TPTP Problem File: SLH0940^1.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/0022_Common_Proof/prob_00329_010342__6715590_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1650 ( 615 unt; 389 typ; 0 def)
% Number of atoms : 3346 (2021 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11518 ( 739 ~; 89 |; 230 &;9052 @)
% ( 0 <=>;1408 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Number of types : 85 ( 84 usr)
% Number of type conns : 705 ( 705 >; 0 *; 0 +; 0 <<)
% Number of symbols : 308 ( 305 usr; 24 con; 0-4 aty)
% Number of variables : 3705 ( 68 ^;3429 !; 208 ?;3705 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:51:48.019
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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% Explicit typings (305)
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push_state_a2: state_a > current_state_a > current_state_a ).
thf(sy_c_Current_Opush_001t__List__Olist_Itf__a_J,type,
push_list_a2: list_a > current_list_a > current_list_a ).
thf(sy_c_Current_Opush_001tf__a,type,
push_a2: a > current_a > current_a ).
thf(sy_c_Current__Aux_Oinvar__current__rel_001t__Common__Ostate_Itf__a_J,type,
curren1694177707811796940tate_a: current_state_a > current_state_a > $o ).
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__Common__Ostate_Itf__a_J,type,
current_list_state_a: current_state_a > list_state_a ).
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_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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Idle_Oidle_OIdle_001tf__a,type,
idle_a3: stack_a > nat > idle_a ).
thf(sy_c_Idle_Opop_001t__Common__Ostate_Itf__a_J,type,
pop_state_a3: idle_state_a > produc1615251267796968561tate_a ).
thf(sy_c_Idle_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a3: idle_list_a > produc7860269014186282413list_a ).
thf(sy_c_Idle_Opop_001tf__a,type,
pop_a3: idle_a > produc7590564867095724333idle_a ).
thf(sy_c_Idle_Opop__rel_001tf__a,type,
pop_rel_a2: idle_a > idle_a > $o ).
thf(sy_c_Idle_Opush_001t__Common__Ostate_Itf__a_J,type,
push_state_a3: state_a > idle_state_a > idle_state_a ).
thf(sy_c_Idle_Opush_001t__List__Olist_Itf__a_J,type,
push_list_a3: list_a > idle_list_a > idle_list_a ).
thf(sy_c_Idle_Opush_001tf__a,type,
push_a3: a > idle_a > idle_a ).
thf(sy_c_Idle_Opush__rel_001tf__a,type,
push_rel_a2: produc7590564867095724333idle_a > produc7590564867095724333idle_a > $o ).
thf(sy_c_Idle__Aux_Oinvar__idle__rel_001tf__a,type,
idle_i6200314614184386870_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Ois__empty__idle__rel_001tf__a,type,
idle_i5927890838958520983_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Olist_001t__Common__Ostate_Itf__a_J,type,
idle_list_state_a: idle_state_a > list_state_a ).
thf(sy_c_Idle__Aux_Olist_001t__List__Olist_Itf__a_J,type,
idle_list_list_a: idle_list_a > list_list_a ).
thf(sy_c_Idle__Aux_Olist_001tf__a,type,
idle_list_a2: idle_a > list_a ).
thf(sy_c_Idle__Aux_Olist__rel_001tf__a,type,
idle_list_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Osize__idle__rel_001tf__a,type,
idle_size_idle_rel_a: idle_a > idle_a > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Common__Ostate_Itf__a_J,type,
append_state_a: list_state_a > list_state_a > list_state_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Odrop_001t__Common__Ostate_Itf__a_J,type,
drop_state_a: nat > list_state_a > list_state_a ).
thf(sy_c_List_Odrop_001t__List__Olist_Itf__a_J,type,
drop_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Ogen__length_001t__Common__Ostate_Itf__a_J,type,
gen_length_state_a: nat > list_state_a > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_Itf__a_J,type,
gen_length_list_a: nat > list_list_a > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001t__Common__Ostate_Itf__a_J,type,
insert_state_a: state_a > list_state_a > list_state_a ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olenlex_001t__Common__Ostate_Itf__a_J,type,
lenlex_state_a: set_Pr7022286013061942721tate_a > set_Pr936312788314626145tate_a ).
thf(sy_c_List_Olenlex_001t__List__Olist_Itf__a_J,type,
lenlex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__Common__Ostate_Itf__a_J,type,
lex_state_a: set_Pr7022286013061942721tate_a > set_Pr936312788314626145tate_a ).
thf(sy_c_List_Olex_001t__List__Olist_Itf__a_J,type,
lex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
cons_list_a_nat: ( list_a > nat ) > list_list_a_nat > list_list_a_nat ).
thf(sy_c_List_Olist_OCons_001_062_Itf__a_Mt__Nat__Onat_J,type,
cons_a_nat: ( a > nat ) > list_a_nat > list_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Common__Ostate_Itf__a_J,type,
cons_state_a: state_a > list_state_a > list_state_a ).
thf(sy_c_List_Olist_OCons_001t__Current__Ocurrent_Itf__a_J,type,
cons_current_a: current_a > list_current_a > list_current_a ).
thf(sy_c_List_Olist_OCons_001t__Idle__Oidle_Itf__a_J,type,
cons_idle_a: idle_a > list_idle_a > list_idle_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
cons_list_state_a: list_state_a > list_list_state_a > list_list_state_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
nil_list_a_nat: list_list_a_nat ).
thf(sy_c_List_Olist_ONil_001_062_Itf__a_Mt__Nat__Onat_J,type,
nil_a_nat: list_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Common__Ostate_Itf__a_J,type,
nil_state_a: list_state_a ).
thf(sy_c_List_Olist_ONil_001t__Current__Ocurrent_Itf__a_J,type,
nil_current_a: list_current_a ).
thf(sy_c_List_Olist_ONil_001t__Idle__Oidle_Itf__a_J,type,
nil_idle_a: list_idle_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
nil_list_state_a: list_list_state_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Common__Ostate_Itf__a_J,type,
hd_state_a: list_state_a > state_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Otl_001t__Common__Ostate_Itf__a_J,type,
tl_state_a: list_state_a > list_state_a ).
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_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrel_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
listrel_a_a_o_list_a: set_Pr7667830121170965149list_a > set_Pr5795302273130163997list_a ).
thf(sy_c_List_Olistrel_001t__Common__Ostate_Itf__a_J_001t__Common__Ostate_Itf__a_J,type,
listre8205798609706990835tate_a: set_Pr7022286013061942721tate_a > set_Pr936312788314626145tate_a ).
thf(sy_c_List_Olistrel_001t__Common__Ostate_Itf__a_J_001t__List__Olist_Itf__a_J,type,
listre1206529783178763665list_a: set_Pr7453869317534491247list_a > set_Pr2507300770693609317list_a ).
thf(sy_c_List_Olistrel_001t__Common__Ostate_Itf__a_J_001tf__a,type,
listrel_state_a_a: set_Pr1095656866282543721te_a_a > set_Pr4116713264744739807list_a ).
thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001t__Common__Ostate_Itf__a_J,type,
listre3686370835355000083tate_a: set_Pr2973361075998125097tate_a > set_Pr1116282471373962303tate_a ).
thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
listre6772471554020304241list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001tf__a,type,
listrel_list_a_a: set_Pr8962057229576493569st_a_a > set_Pr1060768173594829441list_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Common__Ostate_Itf__a_J,type,
listrel_a_state_a: set_Pr324718442235990179tate_a > set_Pr152097031616555065tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
listrel_a_current_a: set_Pr3922190185655878585rent_a > set_Pr7673030520428015033rent_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Idle__Oidle_Itf__a_J,type,
listrel_a_idle_a: set_Pr7225968907286947725idle_a > set_Pr8671745777326831373idle_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__List__Olist_Itf__a_J,type,
listrel_a_list_a: set_Pr2070066670564046349list_a > set_Pr4412185308373534093list_a ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Omeasures_001t__List__Olist_Itf__a_J,type,
measures_list_a: list_list_a_nat > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Omeasures_001tf__a,type,
measures_a: list_a_nat > set_Product_prod_a_a ).
thf(sy_c_List_On__lists_001t__Common__Ostate_Itf__a_J,type,
n_lists_state_a: nat > list_state_a > list_list_state_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_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
nth_a_a_o: list_a_a_o > nat > a > a > $o ).
thf(sy_c_List_Onth_001t__Common__Ostate_Itf__a_J,type,
nth_state_a: list_state_a > nat > state_a ).
thf(sy_c_List_Onth_001t__Current__Ocurrent_Itf__a_J,type,
nth_current_a: list_current_a > nat > current_a ).
thf(sy_c_List_Onth_001t__Idle__Oidle_Itf__a_J,type,
nth_idle_a: list_idle_a > nat > idle_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_Onths_001t__Common__Ostate_Itf__a_J,type,
nths_state_a: list_state_a > set_nat > list_state_a ).
thf(sy_c_List_Onths_001t__List__Olist_Itf__a_J,type,
nths_list_a: list_list_a > set_nat > list_list_a ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oproduct__lists_001t__Common__Ostate_Itf__a_J,type,
produc7194234687627354129tate_a: list_list_state_a > list_list_state_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_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001t__Common__Ostate_Itf__a_J,type,
subseqs_state_a: list_state_a > list_list_state_a ).
thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__a_J,type,
subseqs_list_a: list_list_a > list_list_list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__Common__Ostate_Itf__a_J,type,
take_state_a: nat > list_state_a > list_state_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__Common__Ostate_It__Common__Ostate_Itf__a_J_J,type,
size_s2679102311978592896tate_a: state_state_a > 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_a: state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_It__Common__Ostate_Itf__a_J_J,type,
size_s1806206149518693944tate_a: current_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__Idle__Oidle_It__Common__Ostate_Itf__a_J_J,type,
size_s905445271783748068tate_a: idle_state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Idle__Oidle_It__List__Olist_Itf__a_J_J,type,
size_s4952028384985416054list_a: idle_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Idle__Oidle_Itf__a_J,type,
size_size_idle_a: idle_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_J,type,
size_size_list_a_a_o: list_a_a_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Common__Ostate_Itf__a_J_J,type,
size_s4663437186682225508tate_a: list_state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Current__Ocurrent_Itf__a_J_J,type,
size_s190638326304920994rent_a: list_current_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Idle__Oidle_Itf__a_J_J,type,
size_s8477453896275132790idle_a: list_idle_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__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_It__Common__Ostate_Itf__a_J_J,type,
size_s2616604874200068438tate_a: stack_state_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_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > 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_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J_J,type,
ord_le8856296118373343491tate_a: set_Pr324718442235990179tate_a > set_Pr324718442235990179tate_a > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
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member5812885074326414752tate_a: produc3211089224054848265tate_a > set_Pr1116282471373962303tate_a > $o ).
