int_nat @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s6378993889865489038nt_nat @ Right2 ) ) )
=> ~ ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ ( states_int_nat2 @ Dir2 @ ( reverse_int_nat @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) ) ) )
=> ( ! [Dir2: direction,V2: state_int_nat,Right2: state_int_nat2] :
( ( X
= ( states_int_nat2 @ Dir2 @ ( common_int_nat @ V2 ) @ Right2 ) )
=> ( ( Y
= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ ( common_int_nat @ V2 ) ) @ ( type_s6378993889865489038nt_nat @ Right2 ) ) )
=> ~ ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ ( states_int_nat2 @ Dir2 @ ( common_int_nat @ V2 ) @ Right2 ) ) ) )
=> ( ! [Dir2: direction,Left2: state_int_nat3,V2: current_int_nat,Va2: list_int_nat,Vb2: stack_int_nat,Vc2: list_int_nat,Vd2: nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ Left2 @ ( reverse2_int_nat @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( ( Y
= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ Left2 ) @ ( type_s6378993889865489038nt_nat @ ( reverse2_int_nat @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) )
=> ~ ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ ( states_int_nat2 @ Dir2 @ Left2 @ ( reverse2_int_nat @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir2: direction,Left2: state_int_nat3,V2: state_int_nat] :
( ( X
= ( states_int_nat2 @ Dir2 @ Left2 @ ( common_int_nat2 @ V2 ) ) )
=> ( ( Y
= ( states_int_nat2 @ Dir2 @ ( type_s7127291786553863687nt_nat @ Left2 ) @ ( type_s6378993889865489038nt_nat @ ( common_int_nat2 @ V2 ) ) ) )
=> ~ ( accp_states_int_nat @ step_s6801834171724358809nt_nat @ ( states_int_nat2 @ Dir2 @ Left2 @ ( common_int_nat2 @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% step_states.pelims
thf(fact_370_step__states_Opelims,axiom,
! [X: states_int,Y: states_int] :
( ( ( type_s271243990432398725es_int @ X )
= Y )
=> ( ( accp_states_int @ step_states_rel_int @ X )
=> ( ! [Dir2: direction,CurrentB: current_int,Big2: stack_int,AuxB: list_int,CurrentS: current_int,Uu2: stack_int,AuxS: list_int] :
( ( X
= ( states_int2 @ Dir2 @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int @ CurrentS @ Uu2 @ 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 ) ) )
=> ~ ( accp_states_int @ step_states_rel_int @ ( states_int2 @ Dir2 @ ( reverse_int @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_int @ CurrentS @ Uu2 @ 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 ) )
=> ( ( Y
= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ ( reverse_int @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3231261300627247263te_int @ Right2 ) ) )
=> ~ ( accp_states_int @ step_states_rel_int @ ( 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 ) )
=> ( ( Y
= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ ( common_int @ V2 ) ) @ ( type_s3231261300627247263te_int @ Right2 ) ) )
=> ~ ( accp_states_int @ step_states_rel_int @ ( 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 ) ) )
=> ( ( Y
= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ Left2 ) @ ( type_s3231261300627247263te_int @ ( reverse2_int @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) )
=> ~ ( accp_states_int @ step_states_rel_int @ ( 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 ) ) )
=> ( ( Y
= ( states_int2 @ Dir2 @ ( type_s5405555405925207448te_int @ Left2 ) @ ( type_s3231261300627247263te_int @ ( common_int2 @ V2 ) ) ) )
=> ~ ( accp_states_int @ step_states_rel_int @ ( states_int2 @ Dir2 @ Left2 @ ( common_int2 @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% step_states.pelims
thf(fact_371_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,Uu2: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir2 @ ( reverse_a @ CurrentB @ Big2 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu2 @ 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 @ Uu2 @ 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_372_Cons__lenlex__iff,axiom,
! [M2: int > nat,Ms: list_int_nat,N: int > nat,Ns: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
( ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ ( cons_int_nat @ M2 @ Ms ) @ ( cons_int_nat @ N @ Ns ) ) @ ( lenlex_int_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_s1172059598526859839nt_nat @ Ms ) @ ( size_s1172059598526859839nt_nat @ Ns ) )
| ( ( ( size_s1172059598526859839nt_nat @ Ms )
= ( size_s1172059598526859839nt_nat @ Ns ) )
& ( member4491287985726366656nt_nat @ ( produc7977806053278589903nt_nat @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ Ms @ Ns ) @ ( lenlex_int_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_373_Cons__lenlex__iff,axiom,
! [M2: product_prod_int_int,Ms: list_P5707943133018811711nt_int,N: product_prod_int_int,Ns: list_P5707943133018811711nt_int,R2: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ M2 @ Ms ) @ ( cons_P3334398858971670639nt_int @ N @ Ns ) ) @ ( lenlex6370358691973319492nt_int @ R2 ) )
= ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) )
