TPTP Problem File: SLH0952^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0023_Big_Proof/prob_00209_007750__6791626_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1507 ( 593 unt; 242 typ; 0 def)
% Number of atoms : 3155 (1909 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11398 ( 533 ~; 69 |; 184 &;9214 @)
% ( 0 <=>;1398 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 627 ( 627 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 206 usr; 18 con; 0-4 aty)
% Number of variables : 3662 ( 95 ^;3415 !; 152 ?;3662 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:07.544
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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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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current_list_a: $tType ).
thf(ty_n_t__Common__Ostate_It__List__Olist_Itf__a_J_J,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,
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% Explicit typings (206)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_Itf__a_J,type,
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big_list_current_a: state_a2 > list_a ).
thf(sy_c_Big__Aux_Olist__current__rel_001tf__a,type,
big_li383503880598112847_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_Big__Aux_Oremaining__steps__state__rel_001tf__a,type,
big_re1607094904563243348_rel_a: state_a2 > state_a2 > $o ).
thf(sy_c_Big__Aux_Osize__new__state__rel_001tf__a,type,
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thf(sy_c_Big__Aux_Osize__state__rel_001tf__a,type,
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thf(sy_c_Common_Onormalize_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Common_Onormalize_001tf__a,type,
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thf(sy_c_Common_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a2: state_list_a > produc5879181297811355641list_a ).
thf(sy_c_Common_Opop_001tf__a,type,
pop_a2: state_a > produc3409137331138395373tate_a ).
thf(sy_c_Common_Opush_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Common_Opush_001tf__a,type,
push_a2: a > state_a > state_a ).
thf(sy_c_Common_Ostate_OCopy_001t__List__Olist_Itf__a_J,type,
copy_list_a: current_list_a > list_list_a > list_list_a > nat > state_list_a ).
thf(sy_c_Common_Ostate_OCopy_001tf__a,type,
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thf(sy_c_Common__Aux_Olist_001t__List__Olist_Itf__a_J,type,
common_list_list_a: state_list_a > list_list_a ).
thf(sy_c_Common__Aux_Olist_001tf__a,type,
common_list_a2: state_a > list_a ).
thf(sy_c_Common__Aux_Olist__current_001t__List__Olist_Itf__a_J,type,
common3845368160371929429list_a: state_list_a > list_list_a ).
thf(sy_c_Common__Aux_Olist__current_001tf__a,type,
common1102728217005306191rent_a: state_a > list_a ).
thf(sy_c_Common__Aux_Otake__rev_001t__List__Olist_Itf__a_J,type,
common3049970819459547540list_a: nat > list_list_a > list_list_a ).
thf(sy_c_Common__Aux_Otake__rev_001tf__a,type,
common_take_rev_a: nat > list_a > list_a ).
thf(sy_c_Current_Ocurrent_OCurrent_001t__List__Olist_Itf__a_J,type,
current_list_a2: list_list_a > nat > stack_list_a > nat > current_list_a ).
thf(sy_c_Current_Ocurrent_OCurrent_001tf__a,type,
current_a2: list_a > nat > stack_a > nat > current_a ).
thf(sy_c_Current_Ofirst_001t__List__Olist_Itf__a_J,type,
first_list_a: current_list_a > list_a ).
thf(sy_c_Current_Ofirst_001tf__a,type,
first_a: current_a > a ).
thf(sy_c_Current_Ofirst__rel_001tf__a,type,
first_rel_a: current_a > current_a > $o ).
thf(sy_c_Current_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a3: current_list_a > produc603899091129270873list_a ).
thf(sy_c_Current_Opop_001tf__a,type,
pop_a3: current_a > produc7805042584321970905rent_a ).
thf(sy_c_Current_Opop__rel_001t__List__Olist_Itf__a_J,type,
pop_rel_list_a: current_list_a > current_list_a > $o ).
thf(sy_c_Current_Opop__rel_001tf__a,type,
pop_rel_a: current_a > current_a > $o ).
thf(sy_c_Current_Opush_001t__List__Olist_Itf__a_J,type,
push_list_a3: list_a > current_list_a > current_list_a ).
thf(sy_c_Current_Opush_001tf__a,type,
push_a3: a > current_a > current_a ).
thf(sy_c_Current__Aux_Oinvar__current__rel_001tf__a,type,
curren2598401564740717938_rel_a: current_a > current_a > $o ).
thf(sy_c_Current__Aux_Olist_001t__List__Olist_Itf__a_J,type,
current_list_list_a: current_list_a > list_list_a ).
thf(sy_c_Current__Aux_Olist_001tf__a,type,
current_list_a3: current_a > list_a ).
thf(sy_c_Current__Aux_Olist__rel_001tf__a,type,
current_list_rel_a: current_a > current_a > $o ).
thf(sy_c_Current__Aux_Osize__current__rel_001tf__a,type,
curren1432154427392394533_rel_a: current_a > current_a > $o ).
thf(sy_c_Current__Aux_Osize__new__current__rel_001tf__a,type,
curren2620163519437455445_rel_a: current_a > current_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
bind_list_a_list_a: list_list_a > ( list_a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001tf__a,type,
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thf(sy_c_List_Obind_001tf__a_001t__List__Olist_Itf__a_J,type,
bind_a_list_a: list_a > ( a > list_list_a ) > list_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_Odrop_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Odrop_001tf__a,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_001tf__a,type,
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thf(sy_c_List_Olenlex_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olenlex_001tf__a,type,
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thf(sy_c_List_Olexord_001tf__a,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_Ohd_001tf__a,type,
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thf(sy_c_List_Olist_Osize__list_001t__List__Olist_Itf__a_J,type,
size_list_list_a: ( list_a > nat ) > list_list_a > nat ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
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_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_On__lists_001t__List__Olist_Itf__a_J,type,
n_lists_list_a: nat > list_list_a > list_list_list_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__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__a_J,type,
product_lists_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__List__Olist_Itf__a_J,type,
rev_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__List__Olist_Itf__a_J,type,
rotate1_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > 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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_It__List__Olist_Itf__a_J_J,type,
size_s7734494264869015971list_a: state_list_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_Itf__a_J,type,
size_size_state_a: state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
size_s2742192117134447578list_a: state_list_a > 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_It__List__Olist_Itf__a_J_J,type,
size_s5124551957227789858list_a: current_list_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__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_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__Stack__Ostack_It__List__Olist_Itf__a_J_J,type,
size_s6256786526647004292list_a: stack_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_Itf__a_J,type,
size_size_stack_a: stack_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_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,
ord_le7857023143581076903list_a: set_Pr4048851178543822343list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc3233922274162870707list_a: ( list_a > list_a > $o ) > list_list_a > produc1303580075620398275list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc8643929849434629545list_a: ( a > a ) > produc9164743771328383783list_a > produc1473018763691903991list_a ).
thf(sy_c_Product__Type_OPair_001t__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_Itf__a_J_001t__Big__Ostate_It__List__Olist_Itf__a_J_J,type,
produc3844268483315210058list_a: list_a > state_list_a2 > produc4392152985924360026list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
produc2216564154414103411list_a: list_a > state_list_a > produc5879181297811355641list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
produc1356073977093405001list_a: list_a > current_list_a > produc603899091129270873list_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_001tf__a_001t__Big__Ostate_Itf__a_J,type,
produc8641956578966763338tate_a: a > state_a2 > produc6972303929186420058tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc8263595898873874535tate_a: a > state_a > produc3409137331138395373tate_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_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc6311120620833064345list_a: produc5032551385658279741list_a > a > a > $o ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_Itf__a_J_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
produc8258313757495307061list_a: produc603899091129270873list_a > list_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc3698117735987127555list_a: produc9164743771328383783list_a > list_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Big__Ostate_Itf__a_J,type,
produc736293372669613878tate_a: produc6972303929186420058tate_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc3154331710141225339tate_a: produc3409137331138395373tate_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4952273589686483381rent_a: produc7805042584321970905rent_a > a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
product_fst_a_a: product_prod_a_a > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc3976258275648695771list_a: produc5032551385658279741list_a > list_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_Itf__a_J_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
produc8483681909585661047list_a: produc603899091129270873list_a > current_list_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc8617614985401127493list_a: produc9164743771328383783list_a > list_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Big__Ostate_Itf__a_J,type,
produc7615498795807706488tate_a: produc6972303929186420058tate_a > state_a2 ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc681690970763031737tate_a: produc3409137331138395373tate_a > state_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4695312889421393143rent_a: produc7805042584321970905rent_a > current_a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
product_snd_a_a: product_prod_a_a > a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
collec943055143889122450list_a: ( produc9164743771328383783list_a > $o ) > set_Pr4048851178543822343list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Stack_Ofirst_001t__List__Olist_Itf__a_J,type,
first_list_a2: stack_list_a > list_a ).
thf(sy_c_Stack_Ofirst_001tf__a,type,
first_a2: stack_a > a ).
thf(sy_c_Stack_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a4: stack_list_a > stack_list_a ).
thf(sy_c_Stack_Opop_001tf__a,type,
pop_a4: stack_a > stack_a ).
thf(sy_c_Stack__Aux_Olist_001t__List__Olist_Itf__a_J,type,
stack_list_list_a: stack_list_a > list_list_a ).
thf(sy_c_Stack__Aux_Olist_001tf__a,type,
stack_list_a2: stack_a > list_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_It__List__Olist_Itf__a_J_J,type,
type_i3182563720400918970list_a: state_list_a2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_Itf__a_J,type,
type_i6304938058965754292tate_a: state_a2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
type_i5563415093623196931list_a: state_list_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_Itf__a_J,type,
type_i4669920168676019581tate_a: state_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
type_i8419033571676804025list_a: current_list_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_Itf__a_J,type,
type_i6141643110573041459rent_a: current_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_It__List__Olist_Itf__a_J_J,type,
type_i4148797458121419353list_a: stack_list_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Stack__Ostack_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_a2 > 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_Osize__new__class_Osize__new_001t__Big__Ostate_Itf__a_J,type,
type_s6530235180886170618tate_a: state_a2 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Common__Ostate_Itf__a_J,type,
type_s8424385952999958455tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Current__Ocurrent_Itf__a_J,type,
type_s933026853152659577rent_a: current_a > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_It__List__Olist_Itf__a_J_J,type,
type_s8055303106637973038list_a: state_list_a2 > state_list_a2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Big__Ostate_Itf__a_J,type,
type_s3593206172722485288tate_a: state_a2 > state_a2 ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
type_s3510896519899625359list_a: state_list_a > state_list_a ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Common__Ostate_Itf__a_J,type,
type_s889635741254954505tate_a: state_a > state_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Big__Ostate_It__List__Olist_Itf__a_J_J,type,
accp_state_list_a: ( state_list_a2 > state_list_a2 > $o ) > state_list_a2 > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Big__Ostate_Itf__a_J,type,
accp_state_a: ( state_a2 > state_a2 > $o ) > state_a2 > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
accp_current_list_a: ( current_list_a > current_list_a > $o ) > current_list_a > $o ).
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__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
accp_P1500049460152098787tate_a: ( produc6972303929186420058tate_a > produc6972303929186420058tate_a > $o ) > produc6972303929186420058tate_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
member6427184188916410838list_a: list_P321204300973800749list_a > set_li1395191299911657101list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6824001069763096534od_a_a: list_P1396940483166286381od_a_a > set_li8827807065578854541od_a_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_added____,type,
added: nat ).
thf(sy_v_aux____,type,
aux: list_a ).
thf(sy_v_big_H,type,
big: state_a2 ).
thf(sy_v_biga____,type,
biga: stack_a ).
thf(sy_v_count____,type,
count: nat ).
thf(sy_v_current____,type,
current: current_a ).
thf(sy_v_old____,type,
old: stack_a ).
thf(sy_v_remained____,type,
remained: nat ).
thf(sy_v_xa____,type,
xa: a ).
% Relevant facts (1261)
thf(fact_0__C2_Oprems_C_I2_J,axiom,
type_i6304938058965754292tate_a @ ( reverse_a @ current @ biga @ aux @ count ) ).
% "2.prems"(2)
thf(fact_1__C1_Oprems_C_I2_J,axiom,
type_i6304938058965754292tate_a @ ( reverse_a @ ( current_a2 @ nil_a @ added @ old @ remained ) @ biga @ aux @ count ) ).
% "1.prems"(2)
thf(fact_2_Big__Proof_Oinvar__push,axiom,
! [Big: state_a2,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( push_a @ X @ Big ) ) ) ).
% Big_Proof.invar_push
thf(fact_3_Big__Proof_Oinvar__step,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( type_s3593206172722485288tate_a @ Big ) ) ) ).
% Big_Proof.invar_step
thf(fact_4__C2_Oprems_C_I3_J,axiom,
( ( pop_a @ ( reverse_a @ current @ biga @ aux @ count ) )
= ( produc8641956578966763338tate_a @ xa @ big ) ) ).
% "2.prems"(3)
thf(fact_5__C1_Oprems_C_I3_J,axiom,
( ( pop_a @ ( reverse_a @ ( current_a2 @ nil_a @ added @ old @ remained ) @ biga @ aux @ count ) )
= ( produc8641956578966763338tate_a @ xa @ big ) ) ).
% "1.prems"(3)
thf(fact_6_Big__Proof_Ostep__list__current,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_current_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_current_a @ Big ) ) ) ).
% Big_Proof.step_list_current
thf(fact_7_Big__Proof_Ostep__list,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_a @ Big ) ) ) ).
% Big_Proof.step_list
thf(fact_8_size__list,axiom,
! [Big: state_list_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7734494264869015971list_a @ Big ) )
=> ( ( type_i3182563720400918970list_a @ Big )
=> ( ( big_list_list_a @ Big )
!= nil_list_a ) ) ) ).
% size_list
thf(fact_9_size__list,axiom,
! [Big: state_a2] :
( ( 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_10_Big__Proof_Opop__list,axiom,
! [Big: state_list_a2,X: list_a,Big2: state_list_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7734494264869015971list_a @ Big ) )
=> ( ( type_i3182563720400918970list_a @ Big )
=> ( ( ( pop_list_a @ Big )
= ( produc3844268483315210058list_a @ X @ Big2 ) )
=> ( ( cons_list_a @ X @ ( big_list_list_a @ Big2 ) )
= ( big_list_list_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_11_Big__Proof_Opop__list,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( 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_12_Big__Proof_Opop__list__tl,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( 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_13_Big__Aux_Oinvar__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_i6304938058965754292tate_a @ ( common_a @ State ) )
= ( type_i4669920168676019581tate_a @ State ) ) ).
% Big_Aux.invar_state.simps(1)
thf(fact_14__C2_Oprems_C_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( reverse_a @ current @ biga @ aux @ count ) ) ).
% "2.prems"(1)
thf(fact_15__C1_Oprems_C_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( reverse_a @ ( current_a2 @ nil_a @ added @ old @ remained ) @ biga @ aux @ count ) ) ).
% "1.prems"(1)
thf(fact_16__092_060open_062remained_A_092_060le_062_Alength_A_Itake__rev_Acount_A_IStack__Aux_Olist_Abig_J_A_064_Aaux_J_092_060close_062,axiom,
ord_less_eq_nat @ remained @ ( size_size_list_a @ ( append_a @ ( common_take_rev_a @ count @ ( stack_list_a2 @ biga ) ) @ aux ) ) ).
% \<open>remained \<le> length (take_rev count (Stack_Aux.list big) @ aux)\<close>
thf(fact_17_Big__Proof_Opush__list,axiom,
! [X: list_a,Big: state_list_a2] :
( ( big_list_list_a @ ( push_list_a @ X @ Big ) )
= ( cons_list_a @ X @ ( big_list_list_a @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_18_Big__Proof_Opush__list,axiom,
! [X: a,Big: state_a2] :
( ( big_list_a @ ( push_a @ X @ Big ) )
= ( cons_a @ X @ ( big_list_a @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_19_Big__Aux_Osize__new__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ~ ! [Current: current_a,Uu: stack_a,Uv: list_a,Uw: nat] :
( X
!= ( reverse_a @ Current @ Uu @ Uv @ Uw ) ) ) ).
% Big_Aux.size_new_state.cases
thf(fact_20_Big__Aux_Oremaining__steps__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ~ ! [Uu: list_a,Uv: nat,Uw: stack_a,Remaining: nat,Ux: stack_a,Uy: list_a,Count: nat] :
( X
!= ( reverse_a @ ( current_a2 @ Uu @ Uv @ Uw @ Remaining ) @ Ux @ Uy @ Count ) ) ) ).
% Big_Aux.remaining_steps_state.cases
thf(fact_21_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_22_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_23_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_24_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_25_Big_Opush_Ocases,axiom,
! [X: produc6972303929186420058tate_a] :
( ! [X2: a,State2: state_a] :
( X
!= ( produc8641956578966763338tate_a @ X2 @ ( common_a @ State2 ) ) )
=> ~ ! [X2: a,Current: current_a,Big3: stack_a,Aux: list_a,Count: nat] :
( X
!= ( produc8641956578966763338tate_a @ X2 @ ( reverse_a @ Current @ Big3 @ Aux @ Count ) ) ) ) ).
% Big.push.cases
thf(fact_26_Nil__tl,axiom,
! [Xs: list_list_a] :
( ( nil_list_a
= ( tl_list_a @ Xs ) )
= ( ( Xs = nil_list_a )
| ? [X3: list_a] :
( Xs
= ( cons_list_a @ X3 @ nil_list_a ) ) ) ) ).
% Nil_tl
thf(fact_27_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_28_tl__Nil,axiom,
! [Xs: list_list_a] :
( ( ( tl_list_a @ Xs )
= nil_list_a )
= ( ( Xs = nil_list_a )
| ? [X3: list_a] :
( Xs
= ( cons_list_a @ X3 @ nil_list_a ) ) ) ) ).
% tl_Nil
thf(fact_29_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_30_Current_Opop_Ocases,axiom,
! [X: current_list_a] :
( ! [Added: nat,Old: stack_list_a,Remained: nat] :
( X
!= ( current_list_a2 @ nil_list_a @ Added @ Old @ Remained ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( X
!= ( current_list_a2 @ ( cons_list_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) ) ) ).
% Current.pop.cases
thf(fact_31_Current_Opop_Ocases,axiom,
! [X: current_a] :
( ! [Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( current_a2 @ nil_a @ Added @ Old @ Remained ) )
=> ~ ! [X2: a,Xs2: list_a,Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( current_a2 @ ( cons_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) ) ) ).
% Current.pop.cases
thf(fact_32_Big_Ostate_Oinject_I2_J,axiom,
! [X22: state_a,Y2: state_a] :
( ( ( common_a @ X22 )
= ( common_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% Big.state.inject(2)
thf(fact_33_current_Oinject,axiom,
! [X1: list_a,X22: nat,X32: stack_a,X4: nat,Y1: list_a,Y2: nat,Y3: stack_a,Y4: nat] :
( ( ( current_a2 @ X1 @ X22 @ X32 @ X4 )
= ( current_a2 @ Y1 @ Y2 @ Y3 @ Y4 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 )
& ( X32 = Y3 )
& ( X4 = Y4 ) ) ) ).
