TPTP Problem File: SLH0946^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0025_States_Proof/prob_00523_016978__6994886_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1214 ( 435 unt; 265 typ; 0 def)
% Number of atoms : 2477 (1373 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 9146 ( 459 ~; 63 |; 171 &;7407 @)
% ( 0 <=>;1046 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 48 ( 47 usr)
% Number of type conns : 480 ( 480 >; 0 *; 0 +; 0 <<)
% Number of symbols : 221 ( 218 usr; 21 con; 0-5 aty)
% Number of variables : 2946 ( 51 ^;2780 !; 115 ?;2946 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:35.789
%------------------------------------------------------------------------------
% Could-be-implicit typings (47)
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,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Common__Ostate_Itf__a_J,type,
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thf(ty_n_t__States__Odirection,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (218)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
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thf(sy_c_Big_Opush_001tf__a,type,
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common_a: state_a > state_a3 ).
thf(sy_c_Big_Ostate_OReverse_001tf__a,type,
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thf(sy_c_Fun__Def_Ois__measure_001t__Small__Ostate_Itf__a_J,type,
fun_is6829564153850791975tate_a: ( state_a2 > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__Stack__Ostack_Itf__a_J,type,
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thf(sy_c_Fun__Def_Ois__measure_001t__String__Ochar,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_HOL_Oundefined_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Oenumerate_001tf__a,type,
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thf(sy_c_List_Olast_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_List_Olast_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olast_001t__Nat__Onat,type,
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last_state_a2: list_state_a > state_a2 ).
thf(sy_c_List_Olast_001t__States__Ostates_Itf__a_J,type,
last_states_a: list_states_a > states_a ).
thf(sy_c_List_Olast_001tf__a,type,
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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_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
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_001tf__a,type,
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thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__Big__Ostate_Itf__a_J,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Small__Ostate_Itf__a_J,type,
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thf(sy_c_List_Olist_OCons_001t__States__Ostates_Itf__a_J,type,
cons_states_a: states_a > list_states_a > list_states_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__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__Big__Ostate_Itf__a_J,type,
nil_state_a: list_state_a2 ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_ONil_001t__Small__Ostate_Itf__a_J,type,
nil_state_a2: list_state_a ).
thf(sy_c_List_Olist_ONil_001t__States__Ostates_Itf__a_J,type,
nil_states_a: list_states_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Big__Ostate_Itf__a_J,type,
hd_state_a: list_state_a2 > state_a3 ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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hd_state_a2: list_state_a > state_a2 ).
thf(sy_c_List_Olist_Ohd_001t__States__Ostates_Itf__a_J,type,
hd_states_a: list_states_a > states_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__Big__Ostate_Itf__a_J,type,
list_update_state_a: list_state_a2 > nat > state_a3 > list_state_a2 ).
thf(sy_c_List_Olist__update_001t__List__Olist_Itf__a_J,type,
list_update_list_a: list_list_a > nat > list_a > list_list_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001t__Small__Ostate_Itf__a_J,type,
list_update_state_a2: list_state_a > nat > state_a2 > list_state_a ).
thf(sy_c_List_Olist__update_001t__States__Ostates_Itf__a_J,type,
list_update_states_a: list_states_a > nat > states_a > list_states_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001t__List__Olist_Itf__a_J,type,
listrel1_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
listre6772471554020304241list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
listrel_states_a_nat: set_Pr1464008215722202041_a_nat > set_Pr1437833252353886031st_nat ).
thf(sy_c_List_Olistrel_001tf__a_001t__Big__Ostate_Itf__a_J,type,
listrel_a_state_a: set_Pr4275752383657305402tate_a > set_Pr8989213357517205050tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Small__Ostate_Itf__a_J,type,
listrel_a_state_a2: set_Pr6306228930610421491tate_a > set_Pr6052505092368253171tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Omeasures_001t__List__Olist_Itf__a_J,type,
measures_list_a: list_list_a_nat > set_Pr4048851178543822343list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Big__Ostate_Itf__a_J,type,
nth_state_a: list_state_a2 > nat > state_a3 ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
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thf(sy_c_List_Onth_001t__Small__Ostate_Itf__a_J,type,
nth_state_a2: list_state_a > nat > state_a2 ).
thf(sy_c_List_Onth_001t__States__Ostates_Itf__a_J,type,
nth_states_a: list_states_a > nat > states_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
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thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__List__Olist_Itf__a_J,type,
take_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
zip_list_a_list_a: list_list_a > list_list_a > list_P321204300973800749list_a ).
thf(sy_c_List_Ozip_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
zip_states_a_nat: list_states_a > list_nat > list_P4144771487928076819_a_nat ).
thf(sy_c_List_Ozip_001tf__a_001t__Big__Ostate_Itf__a_J,type,
zip_a_state_a: list_a > list_state_a2 > list_P6747491281921308512tate_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Small__Ostate_Itf__a_J,type,
zip_a_state_a2: list_a > list_state_a > list_P4482599289824689689tate_a ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Stack__Ostack_Itf__a_J_Mt__Stack__Ostack_Itf__a_J_J,type,
compow4264569633760279794tack_a: nat > ( stack_a > stack_a ) > stack_a > stack_a ).
thf(sy_c_Nat_Ocompow_001_062_It__States__Ostates_Itf__a_J_Mt__States__Ostates_Itf__a_J_J,type,
compow495008222514391794ates_a: nat > ( states_a > states_a ) > states_a > states_a ).
thf(sy_c_Nat_Ofunpow_001t__Nat__Onat,type,
funpow_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ofunpow_001t__Stack__Ostack_Itf__a_J,type,
funpow_stack_a: nat > ( stack_a > stack_a ) > stack_a > stack_a ).
thf(sy_c_Nat_Ofunpow_001t__States__Ostates_Itf__a_J,type,
funpow_states_a: nat > ( states_a > states_a ) > states_a > states_a ).
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_It__Big__Ostate_Itf__a_J_J,type,
size_s7859192958365828515tate_a: list_state_a2 > 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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
size_s8463391772401140188tate_a: list_state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__States__Ostates_Itf__a_J_J,type,
size_s3891197933023997302ates_a: list_states_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_Itf__a_J,type,
size_size_state_a3: state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_Itf__a_J,type,
size_size_stack_a: stack_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__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_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
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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 ).
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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__States__Ostates_Itf__a_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc327714349733490451st_nat: list_states_a > list_nat > produc4094504297408651929st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Big__Ostate_Itf__a_J_J,type,
produc3968967176812022602tate_a: list_a > list_state_a2 > produc17304319345593178tate_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
produc1997082749353321475tate_a: list_a > list_state_a > produc7959480069840336147tate_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
product_Pair_nat_a: nat > a > product_prod_nat_a ).
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__Current__Ocurrent_Itf__a_J,type,
produc8503237746132909001rent_a: a > current_a > produc7805042584321970905rent_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_Small_Opop_001tf__a,type,
pop_a3: state_a2 > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opush_001tf__a,type,
push_a2: a > state_a2 > state_a2 ).
thf(sy_c_Small_Ostate_OCommon_001tf__a,type,
common_a2: state_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse1_001tf__a,type,
reverse1_a: current_a > stack_a > list_a > state_a2 ).
thf(sy_c_Small_Ostate_OReverse2_001tf__a,type,
reverse2_a: current_a > list_a > stack_a > list_a > nat > state_a2 ).
thf(sy_c_Small__Aux_Olist_001tf__a,type,
small_list_a: state_a2 > list_a ).
thf(sy_c_Small__Aux_Olist__current_001tf__a,type,
small_list_current_a: state_a2 > list_a ).
thf(sy_c_Stack_Ofirst_001tf__a,type,
first_a: stack_a > a ).
thf(sy_c_Stack_Opop_001tf__a,type,
pop_a4: stack_a > stack_a ).
thf(sy_c_Stack__Aux_Olist_001tf__a,type,
stack_list_a: stack_a > list_a ).
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_001tf__a,type,
step_states_rel_a: states_a > states_a > $o ).
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_001tf__a,type,
states7295096810965389224irst_a: states_a > list_a ).
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_001tf__a,type,
states1596304293096088672irst_a: states_a > list_a ).
thf(sy_c_States__Aux_Olists_001tf__a,type,
states_lists_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_States__Aux_Olists__current_001tf__a,type,
states7719277857994474499rent_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_States__Aux_Olists__current__rel_001tf__a,type,
states5251248496104418302_rel_a: states_a > states_a > $o ).
thf(sy_c_States__Aux_Olists__rel_001tf__a,type,
states_lists_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_Itf__a_J,type,
type_i6304938058965754292tate_a: state_a3 > $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_Itf__a_J,type,
type_i8221491762852169479ates_a: states_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_Itf__a_J,type,
type_i3216275384938974675tack_a: stack_a > $o ).
thf(sy_c_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_Oremaining__steps__class_Oremaining__steps_001t__Common__Ostate_Itf__a_J,type,
type_r2212416260012024137tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__States__Ostates_Itf__a_J,type,
type_r4519047461186610747ates_a: states_a > 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__Small__Ostate_Itf__a_J,type,
type_s6404775287138157491tate_a: state_a2 > nat ).
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_Itf__a_J,type,
type_s3703408523585882337tate_a: state_a2 > state_a2 ).
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type_s4923920245906622843ates_a: states_a > states_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Current__Ocurrent_Itf__a_J,type,
accp_current_a: ( current_a > current_a > $o ) > current_a > $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__List__Olist_Itf__a_J,type,
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member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__States__Ostates_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6795120827800458928st_nat: produc4094504297408651929st_nat > set_Pr1437833252353886031st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Big__Ostate_Itf__a_J_J_J,type,
member6123284207288203267tate_a: produc17304319345593178tate_a > set_Pr8989213357517205050tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Small__Ostate_Itf__a_J_J_J,type,
member4112945611203173692tate_a: produc7959480069840336147tate_a > set_Pr6052505092368253171tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
member6483677129516672026_a_nat: produc1571854377283420419_a_nat > set_Pr1464008215722202041_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
member3175992478928454403tate_a: produc6972303929186420058tate_a > set_Pr4275752383657305402tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
member8547378267715833660tate_a: produc7589950997499123219tate_a > set_Pr6306228930610421491tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_big,type,
big: state_a3 ).
thf(sy_v_big_H,type,
big2: state_a3 ).
thf(sy_v_dir,type,
dir: direction ).
thf(sy_v_dir_H,type,
dir2: direction ).
thf(sy_v_small,type,
small: state_a2 ).
thf(sy_v_small_H,type,
small2: state_a2 ).
thf(sy_v_v______,type,
v: state_a ).
% Relevant facts (945)
thf(fact_0__C2__4_Oprems_C_I1_J,axiom,
type_i8221491762852169479ates_a @ ( states_a2 @ dir @ big @ small ) ).
% "2_4.prems"(1)
thf(fact_1__C2__4_Oprems_C_I2_J,axiom,
( ( type_s4923920245906622843ates_a @ ( states_a2 @ dir @ big @ small ) )
= ( states_a2 @ dir2 @ big2 @ small2 ) ) ).
% "2_4.prems"(2)
thf(fact_2_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_3_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_4_size__neq__size__imp__neq,axiom,
! [X: stack_a,Y: stack_a] :
( ( ( size_size_stack_a @ X )
!= ( size_size_stack_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_5_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_6_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_7_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_8_measure__size,axiom,
fun_is1118958059976982595tate_a @ size_size_state_a2 ).
% measure_size
thf(fact_9_measure__size,axiom,
fun_is4918913982087955905ates_a @ size_size_states_a ).
% measure_size
thf(fact_10_measure__size,axiom,
fun_is2826639914637066575tack_a @ size_size_stack_a ).
% measure_size
thf(fact_11_measure__size,axiom,
fun_is6829564153850791975tate_a @ size_size_state_a3 ).
% measure_size
thf(fact_12_measure__size,axiom,
fun_is4385883845911452270tate_a @ size_size_state_a ).
% measure_size
thf(fact_13_measure__size,axiom,
fun_is_measure_char @ size_size_char ).
% measure_size
thf(fact_14_States__Proof_Oinvar__step,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( type_i8221491762852169479ates_a @ ( type_s4923920245906622843ates_a @ States ) ) ) ).
% States_Proof.invar_step
thf(fact_15_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_16__C2__4_Ohyps_C,axiom,
( ( common_a2 @ v )
= small ) ).
% "2_4.hyps"
thf(fact_17_step__size__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Dir2: direction,Big2: state_a3,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a3 @ Small2 )
= ( size_size_state_a3 @ Small ) ) ) ) ).
% step_size_small
thf(fact_18_is__measure_Osimps,axiom,
( fun_is6829564153850791975tate_a
= ( ^ [A: state_a2 > nat] :
? [X2: state_a2 > nat] :
( ^ [Y2: state_a2 > nat,Z: state_a2 > nat] : ( Y2 = Z )
@ A
@ X2 ) ) ) ).