| ( ( ( size_s5157815400016825771nt_int @ Ms )
= ( size_s5157815400016825771nt_int @ Ns ) )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_374_Cons__lenlex__iff,axiom,
! [M2: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M2 @ Ms ) @ ( cons_list_a @ N @ Ns ) ) @ ( lenlex_list_a @ R2 ) )
= ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) )
| ( ( ( size_s349497388124573686list_a @ Ms )
= ( size_s349497388124573686list_a @ Ns ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_375_Cons__lenlex__iff,axiom,
! [M2: list_int,Ms: list_list_int,N: list_int,Ns: list_list_int,R2: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ ( cons_list_int @ M2 @ 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 @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Ms @ Ns ) @ ( lenlex_list_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_376_Cons__lenlex__iff,axiom,
! [M2: a,Ms: list_a,N: a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M2 @ 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 @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_377_Cons__lenlex__iff,axiom,
! [M2: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ 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 @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_378_Cons__in__lex,axiom,
! [X: int > nat,Xs: list_int_nat,Y: int > nat,Ys4: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
( ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ ( cons_int_nat @ X @ Xs ) @ ( cons_int_nat @ Y @ Ys4 ) ) @ ( lex_int_nat @ R2 ) )
= ( ( ( member4491287985726366656nt_nat @ ( produc7977806053278589903nt_nat @ X @ Y ) @ R2 )
& ( ( size_s1172059598526859839nt_nat @ Xs )
= ( size_s1172059598526859839nt_nat @ Ys4 ) ) )
| ( ( X = Y )
& ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ Xs @ Ys4 ) @ ( lex_int_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_379_Cons__in__lex,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys4: list_P5707943133018811711nt_int,R2: set_Pr2560585780119916871nt_int] :
( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys4 ) ) @ ( lex_Pr5393148144989827363nt_int @ R2 ) )
= ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R2 )
& ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys4 ) ) )
| ( ( X = Y )
& ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys4 ) @ ( lex_Pr5393148144989827363nt_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_380_Cons__in__lex,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys4: list_list_a,R2: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys4 ) ) @ ( lex_list_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R2 )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys4 ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys4 ) @ ( lex_list_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_381_Cons__in__lex,axiom,
! [X: list_int,Xs: list_list_int,Y: list_int,Ys4: list_list_int,R2: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ ( cons_list_int @ X @ Xs ) @ ( cons_list_int @ Y @ Ys4 ) ) @ ( lex_list_int @ R2 ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ R2 )
& ( ( size_s533118279054570080st_int @ Xs )
= ( size_s533118279054570080st_int @ Ys4 ) ) )
| ( ( X = Y )
& ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Xs @ Ys4 ) @ ( lex_list_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_382_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys4: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys4 ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys4 ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys4 ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_383_Cons__in__lex,axiom,
! [X: int,Xs: list_int,Y: int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys4 ) ) @ ( lex_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys4 ) ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( lex_int @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_384_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_385_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_386_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_387_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_388_Nil__lenlex__iff1,axiom,
! [Ns: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
( ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ nil_int_nat @ Ns ) @ ( lenlex_int_nat @ R2 ) )
= ( Ns != nil_int_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_389_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_390_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_391_Nil__notin__lex,axiom,
! [Ys4: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
~ ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ nil_int_nat @ Ys4 ) @ ( lex_int_nat @ R2 ) ) ).
% Nil_notin_lex
thf(fact_392_Nil__notin__lex,axiom,
! [Ys4: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys4 ) @ ( lex_a @ R2 ) ) ).
% Nil_notin_lex
thf(fact_393_Nil__notin__lex,axiom,
! [Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ys4 ) @ ( lex_int @ R2 ) ) ).