% current.inject
thf(fact_34_Big_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X14: nat,Y11: current_a,Y12: stack_a,Y13: list_a,Y14: nat] :
( ( ( reverse_a @ X11 @ X12 @ X13 @ X14 )
= ( reverse_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% Big.state.inject(1)
thf(fact_35_list_Oinject,axiom,
! [X21: list_a,X222: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X222 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_36_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_37_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_38_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_39_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_40_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_41_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_42_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_43_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_44_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_45_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_46_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_47_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_48_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_49_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_50_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_51_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_52_mem__Collect__eq,axiom,
! [A: produc9164743771328383783list_a,P: produc9164743771328383783list_a > $o] :
( ( member8191768239178080336list_a @ A @ ( collec943055143889122450list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_Pr4048851178543822343list_a] :
( ( collec943055143889122450list_a
@ ^ [X3: produc9164743771328383783list_a] : ( member8191768239178080336list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_61_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_62_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_63_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_64_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_65_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_66_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_67_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_68_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_69_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_70_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_71_append1__eq__conv,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_72_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_73_tl__append2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_74_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_75_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_76_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_77_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_78_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_79_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_80_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 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_81_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_82_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_83_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_84_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_85_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_86_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_87_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_88_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_89_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_90_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_91_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_92_append__Cons,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_93_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_94_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_95_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_96_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_97_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_98_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_99_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_100_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_101_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_102_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_103_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_104_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_105_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_106_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_107_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_108_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_109_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_110_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_111_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X2: list_a,Xs3: list_list_a,Y5: list_a,Ys3: list_list_a] :
( ( X2 != Y5 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X2 @ nil_list_a ) @ Xs3 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y5 @ nil_list_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_112_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs3: list_a,Y5: a,Ys3: list_a] :
( ( X2 != Y5 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y5 @ nil_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_113_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_114_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_115_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_116_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_117_size__neq__size__imp__neq,axiom,
! [X: current_a,Y: current_a] :
( ( ( size_size_current_a @ X )
!= ( size_size_current_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_118_size__neq__size__imp__neq,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_119_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_120_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_121_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_122_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_123_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_124_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_125_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_126_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_127_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_128_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_129_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_130_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_131_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_132_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X @ Xs ) ) )
| ? [Ys4: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Ys4 ) )
& ( ( append_list_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_133_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_134_Cons__eq__append__conv,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_list_a] :
( ( ( cons_list_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_list_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_135_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_136_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys5: list_list_a,Y5: list_a] :
( Xs
!= ( append_list_a @ Ys5 @ ( cons_list_a @ Y5 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_137_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys5: list_a,Y5: a] :
( Xs
!= ( append_a @ Ys5 @ ( cons_a @ Y5 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_138_rev__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_139_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_140_list__induct4,axiom,
! [Xs: list_a,Ys: 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 @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_141_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_142_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_143_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_144_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_145_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P: list_list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_146_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P: list_list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_147_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P: list_list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_148_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P: list_a > list_list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a,Z: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_149_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a,Z: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_150_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_151_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_152_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_153_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_154_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_155_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_156_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_157_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( ( size_size_list_a @ Ys5 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_158_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_159_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_160_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys5 ) )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_161_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys5 ) )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_162_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_163_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_164_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_165_tl__append__if,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( tl_list_a @ Ys ) ) )
& ( ( Xs != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_166_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_167_current_Oexhaust,axiom,
! [Y: current_a] :
~ ! [X15: list_a,X23: nat,X33: stack_a,X42: nat] :
( Y
!= ( current_a2 @ X15 @ X23 @ X33 @ X42 ) ) ).
% current.exhaust
thf(fact_168_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_169_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_170_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_171_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_172_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_173_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_174_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys5: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y5 @ Ys5 ) )
=> ( ! [X2: a,Xs2: list_a,Y5: list_a,Ys5: list_list_a] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_175_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y5: a,Ys5: list_a] : ( P @ nil_list_a @ ( cons_a @ Y5 @ Ys5 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: a,Ys5: list_a] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_176_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs2 ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys5: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y5 @ Ys5 ) )
=> ( ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_177_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y5: a,Ys5: list_a] : ( P @ nil_a @ ( cons_a @ Y5 @ Ys5 ) )
=> ( ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_178_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y7: list_a,Ys6: list_list_a] :
( Xs
= ( cons_list_a @ Y7 @ Ys6 ) ) ) ) ).
% neq_Nil_conv
thf(fact_179_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y7: a,Ys6: list_a] :
( Xs
= ( cons_a @ Y7 @ Ys6 ) ) ) ) ).
% neq_Nil_conv
thf(fact_180_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X2: list_a] :
( X
!= ( cons_list_a @ X2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Y5: list_a,Xs2: list_list_a] :
( X
!= ( cons_list_a @ X2 @ ( cons_list_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_181_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y5: a,Xs2: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_182_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_183_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_184_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X223: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_185_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_186_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X222: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X222 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_187_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_188_list_Odistinct_I1_J,axiom,
! [X21: list_a,X222: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_189_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_190_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_191_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_192_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_193_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_194_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_195_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_196_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_197_list_Osel_I3_J,axiom,
! [X21: list_a,X222: list_list_a] :
( ( tl_list_a @ ( cons_list_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_198_list_Osel_I3_J,axiom,
! [X21: a,X222: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_199_list_Osel_I2_J,axiom,
( ( tl_list_a @ nil_list_a )
= nil_list_a ) ).
% list.sel(2)
thf(fact_200_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_201_Big_Ostate_Oexhaust,axiom,
! [Y: state_a2] :
( ! [X112: current_a,X122: stack_a,X132: list_a,X142: nat] :
( Y
!= ( reverse_a @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X23: state_a] :
( Y
!= ( common_a @ X23 ) ) ) ).
% Big.state.exhaust
thf(fact_202_Big_Ostate_Odistinct_I1_J,axiom,
! [X11: current_a,X12: stack_a,X13: list_a,X14: nat,X22: state_a] :
( ( reverse_a @ X11 @ X12 @ X13 @ X14 )
!= ( common_a @ X22 ) ) ).
% Big.state.distinct(1)
thf(fact_203_take__rev__empty,axiom,
! [N: nat] :
( ( common3049970819459547540list_a @ N @ nil_list_a )
= nil_list_a ) ).
% take_rev_empty
thf(fact_204_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_a @ N @ nil_a )
= nil_a ) ).
% take_rev_empty
thf(fact_205_old_Oprod_Oinject,axiom,
! [A: a,B: current_a,A3: a,B2: current_a] :
( ( ( produc8503237746132909001rent_a @ A @ B )
= ( produc8503237746132909001rent_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_206_old_Oprod_Oinject,axiom,
! [A: a > a > $o,B: list_a,A3: a > a > $o,B2: list_a] :
( ( ( produc8111569692950616493list_a @ A @ B )
= ( produc8111569692950616493list_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_207_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_208_old_Oprod_Oinject,axiom,
! [A: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_209_old_Oprod_Oinject,axiom,
! [A: a,B: state_a,A3: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_210_old_Oprod_Oinject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_211_prod_Oinject,axiom,
! [X1: a,X22: current_a,Y1: a,Y2: current_a] :
( ( ( produc8503237746132909001rent_a @ X1 @ X22 )
= ( produc8503237746132909001rent_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_212_prod_Oinject,axiom,
! [X1: a > a > $o,X22: list_a,Y1: a > a > $o,Y2: list_a] :
( ( ( produc8111569692950616493list_a @ X1 @ X22 )
= ( produc8111569692950616493list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_213_prod_Oinject,axiom,
! [X1: list_a,X22: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X22 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_214_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_215_prod_Oinject,axiom,
! [X1: a,X22: state_a,Y1: a,Y2: state_a] :
( ( ( produc8263595898873874535tate_a @ X1 @ X22 )
= ( produc8263595898873874535tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_216_prod_Oinject,axiom,
! [X1: a,X22: state_a2,Y1: a,Y2: state_a2] :
( ( ( produc8641956578966763338tate_a @ X1 @ X22 )
= ( produc8641956578966763338tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_217_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_218_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_219_cons__tl,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_list_a @ Ys ) ) ) ).
% cons_tl
thf(fact_220_cons__tl,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_221_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 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_222_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_223_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_224_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_225_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_226_Current_Opush_Ocases,axiom,
! [X: produc7805042584321970905rent_a] :
~ ! [X2: a,Extra: list_a,Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( produc8503237746132909001rent_a @ X2 @ ( current_a2 @ Extra @ Added @ Old @ Remained ) ) ) ).
% Current.push.cases
thf(fact_227_sorted__wrt_Ocases,axiom,
! [X: produc1303580075620398275list_a] :
( ! [P2: list_a > list_a > $o] :
( X
!= ( produc3233922274162870707list_a @ P2 @ nil_list_a ) )
=> ~ ! [P2: list_a > list_a > $o,X2: list_a,Ys5: list_list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X2 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_228_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ~ ! [P2: a > a > $o,X2: a,Ys5: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X2 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_229_successively_Ocases,axiom,
! [X: produc1303580075620398275list_a] :
( ! [P2: list_a > list_a > $o] :
( X
!= ( produc3233922274162870707list_a @ P2 @ nil_list_a ) )
=> ( ! [P2: list_a > list_a > $o,X2: list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
=> ~ ! [P2: list_a > list_a > $o,X2: list_a,Y5: list_a,Xs2: list_list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X2 @ ( cons_list_a @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_230_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X2: a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X2 @ nil_a ) ) )
=> ~ ! [P2: a > a > $o,X2: a,Y5: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X2 @ ( cons_a @ Y5 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_231_splice_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys5: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys5 ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Ys5: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_232_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys5: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys5 ) )
=> ~ ! [X2: a,Xs2: list_a,Ys5: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_233_shuffles_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys5: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys5 ) )
=> ( ! [Xs2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Y5: list_a,Ys5: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X2 @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_234_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys5: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys5 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X2: a,Xs2: list_a,Y5: a,Ys5: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y5 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_235_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_236_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_237_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_238_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_239_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_240_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_241_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_242_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_243_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_244_dual__order_Oeq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_245_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_247_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_248_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y8: nat,Z3: nat] : ( Y8 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_249_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_250_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_251_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_252_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_253_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_255_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_256_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_257_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_258_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_259_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_260_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_261_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_262_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_263_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_264_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_265_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_266_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_267_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_268_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_269_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_270_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_271_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_272_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_273_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_274_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_275_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_276_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_277_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_278_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_279_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_280_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_281_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_282_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_283_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_284_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_285_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_286_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_287_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_288_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_289_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_290_old_Oprod_Oexhaust,axiom,
! [Y: produc7805042584321970905rent_a] :
~ ! [A4: a,B3: current_a] :
( Y
!= ( produc8503237746132909001rent_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_291_old_Oprod_Oexhaust,axiom,
! [Y: produc5032551385658279741list_a] :
~ ! [A4: a > a > $o,B3: list_a] :
( Y
!= ( produc8111569692950616493list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_292_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A4: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_293_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A4: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_294_old_Oprod_Oexhaust,axiom,
! [Y: produc3409137331138395373tate_a] :
~ ! [A4: a,B3: state_a] :
( Y
!= ( produc8263595898873874535tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_295_old_Oprod_Oexhaust,axiom,
! [Y: produc6972303929186420058tate_a] :
~ ! [A4: a,B3: state_a2] :
( Y
!= ( produc8641956578966763338tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_296_surj__pair,axiom,
! [P5: produc7805042584321970905rent_a] :
? [X2: a,Y5: current_a] :
( P5
= ( produc8503237746132909001rent_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_297_surj__pair,axiom,
! [P5: produc5032551385658279741list_a] :
? [X2: a > a > $o,Y5: list_a] :
( P5
= ( produc8111569692950616493list_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_298_surj__pair,axiom,
! [P5: produc9164743771328383783list_a] :
? [X2: list_a,Y5: list_a] :
( P5
= ( produc6837034575241423639list_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_299_surj__pair,axiom,
! [P5: product_prod_a_a] :
? [X2: a,Y5: a] :
( P5
= ( product_Pair_a_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_300_surj__pair,axiom,
! [P5: produc3409137331138395373tate_a] :
? [X2: a,Y5: state_a] :
( P5
= ( produc8263595898873874535tate_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_301_surj__pair,axiom,
! [P5: produc6972303929186420058tate_a] :
? [X2: a,Y5: state_a2] :
( P5
= ( produc8641956578966763338tate_a @ X2 @ Y5 ) ) ).
% surj_pair
thf(fact_302_prod__cases,axiom,
! [P: produc7805042584321970905rent_a > $o,P5: produc7805042584321970905rent_a] :
( ! [A4: a,B3: current_a] : ( P @ ( produc8503237746132909001rent_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_303_prod__cases,axiom,
! [P: produc5032551385658279741list_a > $o,P5: produc5032551385658279741list_a] :
( ! [A4: a > a > $o,B3: list_a] : ( P @ ( produc8111569692950616493list_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_304_prod__cases,axiom,
! [P: produc9164743771328383783list_a > $o,P5: produc9164743771328383783list_a] :
( ! [A4: list_a,B3: list_a] : ( P @ ( produc6837034575241423639list_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_305_prod__cases,axiom,
! [P: product_prod_a_a > $o,P5: product_prod_a_a] :
( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_306_prod__cases,axiom,
! [P: produc3409137331138395373tate_a > $o,P5: produc3409137331138395373tate_a] :
( ! [A4: a,B3: state_a] : ( P @ ( produc8263595898873874535tate_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_307_prod__cases,axiom,
! [P: produc6972303929186420058tate_a > $o,P5: produc6972303929186420058tate_a] :
( ! [A4: a,B3: state_a2] : ( P @ ( produc8641956578966763338tate_a @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_308_Pair__inject,axiom,
! [A: a,B: current_a,A3: a,B2: current_a] :
( ( ( produc8503237746132909001rent_a @ A @ B )
= ( produc8503237746132909001rent_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_309_Pair__inject,axiom,
! [A: a > a > $o,B: list_a,A3: a > a > $o,B2: list_a] :
( ( ( produc8111569692950616493list_a @ A @ B )
= ( produc8111569692950616493list_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_310_Pair__inject,axiom,
! [A: list_a,B: list_a,A3: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_311_Pair__inject,axiom,
! [A: a,B: a,A3: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_312_Pair__inject,axiom,
! [A: a,B: state_a,A3: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_313_Pair__inject,axiom,
! [A: a,B: state_a2,A3: a,B2: state_a2] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_314_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_315_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_316_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_317_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_318_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_319_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_320_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_321_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ~ ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_322_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_323_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_324_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_325_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_326_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_327_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_328_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_329_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_330_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_331_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_332_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_333_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_334_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_335_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_nat @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_336_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y7: nat] :
( ( ord_less_eq_nat @ X3 @ Y7 )
& ( X3 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_337_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_338_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_339_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_340_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_341_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_342_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_343_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_344_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_345_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_346_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_347_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_nat @ X2 @ Y5 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_348_size__list__length,axiom,
! [Stack: stack_a] :
( ( size_size_list_a @ ( stack_list_a2 @ Stack ) )
= ( size_size_stack_a @ Stack ) ) ).
% size_list_length
thf(fact_349_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6256786526647004292list_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_350_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
!= nil_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_351_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
= nil_list_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s6256786526647004292list_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_352_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_353_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_a @ N @ nil_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_a @ N @ nil_list_a )
= nil_list_list_a ) ) ) ).
% n_lists_Nil
thf(fact_354_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_355_Stack__Proof_Olist__empty__size,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
= nil_list_a )
= ( ( size_s6256786526647004292list_a @ Stack )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_356_Stack__Proof_Olist__empty__size,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
= nil_a )
= ( ( size_size_stack_a @ Stack )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_357_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
=> ( ( size_s6256786526647004292list_a @ Stack )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_358_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
!= nil_a )
=> ( ( size_size_stack_a @ Stack )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_359_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_360_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_a] :
( ( n_lists_list_a @ zero_zero_nat @ Xs )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% n_lists.simps(1)
thf(fact_361_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_362_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_363_list__current__size,axiom,
! [Common: state_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( common3845368160371929429list_a @ Common )
= nil_list_a )
=> ~ ( type_i5563415093623196931list_a @ Common ) ) ) ).
% list_current_size
thf(fact_364_list__current__size,axiom,
! [Common: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( common1102728217005306191rent_a @ Common )
= nil_a )
=> ~ ( type_i4669920168676019581tate_a @ Common ) ) ) ).
% list_current_size
thf(fact_365_Common__Proof_Olist__size,axiom,
! [Common: state_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( common_list_list_a @ Common )
= nil_list_a )
=> ~ ( type_i5563415093623196931list_a @ Common ) ) ) ).
% Common_Proof.list_size
thf(fact_366_Common__Proof_Olist__size,axiom,
! [Common: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( common_list_a2 @ Common )
= nil_a )
=> ~ ( type_i4669920168676019581tate_a @ Common ) ) ) ).
% Common_Proof.list_size
thf(fact_367_Common__Proof_Oinvar__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( type_i4669920168676019581tate_a @ Common2 ) ) ) ) ).
% Common_Proof.invar_pop
thf(fact_368_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_369_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_370_Cons__in__lex,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( lex_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_371_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_372_Nil__lenlex__iff1,axiom,
! [Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Ns ) @ ( lenlex_list_a @ R ) )
= ( Ns != nil_list_a ) ) ).
% Nil_lenlex_iff1
thf(fact_373_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_374_Common__Proof_Opop__list,axiom,
! [Common: state_list_a,X: list_a,Common2: state_list_a] :
( ( type_i5563415093623196931list_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( pop_list_a2 @ Common )
= ( produc2216564154414103411list_a @ X @ Common2 ) )
=> ( ( cons_list_a @ X @ ( common_list_list_a @ Common2 ) )
= ( common_list_list_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_375_Common__Proof_Opop__list,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( cons_a @ X @ ( common_list_a2 @ Common2 ) )
= ( common_list_a2 @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_376_Nil2__notin__lex,axiom,
! [Xs: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ nil_list_a ) @ ( lex_list_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_377_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_378_Nil__notin__lex,axiom,
! [Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Ys ) @ ( lex_list_a @ R ) ) ).
% Nil_notin_lex
thf(fact_379_Nil__notin__lex,axiom,
! [Ys: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R ) ) ).
% Nil_notin_lex
thf(fact_380_lex__append__leftI,axiom,
! [Ys: list_a,Zs: list_a,R: set_Product_prod_a_a,Xs: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) ) ) ).
% lex_append_leftI
thf(fact_381_lenlex__irreflexive,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ! [X2: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ X2 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Xs ) @ ( lenlex_list_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_382_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_383_Nil__lenlex__iff2,axiom,
! [Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ns @ nil_list_a ) @ ( lenlex_list_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_384_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_385_Big__Aux_Olist_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( big_list_a @ ( common_a @ Common ) )
= ( common_list_a2 @ Common ) ) ).