% is_measure.simps
thf(fact_19_is__measure_Osimps,axiom,
( fun_is_measure_char
= ( ^ [A: char > nat] :
? [X2: char > nat] :
( ^ [Y2: char > nat,Z: char > nat] : ( Y2 = Z )
@ A
@ X2 ) ) ) ).
% is_measure.simps
thf(fact_20_is__measure_Osimps,axiom,
( fun_is4385883845911452270tate_a
= ( ^ [A: state_a3 > nat] :
? [X2: state_a3 > nat] :
( ^ [Y2: state_a3 > nat,Z: state_a3 > nat] : ( Y2 = Z )
@ A
@ X2 ) ) ) ).
% is_measure.simps
thf(fact_21_is__measure__trivial,axiom,
! [F: state_a2 > nat] : ( fun_is6829564153850791975tate_a @ F ) ).
% is_measure_trivial
thf(fact_22_is__measure__trivial,axiom,
! [F: char > nat] : ( fun_is_measure_char @ F ) ).
% is_measure_trivial
thf(fact_23_is__measure__trivial,axiom,
! [F: state_a3 > nat] : ( fun_is4385883845911452270tate_a @ F ) ).
% is_measure_trivial
thf(fact_24_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_25_states_Oinject,axiom,
! [X1: direction,X22: state_a3,X3: state_a2,Y1: direction,Y22: state_a3,Y3: state_a2] :
( ( ( states_a2 @ X1 @ X22 @ X3 )
= ( states_a2 @ Y1 @ Y22 @ Y3 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 )
& ( X3 = Y3 ) ) ) ).
% states.inject
thf(fact_26_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_27_step__size__new__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Dir2: direction,Big2: state_a3,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6530235180886170618tate_a @ Big2 )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% step_size_new_big
thf(fact_28_step__size__new__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Dir2: direction,Big2: state_a3,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6404775287138157491tate_a @ Small2 )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% step_size_new_small
thf(fact_29_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_30_step__invars,axiom,
! [States: states_a,Dir: direction,Big: state_a3,Small: state_a2] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_s4923920245906622843ates_a @ States )
= ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
& ( type_i464410347872898157tate_a @ Small ) ) ) ) ).
% step_invars
thf(fact_31_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_32_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_33_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_34_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_35_states_Oexhaust,axiom,
! [Y: states_a] :
~ ! [X12: direction,X23: state_a3,X32: state_a2] :
( Y
!= ( states_a2 @ X12 @ X23 @ X32 ) ) ).
% states.exhaust
thf(fact_36_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_37_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_38_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_39_Small__Aux_Osize__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( size_size_state_a3 @ ( common_a2 @ State ) )
= ( size_size_state_a2 @ State ) ) ).
% Small_Aux.size_state.simps(1)
thf(fact_40_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_41_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_42_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_43_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_44_remaining__steps__states_Ocases,axiom,
! [X: states_a] :
~ ! [Uu: direction,Big3: state_a3,Small3: state_a2] :
( X
!= ( states_a2 @ Uu @ Big3 @ Small3 ) ) ).
% remaining_steps_states.cases
thf(fact_45_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_46_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_47_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_48_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_49_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_50_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_51_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_52_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_53_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_54_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_55_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y222: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_56_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y4: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_57_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y4: a,Ys: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_58_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_59_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_60_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_61_push__small__lists,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Big2: list_a,Small2: list_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big2 @ Small2 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( produc6837034575241423639list_a @ Big2 @ ( cons_a @ X @ Small2 ) ) ) ) ) ).
% push_small_lists
thf(fact_62_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_63_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_64_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_65_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_66_states_Osize__neq,axiom,
! [X: states_a] :
( ( size_size_states_a @ X )
!= zero_zero_nat ) ).
% states.size_neq
thf(fact_67_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_68_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_69_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_70_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_71_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_72_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_73_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X4: nat,Y5: nat] :
( ( P @ X4 @ Y5 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_74_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_75_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_76_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_77_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_78_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_79_lists__current_Oelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ~ ! [Uu: direction,Big3: state_a3,Small3: state_a2] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big3 ) @ ( small_list_current_a @ Small3 ) ) ) ) ) ).
% lists_current.elims
thf(fact_80_lists__current_Osimps,axiom,
! [Uu2: direction,Big: state_a3,Small: state_a2] :
( ( states7719277857994474499rent_a @ ( states_a2 @ Uu2 @ Big @ Small ) )
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big ) @ ( small_list_current_a @ Small ) ) ) ).
% lists_current.simps
thf(fact_81_states_Osize_I2_J,axiom,
! [X1: direction,X22: state_a3,X3: state_a2] :
( ( size_size_states_a @ ( states_a2 @ X1 @ X22 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size(2)
thf(fact_82_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_83_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_84_push__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,Big2: list_a,Small2: list_a,X: a] :
( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big2 @ Small2 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( produc6837034575241423639list_a @ ( cons_a @ X @ Big2 ) @ Small2 ) ) ) ).
% push_big
thf(fact_85_Big_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X14: nat,Y11: current_a,Y12: stack_a,Y13: list_a,Y14: nat] :
( ( ( reverse_a @ X11 @ X122 @ X13 @ X14 )
= ( reverse_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% Big.state.inject(1)
thf(fact_86_states_Osize__gen,axiom,
! [X: a > nat,X1: direction,X22: state_a3,X3: state_a2] :
( ( size_states_a @ X @ ( states_a2 @ X1 @ X22 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size_gen
thf(fact_87_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_88_old_Oprod_Oinject,axiom,
! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A2 @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_89_old_Oprod_Oinject,axiom,
! [A2: states_a,B: nat,A3: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A2 @ B )
= ( produc1877401315875745917_a_nat @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_90_old_Oprod_Oinject,axiom,
! [A2: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A2 @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_91_old_Oprod_Oinject,axiom,
! [A2: a,B: state_a3,A3: a,B2: state_a3] :
( ( ( produc8641956578966763338tate_a @ A2 @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_92_prod_Oinject,axiom,
! [X1: list_a,X22: list_a,Y1: list_a,Y22: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X22 )
= ( produc6837034575241423639list_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_93_prod_Oinject,axiom,
! [X1: states_a,X22: nat,Y1: states_a,Y22: nat] :
( ( ( produc1877401315875745917_a_nat @ X1 @ X22 )
= ( produc1877401315875745917_a_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_94_prod_Oinject,axiom,
! [X1: a,X22: state_a2,Y1: a,Y22: state_a2] :
( ( ( produc1224139502141355779tate_a @ X1 @ X22 )
= ( produc1224139502141355779tate_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_95_prod_Oinject,axiom,
! [X1: a,X22: state_a3,Y1: a,Y22: state_a3] :
( ( ( produc8641956578966763338tate_a @ X1 @ X22 )
= ( produc8641956578966763338tate_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_96_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_97_Cons__in__lex,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys2 ) ) @ ( lex_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( lex_list_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_98_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_99_remaining__steps__0,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_r4519047461186610747ates_a @ States )
= zero_zero_nat )
=> ( ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_0
thf(fact_100_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_101_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_102_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_103_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_104_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_105_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_106_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_107_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_108_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_109_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_110_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_111_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_112_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_113_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_114_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_115_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_116_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_117_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_118_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_119_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_120_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_121_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_122_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_123_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_124_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_125_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_126_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_127_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_128_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_129_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_130_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_131_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_132_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_133_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_134_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_135_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_136_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_137_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_138_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_139_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_140_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_141_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_142_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_143_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_144_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_145_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_146_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_147_size__ok_H_Ocases,axiom,
! [X: produc1571854377283420419_a_nat] :
~ ! [Uu: direction,Big3: state_a3,Small3: state_a2,Steps: nat] :
( X
!= ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Steps ) ) ).
% size_ok'.cases
thf(fact_148_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_149_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_150_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_151_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_152_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_153_remaining__steps__decline__Suc,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( suc @ ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) ) )
= ( type_r4519047461186610747ates_a @ States ) ) ) ) ).
% remaining_steps_decline_Suc
thf(fact_154_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_155_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A4: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_156_old_Oprod_Oexhaust,axiom,
! [Y: produc1571854377283420419_a_nat] :
~ ! [A4: states_a,B3: nat] :
( Y
!= ( produc1877401315875745917_a_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_157_old_Oprod_Oexhaust,axiom,
! [Y: produc7589950997499123219tate_a] :
~ ! [A4: a,B3: state_a2] :
( Y
!= ( produc1224139502141355779tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_158_old_Oprod_Oexhaust,axiom,
! [Y: produc6972303929186420058tate_a] :
~ ! [A4: a,B3: state_a3] :
( Y
!= ( produc8641956578966763338tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_159_surj__pair,axiom,
! [P2: produc9164743771328383783list_a] :
? [X4: list_a,Y5: list_a] :
( P2
= ( produc6837034575241423639list_a @ X4 @ Y5 ) ) ).
% surj_pair
thf(fact_160_surj__pair,axiom,
! [P2: produc1571854377283420419_a_nat] :
? [X4: states_a,Y5: nat] :
( P2
= ( produc1877401315875745917_a_nat @ X4 @ Y5 ) ) ).
% surj_pair
thf(fact_161_surj__pair,axiom,
! [P2: produc7589950997499123219tate_a] :
? [X4: a,Y5: state_a2] :
( P2
= ( produc1224139502141355779tate_a @ X4 @ Y5 ) ) ).
% surj_pair
thf(fact_162_surj__pair,axiom,
! [P2: produc6972303929186420058tate_a] :
? [X4: a,Y5: state_a3] :
( P2
= ( produc8641956578966763338tate_a @ X4 @ Y5 ) ) ).
% surj_pair
thf(fact_163_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_164_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_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: produc6972303929186420058tate_a > $o,P2: produc6972303929186420058tate_a] :
( ! [A4: a,B3: state_a3] : ( P @ ( produc8641956578966763338tate_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_167_Pair__inject,axiom,
! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A2 @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_168_Pair__inject,axiom,
! [A2: states_a,B: nat,A3: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A2 @ B )
= ( produc1877401315875745917_a_nat @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_169_Pair__inject,axiom,
! [A2: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc1224139502141355779tate_a @ A2 @ B )
= ( produc1224139502141355779tate_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_170_Pair__inject,axiom,
! [A2: a,B: state_a3,A3: a,B2: state_a3] :
( ( ( produc8641956578966763338tate_a @ A2 @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_171_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_172_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_173_Small__Proof_Opop__list__current,axiom,
! [Small: state_a2,X: a,Small2: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( small_list_current_a @ Small2 ) )
= ( small_list_current_a @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_174_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_175_Big__Proof_Opop__list__current,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( cons_a @ X @ ( big_list_current_a @ Big2 ) )
= ( big_list_current_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_176_Big__Proof_Opop__list,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( cons_a @ X @ ( big_list_a @ Big2 ) )
= ( big_list_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_177_invars__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
& ( type_i464410347872898157tate_a @ Small2 ) ) ) ) ) ).
% invars_pop_small
thf(fact_178_Small__Proof_Osize__new__pop,axiom,
! [Small: state_a2,X: a,Small2: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( suc @ ( type_s6404775287138157491tate_a @ Small2 ) )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% Small_Proof.size_new_pop
thf(fact_179_invars__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( ( type_i6304938058965754292tate_a @ Big2 )
& ( type_i464410347872898157tate_a @ Small ) ) ) ) ) ).
% invars_pop_big
thf(fact_180_Small__Proof_Osize__pop,axiom,
! [Small: state_a2,X: a,Small2: state_a2] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( suc @ ( size_size_state_a3 @ Small2 ) )
= ( size_size_state_a3 @ Small ) ) ) ) ) ).
% Small_Proof.size_pop
thf(fact_181_Big__Proof_Osize__new__pop,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( suc @ ( type_s6530235180886170618tate_a @ Big2 ) )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% Big_Proof.size_new_pop
thf(fact_182_Big__Proof_Osize__pop,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( suc @ ( size_size_state_a @ Big2 ) )
= ( size_size_state_a @ Big ) ) ) ) ) ).
% Big_Proof.size_pop
thf(fact_183_Big__Proof_Oinvar__pop,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( type_i6304938058965754292tate_a @ Big2 ) ) ) ) ).
% Big_Proof.invar_pop
thf(fact_184_Small__Proof_Oinvar__pop,axiom,
! [Small: state_a2,X: a,Small2: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( type_i464410347872898157tate_a @ Small2 ) ) ) ) ).
% Small_Proof.invar_pop
thf(fact_185_invar__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ).
% invar_pop_big
thf(fact_186_invar__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) ) ) ) ).
% invar_pop_small
thf(fact_187_list__current__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( ( cons_a @ X @ ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_188_list__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( ( cons_a @ X @ ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) )
= ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_189_list__current__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_190_list__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) )
= ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_191_Big__Proof_Oremaining__steps__pop,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ord_less_eq_nat @ ( type_r2494999336194962664tate_a @ Big2 ) @ ( type_r2494999336194962664tate_a @ Big ) ) ) ) ) ).