% Nil_notin_lex
thf(fact_394_Nil2__notin__lex,axiom,
! [Xs: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
~ ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ Xs @ nil_int_nat ) @ ( lex_int_nat @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_395_Nil2__notin__lex,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_396_Nil2__notin__lex,axiom,
! [Xs: list_int,R2: set_Pr958786334691620121nt_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ nil_int ) @ ( lex_int @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_397_lenlex__irreflexive,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a] :
( ! [X4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_398_lenlex__irreflexive,axiom,
! [R2: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ! [X4: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ X4 ) @ R2 )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Xs ) @ ( lenlex_list_a @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_399_lenlex__irreflexive,axiom,
! [R2: set_Pr765067013931698361st_int,Xs: list_list_int] :
( ! [X4: list_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X4 @ X4 ) @ R2 )
=> ~ ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Xs @ Xs ) @ ( lenlex_list_int @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_400_lenlex__irreflexive,axiom,
! [R2: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_401_Nil__lenlex__iff2,axiom,
! [Ns: list_int_nat,R2: set_Pr1394042615296016247nt_nat] :
~ ( member7939632110732153312nt_nat @ ( produc5249521469742256623nt_nat @ Ns @ nil_int_nat ) @ ( lenlex_int_nat @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_402_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_403_Nil__lenlex__iff2,axiom,
! [Ns: list_int,R2: set_Pr958786334691620121nt_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ns @ nil_int ) @ ( lenlex_int @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_404_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_405_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_406_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_407_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_408_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_409_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_410_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_411_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_412_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_413_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_414_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_415_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_416_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_417_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_418_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_419_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_420_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_421_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_422_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_423_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_424_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_425_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_426_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_427_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_428_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_429_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_430_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_431_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_432_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_433_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_434_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_435_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_436_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_437_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_438_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_439_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_440_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_441_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_442_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_443_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_444_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_445_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_446_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_447_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P5 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_457_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_458_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_459_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_460_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_461_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_462_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_463_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_464_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_465_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_466_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_467_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_468_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_469_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_470_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_471_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_472_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_473_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_474_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_475_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_476_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_477_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_478_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_479_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_480_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_481_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_482_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_483_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_484_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_485_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_486_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_487_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_488_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_489_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_490_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_491_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_492_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_493_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_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_nat @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_520_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y6: int] :
( ( ord_less_int @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_521_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y6: nat] :
( ( ord_less_eq_nat @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_522_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y6: int] :
( ( ord_less_eq_int @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_523_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_524_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_525_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_526_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_527_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_528_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_529_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_530_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_531_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_532_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_533_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_534_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_535_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_536_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_537_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_538_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_539_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 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_540_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_541_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_542_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 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_543_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_544_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_545_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_546_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 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_547_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_548_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_549_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_550_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_551_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 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_552_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 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_553_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_554_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_555_lists__current_Opelims,axiom,
! [X: states_int,Y: produc1186641810826059865st_int] :
( ( ( states4446210166810079719nt_int @ X )
= Y )
=> ( ( accp_states_int @ states6576289658430312748el_int @ X )
=> ~ ! [Uu2: direction,Big2: state_int3,Small2: state_int2] :
( ( X
= ( states_int2 @ Uu2 @ Big2 @ Small2 ) )
=> ( ( Y
= ( produc364263696895485585st_int @ ( big_list_current_int @ Big2 ) @ ( small_1709254455575119551nt_int @ Small2 ) ) )
=> ~ ( accp_states_int @ states6576289658430312748el_int @ ( states_int2 @ Uu2 @ Big2 @ Small2 ) ) ) ) ) ) ).
% lists_current.pelims
thf(fact_556_lists__current_Opelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ( ( accp_states_a @ states5251248496104418302_rel_a @ X )
=> ~ ! [Uu2: direction,Big2: state_a3,Small2: state_a2] :
( ( X
= ( states_a2 @ Uu2 @ Big2 @ Small2 ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big2 ) @ ( small_list_current_a @ Small2 ) ) )
=> ~ ( accp_states_a @ states5251248496104418302_rel_a @ ( states_a2 @ Uu2 @ Big2 @ Small2 ) ) ) ) ) ) ).
% lists_current.pelims
thf(fact_557_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_558_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ~ ( ord_less_eq_int @ T @ X7 ) ) ).
% minf(8)
thf(fact_559_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_560_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ord_less_eq_int @ X7 @ T ) ) ).
% minf(6)
thf(fact_561_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_562_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ord_less_eq_int @ T @ X7 ) ) ).
% pinf(8)
thf(fact_563_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_564_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ~ ( ord_less_eq_int @ X7 @ T ) ) ).
% pinf(6)
thf(fact_565_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_566_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_567_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 )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_568_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 )
& ! [X7: int] :
( ( ( ord_less_eq_int @ A @ X7 )
& ( ord_less_int @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_569_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_570_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_571_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_572_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_573_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_574_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_575_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_576_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_577_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_578_minf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_579_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_580_minf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z3 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_581_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_582_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_583_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_584_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_585_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_586_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_587_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_588_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_589_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_590_pinf_I2_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_591_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_592_pinf_I1_J,axiom,
! [P: int > $o,P6: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( P @ X4 )
= ( P6 @ X4 ) ) )
=> ( ? [Z3: int] :
! [X4: int] :
( ( ord_less_int @ Z3 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: int] :
! [X7: int] :
( ( ord_less_int @ Z @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_593_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_594_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_595_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_596_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int_nat @ N @ nil_int_nat )
= ( cons_list_int_nat2 @ nil_int_nat @ nil_list_int_nat2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int_nat @ N @ nil_int_nat )
= nil_list_int_nat2 ) ) ) ).