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member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member4371779931761811402list_a: produc7489172080673977121list_a > set_Pr1060768173594829441list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Common__Ostate_Itf__a_J_J,type,
member1038697612344694410tate_a: produc6211185749192850675tate_a > set_Pr2973361075998125097tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Common__Ostate_Itf__a_J_J_J,type,
member7736023580690228378tate_a: produc3537791446953558659tate_a > set_Pr152097031616555065tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Current__Ocurrent_Itf__a_J_J_J,type,
member1785988321452022018rent_a: produc8183448624056744409rent_a > set_Pr7673030520428015033rent_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Idle__Oidle_Itf__a_J_J_J,type,
member9073500730917845590idle_a: produc2967520842975235501idle_a > set_Pr8671745777326831373idle_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member3917598494194944214list_a: produc7034990643107109933list_a > set_Pr4412185308373534093list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member8006451231845903178st_a_a: produc2579390645249093025st_a_a > set_Pr8962057229576493569st_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J,type,
member7740437332958137860tate_a: produc3409137331138395373tate_a > set_Pr324718442235990179tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Current__Ocurrent_Itf__a_J_J,type,
member8183269688883217154rent_a: produc7805042584321970905rent_a > set_Pr3922190185655878585rent_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
member3794253416837758678idle_a: produc7590564867095724333idle_a > set_Pr7225968907286947725idle_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_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_common_H,type,
common: state_a ).
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_stackSize____,type,
stackSize: nat ).
thf(sy_v_stack____,type,
stack: stack_a ).
thf(sy_v_xa____,type,
xa: a ).
thf(sy_v_xb____,type,
xb: a ).
thf(sy_v_xs____,type,
xs: list_a ).
% Relevant facts (1257)
thf(fact_0__092_060open_062hd_A_IStack__Aux_Olist_Astack_J_A_061_Ax_092_060close_062,axiom,
( ( hd_a @ ( stack_list_a2 @ stack ) )
= xb ) ).
% \<open>hd (Stack_Aux.list stack) = x\<close>
thf(fact_1__C1_Oprems_C_I1_J,axiom,
type_i4669920168676019581tate_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) ).
% "1.prems"(1)
thf(fact_2__C2_Oprems_C_I1_J,axiom,
type_i4669920168676019581tate_a @ ( idle_a2 @ ( current_a2 @ ( cons_a @ xb @ xs ) @ added @ old @ remained ) @ ( idle_a3 @ stack @ stackSize ) ) ).
% "2.prems"(1)
thf(fact_3__C2_Oprems_C_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( idle_a2 @ ( current_a2 @ ( cons_a @ xb @ xs ) @ added @ old @ remained ) @ ( idle_a3 @ stack @ stackSize ) ) ) ).
% "2.prems"(2)
thf(fact_4__C2_Oprems_C_I3_J,axiom,
( ( pop_a @ ( idle_a2 @ ( current_a2 @ ( cons_a @ xb @ xs ) @ added @ old @ remained ) @ ( idle_a3 @ stack @ stackSize ) ) )
= ( produc8263595898873874535tate_a @ xa @ common ) ) ).
% "2.prems"(3)
thf(fact_5__C1_Oprems_C_I3_J,axiom,
( ( pop_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) )
= ( produc8263595898873874535tate_a @ xa @ common ) ) ).
% "1.prems"(3)
thf(fact_6__C1_Oprems_C_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) ) ).
% "1.prems"(2)
thf(fact_7_not__empty_I1_J,axiom,
~ ( type_i3216275384938974675tack_a @ stack ) ).
% not_empty(1)
thf(fact_8_idle_Oinject,axiom,
! [X1: stack_a,X2: nat,Y1: stack_a,Y2: nat] :
( ( ( idle_a3 @ X1 @ X2 )
= ( idle_a3 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% idle.inject
thf(fact_9_current_Oinject,axiom,
! [X1: list_a,X2: nat,X3: stack_a,X4: nat,Y1: list_a,Y2: nat,Y3: stack_a,Y4: nat] :
( ( ( current_a2 @ X1 @ X2 @ X3 @ X4 )
= ( current_a2 @ Y1 @ Y2 @ Y3 @ Y4 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X3 = Y3 )
& ( X4 = Y4 ) ) ) ).
% current.inject
thf(fact_10_state_Oinject_I2_J,axiom,
! [X21: current_a,X22: idle_a,Y21: current_a,Y22: idle_a] :
( ( ( idle_a2 @ X21 @ X22 )
= ( idle_a2 @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% state.inject(2)
thf(fact_11_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_12_list_Oinject,axiom,
! [X21: state_a,X22: list_state_a,Y21: state_a,Y22: list_state_a] :
( ( ( cons_state_a @ X21 @ X22 )
= ( cons_state_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_13_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_14_push__list__current,axiom,
! [X: list_a,Left: state_list_a] :
( ( common3845368160371929429list_a @ ( push_list_a @ X @ Left ) )
= ( cons_list_a @ X @ ( common3845368160371929429list_a @ Left ) ) ) ).
% push_list_current
thf(fact_15_push__list__current,axiom,
! [X: state_a,Left: state_state_a] :
( ( common8656238132076510255tate_a @ ( push_state_a @ X @ Left ) )
= ( cons_state_a @ X @ ( common8656238132076510255tate_a @ Left ) ) ) ).
% push_list_current
thf(fact_16_push__list__current,axiom,
! [X: a,Left: state_a] :
( ( common1102728217005306191rent_a @ ( push_a @ X @ Left ) )
= ( cons_a @ X @ ( common1102728217005306191rent_a @ Left ) ) ) ).
% push_list_current
thf(fact_17_not__empty_I2_J,axiom,
( ( stack_list_a2 @ stack )
!= nil_a ) ).
% not_empty(2)
thf(fact_18_idle_Oexhaust,axiom,
! [Y: idle_a] :
~ ! [X12: stack_a,X23: nat] :
( Y
!= ( idle_a3 @ X12 @ X23 ) ) ).
% idle.exhaust
thf(fact_19_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 ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X5 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_20_transpose_Ocases,axiom,
! [X: list_list_state_a] :
( ( X != nil_list_state_a )
=> ( ! [Xss: list_list_state_a] :
( X
!= ( cons_list_state_a @ nil_state_a @ Xss ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Xss: list_list_state_a] :
( X
!= ( cons_list_state_a @ ( cons_state_a @ X5 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_21_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X5: a,Xs: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X5 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_22_Common__Proof_Oinvar__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( type_i4669920168676019581tate_a @ Common2 ) ) ) ) ).
% Common_Proof.invar_pop
thf(fact_23_Common__Proof_Oinvar__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( push_a @ X @ Common ) ) ) ).
% Common_Proof.invar_push
thf(fact_24_list__nonempty__induct,axiom,
! [Xs2: list_list_a,P: list_list_a > $o] :
( ( Xs2 != nil_list_a )
=> ( ! [X5: list_a] : ( P @ ( cons_list_a @ X5 @ nil_list_a ) )
=> ( ! [X5: list_a,Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_a @ X5 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_25_list__nonempty__induct,axiom,
! [Xs2: list_state_a,P: list_state_a > $o] :
( ( Xs2 != nil_state_a )
=> ( ! [X5: state_a] : ( P @ ( cons_state_a @ X5 @ nil_state_a ) )
=> ( ! [X5: state_a,Xs: list_state_a] :
( ( Xs != nil_state_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_state_a @ X5 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_26_list__nonempty__induct,axiom,
! [Xs2: list_a,P: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X5: a] : ( P @ ( cons_a @ X5 @ nil_a ) )
=> ( ! [X5: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_a @ X5 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_27_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs2: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X5: a,Xs: list_a] : ( P @ ( cons_a @ X5 @ Xs ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ! [X5: a,Xs: list_a,Y5: list_a,Ys2: list_list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_28_list__induct2_H,axiom,
! [P: list_a > list_state_a > $o,Xs2: list_a,Ys: list_state_a] :
( ( P @ nil_a @ nil_state_a )
=> ( ! [X5: a,Xs: list_a] : ( P @ ( cons_a @ X5 @ Xs ) @ nil_state_a )
=> ( ! [Y5: state_a,Ys2: list_state_a] : ( P @ nil_a @ ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ! [X5: a,Xs: list_a,Y5: state_a,Ys2: list_state_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_29_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs2: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X5: list_a,Xs: list_list_a] : ( P @ ( cons_list_a @ X5 @ Xs ) @ nil_a )
=> ( ! [Y5: a,Ys2: list_a] : ( P @ nil_list_a @ ( cons_a @ Y5 @ Ys2 ) )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: a,Ys2: list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_30_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs2: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X5: list_a,Xs: list_list_a] : ( P @ ( cons_list_a @ X5 @ Xs ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys2: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: list_a,Ys2: list_list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_31_list__induct2_H,axiom,
! [P: list_list_a > list_state_a > $o,Xs2: list_list_a,Ys: list_state_a] :
( ( P @ nil_list_a @ nil_state_a )
=> ( ! [X5: list_a,Xs: list_list_a] : ( P @ ( cons_list_a @ X5 @ Xs ) @ nil_state_a )
=> ( ! [Y5: state_a,Ys2: list_state_a] : ( P @ nil_list_a @ ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: state_a,Ys2: list_state_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_32_list__induct2_H,axiom,
! [P: list_state_a > list_a > $o,Xs2: list_state_a,Ys: list_a] :
( ( P @ nil_state_a @ nil_a )
=> ( ! [X5: state_a,Xs: list_state_a] : ( P @ ( cons_state_a @ X5 @ Xs ) @ nil_a )
=> ( ! [Y5: a,Ys2: list_a] : ( P @ nil_state_a @ ( cons_a @ Y5 @ Ys2 ) )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: a,Ys2: list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_33_list__induct2_H,axiom,
! [P: list_state_a > list_list_a > $o,Xs2: list_state_a,Ys: list_list_a] :
( ( P @ nil_state_a @ nil_list_a )
=> ( ! [X5: state_a,Xs: list_state_a] : ( P @ ( cons_state_a @ X5 @ Xs ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys2: list_list_a] : ( P @ nil_state_a @ ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: list_a,Ys2: list_list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_34_list__induct2_H,axiom,
! [P: list_state_a > list_state_a > $o,Xs2: list_state_a,Ys: list_state_a] :
( ( P @ nil_state_a @ nil_state_a )
=> ( ! [X5: state_a,Xs: list_state_a] : ( P @ ( cons_state_a @ X5 @ Xs ) @ nil_state_a )
=> ( ! [Y5: state_a,Ys2: list_state_a] : ( P @ nil_state_a @ ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: state_a,Ys2: list_state_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_35_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs2: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X5: a,Xs: list_a] : ( P @ ( cons_a @ X5 @ Xs ) @ nil_a )
=> ( ! [Y5: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y5 @ Ys2 ) )
=> ( ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_36_neq__Nil__conv,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
= ( ? [Y6: list_a,Ys3: list_list_a] :
( Xs2
= ( cons_list_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_37_neq__Nil__conv,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
= ( ? [Y6: state_a,Ys3: list_state_a] :
( Xs2
= ( cons_state_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_38_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y6: a,Ys3: list_a] :
( Xs2
= ( cons_a @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_39_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X5: list_a] :
( X
!= ( cons_list_a @ X5 @ nil_list_a ) )
=> ~ ! [X5: list_a,Y5: list_a,Xs: list_list_a] :
( X
!= ( cons_list_a @ X5 @ ( cons_list_a @ Y5 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_40_remdups__adj_Ocases,axiom,
! [X: list_state_a] :
( ( X != nil_state_a )
=> ( ! [X5: state_a] :
( X
!= ( cons_state_a @ X5 @ nil_state_a ) )
=> ~ ! [X5: state_a,Y5: state_a,Xs: list_state_a] :
( X
!= ( cons_state_a @ X5 @ ( cons_state_a @ Y5 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_41_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X5: a] :
( X
!= ( cons_a @ X5 @ nil_a ) )
=> ~ ! [X5: a,Y5: a,Xs: list_a] :
( X
!= ( cons_a @ X5 @ ( cons_a @ Y5 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_42_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_43_list_Oexhaust,axiom,
! [Y: list_state_a] :
( ( Y != nil_state_a )
=> ~ ! [X212: state_a,X222: list_state_a] :
( Y
!= ( cons_state_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_44_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_45_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_46_list_OdiscI,axiom,
! [List: list_state_a,X21: state_a,X22: list_state_a] :
( ( List
= ( cons_state_a @ X21 @ X22 ) )
=> ( List != nil_state_a ) ) ).