% Big_Proof.remaining_steps_pop
thf(fact_192_Big__Proof_Opop__list__tl,axiom,
! [Big: state_a3,X: a,Big2: state_a3] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( big_list_a @ Big2 )
= ( tl_a @ ( big_list_a @ Big ) ) ) ) ) ) ).
% Big_Proof.pop_list_tl
thf(fact_193_Cons__lenlex__iff,axiom,
! [M: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M @ Ms ) @ ( cons_list_a @ N @ Ns ) ) @ ( lenlex_list_a @ R ) )
= ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) )
| ( ( ( size_s349497388124573686list_a @ Ms )
= ( size_s349497388124573686list_a @ Ns ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_194_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_195_lists__big__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ) ).
% lists_big_first_pop_big
thf(fact_196_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_197_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_198_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_199_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_200_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_201_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_202_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_203_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_204_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_205_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_206_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_207_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_208_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_209_list_Osel_I3_J,axiom,
! [X21: a,X222: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_210_invar__list__big__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1888450819780863577irst_a @ States )
= ( states7295096810965389224irst_a @ States ) ) ) ).
% invar_list_big_first
thf(fact_211_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_212_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_213_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_214_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_215_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_216_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y5: nat,Z2: nat] :
( ( R2 @ X4 @ Y5 )
=> ( ( R2 @ Y5 @ Z2 )
=> ( R2 @ X4 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_217_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_218_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_219_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_220_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_221_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_222_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_223_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_224_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_225_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_226_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_227_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_228_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_229_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_230_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_231_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_232_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_233_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_234_step__lists__small__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1596304293096088672irst_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states7886008410469471791irst_a @ ( type_s4923920245906622843ates_a @ States ) ) ) ) ).
% step_lists_small_first
thf(fact_235_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_236_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_237_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_238_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_239_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_240_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_241_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_242_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_243_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_244_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_245_impossible__Cons,axiom,
! [Xs: list_a,Ys2: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs
!= ( cons_a @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_246_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs: list_a] :
( ! [X4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_247_lenlex__irreflexive,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ! [X4: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ X4 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Xs ) @ ( lenlex_list_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_248_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_249_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_250_remaining__steps__decline,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) ) @ ( type_r4519047461186610747ates_a @ States ) ) ) ).
% remaining_steps_decline
thf(fact_251_lists__small__first__pop__small,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Small2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a3 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) ) ) ) ) ).
% lists_small_first_pop_small
thf(fact_252_lists__small__first__pop__big,axiom,
! [Dir: direction,Big: state_a3,Small: state_a2,X: a,Big2: 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 @ Big2 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ) ).
% lists_small_first_pop_big
thf(fact_253_cons__tl,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys2 )
=> ( Xs
= ( tl_a @ Ys2 ) ) ) ).
% cons_tl
thf(fact_254_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_255_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_256_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_257_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_258_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_259_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_260_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_261_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_262_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_263_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_264_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_265_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_266_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_267_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_268_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_269_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_270_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_271_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_272_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_273_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_274_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_275_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_276_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_277_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_278_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_279_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_280_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_281_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_282_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P4 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_283_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_284_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_285_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_286_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_287_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X4: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_288_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_289_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_290_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_291_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_292_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_293_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_294_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_295_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_296_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_297_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_298_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_299_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_300_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_301_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B4: nat] :
( ( ord_less_nat @ A @ B4 )
| ( A = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_302_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
& ( A != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_303_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_304_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_305_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B4: nat] :
( ( ord_less_eq_nat @ A @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_306_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A: nat] :
( ( ord_less_nat @ B4 @ A )
| ( A = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_307_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
& ( A != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_308_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_309_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_310_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A: nat] :
( ( ord_less_eq_nat @ B4 @ A )
& ~ ( ord_less_eq_nat @ A @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_311_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_312_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_313_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_314_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
& ( X5 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_315_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_316_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_317_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_318_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_319_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_320_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_321_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_322_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_323_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_324_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_325_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_326_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_327_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_328_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_329_remaining__steps__decline__n__steps,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ States ) @ N )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_decline_n_steps
thf(fact_330_funpow__0,axiom,
! [F: states_a > states_a,X: states_a] :
( ( compow495008222514391794ates_a @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_331_funpow__0,axiom,
! [F: stack_a > stack_a,X: stack_a] :
( ( compow4264569633760279794tack_a @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_332_funpow__0,axiom,
! [F: nat > nat,X: nat] :
( ( compow_nat_nat @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_333_funpow__swap1,axiom,
! [F: states_a > states_a,N: nat,X: states_a] :
( ( F @ ( compow495008222514391794ates_a @ N @ F @ X ) )
= ( compow495008222514391794ates_a @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_334_funpow__swap1,axiom,
! [F: stack_a > stack_a,N: nat,X: stack_a] :
( ( F @ ( compow4264569633760279794tack_a @ N @ F @ X ) )
= ( compow4264569633760279794tack_a @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_335_funpow__swap1,axiom,
! [F: nat > nat,N: nat,X: nat] :
( ( F @ ( compow_nat_nat @ N @ F @ X ) )
= ( compow_nat_nat @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_336_step__consistent__2,axiom,
! [P: states_a > $o,States: states_a,N: nat] :
( ! [States2: states_a] :
( ( type_i8221491762852169479ates_a @ States2 )
=> ( ( P @ States2 )
=> ( P @ ( type_s4923920245906622843ates_a @ States2 ) ) ) )
=> ( ( type_i8221491762852169479ates_a @ States )
=> ( ( P @ States )
=> ( P @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) ) ) ) ).
% step_consistent_2
thf(fact_337_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_338_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_339_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_340_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_341_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_342_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z2 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_343_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_344_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_345_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_346_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_347_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_348_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z2: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z2 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_349_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_350_remaining__steps__0_H,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_r4519047461186610747ates_a @ States )
= zero_zero_nat )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_0'
thf(fact_351_Nat_Ofunpow__code__def,axiom,
funpow_states_a = compow495008222514391794ates_a ).
% Nat.funpow_code_def
thf(fact_352_Nat_Ofunpow__code__def,axiom,
funpow_stack_a = compow4264569633760279794tack_a ).
% Nat.funpow_code_def
thf(fact_353_Nat_Ofunpow__code__def,axiom,
funpow_nat = compow_nat_nat ).
% Nat.funpow_code_def
thf(fact_354_remaining__steps__n__steps__plus,axiom,
! [N: nat,States: states_a] :
( ( ord_less_eq_nat @ N @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( type_i8221491762852169479ates_a @ States )
=> ( ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) @ N )
= ( type_r4519047461186610747ates_a @ States ) ) ) ) ).
% remaining_steps_n_steps_plus
thf(fact_355_remaining__steps__n__steps__sub,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ States ) @ N ) ) ) ).
% remaining_steps_n_steps_sub
thf(fact_356_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_357_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_358_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_359_add__left__cancel,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_360_add__right__cancel,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_361_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_362_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_363_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_364_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_365_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_366_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_367_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_368_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_369_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_370_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_371_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_372_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_373_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_374_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_375_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_376_add__diff__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_377_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_378_add__diff__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_379_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_380_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_381_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_382_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_383_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_384_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_385_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_386_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_387_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_388_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_389_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_390_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_391_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_392_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_393_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_394_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_395_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_396_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_397_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_398_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_399_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_400_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_401_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_402_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_403_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_404_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_405_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_406_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_407_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_408_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_409_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_410_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_411_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_412_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_413_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_414_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_415_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_416_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_417_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_418_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_419_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A2: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_420_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A2: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_421_add_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_422_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A: nat,B4: nat] : ( plus_plus_nat @ B4 @ A ) ) ) ).
% add.commute
thf(fact_423_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_424_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_425_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_426_add__implies__diff,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A2 )
=> ( C2
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_427_diff__right__commute,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_428_diff__diff__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_429_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_430_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_431_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_432_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_433_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_434_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_435_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_436_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_437_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_438_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_439_le__add__diff,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_440_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_441_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_442_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A2 )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_443_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_444_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_445_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_446_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_447_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_448_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_449_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_450_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_451_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_452_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_453_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_454_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_455_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_456_add__mono,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_457_add__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_458_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_459_add__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_460_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A @ C ) ) ) ) ).
% le_iff_add
thf(fact_461_add__le__imp__le__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_462_add__le__imp__le__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_463_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_464_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_465_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_466_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_467_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_468_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_469_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_470_add__strict__mono,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_471_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_472_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_473_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_474_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_475_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_476_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_477_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_478_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_479_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_480_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_481_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_482_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_483_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_484_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_485_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_486_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_487_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_488_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A2: nat] :
( ( A5
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_489_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_490_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_491_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_492_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_493_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_494_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_495_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_496_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_497_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_498_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_499_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_500_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_501_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_502_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_503_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_504_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_505_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_506_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_507_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_508_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_509_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_510_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_511_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a] : ( P @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y5: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y5 @ Ys3 ) )
=> ( ! [X4: a,Xs2: list_a,Y5: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y5 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_512_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_513_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y5: a,Xs2: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_514_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_515_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_516_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_517_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ~ ! [P6: a > a > $o,X4: a,Ys3: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X4 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_518_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ( ! [P6: a > a > $o,X4: a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X4 @ nil_a ) ) )
=> ~ ! [P6: a > a > $o,X4: a,Y5: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X4 @ ( cons_a @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_519_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_520_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_521_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_522_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_523_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_524_add__increasing2,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_525_add__decreasing2,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_526_add__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_increasing
thf(fact_527_add__decreasing,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_528_add__less__le__mono,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_529_add__le__less__mono,axiom,
! [A2: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_530_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_531_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_532_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_533_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_534_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_535_pos__add__strict,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_536_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_537_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_538_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_539_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_540_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_541_diff__less__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_542_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_543_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_544_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_545_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_546_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_547_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_548_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_549_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_550_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_551_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_552_list__induct4,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y5: a,Ys3: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y5 @ Ys3 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_553_list__induct3,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y5: a,Ys3: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y5 @ Ys3 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_554_list__induct2,axiom,
! [Xs: list_a,Ys2: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a,Y5: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y5 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_555_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ~ ! [X4: a,Xs2: list_a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_556_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X4: a,Xs2: list_a,Y5: a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y5 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_557_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_558_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_559_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X5: a] :
( Xs
= ( cons_a @ X5 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_560_Nil2__notin__lex,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_561_Nil__notin__lex,axiom,
! [Ys2: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys2 ) @ ( lex_a @ R ) ) ).
% Nil_notin_lex
thf(fact_562_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_563_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_564_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_565_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_566_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_567_add__strict__increasing,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_568_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_569_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_570_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_571_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_572_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_573_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_574_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_575_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: 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 @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_576_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_577_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_578_remaining__steps__decline__sub,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ States ) @ one_one_nat ) ) ) ).
% remaining_steps_decline_sub
thf(fact_579_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_580_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_581_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_582_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_583_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_584_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_585_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_586_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_587_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_588_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_589_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_590_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_591_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_592_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_593_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_594_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_595_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_596_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_597_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_598_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_599_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_600_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_601_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_602_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_603_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_604_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_605_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_606_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_607_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_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_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_610_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_611_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_612_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_613_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_614_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_615_nth__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_616_nth__tl,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( tl_a @ Xs ) ) )
=> ( ( nth_a @ ( tl_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_617_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_618_lex__take__index,axiom,
! [Xs: list_list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( lex_list_a @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( take_list_a @ I2 @ Xs )
= ( take_list_a @ I2 @ Ys2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_619_lex__take__index,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( lex_a @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys2 ) )
=> ( ( ( take_a @ I2 @ Xs )
= ( take_a @ I2 @ Ys2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_620_listrel__iff__nth,axiom,
! [Xs: list_list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listre6772471554020304241list_a @ R ) )
= ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N4 ) @ ( nth_list_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_621_listrel__iff__nth,axiom,
! [Xs: list_states_a,Ys2: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ Ys2 ) @ ( listrel_states_a_nat @ R ) )
= ( ( ( size_s3891197933023997302ates_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_s3891197933023997302ates_a @ Xs ) )
=> ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ ( nth_states_a @ Xs @ N4 ) @ ( nth_nat @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_622_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s8463391772401140188tate_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ ( nth_a @ Xs @ N4 ) @ ( nth_state_a2 @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_623_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s7859192958365828515tate_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ ( nth_a @ Xs @ N4 ) @ ( nth_state_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_624_listrel__iff__nth,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
& ! [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N4 ) @ ( nth_a @ Ys2 @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_625_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_a @ N @ nil_a )
= nil_Pr1417316670369895453_nat_a ) ).
% enumerate_simps(1)
thf(fact_626_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_627_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_628_take__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_629_take__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_630_enumerate__simps_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( enumerate_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N @ X ) @ ( enumerate_a @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_631_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_632_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_633_take__tl,axiom,
! [N: nat,Xs: list_a] :
( ( take_a @ N @ ( tl_a @ Xs ) )
= ( tl_a @ ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_634_listrel_ONil,axiom,
! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).