% n_lists_Nil
thf(fact_597_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= nil_list_int ) ) ) ).
% n_lists_Nil
thf(fact_598_in__measures_I2_J,axiom,
! [X: list_a,Y: list_a,F: list_a > nat,Fs: list_list_a_nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_599_in__measures_I2_J,axiom,
! [X: list_int,Y: list_int,F: list_int > nat,Fs: list_list_int_nat2] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ ( cons_list_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_600_in__measures_I2_J,axiom,
! [X: int,Y: int,F: int > nat,Fs: list_int_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_601_length__code,axiom,
( size_size_list_int
= ( gen_length_int @ zero_zero_nat ) ) ).
% length_code
thf(fact_602_in__measures_I1_J,axiom,
! [X: list_a,Y: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ nil_list_a_nat ) ) ).
% in_measures(1)
thf(fact_603_in__measures_I1_J,axiom,
! [X: list_int,Y: list_int] :
~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ nil_list_int_nat ) ) ).
% in_measures(1)
thf(fact_604_in__measures_I1_J,axiom,
! [X: int,Y: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ nil_int_nat ) ) ).
% in_measures(1)
thf(fact_605_measures__less,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_606_measures__less,axiom,
! [F: list_int > nat,X: list_int,Y: list_int,Fs: list_list_int_nat2] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ ( cons_list_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_607_measures__less,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_608_measures__lesseq,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_609_measures__lesseq,axiom,
! [F: list_int > nat,X: list_int,Y: list_int,Fs: list_list_int_nat2] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ Fs ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( measures_list_int @ ( cons_list_int_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_610_measures__lesseq,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_611_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_612_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int_nat] :
( ( n_lists_int_nat @ zero_zero_nat @ Xs )
= ( cons_list_int_nat2 @ nil_int_nat @ nil_list_int_nat2 ) ) ).
% n_lists.simps(1)
thf(fact_613_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( n_lists_int @ zero_zero_nat @ Xs )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% n_lists.simps(1)
thf(fact_614_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 ) )
=> ~ ! [X4: int,Y4: int,Xs2: list_int] :
( ( A1
= ( cons_int @ X4 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A22
= ( cons_int @ Y4 @ Ys2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys2 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_615_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 ) )
| ? [X5: int,Y6: int,Xs3: list_int,Ys: list_int] :
( ( A1
= ( cons_int @ X5 @ Xs3 ) )
& ( A22
= ( cons_int @ Y6 @ Ys ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y6 ) @ R2 )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_616_listrel_OCons,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys4: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel_int_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys4 ) ) @ ( listrel_int_int @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_617_listrel__Cons1,axiom,
! [Y: int,Ys4: list_int,Xs: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys4 ) @ Xs ) @ ( listrel_int_int @ R2 ) )
=> ~ ! [Y4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys4 @ Ys2 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_618_listrel__Cons2,axiom,
! [Xs: list_int,Y: int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys4 ) ) @ ( listrel_int_int @ R2 ) )
=> ~ ! [X4: int,Xs2: list_int] :
( ( Xs
= ( cons_int @ X4 @ Xs2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys4 ) @ ( listrel_int_int @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_619_listrel__iff__nth,axiom,
! [Xs: list_int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel_int_int @ R2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys4 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_int @ Xs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N4 ) @ ( nth_int @ Ys4 @ N4 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_620_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_621_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_622_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_623_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_624_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_625_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_626_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_627_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_628_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_629_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_630_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_631_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_632_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_633_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_634_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_635_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_636_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_637_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_638_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_639_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_640_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_641_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_642_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_643_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_644_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_645_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_646_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_647_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_648_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_649_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_650_lexord__irreflexive,axiom,
! [R2: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R2 )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lexord_int @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_651_lexord__linear,axiom,
! [R2: set_Pr958786334691620121nt_int,X: list_int,Y: list_int] :
( ! [A4: int,B3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B3 ) @ R2 )
| ( A4 = B3 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B3 @ A4 ) @ R2 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R2 ) )
| ( X = Y )
| ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Y @ X ) @ ( lexord_int @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_652_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_653_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_654_lex__take__index,axiom,
! [Xs: list_int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( lex_int @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys4 ) )
=> ( ( ( take_int @ I3 @ Xs )
= ( take_int @ I3 @ Ys4 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys4 @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_655_lexord__partial__trans,axiom,
! [Xs: list_int,R2: set_Pr958786334691620121nt_int,Ys4: list_int,Zs: list_int] :