% list.discI
thf(fact_47_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_48_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_49_list_Odistinct_I1_J,axiom,
! [X21: state_a,X22: list_state_a] :
( nil_state_a
!= ( cons_state_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_50_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_51_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 ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( X
!= ( current_list_a2 @ ( cons_list_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ).
% Current.pop.cases
thf(fact_52_Current_Opop_Ocases,axiom,
! [X: current_state_a] :
( ! [Added: nat,Old: stack_state_a,Remained: nat] :
( X
!= ( current_state_a2 @ nil_state_a @ Added @ Old @ Remained ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Added: nat,Old: stack_state_a,Remained: nat] :
( X
!= ( current_state_a2 @ ( cons_state_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ).
% Current.pop.cases
thf(fact_53_Current_Opop_Ocases,axiom,
! [X: current_a] :
( ! [Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( current_a2 @ nil_a @ Added @ Old @ Remained ) )
=> ~ ! [X5: a,Xs: list_a,Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( current_a2 @ ( cons_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ).
% Current.pop.cases
thf(fact_54_list_Osel_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( hd_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_55_list_Osel_I1_J,axiom,
! [X21: state_a,X22: list_state_a] :
( ( hd_state_a @ ( cons_state_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_56_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_57_not__Cons__self2,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( cons_list_a @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_58_not__Cons__self2,axiom,
! [X: state_a,Xs2: list_state_a] :
( ( cons_state_a @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_59_not__Cons__self2,axiom,
! [X: a,Xs2: list_a] :
( ( cons_a @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_60_mem__Collect__eq,axiom,
! [A: produc3409137331138395373tate_a,P: produc3409137331138395373tate_a > $o] :
( ( member7740437332958137860tate_a @ A @ ( collec2179778335344099906tate_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_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_62_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_63_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_Pr324718442235990179tate_a] :
( ( collec2179778335344099906tate_a
@ ^ [X6: produc3409137331138395373tate_a] : ( member7740437332958137860tate_a @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X6: product_prod_a_a] : ( member1426531477525435216od_a_a @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X6: list_a] : ( member_list_a @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X6: nat] : ( member_nat @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X6: a] : ( member_a @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_current_Oexhaust,axiom,
! [Y: current_a] :
~ ! [X12: list_a,X23: nat,X32: stack_a,X42: nat] :
( Y
!= ( current_a2 @ X12 @ X23 @ X32 @ X42 ) ) ).
% current.exhaust
thf(fact_71_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_a @ 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_72_Common__Proof_Opop__list,axiom,
! [Common: state_state_a,X: state_a,Common2: state_state_a] :
( ( type_i5973583926427607063tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2679102311978592896tate_a @ Common ) )
=> ( ( ( pop_state_a @ Common )
= ( produc5199797456077100795tate_a @ X @ Common2 ) )
=> ( ( cons_state_a @ X @ ( common_list_state_a @ Common2 ) )
= ( common_list_state_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_73_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_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( cons_a @ X @ ( common_list_a @ Common2 ) )
= ( common_list_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_74_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_75_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_76_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_77_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_78__092_060open_0620_A_060_Asize_Astack_092_060close_062,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ stack ) ).
% \<open>0 < size stack\<close>
thf(fact_79_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_80_Stack__Proof_Olist__empty,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
= nil_state_a )
= ( type_i6544059586428101441tate_a @ Stack ) ) ).
% Stack_Proof.list_empty
thf(fact_81_Stack__Proof_Olist__empty,axiom,
! [Stack: stack_a] :
( ( ( stack_list_a2 @ Stack )
= nil_a )
= ( type_i3216275384938974675tack_a @ Stack ) ) ).
% Stack_Proof.list_empty
thf(fact_82_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_83_Stack__Proof_Olist__empty__2,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
!= nil_state_a )
=> ~ ( type_i6544059586428101441tate_a @ Stack ) ) ).
% Stack_Proof.list_empty_2
thf(fact_84_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_85_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_86_Stack__Proof_Olist__not__empty,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
!= nil_state_a )
= ( ~ ( type_i6544059586428101441tate_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty
thf(fact_87_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_88_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_89_Stack__Proof_Olist__not__empty__2,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
= nil_state_a )
=> ( type_i6544059586428101441tate_a @ Stack ) ) ).
% Stack_Proof.list_not_empty_2
thf(fact_90_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_91_step__list__current,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common1102728217005306191rent_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ).
% step_list_current
thf(fact_92_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X2 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_93_prod_Oinject,axiom,
! [X1: a > a > $o,X2: list_a,Y1: a > a > $o,Y2: list_a] :
( ( ( produc8111569692950616493list_a @ X1 @ X2 )
= ( produc8111569692950616493list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_94_prod_Oinject,axiom,
! [X1: a,X2: current_a,Y1: a,Y2: current_a] :
( ( ( produc8503237746132909001rent_a @ X1 @ X2 )
= ( produc8503237746132909001rent_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_95_prod_Oinject,axiom,
! [X1: a,X2: idle_a,Y1: a,Y2: idle_a] :
( ( ( produc1265230069547855005idle_a @ X1 @ X2 )
= ( produc1265230069547855005idle_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_96_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_97_prod_Oinject,axiom,
! [X1: a,X2: state_a,Y1: a,Y2: state_a] :
( ( ( produc8263595898873874535tate_a @ X1 @ X2 )
= ( produc8263595898873874535tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_98_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_99_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_100_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_101_old_Oprod_Oinject,axiom,
! [A: a,B: idle_a,A3: a,B2: idle_a] :
( ( ( produc1265230069547855005idle_a @ A @ B )
= ( produc1265230069547855005idle_a @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_102_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_103_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_104_Common__Proof_Opush__list,axiom,
! [X: list_a,Common: state_list_a] :
( ( common_list_list_a @ ( push_list_a @ X @ Common ) )
= ( cons_list_a @ X @ ( common_list_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_105_Common__Proof_Opush__list,axiom,
! [X: state_a,Common: state_state_a] :
( ( common_list_state_a @ ( push_state_a @ X @ Common ) )
= ( cons_state_a @ X @ ( common_list_state_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_106_Common__Proof_Opush__list,axiom,
! [X: a,Common: state_a] :
( ( common_list_a @ ( push_a @ X @ Common ) )
= ( cons_a @ X @ ( common_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_107_step__list,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common_list_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common_list_a @ Common ) ) ) ).
% step_list
thf(fact_108_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F: a > a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F @ ( produc6837034575241423639list_a @ nil_a @ Bs ) ) )
=> ~ ! [F: a > a,A4: a,As: list_a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F @ ( produc6837034575241423639list_a @ ( cons_a @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_109_Idle_Opush_Ocases,axiom,
! [X: produc7590564867095724333idle_a] :
~ ! [X5: a,Stack2: stack_a,StackSize: nat] :
( X
!= ( produc1265230069547855005idle_a @ X5 @ ( idle_a3 @ Stack2 @ StackSize ) ) ) ).
% Idle.push.cases
thf(fact_110_Current_Opush_Ocases,axiom,
! [X: produc7805042584321970905rent_a] :
~ ! [X5: a,Extra: list_a,Added: nat,Old: stack_a,Remained: nat] :
( X
!= ( produc8503237746132909001rent_a @ X5 @ ( current_a2 @ Extra @ Added @ Old @ Remained ) ) ) ).