% listrel.Nil
thf(fact_635_listrel__Nil1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_636_listrel__Nil2,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_637_listrel__eq__len,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_638_tl__take,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( take_a @ N @ Xs ) )
= ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_a @ Xs ) ) ) ).
% tl_take
thf(fact_639_listrel_OCons,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Ys2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_640_listrel_OCons,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys2: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listre6772471554020304241list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys2 ) ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_641_listrel_OCons,axiom,
! [X: states_a,Y: nat,R: set_Pr1464008215722202041_a_nat,Xs: list_states_a,Ys2: list_nat] :
( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X @ Y ) @ R )
=> ( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ Ys2 ) @ ( listrel_states_a_nat @ R ) )
=> ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ ( cons_states_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_states_a_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_642_listrel_OCons,axiom,
! [X: a,Y: state_a2,R: set_Pr6306228930610421491tate_a,Xs: list_a,Ys2: list_state_a] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) @ R )
=> ( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a2 @ R ) )
=> ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a2 @ Y @ Ys2 ) ) @ ( listrel_a_state_a2 @ R ) ) ) ) ).
% listrel.Cons
thf(fact_643_listrel_OCons,axiom,
! [X: a,Y: state_a3,R: set_Pr4275752383657305402tate_a,Xs: list_a,Ys2: list_state_a2] :
( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) @ R )
=> ( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a @ R ) )
=> ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a @ Y @ Ys2 ) ) @ ( listrel_a_state_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_644_listrel__Cons1,axiom,
! [Y: list_a,Ys2: list_list_a,Xs: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ Y @ Ys2 ) @ Xs ) @ ( listre6772471554020304241list_a @ R ) )
=> ~ ! [Y5: list_a,Ys3: list_list_a] :
( ( Xs
= ( cons_list_a @ Y5 @ Ys3 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Y5 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys2 @ Ys3 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_645_listrel__Cons1,axiom,
! [Y: states_a,Ys2: list_states_a,Xs: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ ( cons_states_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_states_a_nat @ R ) )
=> ~ ! [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ Y @ Y5 ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Ys2 @ Ys3 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_646_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [Y5: state_a2,Ys3: list_state_a] :
( ( Xs
= ( cons_state_a2 @ Y5 @ Ys3 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ Y @ Y5 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Ys2 @ Ys3 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_647_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [Y5: state_a3,Ys3: list_state_a2] :
( ( Xs
= ( cons_state_a @ Y5 @ Ys3 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ Y @ Y5 ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Ys2 @ Ys3 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_648_listrel__Cons1,axiom,
! [Y: a,Ys2: list_a,Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys2 ) @ Xs ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_649_listrel__Cons2,axiom,
! [Xs: list_list_a,Y: list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys2 ) ) @ ( listre6772471554020304241list_a @ R ) )
=> ~ ! [X4: list_a,Xs2: list_list_a] :
( ( Xs
= ( cons_list_a @ X4 @ Xs2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys2 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_650_listrel__Cons2,axiom,
! [Xs: list_states_a,Y: nat,Ys2: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_states_a_nat @ R ) )
=> ~ ! [X4: states_a,Xs2: list_states_a] :
( ( Xs
= ( cons_states_a @ X4 @ Xs2 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs2 @ Ys2 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_651_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a2,Ys2: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ ( cons_state_a2 @ Y @ Ys2 ) ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys2 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_652_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a3,Ys2: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ ( cons_state_a @ Y @ Ys2 ) ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs2 @ Ys2 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_653_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X4: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_654_listrel_Ocases,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( listre6772471554020304241list_a @ R ) )
=> ( ( ( A1 = nil_list_a )
=> ( A22 != nil_list_a ) )
=> ~ ! [X4: list_a,Y5: list_a,Xs2: list_list_a] :
( ( A1
= ( cons_list_a @ X4 @ Xs2 ) )
=> ! [Ys3: list_list_a] :
( ( A22
= ( cons_list_a @ Y5 @ Ys3 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y5 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys3 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_655_listrel_Ocases,axiom,
! [A1: list_states_a,A22: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ A1 @ A22 ) @ ( listrel_states_a_nat @ R ) )
=> ( ( ( A1 = nil_states_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X4: states_a,Y5: nat,Xs2: list_states_a] :
( ( A1
= ( cons_states_a @ X4 @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y5 @ Ys3 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y5 ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs2 @ Ys3 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_656_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a2 ) )
=> ~ ! [X4: a,Y5: state_a2,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys3: list_state_a] :
( ( A22
= ( cons_state_a2 @ Y5 @ Ys3 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y5 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys3 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_657_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X4: a,Y5: state_a3,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys3: list_state_a2] :
( ( A22
= ( cons_state_a @ Y5 @ Ys3 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y5 ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs2 @ Ys3 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_658_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X4: a,Y5: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs2 ) )
=> ! [Ys3: list_a] :
( ( A22
= ( cons_a @ Y5 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_659_listrel_Osimps,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( listre6772471554020304241list_a @ R ) )
= ( ( ( A1 = nil_list_a )
& ( A22 = nil_list_a ) )
| ? [X5: list_a,Y4: list_a,Xs3: list_list_a,Ys: list_list_a] :
( ( A1
= ( cons_list_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_list_a @ Y4 @ Ys ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y4 ) @ R )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs3 @ Ys ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_660_listrel_Osimps,axiom,
! [A1: list_states_a,A22: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ A1 @ A22 ) @ ( listrel_states_a_nat @ R ) )
= ( ( ( A1 = nil_states_a )
& ( A22 = nil_nat ) )
| ? [X5: states_a,Y4: nat,Xs3: list_states_a,Ys: list_nat] :
( ( A1
= ( cons_states_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys ) )
& ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X5 @ Y4 ) @ R )
& ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs3 @ Ys ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_661_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a2 ) )
| ? [X5: a,Y4: state_a2,Xs3: list_a,Ys: list_state_a] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_state_a2 @ Y4 @ Ys ) )
& ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y4 ) @ R )
& ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs3 @ Ys ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_662_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a ) )
| ? [X5: a,Y4: state_a3,Xs3: list_a,Ys: list_state_a2] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y4 @ Ys ) )
& ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X5 @ Y4 ) @ R )
& ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs3 @ Ys ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_663_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X5: a,Y4: a,Xs3: list_a,Ys: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs3 ) )
& ( A22
= ( cons_a @ Y4 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_664_take__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_665_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_666_nth__zip,axiom,
! [I: nat,Xs: list_list_a,Ys2: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( nth_Pr5917933638979213230list_a @ ( zip_list_a_list_a @ Xs @ Ys2 ) @ I )
= ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I ) @ ( nth_list_a @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_667_nth__zip,axiom,
! [I: nat,Xs: list_states_a,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_s3891197933023997302ates_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr6783065760942753852_a_nat @ ( zip_states_a_nat @ Xs @ Ys2 ) @ I )
= ( produc1877401315875745917_a_nat @ ( nth_states_a @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_668_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys2: list_state_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8463391772401140188tate_a @ Ys2 ) )
=> ( ( nth_Pr7598562053491293850tate_a @ ( zip_a_state_a2 @ Xs @ Ys2 ) @ I )
= ( produc1224139502141355779tate_a @ ( nth_a @ Xs @ I ) @ ( nth_state_a2 @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_669_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys2: list_state_a2] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s7859192958365828515tate_a @ Ys2 ) )
=> ( ( nth_Pr2816242342134088033tate_a @ ( zip_a_state_a @ Xs @ Ys2 ) @ I )
= ( produc8641956578966763338tate_a @ ( nth_a @ Xs @ I ) @ ( nth_state_a @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_670_same__append__eq,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_671_append__same__eq,axiom,
! [Ys2: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys2 @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_672_append__assoc,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys2 ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_673_append_Oassoc,axiom,
! [A2: list_a,B: list_a,C2: list_a] :
( ( append_a @ ( append_a @ A2 @ B ) @ C2 )
= ( append_a @ A2 @ ( append_a @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_674_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_675_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_676_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_677_append__self__conv2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_678_self__append__conv,axiom,
! [Y: list_a,Ys2: list_a] :
( ( Y
= ( append_a @ Y @ Ys2 ) )
= ( Ys2 = nil_a ) ) ).
% self_append_conv
thf(fact_679_append__self__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_a ) ) ).
% append_self_conv
thf(fact_680_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_681_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_682_append__eq__append__conv,axiom,
! [Xs: list_a,Ys2: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_683_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_684_length__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys2 ) ) ) ).
% length_append
thf(fact_685_zip__append,axiom,
! [Xs: list_a,Us: list_a,Ys2: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Us ) )
=> ( ( zip_a_a @ ( append_a @ Xs @ Ys2 ) @ ( append_a @ Us @ Vs ) )
= ( append5335208819046833346od_a_a @ ( zip_a_a @ Xs @ Us ) @ ( zip_a_a @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_686_tl__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_687_zip__eq__Nil__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( zip_a_a @ Xs @ Ys2 )
= nil_Product_prod_a_a )
= ( ( Xs = nil_a )
| ( Ys2 = nil_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_688_Nil__eq__zip__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( nil_Product_prod_a_a
= ( zip_a_a @ Xs @ Ys2 ) )
= ( ( Xs = nil_a )
| ( Ys2 = nil_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_689_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys2: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys2 ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_690_nth__append__length__plus,axiom,
! [Xs: list_a,Ys2: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_691_take__append,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( take_a @ N @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( take_a @ N @ Xs ) @ ( take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_692_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_693_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys2: list_a] :
( ( zip_a_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) )
= ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( zip_a_a @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_694_zip__Cons__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys2: list_list_a] :
( ( zip_list_a_list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys2 ) )
= ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( zip_list_a_list_a @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_695_zip__Cons__Cons,axiom,
! [X: states_a,Xs: list_states_a,Y: nat,Ys2: list_nat] :
( ( zip_states_a_nat @ ( cons_states_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P1204124355475727053_a_nat @ ( produc1877401315875745917_a_nat @ X @ Y ) @ ( zip_states_a_nat @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_696_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: state_a2,Ys2: list_state_a] :
( ( zip_a_state_a2 @ ( cons_a @ X @ Xs ) @ ( cons_state_a2 @ Y @ Ys2 ) )
= ( cons_P5520042648066362825tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) @ ( zip_a_state_a2 @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_697_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: state_a3,Ys2: list_state_a2] :
( ( zip_a_state_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a @ Y @ Ys2 ) )
= ( cons_P6411715633656068112tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) @ ( zip_a_state_a @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_698_enumerate__append__eq,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( enumerate_a @ N @ ( append_a @ Xs @ Ys2 ) )
= ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_699_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys2: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
( ( ( zip_a_a @ Xs @ Ys2 )
= ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
=> ~ ! [X4: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs4 ) )
=> ! [Y5: a,Ys4: list_a] :
( ( Ys2
= ( cons_a @ Y5 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_a_a @ X4 @ Y5 ) )
=> ( Xys
!= ( zip_a_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_700_zip__eq__ConsE,axiom,
! [Xs: list_list_a,Ys2: list_list_a,Xy: produc9164743771328383783list_a,Xys: list_P321204300973800749list_a] :
( ( ( zip_list_a_list_a @ Xs @ Ys2 )
= ( cons_P5184657343811988189list_a @ Xy @ Xys ) )
=> ~ ! [X4: list_a,Xs4: list_list_a] :
( ( Xs
= ( cons_list_a @ X4 @ Xs4 ) )
=> ! [Y5: list_a,Ys4: list_list_a] :
( ( Ys2
= ( cons_list_a @ Y5 @ Ys4 ) )
=> ( ( Xy
= ( produc6837034575241423639list_a @ X4 @ Y5 ) )
=> ( Xys
!= ( zip_list_a_list_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_701_zip__eq__ConsE,axiom,
! [Xs: list_states_a,Ys2: list_nat,Xy: produc1571854377283420419_a_nat,Xys: list_P4144771487928076819_a_nat] :
( ( ( zip_states_a_nat @ Xs @ Ys2 )
= ( cons_P1204124355475727053_a_nat @ Xy @ Xys ) )
=> ~ ! [X4: states_a,Xs4: list_states_a] :
( ( Xs
= ( cons_states_a @ X4 @ Xs4 ) )
=> ! [Y5: nat,Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ Y5 @ Ys4 ) )
=> ( ( Xy
= ( produc1877401315875745917_a_nat @ X4 @ Y5 ) )
=> ( Xys
!= ( zip_states_a_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_702_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys2: list_state_a,Xy: produc7589950997499123219tate_a,Xys: list_P4482599289824689689tate_a] :
( ( ( zip_a_state_a2 @ Xs @ Ys2 )
= ( cons_P5520042648066362825tate_a @ Xy @ Xys ) )
=> ~ ! [X4: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs4 ) )
=> ! [Y5: state_a2,Ys4: list_state_a] :
( ( Ys2
= ( cons_state_a2 @ Y5 @ Ys4 ) )
=> ( ( Xy
= ( produc1224139502141355779tate_a @ X4 @ Y5 ) )
=> ( Xys
!= ( zip_a_state_a2 @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_703_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys2: list_state_a2,Xy: produc6972303929186420058tate_a,Xys: list_P6747491281921308512tate_a] :
( ( ( zip_a_state_a @ Xs @ Ys2 )
= ( cons_P6411715633656068112tate_a @ Xy @ Xys ) )
=> ~ ! [X4: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X4 @ Xs4 ) )
=> ! [Y5: state_a3,Ys4: list_state_a2] :
( ( Ys2
= ( cons_state_a @ Y5 @ Ys4 ) )
=> ( ( Xy
= ( produc8641956578966763338tate_a @ X4 @ Y5 ) )
=> ( Xys
!= ( zip_a_state_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_704_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_705_append__eq__Cons__conv,axiom,
! [Ys2: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys2 @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys2 = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys2
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_706_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys2 )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_707_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y5: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y5 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_708_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_709_append__Cons,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys2 )
= ( cons_a @ X @ ( append_a @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_710_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys2: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_711_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_712_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys2: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys2 )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_713_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_714_butlast__append,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( ( Ys2 = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys2 != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ Xs @ ( butlast_a @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_715_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_a @ nil_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_716_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_717_append__Nil,axiom,
! [Ys2: list_a] :
( ( append_a @ nil_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_718_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_719_tl__append__if,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( tl_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_720_lex__append__leftI,axiom,
! [Ys2: list_a,Zs: list_a,R: set_Product_prod_a_a,Xs: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys2 ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) ) ) ).