( ! [X4: int,Y4: int,Z: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ Z ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Z ) @ R2 ) ) ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( lexord_int @ R2 ) )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys4 @ Zs ) @ ( lexord_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs ) @ ( lexord_int @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_656_lexord__take__index__conv,axiom,
! [X: list_int,Y: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R2 ) )
= ( ( ( ord_less_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) )
& ( ( take_int @ ( size_size_list_int @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) ) )
& ( ( take_int @ I2 @ X )
= ( take_int @ I2 @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ X @ I2 ) @ ( nth_int @ Y @ I2 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_657_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_658_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_659_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_660_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_661_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_662_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_663_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_664_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_665_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_666_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_667_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_668_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_669_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_679_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_680_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_681_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_682_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_683_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_684_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_685_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_686_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_687_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_688_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_689_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_690_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_691_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_692_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_693_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_694_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_695_lexord__append__leftD,axiom,
! [X: list_int,U: list_int,V: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ X @ U ) @ ( append_int @ X @ V ) ) @ ( lexord_int @ R2 ) )
=> ( ! [A4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ A4 ) @ R2 )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ U @ V ) @ ( lexord_int @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_696_lex__append__left__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys4: list_int,Zs: list_int] :
( ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R2 )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys4 ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R2 ) )
= ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys4 @ Zs ) @ ( lex_int @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_697_lex__append__leftD,axiom,
! [R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys4: list_int,Zs: list_int] :
( ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R2 )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys4 ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R2 ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys4 @ Zs ) @ ( lex_int @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_698_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_699_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_700_lexord__same__pref__iff,axiom,
! [Xs: list_int,Ys4: list_int,Zs: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys4 ) @ ( append_int @ Xs @ Zs ) ) @ ( lexord_int @ R2 ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ ( set_int2 @ Xs ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ X5 ) @ R2 ) )
| ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys4 @ Zs ) @ ( lexord_int @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_701_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_702_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_703_min__less__iff__conj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z2 @ X )
& ( ord_less_nat @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_704_min__less__iff__conj,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ Z2 @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z2 @ X )
& ( ord_less_int @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_705_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_706_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_707_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_708_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_709_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
| ( ord_less_nat @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_710_min__less__iff__disj,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z2 )
= ( ( ord_less_int @ X @ Z2 )
| ( ord_less_int @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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 ) ) )
=> ( ! [Current: current_a] :
( ? [Uu2: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
= ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw @ Ux ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) )
=> ~ ! [Current: current_a] :
( ? [Uy: stack_a,Uz: list_a] :
( X
= ( reverse1_a @ Current @ Uy @ Uz ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ) ) ) ) ).
% Small_Aux.size_state.elims
thf(fact_720_Big__Aux_Osize__state_Oelims,axiom,
! [X: state_a3,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ! [State: state_a] :
( ( X
= ( common_a @ State ) )
=> ( Y
!= ( size_size_state_a2 @ State ) ) )
=> ~ ! [Current: current_a] :
( ? [Uu2: stack_a,Uv2: list_a,Uw: nat] :
( X
= ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ) ) ) ).
% Big_Aux.size_state.elims
thf(fact_721_Big__Aux_Osize__new__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu: stack_a,Uv: list_a,Uw2: nat] :
( ( type_s6530235180886170618tate_a @ ( reverse_a @ Current2 @ Uu @ Uv @ Uw2 ) )
= ( type_s933026853152659577rent_a @ Current2 ) ) ).
% Big_Aux.size_new_state.simps(2)
thf(fact_722_Big__Aux_Osize__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu: stack_a,Uv: list_a,Uw2: nat] :
( ( size_size_state_a @ ( reverse_a @ Current2 @ Uu @ Uv @ Uw2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ).
% Big_Aux.size_state.simps(2)
thf(fact_723_Small__Aux_Osize__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu: list_a,Uv: stack_a,Uw2: list_a,Ux2: nat] :
( ( size_size_state_a3 @ ( reverse2_a @ Current2 @ Uu @ Uv @ Uw2 @ Ux2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ).
% Small_Aux.size_state.simps(2)
thf(fact_724_Small__Aux_Osize__state_Osimps_I3_J,axiom,
! [Current2: current_a,Uy2: stack_a,Uz2: list_a] :
( ( size_size_state_a3 @ ( reverse1_a @ Current2 @ Uy2 @ Uz2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ).