% Big_Aux.list.simps(1)
thf(fact_386_lex__append__leftD,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X2: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ X2 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_387_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_388_lex__append__left__iff,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X2: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ X2 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_389_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_390_pop__list__current,axiom,
! [Common: state_list_a,X: list_a,Common2: state_list_a] :
( ( type_i5563415093623196931list_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( pop_list_a2 @ Common )
= ( produc2216564154414103411list_a @ X @ Common2 ) )
=> ( ( cons_list_a @ X @ ( common3845368160371929429list_a @ Common2 ) )
= ( common3845368160371929429list_a @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_391_pop__list__current,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( cons_a @ X @ ( common1102728217005306191rent_a @ Common2 ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_392_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_393_Big__Aux_Olist__current_Osimps_I1_J,axiom,
! [Common: state_a] :
( ( big_list_current_a @ ( common_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ).
% Big_Aux.list_current.simps(1)
thf(fact_394_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_395_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R2: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys ) ) @ ( lenlex_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_396_remaining__steps__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ord_less_eq_nat @ ( type_r2212416260012024137tate_a @ Common2 ) @ ( type_r2212416260012024137tate_a @ Common ) ) ) ) ) ).
% remaining_steps_pop
thf(fact_397_Common__Proof_Osize__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( suc @ ( size_size_state_a2 @ Common2 ) )
= ( size_size_state_a2 @ Common ) ) ) ) ) ).
% Common_Proof.size_pop
thf(fact_398_size__size__new,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Common ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s8424385952999958455tate_a @ Common ) ) ) ) ).
% size_size_new
thf(fact_399_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_400_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_401_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_402_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_403_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_404_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_405_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_406_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_407_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_408_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_409_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_410_Common__Proof_Osize__new__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s8424385952999958455tate_a @ Common ) )
=> ( ( ( pop_a2 @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( suc @ ( type_s8424385952999958455tate_a @ Common2 ) )
= ( type_s8424385952999958455tate_a @ Common ) ) ) ) ) ).
% Common_Proof.size_new_pop
thf(fact_411_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_412_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_413_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_414_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_415_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_416_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_417_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_418_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X2: nat,Y5: nat] :
( ( P @ X2 @ Y5 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_419_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_420_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_421_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_422_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_423_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_424_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_425_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_426_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_427_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_428_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_429_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_430_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_431_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_432_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_433_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_434_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_435_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_436_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_437_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_438_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_439_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_440_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_441_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_442_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y5: nat,Z: nat] :
( ( R2 @ X2 @ Y5 )
=> ( ( R2 @ Y5 @ Z )
=> ( R2 @ X2 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_443_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_444_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_445_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_446_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_447_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_448_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_449_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_450_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_451_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_452_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_453_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_454_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_455_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_456_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_457_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_458_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_459_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_460_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_461_length__Suc__conv,axiom,
! [Xs: list_list_a,N: nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: list_a,Ys6: list_list_a] :
( ( Xs
= ( cons_list_a @ Y7 @ Ys6 ) )
& ( ( size_s349497388124573686list_a @ Ys6 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_462_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: a,Ys6: list_a] :
( ( Xs
= ( cons_a @ Y7 @ Ys6 ) )
& ( ( size_size_list_a @ Ys6 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_463_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( suc @ N )
= ( size_s349497388124573686list_a @ Xs ) )
= ( ? [Y7: list_a,Ys6: list_list_a] :
( ( Xs
= ( cons_list_a @ Y7 @ Ys6 ) )
& ( ( size_s349497388124573686list_a @ Ys6 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_464_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y7: a,Ys6: list_a] :
( ( Xs
= ( cons_a @ Y7 @ Ys6 ) )
& ( ( size_size_list_a @ Ys6 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_465_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_466_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_467_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_468_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_469_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_470_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_471_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_472_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_473_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_474_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_475_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s349497388124573686list_a @ Xs ) )
= ( ? [X3: list_a,Ys6: list_list_a] :
( ( Xs
= ( cons_list_a @ X3 @ Ys6 ) )
& ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Ys6 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_476_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys6: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys6 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys6 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_477_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_478_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_479_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_480_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_481_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_482_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_483_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P6 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_484_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P6 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_485_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_486_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_487_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_488_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_489_Big_Ostep__state_Ocases,axiom,
! [X: state_a2] :
( ! [State2: state_a] :
( X
!= ( common_a @ State2 ) )
=> ( ! [Current: current_a,Uu: stack_a,Aux: list_a] :
( X
!= ( reverse_a @ Current @ Uu @ Aux @ zero_zero_nat ) )
=> ~ ! [Current: current_a,Big3: stack_a,Aux: list_a,V: nat] :
( X
!= ( reverse_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) ) ) ) ).
% Big.step_state.cases
thf(fact_490_length__Suc__conv__rev,axiom,
! [Xs: list_list_a,N: nat] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: list_a,Ys6: list_list_a] :
( ( Xs
= ( append_list_a @ Ys6 @ ( cons_list_a @ Y7 @ nil_list_a ) ) )
& ( ( size_s349497388124573686list_a @ Ys6 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_491_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ys6 @ ( cons_a @ Y7 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys6 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_492_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_493_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s349497388124573686list_a
= ( ^ [Xs4: list_list_a] : ( if_nat @ ( Xs4 = nil_list_a ) @ zero_zero_nat @ ( suc @ ( size_s349497388124573686list_a @ ( tl_list_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_494_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_495_length__append__singleton,axiom,
! [Xs: list_list_a,X: list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( suc @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_496_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_497_remaining__steps__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r2212416260012024137tate_a @ Common ) )
=> ( ( suc @ ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) ) )
= ( type_r2212416260012024137tate_a @ Common ) ) ) ) ).
% remaining_steps_step
thf(fact_498_length__Cons,axiom,
! [X: list_a,Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( cons_list_a @ X @ Xs ) )
= ( suc @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_499_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_500_Common__Proof_Osize__new__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_s8424385952999958455tate_a @ ( push_a2 @ X @ Common ) )
= ( suc @ ( type_s8424385952999958455tate_a @ Common ) ) ) ) ).
% Common_Proof.size_new_push
thf(fact_501_Common__Proof_Osize__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( size_size_state_a2 @ ( push_a2 @ X @ Common ) )
= ( suc @ ( size_size_state_a2 @ Common ) ) ) ) ).
% Common_Proof.size_push
thf(fact_502_Common__Proof_Opush__list,axiom,
! [X: list_a,Common: state_list_a] :
( ( common_list_list_a @ ( push_list_a2 @ X @ Common ) )
= ( cons_list_a @ X @ ( common_list_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_503_Common__Proof_Opush__list,axiom,
! [X: a,Common: state_a] :
( ( common_list_a2 @ ( push_a2 @ X @ Common ) )
= ( cons_a @ X @ ( common_list_a2 @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_504_push__list__current,axiom,
! [X: list_a,Left: state_list_a] :
( ( common3845368160371929429list_a @ ( push_list_a2 @ X @ Left ) )
= ( cons_list_a @ X @ ( common3845368160371929429list_a @ Left ) ) ) ).
% push_list_current
thf(fact_505_push__list__current,axiom,
! [X: a,Left: state_a] :
( ( common1102728217005306191rent_a @ ( push_a2 @ X @ Left ) )
= ( cons_a @ X @ ( common1102728217005306191rent_a @ Left ) ) ) ).
% push_list_current
thf(fact_506_step__size,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( size_size_state_a2 @ ( type_s889635741254954505tate_a @ Common ) )
= ( size_size_state_a2 @ Common ) ) ) ).
% step_size
thf(fact_507_remaining__steps__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_r2212416260012024137tate_a @ ( push_a2 @ X @ Common ) )
= ( type_r2212416260012024137tate_a @ Common ) ) ) ).
% remaining_steps_push
thf(fact_508_step__size__new,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_s8424385952999958455tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( type_s8424385952999958455tate_a @ Common ) ) ) ).
% step_size_new
thf(fact_509_Common__Proof_Ostep__list,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common_list_a2 @ ( type_s889635741254954505tate_a @ Common ) )
= ( common_list_a2 @ Common ) ) ) ).
% Common_Proof.step_list
thf(fact_510_Common__Proof_Ostep__list__current,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common1102728217005306191rent_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ).
% Common_Proof.step_list_current
thf(fact_511_remaining__steps__step__0,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( type_r2212416260012024137tate_a @ Common )
= zero_zero_nat )
=> ( ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_step_0
thf(fact_512_Common__Proof_Oinvar__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( type_s889635741254954505tate_a @ Common ) ) ) ).
% Common_Proof.invar_step
thf(fact_513_Common__Proof_Oinvar__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( push_a2 @ X @ Common ) ) ) ).
% Common_Proof.invar_push
thf(fact_514_Big_Ostep__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_s3593206172722485288tate_a @ ( common_a @ State ) )
= ( common_a @ ( type_s889635741254954505tate_a @ State ) ) ) ).
% Big.step_state.simps(1)
thf(fact_515_Big_Opush_Osimps_I1_J,axiom,
! [X: a,State: state_a] :
( ( push_a @ X @ ( common_a @ State ) )
= ( common_a @ ( push_a2 @ X @ State ) ) ) ).
% Big.push.simps(1)
thf(fact_516_Big_Opush_Oelims,axiom,
! [X: a,Xa: state_a2,Y: state_a2] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ! [State2: state_a] :
( ( Xa
= ( common_a @ State2 ) )
=> ( Y
!= ( common_a @ ( push_a2 @ X @ State2 ) ) ) )
=> ~ ! [Current: current_a,Big3: stack_a,Aux: list_a,Count: nat] :
( ( Xa
= ( reverse_a @ Current @ Big3 @ Aux @ Count ) )
=> ( Y
!= ( reverse_a @ ( push_a3 @ X @ Current ) @ Big3 @ Aux @ Count ) ) ) ) ) ).
% Big.push.elims
thf(fact_517_Big__Aux_Osize__new__state_Osimps_I1_J,axiom,
! [State: state_a] :
( ( type_s6530235180886170618tate_a @ ( common_a @ State ) )
= ( type_s8424385952999958455tate_a @ State ) ) ).
% Big_Aux.size_new_state.simps(1)
thf(fact_518_SuccD,axiom,
! [K: produc9164743771328383783list_a,Kl: set_li1395191299911657101list_a,Kl2: list_P321204300973800749list_a] :
( ( member8191768239178080336list_a @ K @ ( bNF_Gr5549020744384184130list_a @ Kl @ Kl2 ) )
=> ( member6427184188916410838list_a @ ( append622457529216362434list_a @ Kl2 @ ( cons_P5184657343811988189list_a @ K @ nil_Pr3188421586756112173list_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_519_SuccD,axiom,
! [K: product_prod_a_a,Kl: set_li8827807065578854541od_a_a,Kl2: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ K @ ( bNF_Gr1699325103699555394od_a_a @ Kl @ Kl2 ) )
=> ( member6824001069763096534od_a_a @ ( append5335208819046833346od_a_a @ Kl2 @ ( cons_P7316939126706565853od_a_a @ K @ nil_Product_prod_a_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_520_SuccD,axiom,
! [K: list_a,Kl: set_list_list_a,Kl2: list_list_a] :
( ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl @ Kl2 ) )
=> ( member_list_list_a @ ( append_list_a @ Kl2 @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_521_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_522_SuccI,axiom,
! [Kl2: list_P321204300973800749list_a,K: produc9164743771328383783list_a,Kl: set_li1395191299911657101list_a] :
( ( member6427184188916410838list_a @ ( append622457529216362434list_a @ Kl2 @ ( cons_P5184657343811988189list_a @ K @ nil_Pr3188421586756112173list_a ) ) @ Kl )
=> ( member8191768239178080336list_a @ K @ ( bNF_Gr5549020744384184130list_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_523_SuccI,axiom,
! [Kl2: list_P1396940483166286381od_a_a,K: product_prod_a_a,Kl: set_li8827807065578854541od_a_a] :
( ( member6824001069763096534od_a_a @ ( append5335208819046833346od_a_a @ Kl2 @ ( cons_P7316939126706565853od_a_a @ K @ nil_Product_prod_a_a ) ) @ Kl )
=> ( member1426531477525435216od_a_a @ K @ ( bNF_Gr1699325103699555394od_a_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_524_SuccI,axiom,
! [Kl2: list_list_a,K: list_a,Kl: set_list_list_a] :
( ( member_list_list_a @ ( append_list_a @ Kl2 @ ( cons_list_a @ K @ nil_list_a ) ) @ Kl )
=> ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_525_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_526_Big_Opush_Osimps_I2_J,axiom,
! [X: a,Current2: current_a,Big: stack_a,Aux2: list_a,Count2: nat] :
( ( push_a @ X @ ( reverse_a @ Current2 @ Big @ Aux2 @ Count2 ) )
= ( reverse_a @ ( push_a3 @ X @ Current2 ) @ Big @ Aux2 @ Count2 ) ) ).
% Big.push.simps(2)
thf(fact_527_empty__Shift,axiom,
! [Kl: set_li1395191299911657101list_a,K: produc9164743771328383783list_a] :
( ( member6427184188916410838list_a @ nil_Pr3188421586756112173list_a @ Kl )
=> ( ( member8191768239178080336list_a @ K @ ( bNF_Gr5549020744384184130list_a @ Kl @ nil_Pr3188421586756112173list_a ) )
=> ( member6427184188916410838list_a @ nil_Pr3188421586756112173list_a @ ( bNF_Gr7794995673951123910list_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_528_empty__Shift,axiom,
! [Kl: set_li8827807065578854541od_a_a,K: product_prod_a_a] :
( ( member6824001069763096534od_a_a @ nil_Product_prod_a_a @ Kl )
=> ( ( member1426531477525435216od_a_a @ K @ ( bNF_Gr1699325103699555394od_a_a @ Kl @ nil_Product_prod_a_a ) )
=> ( member6824001069763096534od_a_a @ nil_Product_prod_a_a @ ( bNF_Gr6004602581434028742od_a_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_529_empty__Shift,axiom,
! [Kl: set_list_list_a,K: list_a] :
( ( member_list_list_a @ nil_list_a @ Kl )
=> ( ( member_list_a @ K @ ( bNF_Gr4634511371912843295list_a @ Kl @ nil_list_a ) )
=> ( member_list_list_a @ nil_list_a @ ( bNF_Gr7042794125918077091list_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_530_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_531_Succ__Shift,axiom,
! [Kl: set_list_list_a,K: list_a,Kl2: list_list_a] :
( ( bNF_Gr4634511371912843295list_a @ ( bNF_Gr7042794125918077091list_a @ Kl @ K ) @ Kl2 )
= ( bNF_Gr4634511371912843295list_a @ Kl @ ( cons_list_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_532_Succ__Shift,axiom,
! [Kl: set_list_a,K: a,Kl2: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ_a @ Kl @ ( cons_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_533_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_534_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_535_ShiftD,axiom,
! [Kl2: list_list_a,Kl: set_list_list_a,K: list_a] :
( ( member_list_list_a @ Kl2 @ ( bNF_Gr7042794125918077091list_a @ Kl @ K ) )
=> ( member_list_list_a @ ( cons_list_a @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_536_ShiftD,axiom,
! [Kl2: list_a,Kl: set_list_a,K: a] :
( ( member_list_a @ Kl2 @ ( bNF_Greatest_Shift_a @ Kl @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_537_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_538_Big__Aux_Olist__current_Oelims,axiom,
! [X: state_a2,Y: list_a] :
( ( ( big_list_current_a @ X )
= Y )
=> ( ! [Common3: state_a] :
( ( X
= ( common_a @ Common3 ) )
=> ( Y
!= ( common1102728217005306191rent_a @ Common3 ) ) )
=> ~ ! [Current: current_a] :
( ? [Uu: stack_a,Uv: list_a,Uw: nat] :
( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( current_list_a3 @ Current ) ) ) ) ) ).
% Big_Aux.list_current.elims
thf(fact_539_Big__Aux_Osize__new__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6530235180886170618tate_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( type_s8424385952999958455tate_a @ State2 ) ) )
=> ~ ! [Current: current_a] :
( ? [Uu: stack_a,Uv: list_a,Uw: nat] :
( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current ) ) ) ) ) ).
% Big_Aux.size_new_state.elims
thf(fact_540_product__lists_Osimps_I1_J,axiom,
( ( product_lists_list_a @ nil_list_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% product_lists.simps(1)
thf(fact_541_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_542_Big_Opush_Opelims,axiom,
! [X: a,Xa: state_a2,Y: state_a2] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ( accp_P1500049460152098787tate_a @ push_rel_a @ ( produc8641956578966763338tate_a @ X @ Xa ) )
=> ( ! [State2: state_a] :
( ( Xa
= ( common_a @ State2 ) )
=> ( ( Y
= ( common_a @ ( push_a2 @ X @ State2 ) ) )
=> ~ ( accp_P1500049460152098787tate_a @ push_rel_a @ ( produc8641956578966763338tate_a @ X @ ( common_a @ State2 ) ) ) ) )
=> ~ ! [Current: current_a,Big3: stack_a,Aux: list_a,Count: nat] :
( ( Xa
= ( reverse_a @ Current @ Big3 @ Aux @ Count ) )
=> ( ( Y
= ( reverse_a @ ( push_a3 @ X @ Current ) @ Big3 @ Aux @ Count ) )
=> ~ ( accp_P1500049460152098787tate_a @ push_rel_a @ ( produc8641956578966763338tate_a @ X @ ( reverse_a @ Current @ Big3 @ Aux @ Count ) ) ) ) ) ) ) ) ).
% Big.push.pelims
thf(fact_543_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_544_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_545_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_546_Big__Aux_Olist__current_Osimps_I2_J,axiom,
! [Current2: current_a,Uu2: stack_a,Uv2: list_a,Uw2: nat] :
( ( big_list_current_a @ ( reverse_a @ Current2 @ Uu2 @ Uv2 @ Uw2 ) )
= ( current_list_a3 @ Current2 ) ) ).
% Big_Aux.list_current.simps(2)
thf(fact_547_Big__Aux_Osize__new__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu2: stack_a,Uv2: list_a,Uw2: nat] :
( ( type_s6530235180886170618tate_a @ ( reverse_a @ Current2 @ Uu2 @ Uv2 @ Uw2 ) )
= ( type_s933026853152659577rent_a @ Current2 ) ) ).
% Big_Aux.size_new_state.simps(2)
thf(fact_548_Current__Proof_Opush__list,axiom,
! [X: list_a,Current2: current_list_a] :
( ( current_list_list_a @ ( push_list_a3 @ X @ Current2 ) )
= ( cons_list_a @ X @ ( current_list_list_a @ Current2 ) ) ) ).