% lex_append_leftI
thf(fact_721_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_722_same__length__different,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ? [Pre: list_a,X4: a,Xs4: list_a,Y5: a,Ys4: list_a] :
( ( X4 != Y5 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y5 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_723_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_724_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys2: list_a,Zs: list_a] :
( ! [X4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys2 ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_725_lex__append__leftD,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys2: list_list_a,Zs: list_list_a] :
( ! [X4: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ X4 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys2 ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys2 @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_726_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys2: list_a,Zs: list_a] :
( ! [X4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys2 ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys2 @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_727_lex__append__left__iff,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys2: list_list_a,Zs: list_list_a] :
( ! [X4: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ X4 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys2 ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys2 @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_728_lex__append__rightI,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( lex_a @ R ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys2 @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_729_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R2: set_Product_prod_a_a,Vs: list_a,Ys2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys2 ) ) @ ( lenlex_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_730_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y4: a,Ys: list_a] :
( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y4 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_731_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_732_nth__append,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys2 ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys2 ) @ N )
= ( nth_a @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_733_subrelI,axiom,
! [R: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ! [X4: list_a,Y5: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y5 ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y5 ) @ S ) )
=> ( ord_le7857023143581076903list_a @ R @ S ) ) ).
% subrelI
thf(fact_734_subrelI,axiom,
! [R: set_Pr1464008215722202041_a_nat,S: set_Pr1464008215722202041_a_nat] :
( ! [X4: states_a,Y5: nat] :
( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y5 ) @ R )
=> ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y5 ) @ S ) )
=> ( ord_le6196172653455811609_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_735_subrelI,axiom,
! [R: set_Pr6306228930610421491tate_a,S: set_Pr6306228930610421491tate_a] :
( ! [X4: a,Y5: state_a2] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y5 ) @ R )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y5 ) @ S ) )
=> ( ord_le926351553812985491tate_a @ R @ S ) ) ).
% subrelI
thf(fact_736_subrelI,axiom,
! [R: set_Pr4275752383657305402tate_a,S: set_Pr4275752383657305402tate_a] :
( ! [X4: a,Y5: state_a3] :
( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y5 ) @ R )
=> ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y5 ) @ S ) )
=> ( ord_le7345504482307493082tate_a @ R @ S ) ) ).
% subrelI
thf(fact_737_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_738_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_739_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_740_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_741_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_742_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_743_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_744_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_745_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_746_append__take__drop__id,axiom,
! [N: nat,Xs: list_a] :
( ( append_a @ ( take_a @ N @ Xs ) @ ( drop_a @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_747_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_748_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_749_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_750_drop__append,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_751_drop__Suc,axiom,
! [N: nat,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ Xs )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% drop_Suc
thf(fact_752_tl__drop__2,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( drop_a @ N @ Xs ) )
= ( drop_a @ ( suc @ N ) @ Xs ) ) ).
% tl_drop_2
thf(fact_753_tl__drop,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( drop_a @ N @ Xs ) )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% tl_drop
thf(fact_754_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_755_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys2: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys2 ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_756_take__add,axiom,
! [I: nat,J: nat,Xs: list_a] :
( ( take_a @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( take_a @ J @ ( drop_a @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_757_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys2: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_758_drop__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_759_append__eq__append__conv__if,axiom,
! [Xs_1: list_a,Xs_2: list_a,Ys_1: list_a,Ys_2: list_a] :
( ( ( append_a @ Xs_1 @ Xs_2 )
= ( append_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_a @ ( drop_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( ( take_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_a @ ( drop_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_760_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_761_take__hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_762_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A2: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A2 )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A2 @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_763_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_764_hd__append2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_765_list__update__length,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys2 ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_766_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_767_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_768_take__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% take_hd
thf(fact_769_hd__drop__1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_770_hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ ( drop_a @ ( suc @ N ) @ Xs ) )
= ( drop_a @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_771_hd__zip,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( hd_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys2 ) )
= ( product_Pair_a_a @ ( hd_a @ Xs ) @ ( hd_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_772_hd__zip,axiom,
! [Xs: list_list_a,Ys2: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( Ys2 != nil_list_a )
=> ( ( hd_Pro6433485837049129490list_a @ ( zip_list_a_list_a @ Xs @ Ys2 ) )
= ( produc6837034575241423639list_a @ ( hd_list_a @ Xs ) @ ( hd_list_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_773_hd__zip,axiom,
! [Xs: list_states_a,Ys2: list_nat] :
( ( Xs != nil_states_a )
=> ( ( Ys2 != nil_nat )
=> ( ( hd_Pro4761346146552465112_a_nat @ ( zip_states_a_nat @ Xs @ Ys2 ) )
= ( produc1877401315875745917_a_nat @ ( hd_states_a @ Xs ) @ ( hd_nat @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_774_hd__zip,axiom,
! [Xs: list_a,Ys2: list_state_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_state_a2 )
=> ( ( hd_Pro2196137027883835646tate_a @ ( zip_a_state_a2 @ Xs @ Ys2 ) )
= ( produc1224139502141355779tate_a @ ( hd_a @ Xs ) @ ( hd_state_a2 @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_775_hd__zip,axiom,
! [Xs: list_a,Ys2: list_state_a2] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_state_a )
=> ( ( hd_Pro8463085911274001861tate_a @ ( zip_a_state_a @ Xs @ Ys2 ) )
= ( produc8641956578966763338tate_a @ ( hd_a @ Xs ) @ ( hd_state_a @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_776_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_777_cons__hd,axiom,
! [X: a,Xs: list_a,Ys2: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys2 )
=> ( X
= ( hd_a @ Ys2 ) ) ) ).
% cons_hd
thf(fact_778_hd__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Ys2 ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys2 ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_779_longest__common__prefix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ps: list_a,Xs4: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys2
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_780_list__update_Osimps_I1_J,axiom,
! [I: nat,V: a] :
( ( list_update_a @ nil_a @ I @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_781_list__update__code_I1_J,axiom,
! [I: nat,Y: a] :
( ( list_update_a @ nil_a @ I @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_782_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_783_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_784_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_785_zip__update,axiom,
! [Xs: list_list_a,I: nat,X: list_a,Ys2: list_list_a,Y: list_a] :
( ( zip_list_a_list_a @ ( list_update_list_a @ Xs @ I @ X ) @ ( list_update_list_a @ Ys2 @ I @ Y ) )
= ( list_u6458906768619699605list_a @ ( zip_list_a_list_a @ Xs @ Ys2 ) @ I @ ( produc6837034575241423639list_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_786_zip__update,axiom,
! [Xs: list_states_a,I: nat,X: states_a,Ys2: list_nat,Y: nat] :
( ( zip_states_a_nat @ ( list_update_states_a @ Xs @ I @ X ) @ ( list_update_nat @ Ys2 @ I @ Y ) )
= ( list_u4184495706060574229_a_nat @ ( zip_states_a_nat @ Xs @ Ys2 ) @ I @ ( produc1877401315875745917_a_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_787_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys2: list_state_a,Y: state_a2] :
( ( zip_a_state_a2 @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_state_a2 @ Ys2 @ I @ Y ) )
= ( list_u2256020193368096897tate_a @ ( zip_a_state_a2 @ Xs @ Ys2 ) @ I @ ( produc1224139502141355779tate_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_788_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys2: list_state_a2,Y: state_a3] :
( ( zip_a_state_a @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_state_a @ Ys2 @ I @ Y ) )
= ( list_u2732569323712218824tate_a @ ( zip_a_state_a @ Xs @ Ys2 ) @ I @ ( produc8641956578966763338tate_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_789_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_790_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_791_list__update__append1,axiom,
! [I: nat,Xs: list_a,Ys2: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys2 ) @ I @ X )
= ( append_a @ ( list_update_a @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_792_length__one__hd,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% length_one_hd
thf(fact_793_list__update__append,axiom,
! [N: nat,Xs: list_a,Ys2: list_a,X: a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys2 ) @ N @ X )
= ( append_a @ ( list_update_a @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys2 ) @ N @ X )
= ( append_a @ Xs @ ( list_update_a @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_794_take__hd_H,axiom,
! [Ys2: list_a,X: a,Xs: list_a] :
( ( Ys2 != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys2 ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys2 ) )
=> ( ( hd_a @ Ys2 )
= X ) ) ) ).
% take_hd'
thf(fact_795_take__Suc,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_796_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_797_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F3: a > nat,Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F3 @ ( hd_a @ Xs3 ) ) @ ( size_list_a @ F3 @ ( tl_a @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_798_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_a,Ys2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys2 )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Ys2 ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_799_size__list__append,axiom,
! [F: a > nat,Xs: list_a,Ys2: list_a] :
( ( size_list_a @ F @ ( append_a @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ F @ Ys2 ) ) ) ).
% size_list_append
thf(fact_800_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_801_take__rev__nth,axiom,
! [N: nat,Xs: list_a,X: a,Ys2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( X
= ( nth_a @ Xs @ N ) )
=> ( ( cons_a @ X @ ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys2 ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Ys2 ) ) ) ) ).
% take_rev_nth
thf(fact_802_take__rev__step,axiom,
! [Xs: list_a,N: nat,Acc: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Acc ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_803_listrel1__iff__update,axiom,
! [Xs: list_list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listrel1_list_a @ R ) )
= ( ? [Y4: list_a,N4: nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N4 ) @ Y4 ) @ R )
& ( ord_less_nat @ N4 @ ( size_s349497388124573686list_a @ Xs ) )
& ( Ys2
= ( list_update_list_a @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_804_listrel1__iff__update,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
= ( ? [Y4: a,N4: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N4 ) @ Y4 ) @ R )
& ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs ) )
& ( Ys2
= ( list_update_a @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_805_Cons__listrel1__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys2 ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_806_Cons__listrel1__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_807_listrel1I2,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys2 ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I2
thf(fact_808_not__Nil__listrel1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_809_not__listrel1__Nil,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_810_listrel1__eq__len,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_811_append__listrel1I,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys2 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Vs ) @ ( listrel1_a @ R ) ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys2 @ Vs ) ) @ ( listrel1_a @ R ) ) ) ).
% append_listrel1I
thf(fact_812_Cons__listrel1E2,axiom,
! [Xs: list_list_a,Y: list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys2 ) ) @ ( listrel1_list_a @ R ) )
=> ( ! [X4: list_a] :
( ( Xs
= ( cons_list_a @ X4 @ Ys2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Y @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs2 @ Ys2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_813_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys2 ) ) @ ( listrel1_a @ R ) )
=> ( ! [X4: a] :
( ( Xs
= ( cons_a @ X4 @ Ys2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_814_Cons__listrel1E1,axiom,
! [X: list_a,Xs: list_list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ Ys2 ) @ ( listrel1_list_a @ R ) )
=> ( ! [Y5: list_a] :
( ( Ys2
= ( cons_list_a @ Y5 @ Xs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y5 ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Ys2
= ( cons_list_a @ X @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_815_Cons__listrel1E1,axiom,
! [X: a,Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys2 ) @ ( listrel1_a @ R ) )
=> ( ! [Y5: a] :
( ( Ys2
= ( cons_a @ Y5 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Ys2
= ( cons_a @ X @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_816_listrel1I1,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I1
thf(fact_817_listrel1I1,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Xs ) ) @ ( listrel1_list_a @ R ) ) ) ).