% Small_Aux.size_state.simps(3)
thf(fact_725_listrel1__iff__update,axiom,
! [Xs: list_int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel1_int @ R2 ) )
= ( ? [Y6: int,N4: nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N4 ) @ Y6 ) @ R2 )
& ( ord_less_nat @ N4 @ ( size_size_list_int @ Xs ) )
& ( Ys4
= ( list_update_int @ Xs @ N4 @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_726_Big__Aux_Osize__new__state_Oelims,axiom,
! [X: state_a3,Y: nat] :
( ( ( type_s6530235180886170618tate_a @ X )
= Y )
=> ( ! [State: state_a] :
( ( X
= ( common_a @ State ) )
=> ( Y
!= ( type_s8424385952999958455tate_a @ State ) ) )
=> ~ ! [Current: current_a] :
( ? [Uu2: stack_a,Uv2: list_a,Uw: nat] :
( X
= ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current ) ) ) ) ) ).
% Big_Aux.size_new_state.elims
thf(fact_727_Small__Aux_Osize__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a3 @ X )
= Y )
=> ( ( accp_state_a2 @ small_6459853017724497003_rel_a @ X )
=> ( ! [State: state_a] :
( ( X
= ( common_a2 @ State ) )
=> ( ( Y
= ( size_size_state_a2 @ State ) )
=> ~ ( accp_state_a2 @ small_6459853017724497003_rel_a @ ( common_a2 @ State ) ) ) )
=> ( ! [Current: current_a,Uu2: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( ( X
= ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw @ Ux ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) )
=> ~ ( accp_state_a2 @ small_6459853017724497003_rel_a @ ( reverse2_a @ Current @ Uu2 @ Uv2 @ Uw @ Ux ) ) ) )
=> ~ ! [Current: current_a,Uy: stack_a,Uz: list_a] :
( ( X
= ( reverse1_a @ Current @ Uy @ Uz ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) )
=> ~ ( accp_state_a2 @ small_6459853017724497003_rel_a @ ( reverse1_a @ Current @ Uy @ Uz ) ) ) ) ) ) ) ) ).
% Small_Aux.size_state.pelims
thf(fact_728_Cons__listrel1__Cons,axiom,
! [X: int,Xs: list_int,Y: int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys4 ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
& ( Xs = Ys4 ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_729_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_730_Cons__listrel1E1,axiom,
! [X: int,Xs: list_int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys4 ) @ ( listrel1_int @ R2 ) )
=> ( ! [Y4: int] :
( ( Ys4
= ( cons_int @ Y4 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Ys4
= ( cons_int @ X @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_731_Cons__listrel1E2,axiom,
! [Xs: list_int,Y: int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys4 ) ) @ ( listrel1_int @ R2 ) )
=> ( ! [X4: int] :
( ( Xs
= ( cons_int @ X4 @ Ys4 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_int] :
( ( Xs
= ( cons_int @ Y @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys4 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_732_listrel1I,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys4: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_int @ Us @ ( cons_int @ X @ Vs ) ) )
=> ( ( Ys4
= ( append_int @ Us @ ( cons_int @ Y @ Vs ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel1_int @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_733_listrel1E,axiom,
! [Xs: list_int,Ys4: list_int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel1_int @ R2 ) )
=> ~ ! [X4: int,Y4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y4 ) @ R2 )
=> ! [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ ( cons_int @ X4 @ Vs2 ) ) )
=> ( Ys4
!= ( append_int @ Us2 @ ( cons_int @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_734_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X: int,Ys4: list_int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) @ ( append_int @ Ys4 @ ( cons_int @ Y @ nil_int ) ) ) @ ( listrel1_int @ R2 ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys4 ) @ ( listrel1_int @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys4 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_735_Big__Aux_Osize__state_Opelims,axiom,
! [X: state_a3,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ( accp_state_a @ big_size_state_rel_a @ X )
=> ( ! [State: state_a] :
( ( X
= ( common_a @ State ) )
=> ( ( Y
= ( size_size_state_a2 @ State ) )
=> ~ ( accp_state_a @ big_size_state_rel_a @ ( common_a @ State ) ) ) )
=> ~ ! [Current: current_a,Uu2: stack_a,Uv2: list_a,Uw: nat] :
( ( X
= ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) )
=> ~ ( accp_state_a @ big_size_state_rel_a @ ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) ) ) ) ) ) ) ).
% Big_Aux.size_state.pelims
thf(fact_736_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_737_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_738_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_739_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_740_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_741_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_742_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_743_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_744_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_745_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_746_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_747_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_748_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_749_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_750_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_751_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_752_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_753_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_754_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_755_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_756_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_757_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_758_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_759_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z2: int] :
( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z2 )
= ( ord_min_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% min_diff_distrib_left
thf(fact_760_min__diff,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).