% Current_Proof.push_list
thf(fact_549_Current__Proof_Opush__list,axiom,
! [X: a,Current2: current_a] :
( ( current_list_a3 @ ( push_a3 @ X @ Current2 ) )
= ( cons_a @ X @ ( current_list_a3 @ Current2 ) ) ) ).
% Current_Proof.push_list
thf(fact_550_Current__Aux_Olist_Osimps,axiom,
! [Extra2: list_a,Uu2: nat,Old2: stack_a,Uv2: nat] :
( ( current_list_a3 @ ( current_a2 @ Extra2 @ Uu2 @ Old2 @ Uv2 ) )
= ( append_a @ Extra2 @ ( stack_list_a2 @ Old2 ) ) ) ).
% Current_Aux.list.simps
thf(fact_551_Current__Aux_Olist_Oelims,axiom,
! [X: current_a,Y: list_a] :
( ( ( current_list_a3 @ X )
= Y )
=> ~ ! [Extra: list_a,Uu: nat,Old: stack_a] :
( ? [Uv: nat] :
( X
= ( current_a2 @ Extra @ Uu @ Old @ Uv ) )
=> ( Y
!= ( append_a @ Extra @ ( stack_list_a2 @ Old ) ) ) ) ) ).
% Current_Aux.list.elims
thf(fact_552_bind__simps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,F: list_a > list_a] :
( ( bind_list_a_a @ ( cons_list_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_list_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_553_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_554_take__rev__step,axiom,
! [Xs: list_list_a,N: nat,Acc: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( append_list_a @ ( common3049970819459547540list_a @ N @ ( tl_list_a @ Xs ) ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ Acc ) )
= ( append_list_a @ ( common3049970819459547540list_a @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_555_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_556_bind__simps_I1_J,axiom,
! [F: a > list_list_a] :
( ( bind_a_list_a @ nil_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_557_bind__simps_I1_J,axiom,
! [F: list_a > list_a] :
( ( bind_list_a_a @ nil_list_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_558_bind__simps_I1_J,axiom,
! [F: list_a > list_list_a] :
( ( bind_list_a_list_a @ nil_list_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_559_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_560_hd__append2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ).
% hd_append2
thf(fact_561_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_562_list_Ocollapse,axiom,
! [List: list_list_a] :
( ( List != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_563_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_564_hd__Cons__tl,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs ) @ ( tl_list_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_565_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_566_list_Osel_I1_J,axiom,
! [X21: list_a,X222: list_list_a] :
( ( hd_list_a @ ( cons_list_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_567_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_568_cons__hd,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= Ys )
=> ( X
= ( hd_list_a @ Ys ) ) ) ).
% cons_hd
thf(fact_569_cons__hd,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( X
= ( hd_a @ Ys ) ) ) ).
% cons_hd
thf(fact_570_hd__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( Xs = nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Ys ) ) )
& ( ( Xs != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( hd_list_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_571_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_572_longest__common__prefix,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
? [Ps: list_list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ( Xs
= ( append_list_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_list_a @ Ps @ Ys3 ) )
& ( ( Xs3 = nil_list_a )
| ( Ys3 = nil_list_a )
| ( ( hd_list_a @ Xs3 )
!= ( hd_list_a @ Ys3 ) ) ) ) ).
% longest_common_prefix
thf(fact_573_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys3: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys3 ) )
& ( ( Xs3 = nil_a )
| ( Ys3 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys3 ) ) ) ) ).
% longest_common_prefix
thf(fact_574_list_Oexpand,axiom,
! [List: list_list_a,List2: list_list_a] :
( ( ( List = nil_list_a )
= ( List2 = nil_list_a ) )
=> ( ( ( List != nil_list_a )
=> ( ( List2 != nil_list_a )
=> ( ( ( hd_list_a @ List )
= ( hd_list_a @ List2 ) )
& ( ( tl_list_a @ List )
= ( tl_list_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_575_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_576_list_Oexhaust__sel,axiom,
! [List: list_list_a] :
( ( List != nil_list_a )
=> ( List
= ( cons_list_a @ ( hd_list_a @ List ) @ ( tl_list_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_577_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_578_size__new__current_Ocases,axiom,
! [X: current_a] :
~ ! [Uu: list_a,Added: nat,Uv: stack_a,Remained: nat] :
( X
!= ( current_a2 @ Uu @ Added @ Uv @ Remained ) ) ).
% size_new_current.cases
thf(fact_579_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_list_a,Ys: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_list_a )
=> ( ( append_list_a @ ( common3049970819459547540list_a @ N @ Xs ) @ Ys )
= ( append_list_a @ ( common3049970819459547540list_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_list_a @ Xs ) ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_580_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_581_rotate1__hd__tl,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( rotate1_list_a @ Xs )
= ( append_list_a @ ( tl_list_a @ Xs ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_582_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_583_linarith,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ X @ Y ) @ Z2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ X @ ( suc @ Y ) ) @ Z2 ) ) ).
% linarith
thf(fact_584_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_585_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_586_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_587_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_588_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_589_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_590_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_591_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_592_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rotate1_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_593_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_594_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_595_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_596_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_597_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_598_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_599_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_600_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_601_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_602_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_603_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_604_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_605_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_606_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_607_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_608_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_609_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_610_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_611_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_612_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_613_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_a @ nil_list_a )
= nil_list_a ) ).
% rotate1.simps(1)
thf(fact_614_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_615_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_616_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_617_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_618_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_619_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_620_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_621_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_622_rotate1_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( rotate1_list_a @ ( cons_list_a @ X @ Xs ) )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) ) ).
% rotate1.simps(2)
thf(fact_623_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_624_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_625_remaining__steps__step__sub,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( type_r2212416260012024137tate_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( minus_minus_nat @ ( type_r2212416260012024137tate_a @ Common ) @ one_one_nat ) ) ) ).
% remaining_steps_step_sub
thf(fact_626_b,axiom,
( ( ord_less_eq_nat @ remained @ ( size_size_list_a @ ( append_a @ ( common_take_rev_a @ count @ ( stack_list_a2 @ biga ) ) @ aux ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ old ) )
=> ( ( ord_less_nat @ zero_zero_nat @ remained )
=> ( ( added = zero_zero_nat )
=> ( ( xa
= ( first_a2 @ old ) )
=> ( ( big
= ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ ( pop_a4 @ old ) @ ( minus_minus_nat @ remained @ ( suc @ zero_zero_nat ) ) ) @ biga @ aux @ count ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ remained @ count ) @ ( size_size_list_a @ aux ) )
=> ( ( ord_less_eq_nat @ count @ ( size_size_stack_a @ biga ) )
=> ( ( ( stack_list_a2 @ old )
= ( append_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ aux ) @ ( minus_minus_nat @ ( size_size_stack_a @ old ) @ ( size_size_stack_a @ biga ) ) ) @ ( rev_a @ aux ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_stack_a @ biga ) @ ( size_size_stack_a @ old ) ) @ ( stack_list_a2 @ biga ) ) ) )
=> ( ( ( append_a @ ( take_a @ remained @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ aux ) @ ( minus_minus_nat @ ( size_size_stack_a @ old ) @ ( size_size_stack_a @ biga ) ) ) @ ( rev_a @ aux ) ) ) @ ( take_a @ ( minus_minus_nat @ ( plus_plus_nat @ remained @ ( minus_minus_nat @ ( size_size_list_a @ aux ) @ ( minus_minus_nat @ ( size_size_stack_a @ old ) @ ( size_size_stack_a @ biga ) ) ) ) @ ( size_size_list_a @ aux ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_stack_a @ biga ) @ ( size_size_stack_a @ old ) ) @ ( stack_list_a2 @ biga ) ) ) )
= ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ ( common_take_rev_a @ count @ ( stack_list_a2 @ biga ) ) @ aux ) ) @ remained ) @ ( rev_a @ ( append_a @ ( common_take_rev_a @ count @ ( stack_list_a2 @ biga ) ) @ aux ) ) ) )
=> ( ( tl_a @ ( append_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ aux ) @ ( minus_minus_nat @ ( size_size_stack_a @ old ) @ ( size_size_stack_a @ biga ) ) ) @ ( rev_a @ aux ) ) @ ( drop_a @ ( minus_minus_nat @ ( size_size_stack_a @ biga ) @ ( size_size_stack_a @ old ) ) @ ( stack_list_a2 @ biga ) ) ) )
= ( append_a @ ( drop_a @ ( minus_minus_nat @ ( size_size_list_a @ aux ) @ ( minus_minus_nat @ ( size_size_stack_a @ old ) @ ( suc @ ( size_size_stack_a @ biga ) ) ) ) @ ( rev_a @ aux ) ) @ ( drop_a @ ( minus_minus_nat @ ( suc @ ( size_size_stack_a @ biga ) ) @ ( size_size_stack_a @ old ) ) @ ( stack_list_a2 @ biga ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% b
thf(fact_627_take__rev__nth,axiom,
! [N: nat,Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( X
= ( nth_list_a @ Xs @ N ) )
=> ( ( cons_list_a @ X @ ( append_list_a @ ( common3049970819459547540list_a @ N @ Xs ) @ Ys ) )
= ( append_list_a @ ( common3049970819459547540list_a @ ( suc @ N ) @ Xs ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_628_take__rev__nth,axiom,
! [N: nat,Xs: list_a,X: a,Ys: 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 ) @ Ys ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_629_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_630_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_631_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_632_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_633_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_634_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_635_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_636_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_637_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_638_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_639_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_640_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_641_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_642_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_643_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_644_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_645_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_646_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_647_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_648_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_649_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_650_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_651_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_652_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_653_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_654_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_655_rev__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rev_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rev_is_Nil_conv
thf(fact_656_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_657_Nil__is__rev__conv,axiom,
! [Xs: list_list_a] :
( ( nil_list_a
= ( rev_list_a @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_rev_conv
thf(fact_658_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_659_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_660_drop0,axiom,
( ( drop_a @ zero_zero_nat )
= ( ^ [X3: list_a] : X3 ) ) ).
% drop0
thf(fact_661_rev__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_662_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( drop_a @ N @ ( drop_a @ M @ Xs ) )
= ( drop_a @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_663_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_664_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_665_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_666_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_667_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_668_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_669_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_670_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_671_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_672_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_673_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_674_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_675_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_676_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_677_nth__Cons__0,axiom,
! [X: list_a,Xs: list_list_a] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_678_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_679_nth__Cons__Suc,axiom,
! [X: list_a,Xs: list_list_a,N: nat] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_list_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_680_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_681_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_682_take__Suc__Cons,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( take_list_a @ ( suc @ N ) @ ( cons_list_a @ X @ Xs ) )
= ( cons_list_a @ X @ ( take_list_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_683_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_684_take0,axiom,
( ( take_list_a @ zero_zero_nat )
= ( ^ [Xs4: list_list_a] : nil_list_a ) ) ).
% take0
thf(fact_685_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs4: list_a] : nil_a ) ) ).
% take0
thf(fact_686_take__eq__Nil,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( take_list_a @ N @ Xs )
= nil_list_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_list_a ) ) ) ).
% take_eq_Nil
thf(fact_687_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_688_take__eq__Nil2,axiom,
! [N: nat,Xs: list_list_a] :
( ( nil_list_a
= ( take_list_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_list_a ) ) ) ).
% take_eq_Nil2
thf(fact_689_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_690_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_691_rev__singleton__conv,axiom,
! [Xs: list_list_a,X: list_a] :
( ( ( rev_list_a @ Xs )
= ( cons_list_a @ X @ nil_list_a ) )
= ( Xs
= ( cons_list_a @ X @ nil_list_a ) ) ) ).
% rev_singleton_conv
thf(fact_692_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_693_singleton__rev__conv,axiom,
! [X: list_a,Xs: list_list_a] :
( ( ( cons_list_a @ X @ nil_list_a )
= ( rev_list_a @ Xs ) )
= ( ( cons_list_a @ X @ nil_list_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_694_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_695_take__all__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_696_take__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( take_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_697_drop__Suc__Cons,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( drop_list_a @ ( suc @ N ) @ ( cons_list_a @ X @ Xs ) )
= ( drop_list_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_698_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_699_nth__take,axiom,
! [I: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_a @ ( take_a @ N @ Xs ) @ I )
= ( nth_a @ Xs @ I ) ) ) ).
% nth_take
thf(fact_700_length__drop,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_701_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_702_pop__tl,axiom,
! [Stack: stack_a] :
( ( stack_list_a2 @ ( pop_a4 @ Stack ) )
= ( tl_a @ ( stack_list_a2 @ Stack ) ) ) ).
% pop_tl
thf(fact_703_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_704_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_705_nth__append__length,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( nth_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_706_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_707_rev__eq__Cons__iff,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( rev_list_a @ Xs )
= ( cons_list_a @ Y @ Ys ) )
= ( Xs
= ( append_list_a @ ( rev_list_a @ Ys ) @ ( cons_list_a @ Y @ nil_list_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_708_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_709_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_710_drop__all,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( drop_list_a @ N @ Xs )
= nil_list_a ) ) ).
% drop_all
thf(fact_711_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_712_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( drop_list_a @ N @ Xs )
= nil_list_a )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_713_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_714_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_a] :
( ( nil_list_a
= ( drop_list_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_715_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_716_take__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( take_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( take_a @ N @ Xs ) @ ( take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_717_drop__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_718_hd__take,axiom,
! [J: nat,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_a @ ( take_a @ J @ Xs ) )
= ( hd_a @ Xs ) ) ) ).
% hd_take
thf(fact_719_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_720_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_721_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_722_nth__drop,axiom,
! [N: nat,Xs: list_a,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
= ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_723_nth__Cons__pos,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ N )
= ( nth_list_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_724_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_725_take__hd,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( take_list_a @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) ) ) ).
% take_hd
thf(fact_726_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_727_hd__drop__1,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs ) @ ( drop_list_a @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_728_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_729_hd__drop,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( cons_list_a @ ( hd_list_a @ ( drop_list_a @ N @ Xs ) ) @ ( drop_list_a @ ( suc @ N ) @ Xs ) )
= ( drop_list_a @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_730_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_731_first__take__tl,axiom,
! [Big: stack_list_a,Count2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6256786526647004292list_a @ Big ) )
=> ( ( cons_list_a @ ( first_list_a2 @ Big ) @ ( take_list_a @ Count2 @ ( tl_list_a @ ( stack_list_list_a @ Big ) ) ) )
= ( take_list_a @ ( suc @ Count2 ) @ ( stack_list_list_a @ Big ) ) ) ) ).
% first_take_tl
thf(fact_732_first__take__tl,axiom,
! [Big: stack_a,Count2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Big ) )
=> ( ( cons_a @ ( first_a2 @ Big ) @ ( take_a @ Count2 @ ( tl_a @ ( stack_list_a2 @ Big ) ) ) )
= ( take_a @ ( suc @ Count2 ) @ ( stack_list_a2 @ Big ) ) ) ) ).
% first_take_tl
thf(fact_733_a,axiom,
( ( ord_less_eq_nat @ remained @ ( plus_plus_nat @ count @ ( size_size_list_a @ aux ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ remained )
=> ( ( added = zero_zero_nat )
=> ( ( xa
= ( first_a2 @ old ) )
=> ( ( big
= ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ ( pop_a4 @ old ) @ ( minus_minus_nat @ remained @ ( suc @ zero_zero_nat ) ) ) @ biga @ aux @ count ) )
=> ( ( ord_less_eq_nat @ count @ ( size_size_stack_a @ biga ) )
=> ( ( ( stack_list_a2 @ old )
= ( append_a @ ( rev_a @ aux ) @ ( stack_list_a2 @ biga ) ) )
=> ( ( ( append_a @ ( take_a @ remained @ ( rev_a @ aux ) ) @ ( take_a @ ( minus_minus_nat @ remained @ ( size_size_list_a @ aux ) ) @ ( stack_list_a2 @ biga ) ) )
= ( append_a @ ( drop_a @ ( minus_minus_nat @ ( plus_plus_nat @ count @ ( size_size_list_a @ aux ) ) @ remained ) @ ( rev_a @ aux ) ) @ ( drop_a @ ( minus_minus_nat @ count @ remained ) @ ( take_a @ count @ ( stack_list_a2 @ biga ) ) ) ) )
=> ( ~ ( ord_less_eq_nat @ ( size_size_stack_a @ old ) @ ( plus_plus_nat @ ( size_size_list_a @ aux ) @ ( size_size_stack_a @ biga ) ) )
=> ( ( tl_a @ ( append_a @ ( rev_a @ aux ) @ ( stack_list_a2 @ biga ) ) )
= ( append_a @ ( rev_a @ aux ) @ ( stack_list_a2 @ biga ) ) ) ) ) ) ) ) ) ) ) ) ).
% a
thf(fact_734_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_735_drop__take,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( drop_a @ N @ ( take_a @ M @ Xs ) )
= ( take_a @ ( minus_minus_nat @ M @ N ) @ ( drop_a @ N @ Xs ) ) ) ).
% drop_take
thf(fact_736_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_737_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_738_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_739_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_740_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_741_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_742_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_743_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_744_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_745_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_746_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_747_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_748_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_749_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_750_take__1,axiom,
! [X: nat,Y: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ zero_zero_nat @ X )
& ( ord_less_nat @ zero_zero_nat @ Y ) )
=> ( ( ( take_a @ X @ Xs )
= ( take_a @ Y @ Ys ) )
=> ( ( take_a @ one_one_nat @ Xs )
= ( take_a @ one_one_nat @ Ys ) ) ) ) ).
% take_1
thf(fact_751_pop__drop,axiom,
! [Stack: stack_a] :
( ( stack_list_a2 @ ( pop_a4 @ Stack ) )
= ( drop_a @ one_one_nat @ ( stack_list_a2 @ Stack ) ) ) ).
% pop_drop
thf(fact_752_take__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ! [I2: nat] :
( ( take_a @ I2 @ Xs )
= ( take_a @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_753_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_754_take__drop,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( take_a @ N @ ( drop_a @ M @ Xs ) )
= ( drop_a @ M @ ( take_a @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_755_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_756_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_757_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_758_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_759_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_760_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_761_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_762_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_763_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_764_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_765_nth__via__drop,axiom,
! [N: nat,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( drop_list_a @ N @ Xs )
= ( cons_list_a @ Y @ Ys ) )
=> ( ( nth_list_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_766_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_767_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_768_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_769_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_770_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_771_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_772_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_773_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_774_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_775_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_776_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_777_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_778_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_779_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_780_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_781_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_782_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_783_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_784_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_785_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_786_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_787_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_788_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_789_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_790_tl__drop,axiom,
! [N: nat,Xs: list_a] :
( ( tl_a @ ( drop_a @ N @ Xs ) )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% tl_drop
thf(fact_791_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_792_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_793_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_794_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_795_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_796_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_797_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_798_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_799_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_800_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_801_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_802_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_803_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_804_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_805_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_806_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_807_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_808_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C2: nat] :
( B4
= ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_809_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_810_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_811_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_812_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_813_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_814_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_815_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_816_take__Nil,axiom,
! [N: nat] :
( ( take_list_a @ N @ nil_list_a )
= nil_list_a ) ).