% listrel1I1
thf(fact_818_listrel1E,axiom,
! [Xs: list_list_a,Ys2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listrel1_list_a @ R ) )
=> ~ ! [X4: list_a,Y5: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y5 ) @ R )
=> ! [Us3: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us3 @ ( cons_list_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append_list_a @ Us3 @ ( cons_list_a @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_819_listrel1E,axiom,
! [Xs: list_a,Ys2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
=> ~ ! [X4: a,Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ R )
=> ! [Us3: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us3 @ ( cons_a @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append_a @ Us3 @ ( cons_a @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_820_listrel1I,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_821_listrel1I,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a,Us: list_list_a,Vs: list_list_a,Ys2: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( ( Xs
= ( append_list_a @ Us @ ( cons_list_a @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_list_a @ Us @ ( cons_list_a @ Y @ Vs ) ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_822_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_a,X: list_a,Ys2: list_list_a,Y: list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) @ ( append_list_a @ Ys2 @ ( cons_list_a @ Y @ nil_list_a ) ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys2 ) @ ( listrel1_list_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_823_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys2: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_824_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_825_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_826_last__appendR,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ).
% last_appendR
thf(fact_827_last__appendL,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_828_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_829_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_830_last__append,axiom,
! [Ys2: list_a,Xs: list_a] :
( ( ( Ys2 = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Xs ) ) )
& ( ( Ys2 != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys2 ) )
= ( last_a @ Ys2 ) ) ) ) ).
% last_append
thf(fact_831_longest__common__suffix,axiom,
! [Xs: list_a,Ys2: list_a] :
? [Ss: list_a,Xs4: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys2
= ( append_a @ Ys4 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys4 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_832_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_833_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_834_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_835_last__zip,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( last_P8790725268278465478od_a_a @ ( zip_a_a @ Xs @ Ys2 ) )
= ( product_Pair_a_a @ ( last_a @ Xs ) @ ( last_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_836_last__zip,axiom,
! [Xs: list_list_a,Ys2: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( Ys2 != nil_list_a )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( last_P2616948663241161414list_a @ ( zip_list_a_list_a @ Xs @ Ys2 ) )
= ( produc6837034575241423639list_a @ ( last_list_a @ Xs ) @ ( last_list_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_837_last__zip,axiom,
! [Xs: list_states_a,Ys2: list_nat] :
( ( Xs != nil_states_a )
=> ( ( Ys2 != nil_nat )
=> ( ( ( size_s3891197933023997302ates_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( last_P1521129688936790820_a_nat @ ( zip_states_a_nat @ Xs @ Ys2 ) )
= ( produc1877401315875745917_a_nat @ ( last_states_a @ Xs ) @ ( last_nat @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_838_last__zip,axiom,
! [Xs: list_a,Ys2: list_state_a] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_state_a2 )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s8463391772401140188tate_a @ Ys2 ) )
=> ( ( last_P6377108300570313138tate_a @ ( zip_a_state_a2 @ Xs @ Ys2 ) )
= ( produc1224139502141355779tate_a @ ( last_a @ Xs ) @ ( last_state_a2 @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_839_last__zip,axiom,
! [Xs: list_a,Ys2: list_state_a2] :
( ( Xs != nil_a )
=> ( ( Ys2 != nil_state_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s7859192958365828515tate_a @ Ys2 ) )
=> ( ( last_P7000629880413671545tate_a @ ( zip_a_state_a @ Xs @ Ys2 ) )
= ( produc8641956578966763338tate_a @ ( last_a @ Xs ) @ ( last_state_a @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_840_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_841_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys2: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys2 )
= ( ( Ys2 != nil_a )
& ( ( butlast_a @ Ys2 )
= Xs )
& ( ( last_a @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_842_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_843_take__last__length,axiom,
! [Xs: list_a] :
( ( ( take_a @ ( suc @ zero_zero_nat ) @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ Xs ) ) ) ).
% take_last_length
thf(fact_844_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys2 ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys2 )
& ( ( Xs = nil_a )
| ( Ys2 = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_845_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys2 ) )
= ( Xs = Ys2 ) ) ).
% rev_is_rev_conv
thf(fact_846_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_847_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_848_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_849_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_850_rev__append,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys2 ) )
= ( append_a @ ( rev_a @ Ys2 ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_851_distinct__adj__Cons__Cons,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_852_distinct__adj__rev,axiom,
! [Xs: list_a] :
( ( distinct_adj_a @ ( rev_a @ Xs ) )
= ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_rev
thf(fact_853_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_854_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_855_butlast__rev,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( rev_a @ Xs ) )
= ( rev_a @ ( tl_a @ Xs ) ) ) ).
% butlast_rev
thf(fact_856_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys2 ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys2 ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_857_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_858_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_859_rev__app__single,axiom,
! [Xs: list_a,X: a] :
( ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) )
= ( rev_a @ ( cons_a @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_860_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_861_distinct__adj__ConsD,axiom,
! [X: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_862_distinct__adj__appendD2,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys2 ) )
=> ( distinct_adj_a @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_863_distinct__adj__appendD1,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys2 ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_864_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_865_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_866_rev__swap,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( rev_a @ Xs )
= Ys2 )
= ( Xs
= ( rev_a @ Ys2 ) ) ) ).
% rev_swap
thf(fact_867_zip__rev,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( zip_a_a @ ( rev_a @ Xs ) @ ( rev_a @ Ys2 ) )
= ( rev_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys2 ) ) ) ) ).
% zip_rev
thf(fact_868_take__rev,axiom,
! [N: nat,Xs: list_a] :
( ( take_a @ N @ ( rev_a @ Xs ) )
= ( rev_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_869_rev__take,axiom,
! [I: nat,Xs: list_a] :
( ( rev_a @ ( take_a @ I @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ I ) @ ( rev_a @ Xs ) ) ) ).
% rev_take
thf(fact_870_rev__drop,axiom,
! [I: nat,Xs: list_a] :
( ( rev_a @ ( drop_a @ I @ Xs ) )
= ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ I ) @ ( rev_a @ Xs ) ) ) ).
% rev_drop
thf(fact_871_drop__rev,axiom,
! [N: nat,Xs: list_a] :
( ( drop_a @ N @ ( rev_a @ Xs ) )
= ( rev_a @ ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_872_distinct__adj__Cons,axiom,
! [X: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_873_rev__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_874_rev__update,axiom,
! [K: nat,Xs: list_a,Y: a] :
( ( ord_less_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( rev_a @ ( list_update_a @ Xs @ K @ Y ) )
= ( list_update_a @ ( rev_a @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_875_rev__tl__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( rev_a @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) )
= ( rev_a @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_876_take__last,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ one_one_nat @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) ).
% take_last
thf(fact_877_last__drop__rev,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( last_a @ Xs ) @ ( drop_a @ one_one_nat @ ( rev_a @ Xs ) ) )
= ( rev_a @ Xs ) ) ) ).
% last_drop_rev
thf(fact_878_take__rev__drop,axiom,
! [N: nat,Xs: list_a,Acc: list_a] :
( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Acc )
= ( append_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) @ ( rev_a @ Xs ) ) @ Acc ) ) ).
% take_rev_drop
thf(fact_879_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_880_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_881_Small_Ostate_Oinject_I2_J,axiom,
! [X21: current_a,X222: list_a,X232: stack_a,X24: list_a,X25: nat,Y21: current_a,Y222: list_a,Y23: stack_a,Y24: list_a,Y25: nat] :
( ( ( reverse2_a @ X21 @ X222 @ X232 @ X24 @ X25 )
= ( reverse2_a @ Y21 @ Y222 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 )
& ( X232 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% Small.state.inject(2)
thf(fact_882_Big_Ostate_Oinject_I2_J,axiom,
! [X22: state_a,Y22: state_a] :
( ( ( common_a @ X22 )
= ( common_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% Big.state.inject(2)
thf(fact_883_step__list__reverse2,axiom,
! [Small: state_a2,Current: current_a,Aux: list_a,Big: stack_a,New: list_a,Count: nat] :
( ( Small
= ( reverse2_a @ Current @ Aux @ Big @ New @ Count ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( small_list_a @ ( type_s3703408523585882337tate_a @ Small ) )
= ( small_list_a @ Small ) ) ) ) ).
% step_list_reverse2
thf(fact_884_Big_Ostep__state_Ocases,axiom,
! [X: state_a3] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ( ! [Current2: current_a,Uu: stack_a,Aux2: list_a] :
( X
!= ( reverse_a @ Current2 @ Uu @ Aux2 @ zero_zero_nat ) )
=> ~ ! [Current2: current_a,Big3: stack_a,Aux2: list_a,V2: nat] :
( X
!= ( reverse_a @ Current2 @ Big3 @ Aux2 @ ( suc @ V2 ) ) ) ) ) ).
% Big.step_state.cases
thf(fact_885_Big_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X14: nat,X22: state_a] :
( ( reverse_a @ X11 @ X122 @ X13 @ X14 )
!= ( common_a @ X22 ) ) ).
% Big.state.distinct(1)
thf(fact_886_Big_Ostate_Oexhaust,axiom,
! [Y: state_a3] :
( ! [X112: current_a,X123: stack_a,X132: list_a,X142: nat] :
( Y
!= ( reverse_a @ X112 @ X123 @ X132 @ X142 ) )
=> ~ ! [X23: state_a] :
( Y
!= ( common_a @ X23 ) ) ) ).
% Big.state.exhaust
thf(fact_887_Small_Ostate_Odistinct_I5_J,axiom,
! [X21: current_a,X222: list_a,X232: stack_a,X24: list_a,X25: nat,X3: state_a] :
( ( reverse2_a @ X21 @ X222 @ X232 @ X24 @ X25 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(5)
thf(fact_888_Big_Opush_Ocases,axiom,
! [X: produc6972303929186420058tate_a] :
( ! [X4: a,State2: state_a] :
( X
!= ( produc8641956578966763338tate_a @ X4 @ ( common_a @ State2 ) ) )
=> ~ ! [X4: a,Current2: current_a,Big3: stack_a,Aux2: list_a,Count2: nat] :
( X
!= ( produc8641956578966763338tate_a @ X4 @ ( reverse_a @ Current2 @ Big3 @ Aux2 @ Count2 ) ) ) ) ).
% Big.push.cases
thf(fact_889_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_890_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_891_Big__Aux_Osize__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( size_size_state_a @ ( common_a @ State ) )
= ( size_size_state_a2 @ State ) ) ).
% Big_Aux.size_state.simps(1)
thf(fact_892_step__states_Oelims,axiom,
! [X: states_a,Y: states_a] :
( ( ( type_s4923920245906622843ates_a @ X )
= Y )
=> ( ! [Dir3: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir3 @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( Y
!= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS @ AuxS @ Big3 @ nil_a @ zero_zero_nat ) ) ) )
=> ( ! [Dir3: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir3 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( Y
!= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) ) )
=> ( ! [Dir3: direction,V2: state_a,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir3 @ ( common_a @ V2 ) @ Right2 ) )
=> ( Y
!= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( common_a @ V2 ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) ) )
=> ( ! [Dir3: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Dir3 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir3: direction,Left2: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Dir3 @ Left2 @ ( common_a2 @ V2 ) ) )
=> ( Y
!= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% step_states.elims
thf(fact_893_Small_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,Y11: current_a,Y12: stack_a,Y13: list_a] :
( ( ( reverse1_a @ X11 @ X122 @ X13 )
= ( reverse1_a @ Y11 @ Y12 @ Y13 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 ) ) ) ).
% Small.state.inject(1)
thf(fact_894_Small_Opush_Ocases,axiom,
! [X: produc7589950997499123219tate_a] :
( ! [X4: a,State2: state_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( common_a2 @ State2 ) ) )
=> ( ! [X4: a,Current2: current_a,Small3: stack_a,AuxS: list_a] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse1_a @ Current2 @ Small3 @ AuxS ) ) )
=> ~ ! [X4: a,Current2: current_a,AuxS: list_a,Big3: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( produc1224139502141355779tate_a @ X4 @ ( reverse2_a @ Current2 @ AuxS @ Big3 @ NewS @ Count2 ) ) ) ) ) ).
% Small.push.cases
thf(fact_895_Small__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a2 @ State2 ) )
=> ( ! [Current2: current_a,Uu: list_a,Uv2: stack_a,Uw: list_a,Ux: nat] :
( X
!= ( reverse2_a @ Current2 @ Uu @ Uv2 @ Uw @ Ux ) )
=> ~ ! [Current2: current_a,Uy: stack_a,Uz: list_a] :
( X
!= ( reverse1_a @ Current2 @ Uy @ Uz ) ) ) ) ).