% Current.push.cases
thf(fact_111_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_112_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_113_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,X5: list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X5 @ nil_list_a ) ) )
=> ~ ! [P2: list_a > list_a > $o,X5: list_a,Y5: list_a,Xs: list_list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X5 @ ( cons_list_a @ Y5 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_114_successively_Ocases,axiom,
! [X: produc1289736764964523013tate_a] :
( ! [P2: state_a > state_a > $o] :
( X
!= ( produc8706141295281866623tate_a @ P2 @ nil_state_a ) )
=> ( ! [P2: state_a > state_a > $o,X5: state_a] :
( X
!= ( produc8706141295281866623tate_a @ P2 @ ( cons_state_a @ X5 @ nil_state_a ) ) )
=> ~ ! [P2: state_a > state_a > $o,X5: state_a,Y5: state_a,Xs: list_state_a] :
( X
!= ( produc8706141295281866623tate_a @ P2 @ ( cons_state_a @ X5 @ ( cons_state_a @ Y5 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_115_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X5: a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X5 @ nil_a ) ) )
=> ~ ! [P2: a > a > $o,X5: a,Y5: a,Xs: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X5 @ ( cons_a @ Y5 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_116_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,X5: list_a,Ys2: list_list_a] :
( X
!= ( produc3233922274162870707list_a @ P2 @ ( cons_list_a @ X5 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_117_sorted__wrt_Ocases,axiom,
! [X: produc1289736764964523013tate_a] :
( ! [P2: state_a > state_a > $o] :
( X
!= ( produc8706141295281866623tate_a @ P2 @ nil_state_a ) )
=> ~ ! [P2: state_a > state_a > $o,X5: state_a,Ys2: list_state_a] :
( X
!= ( produc8706141295281866623tate_a @ P2 @ ( cons_state_a @ X5 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_118_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P2: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P2 @ nil_a ) )
=> ~ ! [P2: a > a > $o,X5: a,Ys2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X5 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_119_shuffles_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
=> ( ! [Xs: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Xs @ nil_list_a ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Y5: list_a,Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_120_shuffles_Ocases,axiom,
! [X: produc3809440963839956737tate_a] :
( ! [Ys2: list_state_a] :
( X
!= ( produc5982907372945109049tate_a @ nil_state_a @ Ys2 ) )
=> ( ! [Xs: list_state_a] :
( X
!= ( produc5982907372945109049tate_a @ Xs @ nil_state_a ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Y5: state_a,Ys2: list_state_a] :
( X
!= ( produc5982907372945109049tate_a @ ( cons_state_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_121_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs @ nil_a ) )
=> ~ ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_122_splice_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Ys2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X5 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_123_splice_Ocases,axiom,
! [X: produc3809440963839956737tate_a] :
( ! [Ys2: list_state_a] :
( X
!= ( produc5982907372945109049tate_a @ nil_state_a @ Ys2 ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Ys2: list_state_a] :
( X
!= ( produc5982907372945109049tate_a @ ( cons_state_a @ X5 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_124_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X5: a,Xs: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X5 @ Xs ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_125_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_126_Stack__Proof_Olist__empty__size__2,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
!= nil_state_a )
=> ( ( size_s2616604874200068438tate_a @ Stack )
!= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size_2
thf(fact_127_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_128_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_129_Stack__Proof_Olist__empty__size,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
= nil_state_a )
= ( ( size_s2616604874200068438tate_a @ Stack )
= zero_zero_nat ) ) ).
% Stack_Proof.list_empty_size
thf(fact_130_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_131_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_132_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_133_step__state_Osimps_I1_J,axiom,
! [Current: current_a,Idle: idle_a] :
( ( type_s889635741254954505tate_a @ ( idle_a2 @ Current @ Idle ) )
= ( idle_a2 @ Current @ Idle ) ) ).
% step_state.simps(1)
thf(fact_134_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_135_invar__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( type_s889635741254954505tate_a @ Common ) ) ) ).
% invar_step
thf(fact_136_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_137_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_138_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_139_Pair__inject,axiom,
! [A: a,B: idle_a,A3: a,B2: idle_a] :
( ( ( produc1265230069547855005idle_a @ A @ B )
= ( produc1265230069547855005idle_a @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_140_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_141_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_142_prod__cases,axiom,
! [P: produc9164743771328383783list_a > $o,P3: produc9164743771328383783list_a] :
( ! [A4: list_a,B3: list_a] : ( P @ ( produc6837034575241423639list_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_143_prod__cases,axiom,
! [P: produc5032551385658279741list_a > $o,P3: produc5032551385658279741list_a] :
( ! [A4: a > a > $o,B3: list_a] : ( P @ ( produc8111569692950616493list_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_144_prod__cases,axiom,
! [P: produc7805042584321970905rent_a > $o,P3: produc7805042584321970905rent_a] :
( ! [A4: a,B3: current_a] : ( P @ ( produc8503237746132909001rent_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_145_prod__cases,axiom,
! [P: produc7590564867095724333idle_a > $o,P3: produc7590564867095724333idle_a] :
( ! [A4: a,B3: idle_a] : ( P @ ( produc1265230069547855005idle_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_146_prod__cases,axiom,
! [P: product_prod_a_a > $o,P3: product_prod_a_a] :
( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_147_prod__cases,axiom,
! [P: produc3409137331138395373tate_a > $o,P3: produc3409137331138395373tate_a] :
( ! [A4: a,B3: state_a] : ( P @ ( produc8263595898873874535tate_a @ A4 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_148_surj__pair,axiom,
! [P3: produc9164743771328383783list_a] :
? [X5: list_a,Y5: list_a] :
( P3
= ( produc6837034575241423639list_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_149_surj__pair,axiom,
! [P3: produc5032551385658279741list_a] :
? [X5: a > a > $o,Y5: list_a] :
( P3
= ( produc8111569692950616493list_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_150_surj__pair,axiom,
! [P3: produc7805042584321970905rent_a] :
? [X5: a,Y5: current_a] :
( P3
= ( produc8503237746132909001rent_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_151_surj__pair,axiom,
! [P3: produc7590564867095724333idle_a] :
? [X5: a,Y5: idle_a] :
( P3
= ( produc1265230069547855005idle_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_152_surj__pair,axiom,
! [P3: product_prod_a_a] :
? [X5: a,Y5: a] :
( P3
= ( product_Pair_a_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_153_surj__pair,axiom,
! [P3: produc3409137331138395373tate_a] :
? [X5: a,Y5: state_a] :
( P3
= ( produc8263595898873874535tate_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_154_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A4: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_155_old_Oprod_Oexhaust,axiom,
! [Y: produc5032551385658279741list_a] :
~ ! [A4: a > a > $o,B3: list_a] :
( Y
!= ( produc8111569692950616493list_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_156_old_Oprod_Oexhaust,axiom,
! [Y: produc7805042584321970905rent_a] :
~ ! [A4: a,B3: current_a] :
( Y
!= ( produc8503237746132909001rent_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_157_old_Oprod_Oexhaust,axiom,
! [Y: produc7590564867095724333idle_a] :
~ ! [A4: a,B3: idle_a] :
( Y
!= ( produc1265230069547855005idle_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_158_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_159_old_Oprod_Oexhaust,axiom,
! [Y: produc3409137331138395373tate_a] :
~ ! [A4: a,B3: state_a] :
( Y
!= ( produc8263595898873874535tate_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_160_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_161_Stack__Proof_Olist__not__empty__size__2,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
= nil_state_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s2616604874200068438tate_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size_2
thf(fact_162_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_163_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_164_Stack__Proof_Olist__not__empty__size,axiom,
! [Stack: stack_state_a] :
( ( ( stack_list_state_a @ Stack )
!= nil_state_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s2616604874200068438tate_a @ Stack ) ) ) ).
% Stack_Proof.list_not_empty_size
thf(fact_165_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_166_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_167_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_168_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_169_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_170_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_171_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_172_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_173_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_174_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_175_size__neq__size__imp__neq,axiom,
! [X: idle_a,Y: idle_a] :
( ( ( size_size_idle_a @ X )
!= ( size_size_idle_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_176_size__neq__size__imp__neq,axiom,
! [X: list_state_a,Y: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ X )
!= ( size_s4663437186682225508tate_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_177_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_178_size__neq__size__imp__neq,axiom,
! [X: state_a,Y: state_a] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_179_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_180_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_181_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_182_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_183_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_184_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_185_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_186_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_187_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_188_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_189_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_190_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_191_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_192_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_193_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_194_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_state_a @ N @ nil_state_a )
= ( cons_list_state_a @ nil_state_a @ nil_list_state_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_state_a @ N @ nil_state_a )
= nil_list_state_a ) ) ) ).
% n_lists_Nil
thf(fact_195_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_196_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_list_a] :
( ( n_lists_list_a @ zero_zero_nat @ Xs2 )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% n_lists.simps(1)
thf(fact_197_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_state_a] :
( ( n_lists_state_a @ zero_zero_nat @ Xs2 )
= ( cons_list_state_a @ nil_state_a @ nil_list_state_a ) ) ).
% n_lists.simps(1)
thf(fact_198_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs2 )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_199_cons__hd,axiom,
! [X: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs2 )
= Ys )
=> ( X
= ( hd_list_a @ Ys ) ) ) ).
% cons_hd
thf(fact_200_cons__hd,axiom,
! [X: state_a,Xs2: list_state_a,Ys: list_state_a] :
( ( ( cons_state_a @ X @ Xs2 )
= Ys )
=> ( X
= ( hd_state_a @ Ys ) ) ) ).
% cons_hd
thf(fact_201_cons__hd,axiom,
! [X: a,Xs2: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs2 )
= Ys )
=> ( X
= ( hd_a @ Ys ) ) ) ).
% cons_hd
thf(fact_202_first__list,axiom,
! [Stack: stack_a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ( ( first_a2 @ Stack )
= ( hd_a @ ( stack_list_a2 @ Stack ) ) ) ) ).
% first_list
thf(fact_203_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_204_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_205_product__lists_Osimps_I1_J,axiom,
( ( produc7194234687627354129tate_a @ nil_list_state_a )
= ( cons_list_state_a @ nil_state_a @ nil_list_state_a ) ) ).
% product_lists.simps(1)
thf(fact_206_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_207_in__measures_I2_J,axiom,
! [X: list_a,Y: list_a,F2: list_a > nat,Fs: list_list_a_nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_208_in__measures_I2_J,axiom,
! [X: a,Y: a,F2: a > nat,Fs: list_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F2 @ Fs ) ) )
= ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_209_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_a @ nil_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% subseqs.simps(1)
thf(fact_210_subseqs_Osimps_I1_J,axiom,
( ( subseqs_state_a @ nil_state_a )
= ( cons_list_state_a @ nil_state_a @ nil_list_state_a ) ) ).
% subseqs.simps(1)
thf(fact_211_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_212_Stack__Proof_Opush__list,axiom,
! [X: list_a,Stack: stack_list_a] :
( ( stack_list_list_a @ ( push_list_a4 @ X @ Stack ) )
= ( cons_list_a @ X @ ( stack_list_list_a @ Stack ) ) ) ).