% take_Nil
thf(fact_817_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_818_drop__Nil,axiom,
! [N: nat] :
( ( drop_list_a @ N @ nil_list_a )
= nil_list_a ) ).
% drop_Nil
thf(fact_819_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_820_app__rev,axiom,
! [As2: list_a,Bs2: list_a,Cs: list_a,Ds: list_a] :
( ( ( append_a @ As2 @ ( rev_a @ Bs2 ) )
= ( append_a @ Cs @ ( rev_a @ Ds ) ) )
=> ( ( append_a @ Bs2 @ ( rev_a @ As2 ) )
= ( append_a @ Ds @ ( rev_a @ Cs ) ) ) ) ).
% app_rev
thf(fact_821_drop__0,axiom,
! [Xs: list_a] :
( ( drop_a @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_822_rev_Osimps_I1_J,axiom,
( ( rev_list_a @ nil_list_a )
= nil_list_a ) ).
% rev.simps(1)
thf(fact_823_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_824_nth__take__lemma,axiom,
! [K: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ K @ Xs )
= ( take_a @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_825_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_826_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N @ Xs ) )
= ( nth_a @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_827_id__take__nth__drop,axiom,
! [I: nat,Xs: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( Xs
= ( append_list_a @ ( take_list_a @ I @ Xs ) @ ( cons_list_a @ ( nth_list_a @ Xs @ I ) @ ( drop_list_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_828_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_829_drop__Cons_H,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_list_a @ N @ ( cons_list_a @ X @ Xs ) )
= ( cons_list_a @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_list_a @ N @ ( cons_list_a @ X @ Xs ) )
= ( drop_list_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_830_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_831_nth__Cons_H,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ N )
= ( nth_list_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_832_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_833_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_834_Current_Opush_Osimps,axiom,
! [X: list_a,Extra2: list_list_a,Added2: nat,Old2: stack_list_a,Remained2: nat] :
( ( push_list_a3 @ X @ ( current_list_a2 @ Extra2 @ Added2 @ Old2 @ Remained2 ) )
= ( current_list_a2 @ ( cons_list_a @ X @ Extra2 ) @ ( plus_plus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ).
% Current.push.simps
thf(fact_835_Current_Opush_Osimps,axiom,
! [X: a,Extra2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( ( push_a3 @ X @ ( current_a2 @ Extra2 @ Added2 @ Old2 @ Remained2 ) )
= ( current_a2 @ ( cons_a @ X @ Extra2 ) @ ( plus_plus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ).
% Current.push.simps
thf(fact_836_Current_Opush_Oelims,axiom,
! [X: list_a,Xa: current_list_a,Y: current_list_a] :
( ( ( push_list_a3 @ X @ Xa )
= Y )
=> ~ ! [Extra: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( ( Xa
= ( current_list_a2 @ Extra @ Added @ Old @ Remained ) )
=> ( Y
!= ( current_list_a2 @ ( cons_list_a @ X @ Extra ) @ ( plus_plus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ).
% Current.push.elims
thf(fact_837_Current_Opush_Oelims,axiom,
! [X: a,Xa: current_a,Y: current_a] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ~ ! [Extra: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( Xa
= ( current_a2 @ Extra @ Added @ Old @ Remained ) )
=> ( Y
!= ( current_a2 @ ( cons_a @ X @ Extra ) @ ( plus_plus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ).
% Current.push.elims
thf(fact_838_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_839_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( cons_list_a @ ( nth_list_a @ Xs @ I ) @ ( drop_list_a @ ( suc @ I ) @ Xs ) )
= ( drop_list_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_840_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_841_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_842_take__Cons_H,axiom,
! [N: nat,X: list_a,Xs: list_list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_list_a @ N @ ( cons_list_a @ X @ Xs ) )
= nil_list_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_list_a @ N @ ( cons_list_a @ X @ Xs ) )
= ( cons_list_a @ X @ ( take_list_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_843_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_844_nth__non__equal__first__eq,axiom,
! [X: list_a,Y: list_a,Xs: list_list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_a @ ( cons_list_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_list_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_845_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_846_take__0,axiom,
! [Xs: list_list_a] :
( ( take_list_a @ zero_zero_nat @ Xs )
= nil_list_a ) ).
% take_0
thf(fact_847_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_848_Big__Aux_Oremaining__steps__state_Osimps_I2_J,axiom,
! [Uu2: list_a,Uv2: nat,Uw2: stack_a,Remaining2: nat,Ux2: stack_a,Uy2: list_a,Count2: nat] :
( ( type_r2494999336194962664tate_a @ ( reverse_a @ ( current_a2 @ Uu2 @ Uv2 @ Uw2 @ Remaining2 ) @ Ux2 @ Uy2 @ Count2 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ Count2 @ Remaining2 ) @ one_one_nat ) ) ).
% Big_Aux.remaining_steps_state.simps(2)
thf(fact_849_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_850_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X7: a] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_851_list__eq__iff__nth__eq,axiom,
( ( ^ [Y8: list_a,Z3: list_a] : ( Y8 = Z3 ) )
= ( ^ [Xs4: list_a,Ys6: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys6 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I4 )
= ( nth_a @ Ys6 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_852_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_853_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_854_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_855_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_856_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_857_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_858_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_859_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_860_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_861_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_862_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_863_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_864_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_865_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_866_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_867_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_868_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_869_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_870_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_871_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_872_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_873_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_874_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_875_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_876_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_877_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_878_drop__Suc,axiom,
! [N: nat,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ Xs )
= ( drop_a @ N @ ( tl_a @ Xs ) ) ) ).
% drop_Suc
thf(fact_879_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_880_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_881_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_882_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_883_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_884_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_885_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_886_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_887_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_888_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_889_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_890_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_891_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_892_Big_Ostep__state_Osimps_I3_J,axiom,
! [Current2: current_list_a,Big: stack_list_a,Aux2: list_list_a,V2: nat] :
( ( type_s8055303106637973038list_a @ ( reverse_list_a @ Current2 @ Big @ Aux2 @ ( suc @ V2 ) ) )
= ( reverse_list_a @ Current2 @ ( pop_list_a4 @ Big ) @ ( cons_list_a @ ( first_list_a2 @ Big ) @ Aux2 ) @ ( minus_minus_nat @ ( suc @ V2 ) @ one_one_nat ) ) ) ).
% Big.step_state.simps(3)
thf(fact_893_Big_Ostep__state_Osimps_I3_J,axiom,
! [Current2: current_a,Big: stack_a,Aux2: list_a,V2: nat] :
( ( type_s3593206172722485288tate_a @ ( reverse_a @ Current2 @ Big @ Aux2 @ ( suc @ V2 ) ) )
= ( reverse_a @ Current2 @ ( pop_a4 @ Big ) @ ( cons_a @ ( first_a2 @ Big ) @ Aux2 ) @ ( minus_minus_nat @ ( suc @ V2 ) @ one_one_nat ) ) ) ).
% Big.step_state.simps(3)
thf(fact_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_902_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_a2 @ S2 ) )
= ( take_a @ ( size_size_stack_a @ S2 ) @ ( stack_list_a2 @ S1 ) ) )
=> ( ( first_a2 @ S1 )
= ( first_a2 @ S2 ) ) ) ) ) ).
% take_first
thf(fact_903_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_904_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_905_size__current_Oelims,axiom,
! [X: current_a,Y: nat] :
( ( ( size_size_current_a @ X )
= Y )
=> ~ ! [Uu: list_a,Added: nat,Old: stack_a] :
( ? [Uv: nat] :
( X
= ( current_a2 @ Uu @ Added @ Old @ Uv ) )
=> ( Y
!= ( plus_plus_nat @ Added @ ( size_size_stack_a @ Old ) ) ) ) ) ).
% size_current.elims
thf(fact_906_size__current_Osimps,axiom,
! [Uu2: list_a,Added2: nat,Old2: stack_a,Uv2: nat] :
( ( size_size_current_a @ ( current_a2 @ Uu2 @ Added2 @ Old2 @ Uv2 ) )
= ( plus_plus_nat @ Added2 @ ( size_size_stack_a @ Old2 ) ) ) ).
% size_current.simps
thf(fact_907_lex__take__index,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( take_list_a @ I2 @ Xs )
= ( take_list_a @ I2 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I2 ) @ ( nth_list_a @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_908_lex__take__index,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I2 @ Xs )
= ( take_a @ I2 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_909_size__new__current_Oelims,axiom,
! [X: current_a,Y: nat] :
( ( ( type_s933026853152659577rent_a @ X )
= Y )
=> ~ ! [Uu: list_a,Added: nat,Uv: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ Uu @ Added @ Uv @ Remained ) )
=> ( Y
!= ( plus_plus_nat @ Added @ Remained ) ) ) ) ).
% size_new_current.elims
thf(fact_910_size__new__current_Osimps,axiom,
! [Uu2: list_a,Added2: nat,Uv2: stack_a,Remained2: nat] :
( ( type_s933026853152659577rent_a @ ( current_a2 @ Uu2 @ Added2 @ Uv2 @ Remained2 ) )
= ( plus_plus_nat @ Added2 @ Remained2 ) ) ).
% size_new_current.simps
thf(fact_911_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( take_list_a @ ( suc @ I ) @ Xs )
= ( append_list_a @ ( take_list_a @ I @ Xs ) @ ( cons_list_a @ ( nth_list_a @ Xs @ I ) @ nil_list_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_912_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_913_rev__app__single,axiom,
! [Xs: list_list_a,X: list_a] :
( ( append_list_a @ ( rev_list_a @ Xs ) @ ( cons_list_a @ X @ nil_list_a ) )
= ( rev_list_a @ ( cons_list_a @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_914_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_915_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_916_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_917_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_918_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_919_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_920_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_921_hd__conv__nth,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( hd_list_a @ Xs )
= ( nth_list_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_922_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_923_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_924_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_925_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_926_Big__Aux_Oremaining__steps__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_r2494999336194962664tate_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( type_r2212416260012024137tate_a @ State2 ) ) )
=> ~ ! [Uu: list_a,Uv: nat,Uw: stack_a,Remaining: nat,Ux: stack_a,Uy: list_a,Count: nat] :
( ( X
= ( reverse_a @ ( current_a2 @ Uu @ Uv @ Uw @ Remaining ) @ Ux @ Uy @ Count ) )
=> ( Y
!= ( plus_plus_nat @ ( plus_plus_nat @ Count @ Remaining ) @ one_one_nat ) ) ) ) ) ).
% Big_Aux.remaining_steps_state.elims
thf(fact_927_take__hd__drop,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( append_list_a @ ( take_list_a @ N @ Xs ) @ ( cons_list_a @ ( hd_list_a @ ( drop_list_a @ N @ Xs ) ) @ nil_list_a ) )
= ( take_list_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_928_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_929_first__hd,axiom,
( first_a2
= ( ^ [Stack2: stack_a] : ( hd_a @ ( stack_list_a2 @ Stack2 ) ) ) ) ).
% first_hd
thf(fact_930_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_931_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_932_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_933_list_Osize_I4_J,axiom,
! [X21: list_a,X222: list_list_a] :
( ( size_s349497388124573686list_a @ ( cons_list_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_934_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_935_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_936_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_937_length__one__hd,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) ) ) ).
% length_one_hd
thf(fact_938_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_939_take__hd_H,axiom,
! [Ys: list_list_a,X: list_a,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( ( take_list_a @ ( size_s349497388124573686list_a @ Ys ) @ ( cons_list_a @ X @ Xs ) )
= ( take_list_a @ ( suc @ ( size_s349497388124573686list_a @ Xs ) ) @ Ys ) )
=> ( ( hd_list_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_940_take__hd_H,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys ) )
=> ( ( hd_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_941_rev__tl__hd,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( append_list_a @ ( rev_list_a @ ( tl_list_a @ Xs ) ) @ ( cons_list_a @ ( hd_list_a @ Xs ) @ nil_list_a ) )
= ( rev_list_a @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_942_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_943_take__Suc,axiom,
! [Xs: list_list_a,N: nat] :
( ( Xs != nil_list_a )
=> ( ( take_list_a @ ( suc @ N ) @ Xs )
= ( cons_list_a @ ( hd_list_a @ Xs ) @ ( take_list_a @ N @ ( tl_list_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_944_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_945_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_946_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_947_first__take__pop,axiom,
! [Stack: stack_list_a,X: nat] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( cons_list_a @ ( first_list_a2 @ Stack ) @ ( take_list_a @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( stack_list_list_a @ ( pop_list_a4 @ Stack ) ) ) )
= ( take_list_a @ X @ ( stack_list_list_a @ Stack ) ) ) ) ) ).
% first_take_pop
thf(fact_948_first__take__pop,axiom,
! [Stack: stack_a,X: nat] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( cons_a @ ( first_a2 @ Stack ) @ ( take_a @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( stack_list_a2 @ ( pop_a4 @ Stack ) ) ) )
= ( take_a @ X @ ( stack_list_a2 @ Stack ) ) ) ) ) ).
% first_take_pop
thf(fact_949_Stack__Proof_Opop__list,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( stack_list_a2 @ ( pop_a4 @ Stack ) )
= ( tl_a @ ( stack_list_a2 @ Stack ) ) ) ) ).
% Stack_Proof.pop_list
thf(fact_950_first__list,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( first_a2 @ Stack )
= ( hd_a @ ( stack_list_a2 @ Stack ) ) ) ) ).
% first_list
thf(fact_951_pop__list__length,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( suc @ ( size_size_list_a @ ( stack_list_a2 @ ( pop_a4 @ Stack ) ) ) )
= ( size_size_list_a @ ( stack_list_a2 @ Stack ) ) ) ) ).
% pop_list_length
thf(fact_952_first__pop,axiom,
! [Stack: stack_list_a] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( cons_list_a @ ( first_list_a2 @ Stack ) @ ( stack_list_list_a @ ( pop_list_a4 @ Stack ) ) )
= ( stack_list_list_a @ Stack ) ) ) ).
% first_pop
thf(fact_953_first__pop,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( cons_a @ ( first_a2 @ Stack ) @ ( stack_list_a2 @ ( pop_a4 @ Stack ) ) )
= ( stack_list_a2 @ Stack ) ) ) ).
% first_pop
thf(fact_954_Stack__Proof_Olist__not__empty__2,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
= nil_list_a )
=> ( type_i4148797458121419353list_a @ Stack ) ) ).
% Stack_Proof.list_not_empty_2
thf(fact_955_Stack__Proof_Olist__not__empty__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
= nil_a )
=> ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.list_not_empty_2
thf(fact_956_Stack__Proof_Olist__not__empty,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
= ( ~ ( type_i4148797458121419353list_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty
thf(fact_957_Stack__Proof_Olist__not__empty,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
!= nil_a )
= ( ~ ( type_i3216275384938974675tack_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty
thf(fact_958_Stack__Proof_Olist__empty__2,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
!= nil_list_a )
=> ~ ( type_i4148797458121419353list_a @ Stack ) ) ).
% Stack_Proof.list_empty_2
thf(fact_959_Stack__Proof_Olist__empty__2,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
!= nil_a )
=> ~ ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.list_empty_2
thf(fact_960_Stack__Proof_Olist__empty,axiom,
! [Stack: stack_list_a] :
( ( ( stack_list_list_a @ Stack )
= nil_list_a )
= ( type_i4148797458121419353list_a @ Stack ) ) ).
% Stack_Proof.list_empty
thf(fact_961_Stack__Proof_Olist__empty,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
= nil_a )
= ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.list_empty
thf(fact_962_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_963_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_964_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_965_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_966_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_967_first__take,axiom,
! [Stack: stack_list_a] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( cons_list_a @ ( first_list_a2 @ Stack ) @ nil_list_a )
= ( take_list_a @ one_one_nat @ ( stack_list_list_a @ Stack ) ) ) ) ).
% first_take
thf(fact_968_first__take,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( cons_a @ ( first_a2 @ Stack ) @ nil_a )
= ( take_a @ one_one_nat @ ( stack_list_a2 @ Stack ) ) ) ) ).
% first_take
thf(fact_969_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_970_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_971_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_972_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_973_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_974_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_975_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_976_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_977_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_978_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_979_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_980_take__rev__def,axiom,
( common_take_rev_a
= ( ^ [N2: nat,Xs4: list_a] : ( rev_a @ ( take_a @ N2 @ Xs4 ) ) ) ) ).
% take_rev_def
thf(fact_981_Current_Opop_Oelims,axiom,
! [X: current_list_a,Y: produc603899091129270873list_a] :
( ( ( pop_list_a3 @ X )
= Y )
=> ( ! [Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ nil_list_a @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc1356073977093405001list_a @ ( first_list_a2 @ Old ) @ ( current_list_a2 @ nil_list_a @ Added @ ( pop_list_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ ( cons_list_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc1356073977093405001list_a @ X2 @ ( current_list_a2 @ Xs2 @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_982_Current_Opop_Oelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a3 @ X )
= Y )
=> ( ! [Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ nil_a @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc8503237746132909001rent_a @ ( first_a2 @ Old ) @ ( current_a2 @ nil_a @ Added @ ( pop_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) ) )
=> ~ ! [X2: a,Xs2: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc8503237746132909001rent_a @ X2 @ ( current_a2 @ Xs2 @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_983_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_984_Current_Opop_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,Added2: nat,Old2: stack_list_a,Remained2: nat] :
( ( pop_list_a3 @ ( current_list_a2 @ ( cons_list_a @ X @ Xs ) @ Added2 @ Old2 @ Remained2 ) )
= ( produc1356073977093405001list_a @ X @ ( current_list_a2 @ Xs @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ).
% Current.pop.simps(2)
thf(fact_985_Current_Opop_Osimps_I2_J,axiom,
! [X: a,Xs: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( ( pop_a3 @ ( current_a2 @ ( cons_a @ X @ Xs ) @ Added2 @ Old2 @ Remained2 ) )
= ( produc8503237746132909001rent_a @ X @ ( current_a2 @ Xs @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ).
% Current.pop.simps(2)
thf(fact_986_Current_Opop_Osimps_I1_J,axiom,
! [Added2: nat,Old2: stack_list_a,Remained2: nat] :
( ( pop_list_a3 @ ( current_list_a2 @ nil_list_a @ Added2 @ Old2 @ Remained2 ) )
= ( produc1356073977093405001list_a @ ( first_list_a2 @ Old2 ) @ ( current_list_a2 @ nil_list_a @ Added2 @ ( pop_list_a4 @ Old2 ) @ ( minus_minus_nat @ Remained2 @ one_one_nat ) ) ) ) ).
% Current.pop.simps(1)
thf(fact_987_Current_Opop_Osimps_I1_J,axiom,
! [Added2: nat,Old2: stack_a,Remained2: nat] :
( ( pop_a3 @ ( current_a2 @ nil_a @ Added2 @ Old2 @ Remained2 ) )
= ( produc8503237746132909001rent_a @ ( first_a2 @ Old2 ) @ ( current_a2 @ nil_a @ Added2 @ ( pop_a4 @ Old2 ) @ ( minus_minus_nat @ Remained2 @ one_one_nat ) ) ) ) ).