% Small_Aux.size_new_state.cases
thf(fact_896_Small_Ostep__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a2 @ State2 ) )
=> ( ! [Current2: current_a,Small3: stack_a,AuxS: list_a] :
( X
!= ( reverse1_a @ Current2 @ Small3 @ AuxS ) )
=> ~ ! [Current2: current_a,AuxS: list_a,Big3: stack_a,NewS: list_a,Count2: nat] :
( X
!= ( reverse2_a @ Current2 @ AuxS @ Big3 @ NewS @ Count2 ) ) ) ) ).
% Small.step_state.cases
thf(fact_897_Small_Ostate_Oexhaust,axiom,
! [Y: state_a2] :
( ! [X112: current_a,X123: stack_a,X132: list_a] :
( Y
!= ( reverse1_a @ X112 @ X123 @ X132 ) )
=> ( ! [X212: current_a,X223: list_a,X233: stack_a,X242: list_a,X252: nat] :
( Y
!= ( reverse2_a @ X212 @ X223 @ X233 @ X242 @ X252 ) )
=> ~ ! [X32: state_a] :
( Y
!= ( common_a2 @ X32 ) ) ) ) ).
% Small.state.exhaust
thf(fact_898_Small_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X21: current_a,X222: list_a,X232: stack_a,X24: list_a,X25: nat] :
( ( reverse1_a @ X11 @ X122 @ X13 )
!= ( reverse2_a @ X21 @ X222 @ X232 @ X24 @ X25 ) ) ).
% Small.state.distinct(1)
thf(fact_899_Small_Ostate_Odistinct_I3_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X3: state_a] :
( ( reverse1_a @ X11 @ X122 @ X13 )
!= ( common_a2 @ X3 ) ) ).
% Small.state.distinct(3)
thf(fact_900_States__Aux_Olists_Ocases,axiom,
! [X: states_a] :
( ! [Uu: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,Count2: nat,CurrentS: current_a,Small3: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Uu @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) @ ( reverse1_a @ CurrentS @ Small3 @ AuxS ) ) )
=> ( ! [Uv2: direction,V2: state_a,Small3: state_a2] :
( X
!= ( states_a2 @ Uv2 @ ( common_a @ V2 ) @ Small3 ) )
=> ( ! [Uv2: direction,Big3: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( X
!= ( states_a2 @ Uv2 @ Big3 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Uv2: direction,Big3: state_a3,V2: state_a] :
( X
!= ( states_a2 @ Uv2 @ Big3 @ ( common_a2 @ V2 ) ) ) ) ) ) ).
% States_Aux.lists.cases
thf(fact_901_step__states_Ocases,axiom,
! [X: states_a] :
( ! [Dir3: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( X
!= ( states_a2 @ Dir3 @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( ! [Dir3: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( X
!= ( states_a2 @ Dir3 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ! [Dir3: direction,V2: state_a,Right2: state_a2] :
( X
!= ( states_a2 @ Dir3 @ ( common_a @ V2 ) @ Right2 ) )
=> ( ! [Dir3: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( X
!= ( states_a2 @ Dir3 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ~ ! [Dir3: direction,Left2: state_a3,V2: state_a] :
( X
!= ( states_a2 @ Dir3 @ Left2 @ ( common_a2 @ V2 ) ) ) ) ) ) ) ).
% step_states.cases
thf(fact_902_step__states_Osimps_I1_J,axiom,
! [Dir: direction,CurrentB2: current_a,Big: stack_a,AuxB2: list_a,CurrentS2: current_a,Uu2: stack_a,AuxS2: list_a] :
( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) @ ( reverse1_a @ CurrentS2 @ Uu2 @ AuxS2 ) ) )
= ( states_a2 @ Dir @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS2 @ AuxS2 @ Big @ nil_a @ zero_zero_nat ) ) ) ).
% step_states.simps(1)
thf(fact_903_Big__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a3] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ~ ! [Current2: current_a,Uu: stack_a,Uv2: list_a,Uw: nat] :
( X
!= ( reverse_a @ Current2 @ Uu @ Uv2 @ Uw ) ) ) ).
% Big_Aux.size_new_state.cases
thf(fact_904_step__states_Opelims,axiom,
! [X: states_a,Y: states_a] :
( ( ( type_s4923920245906622843ates_a @ X )
= Y )
=> ( ( accp_states_a @ step_states_rel_a @ X )
=> ( ! [Dir3: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,CurrentS: current_a,Uu: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Dir3 @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) )
=> ( ( Y
= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) ) @ ( reverse2_a @ CurrentS @ AuxS @ Big3 @ nil_a @ zero_zero_nat ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir3 @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ zero_zero_nat ) @ ( reverse1_a @ CurrentS @ Uu @ AuxS ) ) ) ) )
=> ( ! [Dir3: direction,V2: current_a,Va2: stack_a,Vb2: list_a,Vd2: nat,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir3 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) )
=> ( ( Y
= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir3 @ ( reverse_a @ V2 @ Va2 @ Vb2 @ ( suc @ Vd2 ) ) @ Right2 ) ) ) )
=> ( ! [Dir3: direction,V2: state_a,Right2: state_a2] :
( ( X
= ( states_a2 @ Dir3 @ ( common_a @ V2 ) @ Right2 ) )
=> ( ( Y
= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ ( common_a @ V2 ) ) @ ( type_s3703408523585882337tate_a @ Right2 ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir3 @ ( common_a @ V2 ) @ Right2 ) ) ) )
=> ( ! [Dir3: direction,Left2: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Dir3 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( ( Y
= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir3 @ Left2 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Dir3: direction,Left2: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Dir3 @ Left2 @ ( common_a2 @ V2 ) ) )
=> ( ( Y
= ( states_a2 @ Dir3 @ ( type_s3593206172722485288tate_a @ Left2 ) @ ( type_s3703408523585882337tate_a @ ( common_a2 @ V2 ) ) ) )
=> ~ ( accp_states_a @ step_states_rel_a @ ( states_a2 @ Dir3 @ Left2 @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% step_states.pelims
thf(fact_905_lists_Oelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states_lists_a @ X )
= Y )
=> ( ! [Uu: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,Count2: nat,CurrentS: current_a,Small3: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Uu @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) @ ( reverse1_a @ CurrentS @ Small3 @ AuxS ) ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_a @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) ) @ ( small_list_a @ ( reverse2_a @ CurrentS @ ( append_a @ ( common_take_rev_a @ Count2 @ ( stack_list_a @ Small3 ) ) @ AuxS ) @ ( compow4264569633760279794tack_a @ Count2 @ pop_a4 @ Big3 ) @ nil_a @ zero_zero_nat ) ) ) ) )
=> ( ! [Uv2: direction,V2: state_a,Small3: state_a2] :
( ( X
= ( states_a2 @ Uv2 @ ( common_a @ V2 ) @ Small3 ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_a @ ( common_a @ V2 ) ) @ ( small_list_a @ Small3 ) ) ) )
=> ( ! [Uv2: direction,Big3: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Uv2 @ Big3 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_a @ Big3 ) @ ( small_list_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Uv2: direction,Big3: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Uv2 @ Big3 @ ( common_a2 @ V2 ) ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_a @ Big3 ) @ ( small_list_a @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ).
% lists.elims
thf(fact_906_States__Aux_Olists_Osimps_I1_J,axiom,
! [Uu2: direction,CurrentB2: current_a,Big: stack_a,AuxB2: list_a,Count: nat,CurrentS2: current_a,Small: stack_a,AuxS2: list_a] :
( ( states_lists_a @ ( states_a2 @ Uu2 @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ Count ) @ ( reverse1_a @ CurrentS2 @ Small @ AuxS2 ) ) )
= ( produc6837034575241423639list_a @ ( big_list_a @ ( reverse_a @ CurrentB2 @ Big @ AuxB2 @ Count ) ) @ ( small_list_a @ ( reverse2_a @ CurrentS2 @ ( append_a @ ( common_take_rev_a @ Count @ ( stack_list_a @ Small ) ) @ AuxS2 ) @ ( compow4264569633760279794tack_a @ Count @ pop_a4 @ Big ) @ nil_a @ zero_zero_nat ) ) ) ) ).
% States_Aux.lists.simps(1)
thf(fact_907_lists_Opelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states_lists_a @ X )
= Y )
=> ( ( accp_states_a @ states_lists_rel_a @ X )
=> ( ! [Uu: direction,CurrentB: current_a,Big3: stack_a,AuxB: list_a,Count2: nat,CurrentS: current_a,Small3: stack_a,AuxS: list_a] :
( ( X
= ( states_a2 @ Uu @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) @ ( reverse1_a @ CurrentS @ Small3 @ AuxS ) ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_a @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) ) @ ( small_list_a @ ( reverse2_a @ CurrentS @ ( append_a @ ( common_take_rev_a @ Count2 @ ( stack_list_a @ Small3 ) ) @ AuxS ) @ ( compow4264569633760279794tack_a @ Count2 @ pop_a4 @ Big3 ) @ nil_a @ zero_zero_nat ) ) ) )
=> ~ ( accp_states_a @ states_lists_rel_a @ ( states_a2 @ Uu @ ( reverse_a @ CurrentB @ Big3 @ AuxB @ Count2 ) @ ( reverse1_a @ CurrentS @ Small3 @ AuxS ) ) ) ) )
=> ( ! [Uv2: direction,V2: state_a,Small3: state_a2] :
( ( X
= ( states_a2 @ Uv2 @ ( common_a @ V2 ) @ Small3 ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_a @ ( common_a @ V2 ) ) @ ( small_list_a @ Small3 ) ) )
=> ~ ( accp_states_a @ states_lists_rel_a @ ( states_a2 @ Uv2 @ ( common_a @ V2 ) @ Small3 ) ) ) )
=> ( ! [Uv2: direction,Big3: state_a3,V2: current_a,Va2: list_a,Vb2: stack_a,Vc2: list_a,Vd2: nat] :
( ( X
= ( states_a2 @ Uv2 @ Big3 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_a @ Big3 ) @ ( small_list_a @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) )
=> ~ ( accp_states_a @ states_lists_rel_a @ ( states_a2 @ Uv2 @ Big3 @ ( reverse2_a @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) ) ) )
=> ~ ! [Uv2: direction,Big3: state_a3,V2: state_a] :
( ( X
= ( states_a2 @ Uv2 @ Big3 @ ( common_a2 @ V2 ) ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_a @ Big3 ) @ ( small_list_a @ ( common_a2 @ V2 ) ) ) )
=> ~ ( accp_states_a @ states_lists_rel_a @ ( states_a2 @ Uv2 @ Big3 @ ( common_a2 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% lists.pelims
thf(fact_908_Stack__Proof_Osize__pop,axiom,
! [Stack: stack_a] :
( ( size_size_stack_a @ ( pop_a4 @ Stack ) )
= ( minus_minus_nat @ ( size_size_stack_a @ Stack ) @ ( suc @ zero_zero_nat ) ) ) ).
% Stack_Proof.size_pop
thf(fact_909_size__list__length,axiom,
! [Stack: stack_a] :
( ( size_size_list_a @ ( stack_list_a @ Stack ) )
= ( size_size_stack_a @ Stack ) ) ).
% size_list_length
thf(fact_910_popN__size,axiom,
! [N: nat,Stack: stack_a] :
( ( size_size_stack_a @ ( compow4264569633760279794tack_a @ N @ pop_a4 @ Stack ) )
= ( minus_minus_nat @ ( size_size_stack_a @ Stack ) @ N ) ) ).
% popN_size
thf(fact_911_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a @ Stack )
!= nil_a )
=> ( ( size_size_stack_a @ Stack )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_912_Stack__Proof_Olist__empty__size,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a @ Stack )
= nil_a )
= ( ( size_size_stack_a @ Stack )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_913_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a @ Stack )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_914_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a @ Stack )
!= nil_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_915_lists__current_Opelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ( ( accp_states_a @ states5251248496104418302_rel_a @ X )
=> ~ ! [Uu: direction,Big3: state_a3,Small3: state_a2] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( ( Y
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big3 ) @ ( small_list_current_a @ Small3 ) ) )
=> ~ ( accp_states_a @ states5251248496104418302_rel_a @ ( states_a2 @ Uu @ Big3 @ Small3 ) ) ) ) ) ) ).
% lists_current.pelims
thf(fact_916_first__take__tl,axiom,
! [Big: stack_a,Count: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Big ) )
=> ( ( cons_a @ ( first_a @ Big ) @ ( take_a @ Count @ ( tl_a @ ( stack_list_a @ Big ) ) ) )
= ( take_a @ ( suc @ Count ) @ ( stack_list_a @ Big ) ) ) ) ).
% first_take_tl
thf(fact_917_take__first,axiom,
! [S1: stack_a,S2: stack_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ S1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ S2 ) )
=> ( ( ( take_a @ ( size_size_stack_a @ S1 ) @ ( stack_list_a @ S2 ) )
= ( take_a @ ( size_size_stack_a @ S2 ) @ ( stack_list_a @ S1 ) ) )
=> ( ( first_a @ S1 )
= ( first_a @ S2 ) ) ) ) ) ).