% Stack_Proof.push_list
thf(fact_213_Stack__Proof_Opush__list,axiom,
! [X: state_a,Stack: stack_state_a] :
( ( stack_list_state_a @ ( push_state_a4 @ X @ Stack ) )
= ( cons_state_a @ X @ ( stack_list_state_a @ Stack ) ) ) ).
% Stack_Proof.push_list
thf(fact_214_Stack__Proof_Opush__list,axiom,
! [X: a,Stack: stack_a] :
( ( stack_list_a2 @ ( push_a4 @ X @ Stack ) )
= ( cons_a @ X @ ( stack_list_a2 @ Stack ) ) ) ).
% Stack_Proof.push_list
thf(fact_215_insert__Nil,axiom,
! [X: list_a] :
( ( insert_list_a @ X @ nil_list_a )
= ( cons_list_a @ X @ nil_list_a ) ) ).
% insert_Nil
thf(fact_216_insert__Nil,axiom,
! [X: state_a] :
( ( insert_state_a @ X @ nil_state_a )
= ( cons_state_a @ X @ nil_state_a ) ) ).
% insert_Nil
thf(fact_217_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_218_length__0__conv,axiom,
! [Xs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_list_a ) ) ).
% length_0_conv
thf(fact_219_length__0__conv,axiom,
! [Xs2: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_state_a ) ) ).
% length_0_conv
thf(fact_220_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_221_size__list__length,axiom,
! [Stack: stack_state_a] :
( ( size_s4663437186682225508tate_a @ ( stack_list_state_a @ Stack ) )
= ( size_s2616604874200068438tate_a @ Stack ) ) ).
% size_list_length
thf(fact_222_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_223_in__measures_I1_J,axiom,
! [X: list_a,Y: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ nil_list_a_nat ) ) ).
% in_measures(1)
thf(fact_224_in__measures_I1_J,axiom,
! [X: a,Y: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ nil_a_nat ) ) ).
% in_measures(1)
thf(fact_225_length__greater__0__conv,axiom,
! [Xs2: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs2 ) )
= ( Xs2 != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_226_length__greater__0__conv,axiom,
! [Xs2: list_state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s4663437186682225508tate_a @ Xs2 ) )
= ( Xs2 != nil_state_a ) ) ).
% length_greater_0_conv
thf(fact_227_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_228_length__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ! [Xs: list_a] :
( ! [Ys4: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_229_length__induct,axiom,
! [P: list_state_a > $o,Xs2: list_state_a] :
( ! [Xs: list_state_a] :
( ! [Ys4: list_state_a] :
( ( ord_less_nat @ ( size_s4663437186682225508tate_a @ Ys4 ) @ ( size_s4663437186682225508tate_a @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_230_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_231_list_Osize_I3_J,axiom,
( ( size_s4663437186682225508tate_a @ nil_state_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_232_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_233_list__induct2,axiom,
! [Xs2: list_list_a,Ys: list_list_a,P: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_234_list__induct2,axiom,
! [Xs2: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_235_list__induct2,axiom,
! [Xs2: list_list_a,Ys: list_state_a,P: list_list_a > list_state_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_state_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: state_a,Ys2: list_state_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s4663437186682225508tate_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_236_list__induct2,axiom,
! [Xs2: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X5: a,Xs: list_a,Y5: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_237_list__induct2,axiom,
! [Xs2: list_a,Ys: list_state_a,P: list_a > list_state_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ( P @ nil_a @ nil_state_a )
=> ( ! [X5: a,Xs: list_a,Y5: state_a,Ys2: list_state_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s4663437186682225508tate_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_238_list__induct2,axiom,
! [Xs2: list_state_a,Ys: list_list_a,P: list_state_a > list_list_a > $o] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_state_a @ nil_list_a )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: list_a,Ys2: list_list_a] :
( ( ( size_s4663437186682225508tate_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_239_list__induct2,axiom,
! [Xs2: list_state_a,Ys: list_a,P: list_state_a > list_a > $o] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_state_a @ nil_a )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: a,Ys2: list_a] :
( ( ( size_s4663437186682225508tate_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_240_list__induct2,axiom,
! [Xs2: list_state_a,Ys: list_state_a,P: list_state_a > list_state_a > $o] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ( P @ nil_state_a @ nil_state_a )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: state_a,Ys2: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs )
= ( size_s4663437186682225508tate_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_241_list__induct2,axiom,
! [Xs2: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_242_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_243_list__induct3,axiom,
! [Xs2: list_list_a,Ys: list_a,Zs: list_a,P: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_244_list__induct3,axiom,
! [Xs2: list_a,Ys: list_list_a,Zs: list_a,P: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X5: a,Xs: list_a,Y5: list_a,Ys2: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_245_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_list_a,P: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_246_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_state_a,P: list_a > list_a > list_state_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s4663437186682225508tate_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_state_a )
=> ( ! [X5: a,Xs: list_a,Y5: a,Ys2: list_a,Z: state_a,Zs2: list_state_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s4663437186682225508tate_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_state_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_247_list__induct3,axiom,
! [Xs2: list_a,Ys: list_state_a,Zs: list_a,P: list_a > list_state_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ( ( size_s4663437186682225508tate_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_state_a @ nil_a )
=> ( ! [X5: a,Xs: list_a,Y5: state_a,Ys2: list_state_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s4663437186682225508tate_a @ Ys2 ) )
=> ( ( ( size_s4663437186682225508tate_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs ) @ ( cons_state_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_248_list__induct3,axiom,
! [Xs2: list_state_a,Ys: list_a,Zs: list_a,P: list_state_a > list_a > list_a > $o] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_state_a @ nil_a @ nil_a )
=> ( ! [X5: state_a,Xs: list_state_a,Y5: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_s4663437186682225508tate_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_state_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_249_list__induct3,axiom,
! [Xs2: list_list_a,Ys: list_list_a,Zs: list_a,P: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: list_a,Ys2: list_list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_250_list__induct3,axiom,
! [Xs2: list_list_a,Ys: list_a,Zs: list_list_a,P: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: a,Ys2: list_a,Z: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_251_list__induct3,axiom,
! [Xs2: list_list_a,Ys: list_a,Zs: list_state_a,P: list_list_a > list_a > list_state_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s4663437186682225508tate_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_state_a )
=> ( ! [X5: list_a,Xs: list_list_a,Y5: a,Ys2: list_a,Z: state_a,Zs2: list_state_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s4663437186682225508tate_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X5 @ Xs ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_state_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_252_push__not__empty,axiom,
! [Stack: stack_a,X: a] :
( ~ ( type_i3216275384938974675tack_a @ Stack )
=> ~ ( type_i3216275384938974675tack_a @ ( push_a4 @ X @ Stack ) ) ) ).
% push_not_empty
thf(fact_253_first__hd,axiom,
( first_a2
= ( ^ [Stack3: stack_a] : ( hd_a @ ( stack_list_a2 @ Stack3 ) ) ) ) ).
% first_hd
thf(fact_254_measures__less,axiom,
! [F2: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_255_measures__less,axiom,
! [F2: a > nat,X: a,Y: a,Fs: list_a_nat] :
( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_256_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_257_first__pop,axiom,
! [Stack: stack_list_a] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( cons_list_a @ ( first_list_a @ Stack ) @ ( stack_list_list_a @ ( pop_list_a4 @ Stack ) ) )
= ( stack_list_list_a @ Stack ) ) ) ).
% first_pop
thf(fact_258_first__pop,axiom,
! [Stack: stack_state_a] :
( ~ ( type_i6544059586428101441tate_a @ Stack )
=> ( ( cons_state_a @ ( first_state_a @ Stack ) @ ( stack_list_state_a @ ( pop_state_a4 @ Stack ) ) )
= ( stack_list_state_a @ Stack ) ) ) ).
% first_pop
thf(fact_259_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_260_size__idle_Oelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ~ ! [Stack2: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( Y
!= ( size_size_stack_a @ Stack2 ) ) ) ) ).
% size_idle.elims
thf(fact_261_size__idle_Osimps,axiom,
! [Stack: stack_a,Uu2: nat] :
( ( size_size_idle_a @ ( idle_a3 @ Stack @ Uu2 ) )
= ( size_size_stack_a @ Stack ) ) ).
% size_idle.simps
thf(fact_262_Cons__lenlex__iff,axiom,
! [M2: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M2 @ 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 @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_263_Cons__lenlex__iff,axiom,
! [M2: state_a,Ms: list_state_a,N: state_a,Ns: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( cons_state_a @ M2 @ Ms ) @ ( cons_state_a @ N @ Ns ) ) @ ( lenlex_state_a @ R ) )
= ( ( ord_less_nat @ ( size_s4663437186682225508tate_a @ Ms ) @ ( size_s4663437186682225508tate_a @ Ns ) )
| ( ( ( size_s4663437186682225508tate_a @ Ms )
= ( size_s4663437186682225508tate_a @ Ns ) )
& ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Ms @ Ns ) @ ( lenlex_state_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_264_Cons__lenlex__iff,axiom,
! [M2: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M2 @ 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 @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_265_Cons__in__lex,axiom,
! [X: list_a,Xs2: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y @ Ys ) ) @ ( lex_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys ) @ ( lex_list_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_266_Cons__in__lex,axiom,
! [X: state_a,Xs2: list_state_a,Y: state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( cons_state_a @ X @ Xs2 ) @ ( cons_state_a @ Y @ Ys ) ) @ ( lex_state_a @ R ) )
= ( ( ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ X @ Y ) @ R )
& ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) ) )
| ( ( X = Y )
& ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( lex_state_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_267_Cons__in__lex,axiom,
! [X: a,Xs2: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_268_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_269_nths__singleton,axiom,
! [A2: set_nat,X: list_a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_a @ ( cons_list_a @ X @ nil_list_a ) @ A2 )
= ( cons_list_a @ X @ nil_list_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_a @ ( cons_list_a @ X @ nil_list_a ) @ A2 )
= nil_list_a ) ) ) ).
% nths_singleton
thf(fact_270_nths__singleton,axiom,
! [A2: set_nat,X: state_a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_state_a @ ( cons_state_a @ X @ nil_state_a ) @ A2 )
= ( cons_state_a @ X @ nil_state_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_state_a @ ( cons_state_a @ X @ nil_state_a ) @ A2 )
= nil_state_a ) ) ) ).