% min_diff
thf(fact_761_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_762_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_763_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_764_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_765_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_766_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_767_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_768_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_769_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_770_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_771_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_772_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_773_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_774_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_775_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_776_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_777_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_787_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_788_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_789_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_790_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_791_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_792_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_793_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_794_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_795_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_796_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_797_Big__Aux_Osize__new__state_Opelims,axiom,
! [X: state_a3,Y: nat] :
( ( ( type_s6530235180886170618tate_a @ X )
= Y )
=> ( ( accp_state_a @ big_si5937185285519891526_rel_a @ X )
=> ( ! [State: state_a] :
( ( X
= ( common_a @ State ) )
=> ( ( Y
= ( type_s8424385952999958455tate_a @ State ) )
=> ~ ( accp_state_a @ big_si5937185285519891526_rel_a @ ( common_a @ State ) ) ) )
=> ~ ! [Current: current_a,Uu2: stack_a,Uv2: list_a,Uw: nat] :
( ( X
= ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current ) )
=> ~ ( accp_state_a @ big_si5937185285519891526_rel_a @ ( reverse_a @ Current @ Uu2 @ Uv2 @ Uw ) ) ) ) ) ) ) ).
% Big_Aux.size_new_state.pelims
thf(fact_798_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_799_min__0__1_I2_J,axiom,
( ( ord_min_int @ one_one_int @ zero_zero_int )
= zero_zero_int ) ).
% min_0_1(2)
thf(fact_800_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_801_min__0__1_I1_J,axiom,
( ( ord_min_int @ zero_zero_int @ one_one_int )
= zero_zero_int ) ).
% min_0_1(1)
thf(fact_802_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_803_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_804_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_805_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_806_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_807_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_808_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_809_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_810_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_811_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_812_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_813_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_814_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_815_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_816_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_817_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_818_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_819_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_820_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_821_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_822_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_823_Suc__sub,axiom,
! [N: nat,M2: nat] :
( ( ( suc @ N )
= M2 )
=> ( N
= ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_824_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_825_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_826_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_827_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_828_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_829_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_830_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_831_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_832_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_833_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_834_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_835_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_836_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_837_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_838_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_839_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_840_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_841_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_842_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_843_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_844_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_845_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_846_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_847_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_848_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_849_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_850_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_851_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_852_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_853_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_854_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_855_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_856_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_857_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_858_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_859_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_860_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_861_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_862_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_863_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_864_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_865_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_866_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M2 @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_883_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_884_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_885_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_886_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_887_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_888_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_889_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_890_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_891_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_892_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_893_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_894_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_895_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_896_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_908_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_922_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_923_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_924_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_925_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_926_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_927_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_928_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_929_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_930_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_931_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_932_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_955_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_956_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_957_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_958_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_959_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_960_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_961_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_962_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_963_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_964_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_965_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_966_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_967_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_968_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_969_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_970_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_971_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_972_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_973_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_974_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_975_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_976_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_977_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_978_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_979_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_980_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_981_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_982_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_983_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_984_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_985_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_986_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_987_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_988_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_989_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_990_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_991_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_992_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_993_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_994_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_995_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_996_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_997_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_999_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1000_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1001_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1002_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1003_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1004_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1005_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_1006_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
thf(fact_1007_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_1008_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_1009_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_1010_min__add__distrib__left,axiom,
! [X: int,Y: int,Z2: int] :
( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z2 )
= ( ord_min_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).