% take_first
thf(fact_918_Big_Ostep__state_Osimps_I3_J,axiom,
! [Current: current_a,Big: stack_a,Aux: list_a,V: nat] :
( ( type_s3593206172722485288tate_a @ ( reverse_a @ Current @ Big @ Aux @ ( suc @ V ) ) )
= ( reverse_a @ Current @ ( pop_a4 @ Big ) @ ( cons_a @ ( first_a @ Big ) @ Aux ) @ ( minus_minus_nat @ ( suc @ V ) @ one_one_nat ) ) ) ).
% Big.step_state.simps(3)
thf(fact_919_first__take__pop,axiom,
! [Stack: stack_a,X: nat] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( cons_a @ ( first_a @ Stack ) @ ( take_a @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( stack_list_a @ ( pop_a4 @ Stack ) ) ) )
= ( take_a @ X @ ( stack_list_a @ Stack ) ) ) ) ) ).
% first_take_pop
thf(fact_920_Small__Aux_Olist_Osimps_I2_J,axiom,
! [Extra: list_a,Uu2: nat,Uv: stack_a,Remained: nat,Aux: list_a,Big: stack_a,New: list_a,Count: nat] :
( ( small_list_a @ ( reverse2_a @ ( current_a2 @ Extra @ Uu2 @ Uv @ Remained ) @ Aux @ Big @ New @ Count ) )
= ( append_a @ Extra @ ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ Remained @ ( plus_plus_nat @ Count @ ( size_size_stack_a @ Big ) ) ) @ Aux ) @ ( append_a @ ( rev_a @ ( stack_list_a @ Big ) ) @ New ) ) ) ) ).
% Small_Aux.list.simps(2)
thf(fact_921_pop__list__length,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( suc @ ( size_size_list_a @ ( stack_list_a @ ( pop_a4 @ Stack ) ) ) )
= ( size_size_list_a @ ( stack_list_a @ Stack ) ) ) ) ).
% pop_list_length
thf(fact_922_first__pop,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( cons_a @ ( first_a @ Stack ) @ ( stack_list_a @ ( pop_a4 @ Stack ) ) )
= ( stack_list_a @ Stack ) ) ) ).
% first_pop
thf(fact_923_Big__Aux_Oremaining__steps__state_Ocases,axiom,
! [X: state_a3] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ~ ! [Uu: list_a,Uv2: nat,Uw: stack_a,Remaining: nat,Ux: stack_a,Uy: list_a,Count2: nat] :
( X
!= ( reverse_a @ ( current_a2 @ Uu @ Uv2 @ Uw @ Remaining ) @ Ux @ Uy @ Count2 ) ) ) ).
% Big_Aux.remaining_steps_state.cases
thf(fact_924_Small__Aux_Olist_Ocases,axiom,
! [X: state_a2] :
( ! [Common2: state_a] :
( X
!= ( common_a2 @ Common2 ) )
=> ( ! [Extra2: list_a,Uu: nat,Uv2: stack_a,Remained2: nat,Aux2: list_a,Big3: stack_a,New2: list_a,Count2: nat] :
( X
!= ( reverse2_a @ ( current_a2 @ Extra2 @ Uu @ Uv2 @ Remained2 ) @ Aux2 @ Big3 @ New2 @ Count2 ) )
=> ~ ! [V2: current_a,Va2: stack_a,Vb2: list_a] :
( X
!= ( reverse1_a @ V2 @ Va2 @ Vb2 ) ) ) ) ).
% Small_Aux.list.cases
thf(fact_925_Stack__Proof_Osize__empty,axiom,
! [Stack: stack_a] :
( ( ( size_size_stack_a @ Stack )
= zero_zero_nat )
= ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.size_empty
thf(fact_926_Stack__Proof_Osize__empty__2,axiom,
! [Stack: stack_a] :
( ( ( size_size_stack_a @ Stack )
= zero_zero_nat )
=> ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.size_empty_2
thf(fact_927_Stack__Proof_Osize__not__empty__2,axiom,
! [Stack: stack_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) )
=> ~ ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.size_not_empty_2
thf(fact_928_Stack__Proof_Osize__not__empty,axiom,
! [Stack: stack_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) )
= ( ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ).
% Stack_Proof.size_not_empty
thf(fact_929_Big__Aux_Oremaining__steps__state_Osimps_I2_J,axiom,
! [Uu2: list_a,Uv: nat,Uw2: stack_a,Remaining2: nat,Ux2: stack_a,Uy2: list_a,Count: nat] :
( ( type_r2494999336194962664tate_a @ ( reverse_a @ ( current_a2 @ Uu2 @ Uv @ Uw2 @ Remaining2 ) @ Ux2 @ Uy2 @ Count ) )
= ( plus_plus_nat @ ( plus_plus_nat @ Count @ Remaining2 ) @ one_one_nat ) ) ).
% Big_Aux.remaining_steps_state.simps(2)
thf(fact_930_Small_Ostep__state_Osimps_I2_J,axiom,
! [Small: stack_a,Current: current_a,AuxS2: list_a] :
( ( ( type_i3216275384938974675tack_a @ Small )
=> ( ( type_s3703408523585882337tate_a @ ( reverse1_a @ Current @ Small @ AuxS2 ) )
= ( reverse1_a @ Current @ Small @ AuxS2 ) ) )
& ( ~ ( type_i3216275384938974675tack_a @ Small )
=> ( ( type_s3703408523585882337tate_a @ ( reverse1_a @ Current @ Small @ AuxS2 ) )
= ( reverse1_a @ Current @ ( pop_a4 @ Small ) @ ( cons_a @ ( first_a @ Small ) @ AuxS2 ) ) ) ) ) ).
% Small.step_state.simps(2)
thf(fact_931_first__take,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( cons_a @ ( first_a @ Stack ) @ nil_a )
= ( take_a @ one_one_nat @ ( stack_list_a @ Stack ) ) ) ) ).
% first_take
thf(fact_932_size__current_Osimps,axiom,
! [Uu2: list_a,Added: nat,Old: stack_a,Uv: nat] :
( ( size_size_current_a @ ( current_a2 @ Uu2 @ Added @ Old @ Uv ) )
= ( plus_plus_nat @ Added @ ( size_size_stack_a @ Old ) ) ) ).
% size_current.simps
thf(fact_933_size__current_Oelims,axiom,
! [X: current_a,Y: nat] :
( ( ( size_size_current_a @ X )
= Y )
=> ~ ! [Uu: list_a,Added2: nat,Old2: stack_a] :
( ? [Uv2: nat] :
( X
= ( current_a2 @ Uu @ Added2 @ Old2 @ Uv2 ) )
=> ( Y
!= ( plus_plus_nat @ Added2 @ ( size_size_stack_a @ Old2 ) ) ) ) ) ).
% size_current.elims
thf(fact_934_Current_Opop_Ocases,axiom,
! [X: current_a] :
( ! [Added2: nat,Old2: stack_a,Remained2: nat] :
( X
!= ( current_a2 @ nil_a @ Added2 @ Old2 @ Remained2 ) )
=> ~ ! [X4: a,Xs2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( X
!= ( current_a2 @ ( cons_a @ X4 @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) ) ) ).
% Current.pop.cases
thf(fact_935_Current_Opush_Ocases,axiom,
! [X: produc7805042584321970905rent_a] :
~ ! [X4: a,Extra2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( X
!= ( produc8503237746132909001rent_a @ X4 @ ( current_a2 @ Extra2 @ Added2 @ Old2 @ Remained2 ) ) ) ).
% Current.push.cases
thf(fact_936_Big__Aux_Oremaining__steps__state_Oelims,axiom,
! [X: state_a3,Y: nat] :
( ( ( type_r2494999336194962664tate_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( type_r2212416260012024137tate_a @ State2 ) ) )
=> ~ ! [Uu: list_a,Uv2: nat,Uw: stack_a,Remaining: nat,Ux: stack_a,Uy: list_a,Count2: nat] :
( ( X
= ( reverse_a @ ( current_a2 @ Uu @ Uv2 @ Uw @ Remaining ) @ Ux @ Uy @ Count2 ) )
=> ( Y
!= ( plus_plus_nat @ ( plus_plus_nat @ Count2 @ Remaining ) @ one_one_nat ) ) ) ) ) ).
% Big_Aux.remaining_steps_state.elims
thf(fact_937_Current_Opop_Oelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a2 @ X )
= Y )
=> ( ! [Added2: nat,Old2: stack_a,Remained2: nat] :
( ( X
= ( current_a2 @ nil_a @ Added2 @ Old2 @ Remained2 ) )
=> ( Y
!= ( produc8503237746132909001rent_a @ ( first_a @ Old2 ) @ ( current_a2 @ nil_a @ Added2 @ ( pop_a4 @ Old2 ) @ ( minus_minus_nat @ Remained2 @ one_one_nat ) ) ) ) )
=> ~ ! [X4: a,Xs2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X4 @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) )
=> ( Y
!= ( produc8503237746132909001rent_a @ X4 @ ( current_a2 @ Xs2 @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_938_Big__Aux_Oremaining__steps__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_r2494999336194962664tate_a @ ( common_a @ State ) )
= ( type_r2212416260012024137tate_a @ State ) ) ).
% Big_Aux.remaining_steps_state.simps(1)
thf(fact_939_Current_Opop_Osimps_I2_J,axiom,
! [X: a,Xs: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( pop_a2 @ ( current_a2 @ ( cons_a @ X @ Xs ) @ Added @ Old @ Remained ) )
= ( produc8503237746132909001rent_a @ X @ ( current_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ).
% Current.pop.simps(2)
thf(fact_940_Current_Opop_Osimps_I1_J,axiom,
! [Added: nat,Old: stack_a,Remained: nat] :
( ( pop_a2 @ ( current_a2 @ nil_a @ Added @ Old @ Remained ) )
= ( produc8503237746132909001rent_a @ ( first_a @ Old ) @ ( current_a2 @ nil_a @ Added @ ( pop_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) ) ).
% Current.pop.simps(1)
thf(fact_941_Small__Aux_Olist_Oelims,axiom,
! [X: state_a2,Y: list_a] :
( ( ( small_list_a @ X )
= Y )
=> ( ! [Common2: state_a] :
( ( X
= ( common_a2 @ Common2 ) )
=> ( Y
!= ( common_list_a @ Common2 ) ) )
=> ( ! [Extra2: list_a,Uu: nat,Uv2: stack_a,Remained2: nat,Aux2: list_a,Big3: stack_a,New2: list_a,Count2: nat] :
( ( X
= ( reverse2_a @ ( current_a2 @ Extra2 @ Uu @ Uv2 @ Remained2 ) @ Aux2 @ Big3 @ New2 @ Count2 ) )
=> ( Y
!= ( append_a @ Extra2 @ ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ Remained2 @ ( plus_plus_nat @ Count2 @ ( size_size_stack_a @ Big3 ) ) ) @ Aux2 ) @ ( append_a @ ( rev_a @ ( stack_list_a @ Big3 ) ) @ New2 ) ) ) ) )
=> ~ ( ? [V2: current_a,Va2: stack_a,Vb2: list_a] :
( X
= ( reverse1_a @ V2 @ Va2 @ Vb2 ) )
=> ( Y != undefined_list_a ) ) ) ) ) ).
% Small_Aux.list.elims
thf(fact_942_Current_Opop_Opelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a2 @ X )
= Y )
=> ( ( accp_current_a @ pop_rel_a @ X )
=> ( ! [Added2: nat,Old2: stack_a,Remained2: nat] :
( ( X
= ( current_a2 @ nil_a @ Added2 @ Old2 @ Remained2 ) )
=> ( ( Y
= ( produc8503237746132909001rent_a @ ( first_a @ Old2 ) @ ( current_a2 @ nil_a @ Added2 @ ( pop_a4 @ Old2 ) @ ( minus_minus_nat @ Remained2 @ one_one_nat ) ) ) )
=> ~ ( accp_current_a @ pop_rel_a @ ( current_a2 @ nil_a @ Added2 @ Old2 @ Remained2 ) ) ) )
=> ~ ! [X4: a,Xs2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X4 @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) )
=> ( ( Y
= ( produc8503237746132909001rent_a @ X4 @ ( current_a2 @ Xs2 @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) )
=> ~ ( accp_current_a @ pop_rel_a @ ( current_a2 @ ( cons_a @ X4 @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_943_Big__Aux_Olist_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( big_list_a @ ( common_a @ Common ) )
= ( common_list_a @ Common ) ) ).
% Big_Aux.list.simps(1)
thf(fact_944_Small__Aux_Olist_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( small_list_a @ ( common_a2 @ Common ) )
= ( common_list_a @ Common ) ) ).
% Small_Aux.list.simps(1)
% Helper facts (3)
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,
( ( size_size_state_a @ big2 )
= ( size_size_state_a @ big ) ) ).
%------------------------------------------------------------------------------