% nths_singleton
thf(fact_271_nths__singleton,axiom,
! [A2: set_nat,X: a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= ( cons_a @ X @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_272_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X5: a,Y5: a,Xs: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_273_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X5: a,Y5: state_a,Xs: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs ) )
=> ! [Ys2: list_state_a] :
( ( A22
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X5 @ Y5 ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs @ Ys2 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_274_listrel_Ocases,axiom,
! [A1: list_a,A22: list_list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ A1 @ A22 ) @ ( listrel_a_list_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_list_a ) )
=> ~ ! [X5: a,Y5: list_a,Xs: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs ) )
=> ! [Ys2: list_list_a] :
( ( A22
= ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X5 @ Y5 ) @ R )
=> ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ Ys2 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_275_listrel_Ocases,axiom,
! [A1: list_list_a,A22: list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ A1 @ A22 ) @ ( listrel_list_a_a @ R ) )
=> ( ( ( A1 = nil_list_a )
=> ( A22 != nil_a ) )
=> ~ ! [X5: list_a,Y5: a,Xs: list_list_a] :
( ( A1
= ( cons_list_a @ X5 @ Xs ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X5 @ Y5 ) @ R )
=> ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ Ys2 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_276_listrel_Ocases,axiom,
! [A1: list_state_a,A22: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ A1 @ A22 ) @ ( listrel_state_a_a @ R ) )
=> ( ( ( A1 = nil_state_a )
=> ( A22 != nil_a ) )
=> ~ ! [X5: state_a,Y5: a,Xs: list_state_a] :
( ( A1
= ( cons_state_a @ X5 @ Xs ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ X5 @ Y5 ) @ R )
=> ~ ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs @ Ys2 ) @ ( listrel_state_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_277_listrel_Ocases,axiom,
! [A1: list_a,A22: list_current_a,R: set_Pr3922190185655878585rent_a] :
( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ A1 @ A22 ) @ ( listrel_a_current_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_current_a ) )
=> ~ ! [X5: a,Y5: current_a,Xs: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs ) )
=> ! [Ys2: list_current_a] :
( ( A22
= ( cons_current_a @ Y5 @ Ys2 ) )
=> ( ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ X5 @ Y5 ) @ R )
=> ~ ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs @ Ys2 ) @ ( listrel_a_current_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_278_listrel_Ocases,axiom,
! [A1: list_a,A22: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ A1 @ A22 ) @ ( listrel_a_idle_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_idle_a ) )
=> ~ ! [X5: a,Y5: idle_a,Xs: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs ) )
=> ! [Ys2: list_idle_a] :
( ( A22
= ( cons_idle_a @ Y5 @ Ys2 ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X5 @ Y5 ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs @ Ys2 ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_279_listrel_Ocases,axiom,
! [A1: list_list_a,A22: list_state_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ A1 @ A22 ) @ ( listre3686370835355000083tate_a @ R ) )
=> ( ( ( A1 = nil_list_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X5: list_a,Y5: state_a,Xs: list_list_a] :
( ( A1
= ( cons_list_a @ X5 @ Xs ) )
=> ! [Ys2: list_state_a] :
( ( A22
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member1038697612344694410tate_a @ ( produc4951958046122062829tate_a @ X5 @ Y5 ) @ R )
=> ~ ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs @ Ys2 ) @ ( listre3686370835355000083tate_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_280_listrel_Ocases,axiom,
! [A1: list_state_a,A22: list_list_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ A1 @ A22 ) @ ( listre1206529783178763665list_a @ R ) )
=> ( ( ( A1 = nil_state_a )
=> ( A22 != nil_list_a ) )
=> ~ ! [X5: state_a,Y5: list_a,Xs: list_state_a] :
( ( A1
= ( cons_state_a @ X5 @ Xs ) )
=> ! [Ys2: list_list_a] :
( ( A22
= ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ( member6025650317227102160list_a @ ( produc2472116993945826411list_a @ X5 @ Y5 ) @ R )
=> ~ ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs @ Ys2 ) @ ( listre1206529783178763665list_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_281_listrel_Ocases,axiom,
! [A1: list_state_a,A22: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ A1 @ A22 ) @ ( listre8205798609706990835tate_a @ R ) )
=> ( ( ( A1 = nil_state_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X5: state_a,Y5: state_a,Xs: list_state_a] :
( ( A1
= ( cons_state_a @ X5 @ Xs ) )
=> ! [Ys2: list_state_a] :
( ( A22
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ X5 @ Y5 ) @ R )
=> ~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs @ Ys2 ) @ ( listre8205798609706990835tate_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_282_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_list_a @ nil_list_a @ A2 )
= nil_list_a ) ).
% nths_nil
thf(fact_283_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_state_a @ nil_state_a @ A2 )
= nil_state_a ) ).
% nths_nil
thf(fact_284_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_a @ nil_a @ A2 )
= nil_a ) ).
% nths_nil
thf(fact_285_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_list_a] :
( ( nil_list_a
= ( take_list_a @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_list_a ) ) ) ).
% take_eq_Nil2
thf(fact_286_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_state_a] :
( ( nil_state_a
= ( take_state_a @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_state_a ) ) ) ).
% take_eq_Nil2
thf(fact_287_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_a] :
( ( nil_a
= ( take_a @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_288_take__eq__Nil,axiom,
! [N: nat,Xs2: list_list_a] :
( ( ( take_list_a @ N @ Xs2 )
= nil_list_a )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_list_a ) ) ) ).
% take_eq_Nil
thf(fact_289_take__eq__Nil,axiom,
! [N: nat,Xs2: list_state_a] :
( ( ( take_state_a @ N @ Xs2 )
= nil_state_a )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_state_a ) ) ) ).
% take_eq_Nil
thf(fact_290_take__eq__Nil,axiom,
! [N: nat,Xs2: list_a] :
( ( ( take_a @ N @ Xs2 )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_291_take0,axiom,
( ( take_list_a @ zero_zero_nat )
= ( ^ [Xs3: list_list_a] : nil_list_a ) ) ).
% take0
thf(fact_292_take0,axiom,
( ( take_state_a @ zero_zero_nat )
= ( ^ [Xs3: list_state_a] : nil_state_a ) ) ).
% take0
thf(fact_293_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_294_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_295_Nil__lenlex__iff1,axiom,
! [Ns: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ nil_state_a @ Ns ) @ ( lenlex_state_a @ R ) )
= ( Ns != nil_state_a ) ) ).
% Nil_lenlex_iff1
thf(fact_296_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_297_hd__take,axiom,
! [J: nat,Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_a @ ( take_a @ J @ Xs2 ) )
= ( hd_a @ Xs2 ) ) ) ).
% hd_take
thf(fact_298_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_a] :
( ( size_size_list_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_299_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_state_a] :
( ( size_s4663437186682225508tate_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_300_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_301_neq__if__length__neq,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
!= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_302_take__equalityI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ! [I: nat] :
( ( take_a @ I @ Xs2 )
= ( take_a @ I @ Ys ) )
=> ( Xs2 = Ys ) ) ).
% take_equalityI
thf(fact_303_take__Nil,axiom,
! [N: nat] :
( ( take_list_a @ N @ nil_list_a )
= nil_list_a ) ).
% take_Nil
thf(fact_304_take__Nil,axiom,
! [N: nat] :
( ( take_state_a @ N @ nil_state_a )
= nil_state_a ) ).
% take_Nil
thf(fact_305_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_306_listrel__eq__len,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_307_listrel__eq__len,axiom,
! [Xs2: list_a,Ys: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ Ys ) @ ( listrel_a_state_a @ R ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_308_listrel__eq__len,axiom,
! [Xs2: list_state_a,Ys: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs2 @ Ys ) @ ( listrel_state_a_a @ R ) )
=> ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_309_listrel__eq__len,axiom,
! [Xs2: list_state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( listre8205798609706990835tate_a @ R ) )
=> ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_310_nths__all,axiom,
! [Xs2: list_a,I2: set_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( member_nat @ I @ I2 ) )
=> ( ( nths_a @ Xs2 @ I2 )
= Xs2 ) ) ).
% nths_all
thf(fact_311_nths__all,axiom,
! [Xs2: list_state_a,I2: set_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( member_nat @ I @ I2 ) )
=> ( ( nths_state_a @ Xs2 @ I2 )
= Xs2 ) ) ).
% nths_all
thf(fact_312_take__0,axiom,
! [Xs2: list_list_a] :
( ( take_list_a @ zero_zero_nat @ Xs2 )
= nil_list_a ) ).
% take_0
thf(fact_313_take__0,axiom,
! [Xs2: list_state_a] :
( ( take_state_a @ zero_zero_nat @ Xs2 )
= nil_state_a ) ).
% take_0
thf(fact_314_take__0,axiom,
! [Xs2: list_a] :
( ( take_a @ zero_zero_nat @ Xs2 )
= nil_a ) ).
% take_0
thf(fact_315_listrel__Nil2,axiom,
! [Xs2: list_list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs2 @ nil_a ) @ ( listrel_list_a_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil2
thf(fact_316_listrel__Nil2,axiom,
! [Xs2: list_state_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs2 @ nil_a ) @ ( listrel_state_a_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil2
thf(fact_317_listrel__Nil2,axiom,
! [Xs2: list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs2 @ nil_list_a ) @ ( listrel_a_list_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil2
thf(fact_318_listrel__Nil2,axiom,
! [Xs2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) @ ( listre6772471554020304241list_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil2
thf(fact_319_listrel__Nil2,axiom,
! [Xs2: list_state_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs2 @ nil_list_a ) @ ( listre1206529783178763665list_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil2
thf(fact_320_listrel__Nil2,axiom,
! [Xs2: list_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ nil_state_a ) @ ( listrel_a_state_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil2
thf(fact_321_listrel__Nil2,axiom,
! [Xs2: list_list_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs2 @ nil_state_a ) @ ( listre3686370835355000083tate_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil2
thf(fact_322_listrel__Nil2,axiom,
! [Xs2: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ nil_state_a ) @ ( listre8205798609706990835tate_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil2
thf(fact_323_listrel__Nil2,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) @ ( listrel_a_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil2
thf(fact_324_listrel__Nil1,axiom,
! [Xs2: list_list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ nil_a @ Xs2 ) @ ( listrel_a_list_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil1
thf(fact_325_listrel__Nil1,axiom,
! [Xs2: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ nil_a @ Xs2 ) @ ( listrel_a_state_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil1
thf(fact_326_listrel__Nil1,axiom,
! [Xs2: list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ nil_list_a @ Xs2 ) @ ( listrel_list_a_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil1
thf(fact_327_listrel__Nil1,axiom,
! [Xs2: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Xs2 ) @ ( listre6772471554020304241list_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil1
thf(fact_328_listrel__Nil1,axiom,
! [Xs2: list_state_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ nil_list_a @ Xs2 ) @ ( listre3686370835355000083tate_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil1
thf(fact_329_listrel__Nil1,axiom,
! [Xs2: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ nil_state_a @ Xs2 ) @ ( listrel_state_a_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil1
thf(fact_330_listrel__Nil1,axiom,
! [Xs2: list_list_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ nil_state_a @ Xs2 ) @ ( listre1206529783178763665list_a @ R ) )
=> ( Xs2 = nil_list_a ) ) ).