% Current.pop.simps(1)
thf(fact_988_size__pop__suc,axiom,
! [Current2: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( ( pop_a3 @ Current2 )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( suc @ ( size_size_current_a @ Current3 ) )
= ( size_size_current_a @ Current2 ) ) ) ) ) ).
% size_pop_suc
thf(fact_989_take__last__length,axiom,
! [Xs: list_list_a] :
( ( ( take_list_a @ ( suc @ zero_zero_nat ) @ ( rev_list_a @ Xs ) )
= ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% take_last_length
thf(fact_990_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_991_last__appendL,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys = nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Xs ) ) ) ).
% last_appendL
thf(fact_992_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_993_last__appendR,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ).
% last_appendR
thf(fact_994_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_995_last__snoc,axiom,
! [Xs: list_list_a,X: list_a] :
( ( last_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= X ) ).
% last_snoc
thf(fact_996_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_997_last__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( last_a @ ( drop_a @ N @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_drop
thf(fact_998_last__ConsR,axiom,
! [Xs: list_list_a,X: list_a] :
( ( Xs != nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
= ( last_list_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_999_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_1000_last__ConsL,axiom,
! [Xs: list_list_a,X: list_a] :
( ( Xs = nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1001_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1002_last_Osimps,axiom,
! [Xs: list_list_a,X: list_a] :
( ( ( Xs = nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_list_a )
=> ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
= ( last_list_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_1003_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_1004_longest__common__suffix,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
? [Ss: list_list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ( Xs
= ( append_list_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_list_a @ Ys3 @ Ss ) )
& ( ( Xs3 = nil_list_a )
| ( Ys3 = nil_list_a )
| ( ( last_list_a @ Xs3 )
!= ( last_list_a @ Ys3 ) ) ) ) ).
% longest_common_suffix
thf(fact_1005_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys3: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys3 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys3 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys3 ) ) ) ) ).
% longest_common_suffix
thf(fact_1006_last__append,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( ( Ys = nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Xs ) ) )
& ( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ) ).
% last_append
thf(fact_1007_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_1008_last__tl,axiom,
! [Xs: list_list_a] :
( ( ( Xs = nil_list_a )
| ( ( tl_list_a @ Xs )
!= nil_list_a ) )
=> ( ( last_list_a @ ( tl_list_a @ Xs ) )
= ( last_list_a @ Xs ) ) ) ).
% last_tl
thf(fact_1009_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_1010_hd__Nil__eq__last,axiom,
( ( hd_list_a @ nil_list_a )
= ( last_list_a @ nil_list_a ) ) ).
% hd_Nil_eq_last
thf(fact_1011_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_1012_invar__current_Osimps,axiom,
! [Extra2: list_a,Added2: nat,Uu2: stack_a,Uv2: nat] :
( ( type_i6141643110573041459rent_a @ ( current_a2 @ Extra2 @ Added2 @ Uu2 @ Uv2 ) )
= ( ( size_size_list_a @ Extra2 )
= Added2 ) ) ).
% invar_current.simps
thf(fact_1013_invar__current_Oelims_I1_J,axiom,
! [X: current_a,Y: $o] :
( ( ( type_i6141643110573041459rent_a @ X )
= Y )
=> ~ ! [Extra: list_a,Added: nat] :
( ? [Uu: stack_a,Uv: nat] :
( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( Y
= ( ( size_size_list_a @ Extra )
!= Added ) ) ) ) ).
% invar_current.elims(1)
thf(fact_1014_invar__current_Oelims_I2_J,axiom,
! [X: current_a] :
( ( type_i6141643110573041459rent_a @ X )
=> ~ ! [Extra: list_a,Added: nat] :
( ? [Uu: stack_a,Uv: nat] :
( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_size_list_a @ Extra )
!= Added ) ) ) ).
% invar_current.elims(2)
thf(fact_1015_invar__current_Oelims_I3_J,axiom,
! [X: current_a] :
( ~ ( type_i6141643110573041459rent_a @ X )
=> ~ ! [Extra: list_a,Added: nat] :
( ? [Uu: stack_a,Uv: nat] :
( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_size_list_a @ Extra )
= Added ) ) ) ).
% invar_current.elims(3)
thf(fact_1016_last__rev,axiom,
! [Xs: list_a] :
( ( last_a @ ( rev_a @ Xs ) )
= ( hd_a @ Xs ) ) ).
% last_rev
thf(fact_1017_hd__rev,axiom,
! [Xs: list_a] :
( ( hd_a @ ( rev_a @ Xs ) )
= ( last_a @ Xs ) ) ).
% hd_rev
thf(fact_1018_last__conv__nth,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( last_list_a @ Xs )
= ( nth_list_a @ Xs @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1019_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_1020_take__last,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( take_list_a @ one_one_nat @ ( rev_list_a @ Xs ) )
= ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) ) ) ).
% take_last
thf(fact_1021_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_1022_last__drop__rev,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( cons_list_a @ ( last_list_a @ Xs ) @ ( drop_list_a @ one_one_nat @ ( rev_list_a @ Xs ) ) )
= ( rev_list_a @ Xs ) ) ) ).
% last_drop_rev
thf(fact_1023_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_1024_Current__Proof_Olist__size,axiom,
! [Current2: current_list_a] :
( ( type_i8419033571676804025list_a @ Current2 )
=> ( ( ( current_list_list_a @ Current2 )
= nil_list_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s5124551957227789858list_a @ Current2 ) ) ) ) ).
% Current_Proof.list_size
thf(fact_1025_Current__Proof_Olist__size,axiom,
! [Current2: current_a] :
( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( ( current_list_a3 @ Current2 )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) ) ) ) ).
% Current_Proof.list_size
thf(fact_1026_Current__Proof_Oinvar__pop,axiom,
! [Current2: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( ( pop_a3 @ Current2 )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( type_i6141643110573041459rent_a @ Current3 ) ) ) ) ).
% Current_Proof.invar_pop
thf(fact_1027_Current__Proof_Osize__pop__sub,axiom,
! [Current2: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( ( pop_a3 @ Current2 )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( size_size_current_a @ Current3 )
= ( minus_minus_nat @ ( size_size_current_a @ Current2 ) @ one_one_nat ) ) ) ) ) ).
% Current_Proof.size_pop_sub
thf(fact_1028_Current__Proof_Opop__list,axiom,
! [Current2: current_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5124551957227789858list_a @ Current2 ) )
=> ( ( type_i8419033571676804025list_a @ Current2 )
=> ( ( cons_list_a @ ( produc8258313757495307061list_a @ ( pop_list_a3 @ Current2 ) ) @ ( tl_list_a @ ( current_list_list_a @ Current2 ) ) )
= ( current_list_list_a @ Current2 ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_1029_Current__Proof_Opop__list,axiom,
! [Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( cons_a @ ( produc4952273589686483381rent_a @ ( pop_a3 @ Current2 ) ) @ ( tl_a @ ( current_list_a3 @ Current2 ) ) )
= ( current_list_a3 @ Current2 ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_1030_drop__first__list,axiom,
! [Current2: current_a] :
( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( current_list_a3 @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) )
= ( tl_a @ ( current_list_a3 @ Current2 ) ) ) ) ) ).
% drop_first_list
thf(fact_1031_prod_Ocollapse,axiom,
! [Prod: produc5032551385658279741list_a] :
( ( produc8111569692950616493list_a @ ( produc6311120620833064345list_a @ Prod ) @ ( produc3976258275648695771list_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1032_prod_Ocollapse,axiom,
! [Prod: produc9164743771328383783list_a] :
( ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ Prod ) @ ( produc8617614985401127493list_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1033_prod_Ocollapse,axiom,
! [Prod: product_prod_a_a] :
( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1034_prod_Ocollapse,axiom,
! [Prod: produc3409137331138395373tate_a] :
( ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1035_prod_Ocollapse,axiom,
! [Prod: produc7805042584321970905rent_a] :
( ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ Prod ) @ ( produc4695312889421393143rent_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1036_prod_Ocollapse,axiom,
! [Prod: produc6972303929186420058tate_a] :
( ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_1037_Current__Proof_Osize__pop,axiom,
! [Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( suc @ ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) ) )
= ( size_size_current_a @ Current2 ) ) ) ) ).
% Current_Proof.size_pop
thf(fact_1038_Current__Proof_Osize__new__pop,axiom,
! [Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( type_s933026853152659577rent_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( suc @ ( type_s933026853152659577rent_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) ) )
= ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ).
% Current_Proof.size_new_pop
thf(fact_1039_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( a > a > $o ) > list_a > $o,X: a > a > $o,Y: list_a,A: produc5032551385658279741list_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8111569692950616493list_a @ X @ Y ) )
=> ( P @ ( produc6311120620833064345list_a @ A ) @ ( produc3976258275648695771list_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1040_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_a > list_a > $o,X: list_a,Y: list_a,A: produc9164743771328383783list_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc6837034575241423639list_a @ X @ Y ) )
=> ( P @ ( produc3698117735987127555list_a @ A ) @ ( produc8617614985401127493list_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1041_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_a_a @ X @ Y ) )
=> ( P @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1042_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > state_a > $o,X: a,Y: state_a,A: produc3409137331138395373tate_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8263595898873874535tate_a @ X @ Y ) )
=> ( P @ ( produc3154331710141225339tate_a @ A ) @ ( produc681690970763031737tate_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1043_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > current_a > $o,X: a,Y: current_a,A: produc7805042584321970905rent_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8503237746132909001rent_a @ X @ Y ) )
=> ( P @ ( produc4952273589686483381rent_a @ A ) @ ( produc4695312889421393143rent_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1044_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: a > state_a2 > $o,X: a,Y: state_a2,A: produc6972303929186420058tate_a] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8641956578966763338tate_a @ X @ Y ) )
=> ( P @ ( produc736293372669613878tate_a @ A ) @ ( produc7615498795807706488tate_a @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1045_snd__eqD,axiom,
! [X: a > a > $o,Y: list_a,A: list_a] :
( ( ( produc3976258275648695771list_a @ ( produc8111569692950616493list_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1046_snd__eqD,axiom,
! [X: list_a,Y: list_a,A: list_a] :
( ( ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1047_snd__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1048_snd__eqD,axiom,
! [X: a,Y: state_a,A: state_a] :
( ( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1049_snd__eqD,axiom,
! [X: a,Y: current_a,A: current_a] :
( ( ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1050_snd__eqD,axiom,
! [X: a,Y: state_a2,A: state_a2] :
( ( ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_1051_snd__conv,axiom,
! [X1: a > a > $o,X22: list_a] :
( ( produc3976258275648695771list_a @ ( produc8111569692950616493list_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1052_snd__conv,axiom,
! [X1: list_a,X22: list_a] :
( ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1053_snd__conv,axiom,
! [X1: a,X22: a] :
( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1054_snd__conv,axiom,
! [X1: a,X22: state_a] :
( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1055_snd__conv,axiom,
! [X1: a,X22: current_a] :
( ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1056_snd__conv,axiom,
! [X1: a,X22: state_a2] :
( ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_1057_fst__eqD,axiom,
! [X: a > a > $o,Y: list_a,A: a > a > $o] :
( ( ( produc6311120620833064345list_a @ ( produc8111569692950616493list_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1058_fst__eqD,axiom,
! [X: list_a,Y: list_a,A: list_a] :
( ( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1059_fst__eqD,axiom,
! [X: a,Y: a,A: a] :
( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1060_fst__eqD,axiom,
! [X: a,Y: state_a,A: a] :
( ( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1061_fst__eqD,axiom,
! [X: a,Y: current_a,A: a] :
( ( ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1062_fst__eqD,axiom,
! [X: a,Y: state_a2,A: a] :
( ( ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_1063_fst__conv,axiom,
! [X1: a > a > $o,X22: list_a] :
( ( produc6311120620833064345list_a @ ( produc8111569692950616493list_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1064_fst__conv,axiom,
! [X1: list_a,X22: list_a] :
( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1065_fst__conv,axiom,
! [X1: a,X22: a] :
( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1066_fst__conv,axiom,
! [X1: a,X22: state_a] :
( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1067_fst__conv,axiom,
! [X1: a,X22: current_a] :
( ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1068_fst__conv,axiom,
! [X1: a,X22: state_a2] :
( ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_1069_surjective__pairing,axiom,
! [T: produc5032551385658279741list_a] :
( T
= ( produc8111569692950616493list_a @ ( produc6311120620833064345list_a @ T ) @ ( produc3976258275648695771list_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1070_surjective__pairing,axiom,
! [T: produc9164743771328383783list_a] :
( T
= ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ T ) @ ( produc8617614985401127493list_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1071_surjective__pairing,axiom,
! [T: product_prod_a_a] :
( T
= ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1072_surjective__pairing,axiom,
! [T: produc3409137331138395373tate_a] :
( T
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ T ) @ ( produc681690970763031737tate_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1073_surjective__pairing,axiom,
! [T: produc7805042584321970905rent_a] :
( T
= ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ T ) @ ( produc4695312889421393143rent_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1074_surjective__pairing,axiom,
! [T: produc6972303929186420058tate_a] :
( T
= ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ T ) @ ( produc7615498795807706488tate_a @ T ) ) ) ).
% surjective_pairing
thf(fact_1075_prod_Oexhaust__sel,axiom,
! [Prod: produc5032551385658279741list_a] :
( Prod
= ( produc8111569692950616493list_a @ ( produc6311120620833064345list_a @ Prod ) @ ( produc3976258275648695771list_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1076_prod_Oexhaust__sel,axiom,
! [Prod: produc9164743771328383783list_a] :
( Prod
= ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ Prod ) @ ( produc8617614985401127493list_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1077_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_a_a] :
( Prod
= ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1078_prod_Oexhaust__sel,axiom,
! [Prod: produc3409137331138395373tate_a] :
( Prod
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1079_prod_Oexhaust__sel,axiom,
! [Prod: produc7805042584321970905rent_a] :
( Prod
= ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ Prod ) @ ( produc4695312889421393143rent_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1080_prod_Oexhaust__sel,axiom,
! [Prod: produc6972303929186420058tate_a] :
( Prod
= ( produc8641956578966763338tate_a @ ( produc736293372669613878tate_a @ Prod ) @ ( produc7615498795807706488tate_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_1081_prod_Oexpand,axiom,
! [Prod: produc7805042584321970905rent_a,Prod2: produc7805042584321970905rent_a] :
( ( ( ( produc4952273589686483381rent_a @ Prod )
= ( produc4952273589686483381rent_a @ Prod2 ) )
& ( ( produc4695312889421393143rent_a @ Prod )
= ( produc4695312889421393143rent_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_1082_prod_Oexpand,axiom,
! [Prod: produc6972303929186420058tate_a,Prod2: produc6972303929186420058tate_a] :
( ( ( ( produc736293372669613878tate_a @ Prod )
= ( produc736293372669613878tate_a @ Prod2 ) )
& ( ( produc7615498795807706488tate_a @ Prod )
= ( produc7615498795807706488tate_a @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_1083_prod__eqI,axiom,
! [P5: produc7805042584321970905rent_a,Q4: produc7805042584321970905rent_a] :
( ( ( produc4952273589686483381rent_a @ P5 )
= ( produc4952273589686483381rent_a @ Q4 ) )
=> ( ( ( produc4695312889421393143rent_a @ P5 )
= ( produc4695312889421393143rent_a @ Q4 ) )
=> ( P5 = Q4 ) ) ) ).
% prod_eqI
thf(fact_1084_prod__eqI,axiom,
! [P5: produc6972303929186420058tate_a,Q4: produc6972303929186420058tate_a] :
( ( ( produc736293372669613878tate_a @ P5 )
= ( produc736293372669613878tate_a @ Q4 ) )
=> ( ( ( produc7615498795807706488tate_a @ P5 )
= ( produc7615498795807706488tate_a @ Q4 ) )
=> ( P5 = Q4 ) ) ) ).
% prod_eqI
thf(fact_1085_prod__eq__iff,axiom,
( ( ^ [Y8: produc7805042584321970905rent_a,Z3: produc7805042584321970905rent_a] : ( Y8 = Z3 ) )
= ( ^ [S3: produc7805042584321970905rent_a,T2: produc7805042584321970905rent_a] :
( ( ( produc4952273589686483381rent_a @ S3 )
= ( produc4952273589686483381rent_a @ T2 ) )
& ( ( produc4695312889421393143rent_a @ S3 )
= ( produc4695312889421393143rent_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_1086_prod__eq__iff,axiom,
( ( ^ [Y8: produc6972303929186420058tate_a,Z3: produc6972303929186420058tate_a] : ( Y8 = Z3 ) )
= ( ^ [S3: produc6972303929186420058tate_a,T2: produc6972303929186420058tate_a] :
( ( ( produc736293372669613878tate_a @ S3 )
= ( produc736293372669613878tate_a @ T2 ) )
& ( ( produc7615498795807706488tate_a @ S3 )
= ( produc7615498795807706488tate_a @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_1087_invar__drop__first,axiom,
! [Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( type_i6141643110573041459rent_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) ) ) ) ).
% invar_drop_first
thf(fact_1088_size__drop__first__sub,axiom,
! [Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current2 ) )
=> ( ( type_i6141643110573041459rent_a @ Current2 )
=> ( ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) )
= ( minus_minus_nat @ ( size_size_current_a @ Current2 ) @ one_one_nat ) ) ) ) ).
% size_drop_first_sub
thf(fact_1089_list__Reverse,axiom,
! [Current2: current_list_a,Big: stack_list_a,Aux2: list_list_a,Count2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s7734494264869015971list_a @ ( reverse_list_a @ Current2 @ Big @ Aux2 @ Count2 ) ) )
=> ( ( type_i3182563720400918970list_a @ ( reverse_list_a @ Current2 @ Big @ Aux2 @ Count2 ) )
=> ( ( cons_list_a @ ( first_list_a @ Current2 ) @ ( big_list_list_a @ ( reverse_list_a @ ( produc8483681909585661047list_a @ ( pop_list_a3 @ Current2 ) ) @ Big @ Aux2 @ Count2 ) ) )
= ( big_list_list_a @ ( reverse_list_a @ Current2 @ Big @ Aux2 @ Count2 ) ) ) ) ) ).
% list_Reverse
thf(fact_1090_list__Reverse,axiom,
! [Current2: current_a,Big: stack_a,Aux2: list_a,Count2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( reverse_a @ Current2 @ Big @ Aux2 @ Count2 ) ) )
=> ( ( type_i6304938058965754292tate_a @ ( reverse_a @ Current2 @ Big @ Aux2 @ Count2 ) )
=> ( ( cons_a @ ( first_a @ Current2 ) @ ( big_list_a @ ( reverse_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) @ Big @ Aux2 @ Count2 ) ) )
= ( big_list_a @ ( reverse_a @ Current2 @ Big @ Aux2 @ Count2 ) ) ) ) ) ).