% min_add_distrib_left
thf(fact_1011_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% min_add_distrib_left
thf(fact_1012_min__add__distrib__right,axiom,
! [X: int,Y: int,Z2: int] :
( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z2 ) )
= ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% min_add_distrib_right
thf(fact_1013_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z2 ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% min_add_distrib_right
thf(fact_1014_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1015_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1016_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1017_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1018_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1019_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J2 )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1033_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1034_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1035_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1036_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1037_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_1038_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1039_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1040_trans__le__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1041_trans__le__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1042_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_1043_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1044_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1045_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1046_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1047_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1048_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1049_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1050_trans__less__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1051_trans__less__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1052_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1053_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1054_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1055_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1056_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_1057_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_1058_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1059_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1071_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1072_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1073_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1074_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1075_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1076_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_1077_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1078_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1079_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J2: int,K: int,L: int] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1080_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_1081_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1082_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1083_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1084_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_1085_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_1086_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_1087_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_1088_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1089_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1090_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_1091_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_1092_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_1093_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_1094_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_1095_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_1096_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_1097_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1098_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1099_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1100_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1101_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1102_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1103_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1128_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1129_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1130_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1131_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_1132_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_1133_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_1134_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_1135_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1136_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1137_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1138_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1139_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1140_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1141_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_1142_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_1143_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_1144_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1145_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1146_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1147_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1148_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1149_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1150_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1151_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1152_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1153_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1154_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1155_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1156_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1157_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1158_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1159_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1160_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1161_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1162_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1163_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1164_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1165_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1166_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1167_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1168_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1169_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1170_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1171_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1172_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1173_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1174_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1175_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1176_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1177_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1178_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1179_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1180_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_1181_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C2: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1182_abs__diff__less__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1183_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1184_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1185_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1186_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1187_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1188_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1189_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1190_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1191_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1192_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1193_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1194_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1195_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1196_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1197_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1198_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1199_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1200_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1201_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1202_nth__zip,axiom,
! [I: nat,Xs: list_int,Ys4: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys4 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys4 ) @ I )
= ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys4 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1203_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_1204_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_1205_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_1206_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_1207_zip__replicate,axiom,
! [I: nat,X: int,J2: nat,Y: int] :
( ( zip_int_int @ ( replicate_int @ I @ X ) @ ( replicate_int @ J2 @ Y ) )
= ( replic1057375728873637753nt_int @ ( ord_min_nat @ I @ J2 ) @ ( product_Pair_int_int @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_1208_zip__Cons__Cons,axiom,
! [X: int,Xs: list_int,Y: int,Ys4: list_int] :
( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys4 ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys4 ) ) ) ).
% zip_Cons_Cons
thf(fact_1209_zip__update,axiom,
! [Xs: list_int,I: nat,X: int,Ys4: list_int,Y: int] :
( ( zip_int_int @ ( list_update_int @ Xs @ I @ X ) @ ( list_update_int @ Ys4 @ I @ Y ) )
= ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys4 ) @ I @ ( product_Pair_int_int @ X @ Y ) ) ) ).
% zip_update
thf(fact_1210_zip__same,axiom,
! [A: int,B: int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
= ( ( member_int @ A @ ( set_int2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_1211_in__set__zipE,axiom,
! [X: int,Y: int,Xs: list_int,Ys4: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys4 ) ) )
=> ~ ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ~ ( member_int @ Y @ ( set_int2 @ Ys4 ) ) ) ) ).
% in_set_zipE
thf(fact_1212_set__zip__leftD,axiom,
! [X: int,Y: int,Xs: list_int,Ys4: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys4 ) ) )
=> ( member_int @ X @ ( set_int2 @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1213_set__zip__rightD,axiom,
! [X: int,Y: int,Xs: list_int,Ys4: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys4 ) ) )
=> ( member_int @ Y @ ( set_int2 @ Ys4 ) ) ) ).
% set_zip_rightD
thf(fact_1214_zip__eq__ConsE,axiom,
! [Xs: list_int,Ys4: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs @ Ys4 )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y4: int,Ys5: list_int] :
( ( Ys4
= ( cons_int @ Y4 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_int_int @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1215_hd__zip,axiom,
! [Xs: list_int,Ys4: list_int] :
( ( Xs != nil_int )
=> ( ( Ys4 != nil_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( zip_int_int @ Xs @ Ys4 ) )
= ( product_Pair_int_int @ ( hd_int @ Xs ) @ ( hd_int @ Ys4 ) ) ) ) ) ).
% hd_zip
thf(fact_1216_in__set__impl__in__set__zip1,axiom,
! [Xs: list_int,Ys4: list_int,X: int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys4 ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ~ ! [Y4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys4 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1217_in__set__impl__in__set__zip2,axiom,
! [Xs: list_int,Ys4: list_int,Y: int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys4 ) )
=> ( ( member_int @ Y @ ( set_int2 @ Ys4 ) )
=> ~ ! [X4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys4 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1218_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_1219_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1220_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1221_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1222_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1223_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1224_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1225_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1226_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1227_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1228_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1229_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1230_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1231_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1232_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1233_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1234_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1235_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1236_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1237_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1238_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1239_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1240_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1241_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1242_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1243_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1244_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1245_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1246_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_1247_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1248_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1249_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1250_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1251_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1252_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1253_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_1254_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_1255_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1256_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_1257_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1258_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1259_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_1260_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_1261_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1262_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_1263_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1264_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1265_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
% Helper facts (5)
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_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( type_i6304938058965754292tate_a @ big )
& ( type_i464410347872898157tate_a @ small ) ) ).
%------------------------------------------------------------------------------