% listrel_Nil1
thf(fact_331_listrel__Nil1,axiom,
! [Xs2: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ nil_state_a @ Xs2 ) @ ( listre8205798609706990835tate_a @ R ) )
=> ( Xs2 = nil_state_a ) ) ).
% listrel_Nil1
thf(fact_332_listrel__Nil1,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs2 ) @ ( listrel_a_a @ R ) )
=> ( Xs2 = nil_a ) ) ).
% listrel_Nil1
thf(fact_333_listrel_ONil,axiom,
! [R: set_Pr2070066670564046349list_a] : ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ nil_a @ nil_list_a ) @ ( listrel_a_list_a @ R ) ) ).
% listrel.Nil
thf(fact_334_listrel_ONil,axiom,
! [R: set_Pr324718442235990179tate_a] : ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ nil_a @ nil_state_a ) @ ( listrel_a_state_a @ R ) ) ).
% listrel.Nil
thf(fact_335_listrel_ONil,axiom,
! [R: set_Pr8962057229576493569st_a_a] : ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ nil_list_a @ nil_a ) @ ( listrel_list_a_a @ R ) ) ).
% listrel.Nil
thf(fact_336_listrel_ONil,axiom,
! [R: set_Pr4048851178543822343list_a] : ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ nil_list_a ) @ ( listre6772471554020304241list_a @ R ) ) ).
% listrel.Nil
thf(fact_337_listrel_ONil,axiom,
! [R: set_Pr2973361075998125097tate_a] : ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ nil_list_a @ nil_state_a ) @ ( listre3686370835355000083tate_a @ R ) ) ).
% listrel.Nil
thf(fact_338_listrel_ONil,axiom,
! [R: set_Pr1095656866282543721te_a_a] : ( member491016310653805376list_a @ ( produc69382289565035227list_a @ nil_state_a @ nil_a ) @ ( listrel_state_a_a @ R ) ) ).
% listrel.Nil
thf(fact_339_listrel_ONil,axiom,
! [R: set_Pr7453869317534491247list_a] : ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ nil_state_a @ nil_list_a ) @ ( listre1206529783178763665list_a @ R ) ) ).
% listrel.Nil
thf(fact_340_listrel_ONil,axiom,
! [R: set_Pr7022286013061942721tate_a] : ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ nil_state_a @ nil_state_a ) @ ( listre8205798609706990835tate_a @ R ) ) ).
% listrel.Nil
thf(fact_341_listrel_ONil,axiom,
! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).
% listrel.Nil
thf(fact_342_Nil2__notin__lex,axiom,
! [Xs2: list_list_a,R: set_Pr4048851178543822343list_a] :
~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) @ ( lex_list_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_343_Nil2__notin__lex,axiom,
! [Xs2: list_state_a,R: set_Pr7022286013061942721tate_a] :
~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ nil_state_a ) @ ( lex_state_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_344_Nil2__notin__lex,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) @ ( lex_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_345_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_346_Nil__notin__lex,axiom,
! [Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ nil_state_a @ Ys ) @ ( lex_state_a @ R ) ) ).
% Nil_notin_lex
thf(fact_347_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_348_lenlex__irreflexive,axiom,
! [R: set_Pr4048851178543822343list_a,Xs2: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Xs2 ) @ ( lenlex_list_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_349_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Xs2 ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_350_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_351_Nil__lenlex__iff2,axiom,
! [Ns: list_state_a,R: set_Pr7022286013061942721tate_a] :
~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Ns @ nil_state_a ) @ ( lenlex_state_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_352_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_353_listrel__Cons2,axiom,
! [Xs2: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X5: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X5 @ Xs ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_354_listrel__Cons2,axiom,
! [Xs2: list_a,Y: state_a,Ys: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ ( cons_state_a @ Y @ Ys ) ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [X5: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X5 @ Xs ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X5 @ Y ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs @ Ys ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_355_listrel__Cons2,axiom,
! [Xs2: list_list_a,Y: a,Ys: list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) @ ( listrel_list_a_a @ R ) )
=> ~ ! [X5: list_a,Xs: list_list_a] :
( ( Xs2
= ( cons_list_a @ X5 @ Xs ) )
=> ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X5 @ Y ) @ R )
=> ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ Ys ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_356_listrel__Cons2,axiom,
! [Xs2: list_state_a,Y: a,Ys: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) @ ( listrel_state_a_a @ R ) )
=> ~ ! [X5: state_a,Xs: list_state_a] :
( ( Xs2
= ( cons_state_a @ X5 @ Xs ) )
=> ( ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ X5 @ Y ) @ R )
=> ~ ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs @ Ys ) @ ( listrel_state_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_357_listrel__Cons2,axiom,
! [Xs2: list_a,Y: list_a,Ys: list_list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs2 @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_a_list_a @ R ) )
=> ~ ! [X5: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X5 @ Xs ) )
=> ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X5 @ Y ) @ R )
=> ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ Ys ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_358_listrel__Cons2,axiom,
! [Xs2: list_a,Y: current_a,Ys: list_current_a,R: set_Pr3922190185655878585rent_a] :
( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs2 @ ( cons_current_a @ Y @ Ys ) ) @ ( listrel_a_current_a @ R ) )
=> ~ ! [X5: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X5 @ Xs ) )
=> ( ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ X5 @ Y ) @ R )
=> ~ ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs @ Ys ) @ ( listrel_a_current_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_359_listrel__Cons2,axiom,
! [Xs2: list_a,Y: idle_a,Ys: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs2 @ ( cons_idle_a @ Y @ Ys ) ) @ ( listrel_a_idle_a @ R ) )
=> ~ ! [X5: a,Xs: list_a] :
( ( Xs2
= ( cons_a @ X5 @ Xs ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X5 @ Y ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs @ Ys ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_360_listrel__Cons2,axiom,
! [Xs2: list_state_a,Y: list_a,Ys: list_list_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs2 @ ( cons_list_a @ Y @ Ys ) ) @ ( listre1206529783178763665list_a @ R ) )
=> ~ ! [X5: state_a,Xs: list_state_a] :
( ( Xs2
= ( cons_state_a @ X5 @ Xs ) )
=> ( ( member6025650317227102160list_a @ ( produc2472116993945826411list_a @ X5 @ Y ) @ R )
=> ~ ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs @ Ys ) @ ( listre1206529783178763665list_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_361_listrel__Cons2,axiom,
! [Xs2: list_list_a,Y: state_a,Ys: list_state_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs2 @ ( cons_state_a @ Y @ Ys ) ) @ ( listre3686370835355000083tate_a @ R ) )
=> ~ ! [X5: list_a,Xs: list_list_a] :
( ( Xs2
= ( cons_list_a @ X5 @ Xs ) )
=> ( ( member1038697612344694410tate_a @ ( produc4951958046122062829tate_a @ X5 @ Y ) @ R )
=> ~ ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs @ Ys ) @ ( listre3686370835355000083tate_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_362_listrel__Cons2,axiom,
! [Xs2: list_state_a,Y: state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ ( cons_state_a @ Y @ Ys ) ) @ ( listre8205798609706990835tate_a @ R ) )
=> ~ ! [X5: state_a,Xs: list_state_a] :
( ( Xs2
= ( cons_state_a @ X5 @ Xs ) )
=> ( ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ X5 @ Y ) @ R )
=> ~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs @ Ys ) @ ( listre8205798609706990835tate_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_363_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs2: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y5: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_364_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs2: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [Y5: state_a,Ys2: list_state_a] :
( ( Xs2
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ Y @ Y5 ) @ R )
=> ~ ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Ys @ Ys2 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_365_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs2: list_list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_list_a @ R ) )
=> ~ ! [Y5: list_a,Ys2: list_list_a] :
( ( Xs2
= ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ Y @ Y5 ) @ R )
=> ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Ys @ Ys2 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_366_listrel__Cons1,axiom,
! [Y: list_a,Ys: list_list_a,Xs2: list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ ( cons_list_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_list_a_a @ R ) )
=> ~ ! [Y5: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ Y @ Y5 ) @ R )
=> ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Ys @ Ys2 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_367_listrel__Cons1,axiom,
! [Y: state_a,Ys: list_state_a,Xs2: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ ( cons_state_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_state_a_a @ R ) )
=> ~ ! [Y5: a,Ys2: list_a] :
( ( Xs2
= ( cons_a @ Y5 @ Ys2 ) )
=> ( ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ Y @ Y5 ) @ R )
=> ~ ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Ys @ Ys2 ) @ ( listrel_state_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_368_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs2: list_current_a,R: set_Pr3922190185655878585rent_a] :
( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_current_a @ R ) )
=> ~ ! [Y5: current_a,Ys2: list_current_a] :
( ( Xs2
= ( cons_current_a @ Y5 @ Ys2 ) )
=> ( ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ Y @ Y5 ) @ R )
=> ~ ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Ys @ Ys2 ) @ ( listrel_a_current_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_369_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs2: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_idle_a @ R ) )
=> ~ ! [Y5: idle_a,Ys2: list_idle_a] :
( ( Xs2
= ( cons_idle_a @ Y5 @ Ys2 ) )
=> ( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ Y @ Y5 ) @ R )
=> ~ ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Ys @ Ys2 ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_370_listrel__Cons1,axiom,
! [Y: list_a,Ys: list_list_a,Xs2: list_state_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ ( cons_list_a @ Y @ Ys ) @ Xs2 ) @ ( listre3686370835355000083tate_a @ R ) )
=> ~ ! [Y5: state_a,Ys2: list_state_a] :
( ( Xs2
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member1038697612344694410tate_a @ ( produc4951958046122062829tate_a @ Y @ Y5 ) @ R )
=> ~ ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Ys @ Ys2 ) @ ( listre3686370835355000083tate_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_371_listrel__Cons1,axiom,