% list_Reverse
thf(fact_1091_size__prod__simp,axiom,
( basic_2002407943979918054rent_a
= ( ^ [F3: a > nat,G: current_a > nat,P7: produc7805042584321970905rent_a] : ( plus_plus_nat @ ( plus_plus_nat @ ( F3 @ ( produc4952273589686483381rent_a @ P7 ) ) @ ( G @ ( produc4695312889421393143rent_a @ P7 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% size_prod_simp
thf(fact_1092_size__prod__simp,axiom,
( basic_1521233549759257063tate_a
= ( ^ [F3: a > nat,G: state_a2 > nat,P7: produc6972303929186420058tate_a] : ( plus_plus_nat @ ( plus_plus_nat @ ( F3 @ ( produc736293372669613878tate_a @ P7 ) ) @ ( G @ ( produc7615498795807706488tate_a @ P7 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% size_prod_simp
thf(fact_1093_Current_Ofirst_Oelims,axiom,
! [X: current_a,Y: a] :
( ( ( first_a @ X )
= Y )
=> ( Y
= ( produc4952273589686483381rent_a @ ( pop_a3 @ X ) ) ) ) ).
% Current.first.elims
thf(fact_1094_Current_Ofirst_Osimps,axiom,
( first_a
= ( ^ [Current4: current_a] : ( produc4952273589686483381rent_a @ ( pop_a3 @ Current4 ) ) ) ) ).
% Current.first.simps
thf(fact_1095_Big_Opop_Osimps_I2_J,axiom,
! [Current2: current_a,Big: stack_a,Aux2: list_a,Count2: nat] :
( ( pop_a @ ( reverse_a @ Current2 @ Big @ Aux2 @ Count2 ) )
= ( produc8641956578966763338tate_a @ ( first_a @ Current2 ) @ ( reverse_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) @ Big @ Aux2 @ Count2 ) ) ) ).
% Big.pop.simps(2)
thf(fact_1096_exI__realizer,axiom,
! [P: list_a > ( a > a > $o ) > $o,Y: list_a,X: a > a > $o] :
( ( P @ Y @ X )
=> ( P @ ( produc3976258275648695771list_a @ ( produc8111569692950616493list_a @ X @ Y ) ) @ ( produc6311120620833064345list_a @ ( produc8111569692950616493list_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1097_exI__realizer,axiom,
! [P: list_a > list_a > $o,Y: list_a,X: list_a] :
( ( P @ Y @ X )
=> ( P @ ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ X @ Y ) ) @ ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1098_exI__realizer,axiom,
! [P: a > a > $o,Y: a,X: a] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1099_exI__realizer,axiom,
! [P: state_a > a > $o,Y: state_a,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) ) @ ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1100_exI__realizer,axiom,
! [P: current_a > a > $o,Y: current_a,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) ) @ ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1101_exI__realizer,axiom,
! [P: state_a2 > a > $o,Y: state_a2,X: a] :
( ( P @ Y @ X )
=> ( P @ ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) ) @ ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1102_conjI__realizer,axiom,
! [P: ( a > a > $o ) > $o,P5: a > a > $o,Q: list_a > $o,Q4: list_a] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( produc6311120620833064345list_a @ ( produc8111569692950616493list_a @ P5 @ Q4 ) ) )
& ( Q @ ( produc3976258275648695771list_a @ ( produc8111569692950616493list_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1103_conjI__realizer,axiom,
! [P: list_a > $o,P5: list_a,Q: list_a > $o,Q4: list_a] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ P5 @ Q4 ) ) )
& ( Q @ ( produc8617614985401127493list_a @ ( produc6837034575241423639list_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1104_conjI__realizer,axiom,
! [P: a > $o,P5: a,Q: a > $o,Q4: a] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( product_fst_a_a @ ( product_Pair_a_a @ P5 @ Q4 ) ) )
& ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1105_conjI__realizer,axiom,
! [P: a > $o,P5: a,Q: state_a > $o,Q4: state_a] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ P5 @ Q4 ) ) )
& ( Q @ ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1106_conjI__realizer,axiom,
! [P: a > $o,P5: a,Q: current_a > $o,Q4: current_a] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( produc4952273589686483381rent_a @ ( produc8503237746132909001rent_a @ P5 @ Q4 ) ) )
& ( Q @ ( produc4695312889421393143rent_a @ ( produc8503237746132909001rent_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1107_conjI__realizer,axiom,
! [P: a > $o,P5: a,Q: state_a2 > $o,Q4: state_a2] :
( ( P @ P5 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( produc736293372669613878tate_a @ ( produc8641956578966763338tate_a @ P5 @ Q4 ) ) )
& ( Q @ ( produc7615498795807706488tate_a @ ( produc8641956578966763338tate_a @ P5 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1108_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_list_a
= ( ^ [F3: list_a > nat,Xs4: list_list_a] : ( if_nat @ ( Xs4 = nil_list_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F3 @ ( hd_list_a @ Xs4 ) ) @ ( size_list_list_a @ F3 @ ( tl_list_a @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1109_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F3: a > nat,Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F3 @ ( hd_a @ Xs4 ) ) @ ( size_list_a @ F3 @ ( tl_a @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1110_Current_Opop_Opelims,axiom,
! [X: current_list_a,Y: produc603899091129270873list_a] :
( ( ( pop_list_a3 @ X )
= Y )
=> ( ( accp_current_list_a @ pop_rel_list_a @ X )
=> ( ! [Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ nil_list_a @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc1356073977093405001list_a @ ( first_list_a2 @ Old ) @ ( current_list_a2 @ nil_list_a @ Added @ ( pop_list_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) )
=> ~ ( accp_current_list_a @ pop_rel_list_a @ ( current_list_a2 @ nil_list_a @ Added @ Old @ Remained ) ) ) )
=> ~ ! [X2: list_a,Xs2: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ ( cons_list_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc1356073977093405001list_a @ X2 @ ( current_list_a2 @ Xs2 @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) )
=> ~ ( accp_current_list_a @ pop_rel_list_a @ ( current_list_a2 @ ( cons_list_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_1111_Current_Opop_Opelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a3 @ X )
= Y )
=> ( ( accp_current_a @ pop_rel_a @ X )
=> ( ! [Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ nil_a @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc8503237746132909001rent_a @ ( first_a2 @ Old ) @ ( current_a2 @ nil_a @ Added @ ( pop_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) )
=> ~ ( accp_current_a @ pop_rel_a @ ( current_a2 @ nil_a @ Added @ Old @ Remained ) ) ) )
=> ~ ! [X2: a,Xs2: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc8503237746132909001rent_a @ X2 @ ( current_a2 @ Xs2 @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) )
=> ~ ( accp_current_a @ pop_rel_a @ ( current_a2 @ ( cons_a @ X2 @ Xs2 ) @ Added @ Old @ Remained ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_1112_size__list__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( size_list_a @ F @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1113_list_Osize__gen_I1_J,axiom,
! [X: list_a > nat] :
( ( size_list_list_a @ X @ nil_list_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1114_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1115_Current__Aux_Olist_Opelims,axiom,
! [X: current_a,Y: list_a] :
( ( ( current_list_a3 @ X )
= Y )
=> ( ( accp_current_a @ current_list_rel_a @ X )
=> ~ ! [Extra: list_a,Uu: nat,Old: stack_a,Uv: nat] :
( ( X
= ( current_a2 @ Extra @ Uu @ Old @ Uv ) )
=> ( ( Y
= ( append_a @ Extra @ ( stack_list_a2 @ Old ) ) )
=> ~ ( accp_current_a @ current_list_rel_a @ ( current_a2 @ Extra @ Uu @ Old @ Uv ) ) ) ) ) ) ).
% Current_Aux.list.pelims
thf(fact_1116_size__current_Opelims,axiom,
! [X: current_a,Y: nat] :
( ( ( size_size_current_a @ X )
= Y )
=> ( ( accp_current_a @ curren1432154427392394533_rel_a @ X )
=> ~ ! [Uu: list_a,Added: nat,Old: stack_a,Uv: nat] :
( ( X
= ( current_a2 @ Uu @ Added @ Old @ Uv ) )
=> ( ( Y
= ( plus_plus_nat @ Added @ ( size_size_stack_a @ Old ) ) )
=> ~ ( accp_current_a @ curren1432154427392394533_rel_a @ ( current_a2 @ Uu @ Added @ Old @ Uv ) ) ) ) ) ) ).
% size_current.pelims
thf(fact_1117_Current_Ofirst_Opelims,axiom,
! [X: current_a,Y: a] :
( ( ( first_a @ X )
= Y )
=> ( ( accp_current_a @ first_rel_a @ X )
=> ~ ( ( Y
= ( produc4952273589686483381rent_a @ ( pop_a3 @ X ) ) )
=> ~ ( accp_current_a @ first_rel_a @ X ) ) ) ) ).
% Current.first.pelims
thf(fact_1118_size__new__current_Opelims,axiom,
! [X: current_a,Y: nat] :
( ( ( type_s933026853152659577rent_a @ X )
= Y )
=> ( ( accp_current_a @ curren2620163519437455445_rel_a @ X )
=> ~ ! [Uu: list_a,Added: nat,Uv: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ Uu @ Added @ Uv @ Remained ) )
=> ( ( Y
= ( plus_plus_nat @ Added @ Remained ) )
=> ~ ( accp_current_a @ curren2620163519437455445_rel_a @ ( current_a2 @ Uu @ Added @ Uv @ Remained ) ) ) ) ) ) ).
% size_new_current.pelims
thf(fact_1119_invar__current_Opelims_I3_J,axiom,
! [X: current_a] :
( ~ ( type_i6141643110573041459rent_a @ X )
=> ( ( accp_current_a @ curren2598401564740717938_rel_a @ X )
=> ~ ! [Extra: list_a,Added: nat,Uu: stack_a,Uv: nat] :
( ( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( accp_current_a @ curren2598401564740717938_rel_a @ ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_size_list_a @ Extra )
= Added ) ) ) ) ) ).
% invar_current.pelims(3)
thf(fact_1120_invar__current_Opelims_I2_J,axiom,
! [X: current_a] :
( ( type_i6141643110573041459rent_a @ X )
=> ( ( accp_current_a @ curren2598401564740717938_rel_a @ X )
=> ~ ! [Extra: list_a,Added: nat,Uu: stack_a,Uv: nat] :
( ( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( accp_current_a @ curren2598401564740717938_rel_a @ ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_size_list_a @ Extra )
!= Added ) ) ) ) ) ).
% invar_current.pelims(2)
thf(fact_1121_invar__current_Opelims_I1_J,axiom,
! [X: current_a,Y: $o] :
( ( ( type_i6141643110573041459rent_a @ X )
= Y )
=> ( ( accp_current_a @ curren2598401564740717938_rel_a @ X )
=> ~ ! [Extra: list_a,Added: nat,Uu: stack_a,Uv: nat] :
( ( X
= ( current_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( Y
= ( ( size_size_list_a @ Extra )
= Added ) )
=> ~ ( accp_current_a @ curren2598401564740717938_rel_a @ ( current_a2 @ Extra @ Added @ Uu @ Uv ) ) ) ) ) ) ).
% invar_current.pelims(1)
thf(fact_1122_Big_Ostep__state_Oelims,axiom,
! [X: state_list_a2,Y: state_list_a2] :
( ( ( type_s8055303106637973038list_a @ X )
= Y )
=> ( ! [State2: state_list_a] :
( ( X
= ( common_list_a @ State2 ) )
=> ( Y
!= ( common_list_a @ ( type_s3510896519899625359list_a @ State2 ) ) ) )
=> ( ! [Current: current_list_a,Uu: stack_list_a,Aux: list_list_a] :
( ( X
= ( reverse_list_a @ Current @ Uu @ Aux @ zero_zero_nat ) )
=> ( Y
!= ( common_list_a @ ( normalize_list_a @ ( copy_list_a @ Current @ Aux @ nil_list_a @ zero_zero_nat ) ) ) ) )
=> ~ ! [Current: current_list_a,Big3: stack_list_a,Aux: list_list_a,V: nat] :
( ( X
= ( reverse_list_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) )
=> ( Y
!= ( reverse_list_a @ Current @ ( pop_list_a4 @ Big3 ) @ ( cons_list_a @ ( first_list_a2 @ Big3 ) @ Aux ) @ ( minus_minus_nat @ ( suc @ V ) @ one_one_nat ) ) ) ) ) ) ) ).
% Big.step_state.elims
thf(fact_1123_Big_Ostep__state_Oelims,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( type_s3593206172722485288tate_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( common_a @ ( type_s889635741254954505tate_a @ State2 ) ) ) )
=> ( ! [Current: current_a,Uu: stack_a,Aux: list_a] :
( ( X
= ( reverse_a @ Current @ Uu @ Aux @ zero_zero_nat ) )
=> ( Y
!= ( common_a @ ( normalize_a @ ( copy_a @ Current @ Aux @ nil_a @ zero_zero_nat ) ) ) ) )
=> ~ ! [Current: current_a,Big3: stack_a,Aux: list_a,V: nat] :
( ( X
= ( reverse_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) )
=> ( Y
!= ( reverse_a @ Current @ ( pop_a4 @ Big3 ) @ ( cons_a @ ( first_a2 @ Big3 ) @ Aux ) @ ( minus_minus_nat @ ( suc @ V ) @ one_one_nat ) ) ) ) ) ) ) ).
% Big.step_state.elims
thf(fact_1124_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_list_a,A: list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( list_update_list_a @ Xs @ I @ A )
= ( append_list_a @ ( take_list_a @ I @ Xs ) @ ( cons_list_a @ A @ ( drop_list_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1125_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1126_list__update__overwrite,axiom,
! [Xs: list_a,I: nat,X: a,Y: a] :
( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I @ Y )
= ( list_update_a @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_1127_list__update__nonempty,axiom,
! [Xs: list_list_a,K: nat,X: list_a] :
( ( ( list_update_list_a @ Xs @ K @ X )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% list_update_nonempty
thf(fact_1128_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_1129_length__list__update,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_1130_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1131_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1132_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1133_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_a,Y: a] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_a @ N @ ( list_update_a @ Xs @ M @ Y ) )
= ( take_a @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1134_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_a @ M @ ( list_update_a @ Xs @ N @ X ) )
= ( drop_a @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_1135_list__update__length,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( list_update_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs ) @ Y )
= ( append_list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1136_list__update__length,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1137_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1138_take__update__swap,axiom,
! [M: nat,Xs: list_a,N: nat,X: a] :
( ( take_a @ M @ ( list_update_a @ Xs @ N @ X ) )
= ( list_update_a @ ( take_a @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1139_list__update__code_I2_J,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a] :
( ( list_update_list_a @ ( cons_list_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_list_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1140_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_1141_list__update__code_I3_J,axiom,
! [X: list_a,Xs: list_list_a,I: nat,Y: list_a] :
( ( list_update_list_a @ ( cons_list_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_list_a @ X @ ( list_update_list_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1142_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_1143_list__update__swap,axiom,
! [I: nat,I5: nat,Xs: list_a,X: a,X8: a] :
( ( I != I5 )
=> ( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I5 @ X8 )
= ( list_update_a @ ( list_update_a @ Xs @ I5 @ X8 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_1144_list__update_Osimps_I1_J,axiom,
! [I: nat,V2: list_a] :
( ( list_update_list_a @ nil_list_a @ I @ V2 )
= nil_list_a ) ).
% list_update.simps(1)
thf(fact_1145_list__update_Osimps_I1_J,axiom,
! [I: nat,V2: a] :
( ( list_update_a @ nil_a @ I @ V2 )
= nil_a ) ).
% list_update.simps(1)
thf(fact_1146_list__update__code_I1_J,axiom,
! [I: nat,Y: list_a] :
( ( list_update_list_a @ nil_list_a @ I @ Y )
= nil_list_a ) ).
% list_update_code(1)
thf(fact_1147_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_1148_Common__Aux_Olist__current_Osimps_I2_J,axiom,
! [Current2: current_a,Uv2: list_a,Uw2: list_a,Ux2: nat] :
( ( common1102728217005306191rent_a @ ( copy_a @ Current2 @ Uv2 @ Uw2 @ Ux2 ) )
= ( current_list_a3 @ Current2 ) ) ).
% Common_Aux.list_current.simps(2)
thf(fact_1149_list__update__append1,axiom,
! [I: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I @ X )
= ( append_a @ ( list_update_a @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1150_Common__Aux_Oremaining__steps__state_Osimps_I2_J,axiom,
! [Uw2: list_a,Ux2: nat,Uy2: stack_a,Remained2: nat,Aux2: list_a,New: list_a,Moved: nat] :
( ( type_r2212416260012024137tate_a @ ( copy_a @ ( current_a2 @ Uw2 @ Ux2 @ Uy2 @ Remained2 ) @ Aux2 @ New @ Moved ) )
= ( minus_minus_nat @ Remained2 @ Moved ) ) ).
% Common_Aux.remaining_steps_state.simps(2)
thf(fact_1151_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1152_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X )
= Xs )
= ( ( nth_a @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1153_Big_Ostep__state_Osimps_I2_J,axiom,
! [Current2: current_list_a,Uu2: stack_list_a,Aux2: list_list_a] :
( ( type_s8055303106637973038list_a @ ( reverse_list_a @ Current2 @ Uu2 @ Aux2 @ zero_zero_nat ) )
= ( common_list_a @ ( normalize_list_a @ ( copy_list_a @ Current2 @ Aux2 @ nil_list_a @ zero_zero_nat ) ) ) ) ).
% Big.step_state.simps(2)
thf(fact_1154_Big_Ostep__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu2: stack_a,Aux2: list_a] :
( ( type_s3593206172722485288tate_a @ ( reverse_a @ Current2 @ Uu2 @ Aux2 @ zero_zero_nat ) )
= ( common_a @ ( normalize_a @ ( copy_a @ Current2 @ Aux2 @ nil_a @ zero_zero_nat ) ) ) ) ).
% Big.step_state.simps(2)
thf(fact_1155_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_a,X: a] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_a @ M @ ( list_update_a @ Xs @ N @ X ) )
= ( list_update_a @ ( drop_a @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1156_list__update__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X )
= ( append_a @ ( list_update_a @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1157_Common__Aux_Olist_Osimps_I2_J,axiom,
! [Extra2: list_a,Uv2: nat,Uw2: stack_a,Remained2: nat,Old2: list_a,New: list_a,Moved: nat] :
( ( common_list_a2 @ ( copy_a @ ( current_a2 @ Extra2 @ Uv2 @ Uw2 @ Remained2 ) @ Old2 @ New @ Moved ) )
= ( append_a @ Extra2 @ ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ Remained2 @ Moved ) @ Old2 ) @ New ) ) ) ).
% Common_Aux.list.simps(2)
thf(fact_1158_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_1159_last__list__update,axiom,
! [Xs: list_list_a,K: nat,X: list_a] :
( ( Xs != nil_list_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_list_a @ ( list_update_list_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s349497388124573686list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_list_a @ ( list_update_list_a @ Xs @ K @ X ) )
= ( last_list_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1160_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_1161_Common_Opop_Osimps_I2_J,axiom,
! [Current2: current_a,Aux2: list_a,New: list_a,Moved: nat] :
( ( pop_a2 @ ( copy_a @ Current2 @ Aux2 @ New @ Moved ) )
= ( produc8263595898873874535tate_a @ ( first_a @ Current2 ) @ ( normalize_a @ ( copy_a @ ( produc4695312889421393143rent_a @ ( pop_a3 @ Current2 ) ) @ Aux2 @ New @ Moved ) ) ) ) ).
% Common.pop.simps(2)
thf(fact_1162_Big_Ostep__state_Opelims,axiom,
! [X: state_list_a2,Y: state_list_a2] :
( ( ( type_s8055303106637973038list_a @ X )
= Y )
=> ( ( accp_state_list_a @ step_s2698489867611947387list_a @ X )
=> ( ! [State2: state_list_a] :
( ( X
= ( common_list_a @ State2 ) )
=> ( ( Y
= ( common_list_a @ ( type_s3510896519899625359list_a @ State2 ) ) )
=> ~ ( accp_state_list_a @ step_s2698489867611947387list_a @ ( common_list_a @ State2 ) ) ) )
=> ( ! [Current: current_list_a,Uu: stack_list_a,Aux: list_list_a] :
( ( X
= ( reverse_list_a @ Current @ Uu @ Aux @ zero_zero_nat ) )
=> ( ( Y
= ( common_list_a @ ( normalize_list_a @ ( copy_list_a @ Current @ Aux @ nil_list_a @ zero_zero_nat ) ) ) )
=> ~ ( accp_state_list_a @ step_s2698489867611947387list_a @ ( reverse_list_a @ Current @ Uu @ Aux @ zero_zero_nat ) ) ) )
=> ~ ! [Current: current_list_a,Big3: stack_list_a,Aux: list_list_a,V: nat] :
( ( X
= ( reverse_list_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) )
=> ( ( Y
= ( reverse_list_a @ Current @ ( pop_list_a4 @ Big3 ) @ ( cons_list_a @ ( first_list_a2 @ Big3 ) @ Aux ) @ ( minus_minus_nat @ ( suc @ V ) @ one_one_nat ) ) )
=> ~ ( accp_state_list_a @ step_s2698489867611947387list_a @ ( reverse_list_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) ) ) ) ) ) ) ) ).
% Big.step_state.pelims
thf(fact_1163_Big_Ostep__state_Opelims,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( type_s3593206172722485288tate_a @ X )
= Y )
=> ( ( accp_state_a @ step_state_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( common_a @ ( type_s889635741254954505tate_a @ State2 ) ) )
=> ~ ( accp_state_a @ step_state_rel_a @ ( common_a @ State2 ) ) ) )
=> ( ! [Current: current_a,Uu: stack_a,Aux: list_a] :
( ( X
= ( reverse_a @ Current @ Uu @ Aux @ zero_zero_nat ) )
=> ( ( Y
= ( common_a @ ( normalize_a @ ( copy_a @ Current @ Aux @ nil_a @ zero_zero_nat ) ) ) )
=> ~ ( accp_state_a @ step_state_rel_a @ ( reverse_a @ Current @ Uu @ Aux @ zero_zero_nat ) ) ) )
=> ~ ! [Current: current_a,Big3: stack_a,Aux: list_a,V: nat] :
( ( X
= ( reverse_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) )
=> ( ( Y
= ( reverse_a @ Current @ ( pop_a4 @ Big3 ) @ ( cons_a @ ( first_a2 @ Big3 ) @ Aux ) @ ( minus_minus_nat @ ( suc @ V ) @ one_one_nat ) ) )
=> ~ ( accp_state_a @ step_state_rel_a @ ( reverse_a @ Current @ Big3 @ Aux @ ( suc @ V ) ) ) ) ) ) ) ) ) ).
% Big.step_state.pelims
thf(fact_1164_Big__Aux_Oremaining__steps__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_r2494999336194962664tate_a @ X )
= Y )
=> ( ( accp_state_a @ big_re1607094904563243348_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( type_r2212416260012024137tate_a @ State2 ) )
=> ~ ( accp_state_a @ big_re1607094904563243348_rel_a @ ( common_a @ State2 ) ) ) )
=> ~ ! [Uu: list_a,Uv: nat,Uw: stack_a,Remaining: nat,Ux: stack_a,Uy: list_a,Count: nat] :
( ( X
= ( reverse_a @ ( current_a2 @ Uu @ Uv @ Uw @ Remaining ) @ Ux @ Uy @ Count ) )
=> ( ( Y
= ( plus_plus_nat @ ( plus_plus_nat @ Count @ Remaining ) @ one_one_nat ) )
=> ~ ( accp_state_a @ big_re1607094904563243348_rel_a @ ( reverse_a @ ( current_a2 @ Uu @ Uv @ Uw @ Remaining ) @ Ux @ Uy @ Count ) ) ) ) ) ) ) ).
% Big_Aux.remaining_steps_state.pelims
thf(fact_1165_Big__Aux_Osize__new__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( type_s6530235180886170618tate_a @ X )
= Y )
=> ( ( accp_state_a @ big_si5937185285519891526_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( type_s8424385952999958455tate_a @ State2 ) )
=> ~ ( accp_state_a @ big_si5937185285519891526_rel_a @ ( common_a @ State2 ) ) ) )
=> ~ ! [Current: current_a,Uu: stack_a,Uv: list_a,Uw: nat] :
( ( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current ) )
=> ~ ( accp_state_a @ big_si5937185285519891526_rel_a @ ( reverse_a @ Current @ Uu @ Uv @ Uw ) ) ) ) ) ) ) ).
% Big_Aux.size_new_state.pelims
thf(fact_1166_Big__Aux_Olist__current_Opelims,axiom,
! [X: state_a2,Y: list_a] :
( ( ( big_list_current_a @ X )
= Y )
=> ( ( accp_state_a @ big_li383503880598112847_rel_a @ X )
=> ( ! [Common3: state_a] :
( ( X
= ( common_a @ Common3 ) )
=> ( ( Y
= ( common1102728217005306191rent_a @ Common3 ) )
=> ~ ( accp_state_a @ big_li383503880598112847_rel_a @ ( common_a @ Common3 ) ) ) )
=> ~ ! [Current: current_a,Uu: stack_a,Uv: list_a,Uw: nat] :
( ( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( current_list_a3 @ Current ) )
=> ~ ( accp_state_a @ big_li383503880598112847_rel_a @ ( reverse_a @ Current @ Uu @ Uv @ Uw ) ) ) ) ) ) ) ).
% Big_Aux.list_current.pelims
thf(fact_1167_eq__snd__iff,axiom,
! [B: list_a,P5: produc5032551385658279741list_a] :
( ( B
= ( produc3976258275648695771list_a @ P5 ) )
= ( ? [A5: a > a > $o] :
( P5
= ( produc8111569692950616493list_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1168_eq__snd__iff,axiom,
! [B: list_a,P5: produc9164743771328383783list_a] :
( ( B
= ( produc8617614985401127493list_a @ P5 ) )
= ( ? [A5: list_a] :
( P5
= ( produc6837034575241423639list_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1169_eq__snd__iff,axiom,
! [B: a,P5: product_prod_a_a] :
( ( B
= ( product_snd_a_a @ P5 ) )
= ( ? [A5: a] :
( P5
= ( product_Pair_a_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1170_eq__snd__iff,axiom,
! [B: state_a,P5: produc3409137331138395373tate_a] :
( ( B
= ( produc681690970763031737tate_a @ P5 ) )
= ( ? [A5: a] :
( P5
= ( produc8263595898873874535tate_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1171_eq__snd__iff,axiom,
! [B: current_a,P5: produc7805042584321970905rent_a] :
( ( B
= ( produc4695312889421393143rent_a @ P5 ) )
= ( ? [A5: a] :
( P5
= ( produc8503237746132909001rent_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1172_eq__snd__iff,axiom,
! [B: state_a2,P5: produc6972303929186420058tate_a] :
( ( B
= ( produc7615498795807706488tate_a @ P5 ) )
= ( ? [A5: a] :
( P5
= ( produc8641956578966763338tate_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_1173_eq__fst__iff,axiom,
! [A: a > a > $o,P5: produc5032551385658279741list_a] :
( ( A
= ( produc6311120620833064345list_a @ P5 ) )
= ( ? [B4: list_a] :
( P5
= ( produc8111569692950616493list_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1174_eq__fst__iff,axiom,
! [A: list_a,P5: produc9164743771328383783list_a] :
( ( A
= ( produc3698117735987127555list_a @ P5 ) )
= ( ? [B4: list_a] :
( P5
= ( produc6837034575241423639list_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1175_eq__fst__iff,axiom,
! [A: a,P5: product_prod_a_a] :
( ( A
= ( product_fst_a_a @ P5 ) )
= ( ? [B4: a] :
( P5
= ( product_Pair_a_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1176_eq__fst__iff,axiom,
! [A: a,P5: produc3409137331138395373tate_a] :
( ( A
= ( produc3154331710141225339tate_a @ P5 ) )
= ( ? [B4: state_a] :
( P5
= ( produc8263595898873874535tate_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1177_eq__fst__iff,axiom,
! [A: a,P5: produc7805042584321970905rent_a] :
( ( A
= ( produc4952273589686483381rent_a @ P5 ) )
= ( ? [B4: current_a] :
( P5
= ( produc8503237746132909001rent_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1178_eq__fst__iff,axiom,
! [A: a,P5: produc6972303929186420058tate_a] :
( ( A
= ( produc736293372669613878tate_a @ P5 ) )
= ( ? [B4: state_a2] :
( P5
= ( produc8641956578966763338tate_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_1179_sndI,axiom,
! [X: produc5032551385658279741list_a,Y: a > a > $o,Z2: list_a] :
( ( X
= ( produc8111569692950616493list_a @ Y @ Z2 ) )
=> ( ( produc3976258275648695771list_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1180_sndI,axiom,
! [X: produc9164743771328383783list_a,Y: list_a,Z2: list_a] :
( ( X
= ( produc6837034575241423639list_a @ Y @ Z2 ) )
=> ( ( produc8617614985401127493list_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1181_sndI,axiom,
! [X: product_prod_a_a,Y: a,Z2: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z2 ) )
=> ( ( product_snd_a_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1182_sndI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z2: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z2 ) )
=> ( ( produc681690970763031737tate_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1183_sndI,axiom,
! [X: produc7805042584321970905rent_a,Y: a,Z2: current_a] :
( ( X
= ( produc8503237746132909001rent_a @ Y @ Z2 ) )
=> ( ( produc4695312889421393143rent_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1184_sndI,axiom,
! [X: produc6972303929186420058tate_a,Y: a,Z2: state_a2] :
( ( X
= ( produc8641956578966763338tate_a @ Y @ Z2 ) )
=> ( ( produc7615498795807706488tate_a @ X )
= Z2 ) ) ).
% sndI
thf(fact_1185_fstI,axiom,
! [X: produc5032551385658279741list_a,Y: a > a > $o,Z2: list_a] :
( ( X
= ( produc8111569692950616493list_a @ Y @ Z2 ) )
=> ( ( produc6311120620833064345list_a @ X )
= Y ) ) ).
% fstI
thf(fact_1186_fstI,axiom,
! [X: produc9164743771328383783list_a,Y: list_a,Z2: list_a] :
( ( X
= ( produc6837034575241423639list_a @ Y @ Z2 ) )
=> ( ( produc3698117735987127555list_a @ X )
= Y ) ) ).
% fstI
thf(fact_1187_fstI,axiom,
! [X: product_prod_a_a,Y: a,Z2: a] :
( ( X
= ( product_Pair_a_a @ Y @ Z2 ) )
=> ( ( product_fst_a_a @ X )
= Y ) ) ).
% fstI
thf(fact_1188_fstI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z2: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z2 ) )
=> ( ( produc3154331710141225339tate_a @ X )
= Y ) ) ).
% fstI
thf(fact_1189_fstI,axiom,
! [X: produc7805042584321970905rent_a,Y: a,Z2: current_a] :
( ( X
= ( produc8503237746132909001rent_a @ Y @ Z2 ) )
=> ( ( produc4952273589686483381rent_a @ X )
= Y ) ) ).
% fstI
thf(fact_1190_fstI,axiom,
! [X: produc6972303929186420058tate_a,Y: a,Z2: state_a2] :
( ( X
= ( produc8641956578966763338tate_a @ Y @ Z2 ) )
=> ( ( produc736293372669613878tate_a @ X )
= Y ) ) ).
% fstI
thf(fact_1191_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1192_listrel1__iff__update,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
= ( ? [Y7: list_a,N2: nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N2 ) @ Y7 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( Ys
= ( list_update_list_a @ Xs @ N2 @ Y7 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1193_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
= ( ? [Y7: a,N2: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ Y7 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N2 @ Y7 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1194_Cons__listrel1__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1195_Cons__listrel1__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1196_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_1197_listrel1__mono,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R @ S )
=> ( ord_le7857023143581076903list_a @ ( listrel1_a @ R ) @ ( listrel1_a @ S ) ) ) ).
% listrel1_mono
thf(fact_1198_append__listrel1I,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Vs ) @ ( listrel1_a @ R ) ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( listrel1_a @ R ) ) ) ).
% append_listrel1I
thf(fact_1199_not__listrel1__Nil,axiom,
! [Xs: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ nil_list_a ) @ ( listrel1_list_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_1200_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_1201_not__Nil__listrel1,axiom,
! [Xs: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Xs ) @ ( listrel1_list_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_1202_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_1203_listrel1I2,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a,X: list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ X @ Ys ) ) @ ( listrel1_list_a @ R ) ) ) ).
% listrel1I2
thf(fact_1204_listrel1I2,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I2
thf(fact_1205_Cons__listrel1E2,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
=> ( ! [X2: list_a] :
( ( Xs
= ( cons_list_a @ X2 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Y @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs2 @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1206_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
=> ( ! [X2: a] :
( ( Xs
= ( cons_a @ X2 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1207_Cons__listrel1E1,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_a @ R ) )
=> ( ! [Y5: list_a] :
( ( Ys
= ( cons_list_a @ Y5 @ Xs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y5 ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1208_Cons__listrel1E1,axiom,
! [X: a,Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys ) @ ( listrel1_a @ R ) )
=> ( ! [Y5: a] :
( ( Ys
= ( cons_a @ Y5 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1209_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_1210_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_1211_listrel1I,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1212_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ~ ! [X2: a,Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y5 ) @ R )
=> ! [Us3: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us3 @ ( cons_a @ X2 @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us3 @ ( cons_a @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1213_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1214_Big__Aux_Osize__state_Oelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( Y
!= ( size_size_state_a2 @ State2 ) ) )
=> ~ ! [Current: current_a] :
( ? [Uu: stack_a,Uv: list_a,Uw: nat] :
( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ) ) ) ).
% Big_Aux.size_state.elims
thf(fact_1215_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1216_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1217_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1218_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_1219_take__take,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( take_a @ N @ ( take_a @ M @ Xs ) )
= ( take_a @ ( ord_min_nat @ N @ M ) @ Xs ) ) ).
% take_take
thf(fact_1220_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_1221_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_1222_length__take,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( take_a @ N @ Xs ) )
= ( ord_min_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_take
thf(fact_1223_Suc__min,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( suc @ ( ord_min_nat @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( minus_minus_nat @ Y @ ( suc @ zero_zero_nat ) ) ) )
= ( ord_min_nat @ X @ Y ) ) ) ) ).
% Suc_min
thf(fact_1224_min__add__distrib__right,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z2 ) )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% min_add_distrib_right
thf(fact_1225_min__add__distrib__left,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ord_min_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).
% min_add_distrib_left
thf(fact_1226_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1227_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_1228_min__def,axiom,
( ord_min_nat
= ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ A5 @ B4 ) ) ) ).
% min_def
thf(fact_1229_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_1230_Big__Aux_Osize__state_Osimps_I2_J,axiom,
! [Current2: current_a,Uu2: stack_a,Uv2: list_a,Uw2: nat] :
( ( size_size_state_a @ ( reverse_a @ Current2 @ Uu2 @ Uv2 @ Uw2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ).
% Big_Aux.size_state.simps(2)
thf(fact_1231_Big__Aux_Osize__state_Opelims,axiom,
! [X: state_a2,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ( accp_state_a @ big_size_state_rel_a @ X )
=> ( ! [State2: state_a] :
( ( X
= ( common_a @ State2 ) )
=> ( ( Y
= ( size_size_state_a2 @ State2 ) )
=> ~ ( accp_state_a @ big_size_state_rel_a @ ( common_a @ State2 ) ) ) )
=> ~ ! [Current: current_a,Uu: stack_a,Uv: list_a,Uw: nat] :
( ( X
= ( reverse_a @ Current @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) )
=> ~ ( accp_state_a @ big_size_state_rel_a @ ( reverse_a @ Current @ Uu @ Uv @ Uw ) ) ) ) ) ) ) ).
% Big_Aux.size_state.pelims
thf(fact_1232_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_1233_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_1234_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_1235_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_1236_min__less__iff__conj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z2 @ X )
& ( ord_less_nat @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_1237_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_1238_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
| ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_1239_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_1240_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_1241_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_min_nat @ A5 @ B4 )
= B4 ) ) ) ).
% min.absorb_iff2
thf(fact_1242_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_min_nat @ A5 @ B4 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_1243_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_1244_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_1245_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( ord_min_nat @ A5 @ B4 ) ) ) ) ).
% min.order_iff
thf(fact_1246_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_1247_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_1248_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_1249_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_1250_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_1251_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_1252_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_1253_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1254_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_1255_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
| ( ord_less_nat @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_1256_lexord__take__index__conv,axiom,
! [X: list_a,Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) )
& ( ( take_a @ ( size_size_list_a @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) ) )
& ( ( take_a @ I4 @ X )
= ( take_a @ I4 @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ X @ I4 ) @ ( nth_a @ Y @ I4 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_1257_lexord__cons__cons,axiom,
! [A: a,X: list_a,B: a,Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
| ( ( A = B )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1258_lexord__Nil__left,axiom,
! [Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R ) )
= ( ? [A5: a,X3: list_a] :
( Y
= ( cons_a @ A5 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_1259_lexord__Nil__right,axiom,
! [X: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ nil_a ) @ ( lexord_a @ R ) ) ).
% lexord_Nil_right
thf(fact_1260_lexord__append__leftI,axiom,
! [U: list_a,V2: list_a,R: set_Product_prod_a_a,X: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ V2 ) @ ( lexord_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ X @ U ) @ ( append_a @ X @ V2 ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_leftI
% 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,
type_i6304938058965754292tate_a @ big ).
%------------------------------------------------------------------------------