! [Y: state_a,Ys: list_state_a,Xs2: list_list_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ ( cons_state_a @ Y @ Ys ) @ Xs2 ) @ ( listre1206529783178763665list_a @ R ) )
=> ~ ! [Y5: list_a,Ys2: list_list_a] :
( ( Xs2
= ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ( member6025650317227102160list_a @ ( produc2472116993945826411list_a @ Y @ Y5 ) @ R )
=> ~ ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Ys @ Ys2 ) @ ( listre1206529783178763665list_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_372_listrel__Cons1,axiom,
! [Y: state_a,Ys: list_state_a,Xs2: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( cons_state_a @ Y @ Ys ) @ Xs2 ) @ ( listre8205798609706990835tate_a @ R ) )
=> ~ ! [Y5: state_a,Ys2: list_state_a] :
( ( Xs2
= ( cons_state_a @ Y5 @ Ys2 ) )
=> ( ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ Y @ Y5 ) @ R )
=> ~ ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Ys @ Ys2 ) @ ( listre8205798609706990835tate_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_373_listrel_OCons,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_374_listrel_OCons,axiom,
! [X: a,Y: state_a,R: set_Pr324718442235990179tate_a,Xs2: list_a,Ys: list_state_a] :
( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) @ R )
=> ( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ Ys ) @ ( listrel_a_state_a @ R ) )
=> ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ ( cons_a @ X @ Xs2 ) @ ( cons_state_a @ Y @ Ys ) ) @ ( listrel_a_state_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_375_listrel_OCons,axiom,
! [X: a,Y: list_a,R: set_Pr2070066670564046349list_a,Xs2: list_a,Ys: list_list_a] :
( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X @ Y ) @ R )
=> ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs2 @ Ys ) @ ( listrel_a_list_a @ R ) )
=> ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_a_list_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_376_listrel_OCons,axiom,
! [X: list_a,Y: a,R: set_Pr8962057229576493569st_a_a,Xs2: list_list_a,Ys: list_a] :
( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X @ Y ) @ R )
=> ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs2 @ Ys ) @ ( listrel_list_a_a @ R ) )
=> ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_list_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_377_listrel_OCons,axiom,
! [X: state_a,Y: a,R: set_Pr1095656866282543721te_a_a,Xs2: list_state_a,Ys: list_a] :
( ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ X @ Y ) @ R )
=> ( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs2 @ Ys ) @ ( listrel_state_a_a @ R ) )
=> ( member491016310653805376list_a @ ( produc69382289565035227list_a @ ( cons_state_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_state_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_378_listrel_OCons,axiom,
! [X: a,Y: current_a,R: set_Pr3922190185655878585rent_a,Xs2: list_a,Ys: list_current_a] :
( ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ X @ Y ) @ R )
=> ( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs2 @ Ys ) @ ( listrel_a_current_a @ R ) )
=> ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ ( cons_a @ X @ Xs2 ) @ ( cons_current_a @ Y @ Ys ) ) @ ( listrel_a_current_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_379_listrel_OCons,axiom,
! [X: a,Y: idle_a,R: set_Pr7225968907286947725idle_a,Xs2: list_a,Ys: list_idle_a] :
( ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X @ Y ) @ R )
=> ( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs2 @ Ys ) @ ( listrel_a_idle_a @ R ) )
=> ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ ( cons_a @ X @ Xs2 ) @ ( cons_idle_a @ Y @ Ys ) ) @ ( listrel_a_idle_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_380_listrel_OCons,axiom,
! [X: list_a,Y: state_a,R: set_Pr2973361075998125097tate_a,Xs2: list_list_a,Ys: list_state_a] :
( ( member1038697612344694410tate_a @ ( produc4951958046122062829tate_a @ X @ Y ) @ R )
=> ( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs2 @ Ys ) @ ( listre3686370835355000083tate_a @ R ) )
=> ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_state_a @ Y @ Ys ) ) @ ( listre3686370835355000083tate_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_381_listrel_OCons,axiom,
! [X: state_a,Y: list_a,R: set_Pr7453869317534491247list_a,Xs2: list_state_a,Ys: list_list_a] :
( ( member6025650317227102160list_a @ ( produc2472116993945826411list_a @ X @ Y ) @ R )
=> ( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs2 @ Ys ) @ ( listre1206529783178763665list_a @ R ) )
=> ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ ( cons_state_a @ X @ Xs2 ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listre1206529783178763665list_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_382_listrel_OCons,axiom,
! [X: state_a,Y: state_a,R: set_Pr7022286013061942721tate_a,Xs2: list_state_a,Ys: list_state_a] :
( ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ X @ Y ) @ R )
=> ( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( listre8205798609706990835tate_a @ R ) )
=> ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( cons_state_a @ X @ Xs2 ) @ ( cons_state_a @ Y @ Ys ) ) @ ( listre8205798609706990835tate_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_383_is__empty__idle_Ocases,axiom,
! [X: idle_a] :
~ ! [Stack2: stack_a,Uu: nat] :
( X
!= ( idle_a3 @ Stack2 @ Uu ) ) ).
% is_empty_idle.cases
thf(fact_384_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X6: a,Y6: a,Xs3: list_a,Ys3: list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_a @ Y6 @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y6 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_385_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a ) )
| ? [X6: a,Y6: state_a,Xs3: list_a,Ys3: list_state_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y6 @ Ys3 ) )
& ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X6 @ Y6 ) @ R )
& ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs3 @ Ys3 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_386_listrel_Osimps,axiom,
! [A1: list_a,A22: list_list_a,R: set_Pr2070066670564046349list_a] :
( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ A1 @ A22 ) @ ( listrel_a_list_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_list_a ) )
| ? [X6: a,Y6: list_a,Xs3: list_a,Ys3: list_list_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_list_a @ Y6 @ Ys3 ) )
& ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X6 @ Y6 ) @ R )
& ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs3 @ Ys3 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_387_listrel_Osimps,axiom,
! [A1: list_list_a,A22: list_a,R: set_Pr8962057229576493569st_a_a] :
( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ A1 @ A22 ) @ ( listrel_list_a_a @ R ) )
= ( ( ( A1 = nil_list_a )
& ( A22 = nil_a ) )
| ? [X6: list_a,Y6: a,Xs3: list_list_a,Ys3: list_a] :
( ( A1
= ( cons_list_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_a @ Y6 @ Ys3 ) )
& ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X6 @ Y6 ) @ R )
& ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs3 @ Ys3 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_388_listrel_Osimps,axiom,
! [A1: list_state_a,A22: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ A1 @ A22 ) @ ( listrel_state_a_a @ R ) )
= ( ( ( A1 = nil_state_a )
& ( A22 = nil_a ) )
| ? [X6: state_a,Y6: a,Xs3: list_state_a,Ys3: list_a] :
( ( A1
= ( cons_state_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_a @ Y6 @ Ys3 ) )
& ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ X6 @ Y6 ) @ R )
& ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs3 @ Ys3 ) @ ( listrel_state_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_389_listrel_Osimps,axiom,
! [A1: list_a,A22: list_current_a,R: set_Pr3922190185655878585rent_a] :
( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ A1 @ A22 ) @ ( listrel_a_current_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_current_a ) )
| ? [X6: a,Y6: current_a,Xs3: list_a,Ys3: list_current_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_current_a @ Y6 @ Ys3 ) )
& ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ X6 @ Y6 ) @ R )
& ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs3 @ Ys3 ) @ ( listrel_a_current_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_390_listrel_Osimps,axiom,
! [A1: list_a,A22: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ A1 @ A22 ) @ ( listrel_a_idle_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_idle_a ) )
| ? [X6: a,Y6: idle_a,Xs3: list_a,Ys3: list_idle_a] :
( ( A1
= ( cons_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_idle_a @ Y6 @ Ys3 ) )
& ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ X6 @ Y6 ) @ R )
& ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs3 @ Ys3 ) @ ( listrel_a_idle_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_391_listrel_Osimps,axiom,
! [A1: list_list_a,A22: list_state_a,R: set_Pr2973361075998125097tate_a] :
( ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ A1 @ A22 ) @ ( listre3686370835355000083tate_a @ R ) )
= ( ( ( A1 = nil_list_a )
& ( A22 = nil_state_a ) )
| ? [X6: list_a,Y6: state_a,Xs3: list_list_a,Ys3: list_state_a] :
( ( A1
= ( cons_list_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y6 @ Ys3 ) )
& ( member1038697612344694410tate_a @ ( produc4951958046122062829tate_a @ X6 @ Y6 ) @ R )
& ( member5812885074326414752tate_a @ ( produc1026893858372407043tate_a @ Xs3 @ Ys3 ) @ ( listre3686370835355000083tate_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_392_listrel_Osimps,axiom,
! [A1: list_state_a,A22: list_list_a,R: set_Pr7453869317534491247list_a] :
( ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ A1 @ A22 ) @ ( listre1206529783178763665list_a @ R ) )
= ( ( ( A1 = nil_state_a )
& ( A22 = nil_list_a ) )
| ? [X6: state_a,Y6: list_a,Xs3: list_state_a,Ys3: list_list_a] :
( ( A1
= ( cons_state_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_list_a @ Y6 @ Ys3 ) )
& ( member6025650317227102160list_a @ ( produc2472116993945826411list_a @ X6 @ Y6 ) @ R )
& ( member6299280713945060294list_a @ ( produc3561183582102369list_a @ Xs3 @ Ys3 ) @ ( listre1206529783178763665list_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_393_listrel_Osimps,axiom,
! [A1: list_state_a,A22: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ A1 @ A22 ) @ ( listre8205798609706990835tate_a @ R ) )
= ( ( ( A1 = nil_state_a )
& ( A22 = nil_state_a ) )
| ? [X6: state_a,Y6: state_a,Xs3: list_state_a,Ys3: list_state_a] :
( ( A1
= ( cons_state_a @ X6 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y6 @ Ys3 ) )
& ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ X6 @ Y6 ) @ R )
& ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs3 @ Ys3 ) @ ( listre8205798609706990835tate_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_394_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_395_first__take,axiom,
! [Stack: stack_list_a] :
( ~ ( type_i4148797458121419353list_a @ Stack )
=> ( ( cons_list_a @ ( first_list_a @ Stack ) @ nil_list_a )
= ( take_list_a @ one_one_nat @ ( stack_list_list_a @ Stack ) ) ) ) ).
% first_take
thf(fact_396_first__take,axiom,
! [Stack: stack_state_a] :
( ~ ( type_i6544059586428101441tate_a @ Stack )
=> ( ( cons_state_a @ ( first_state_a @ Stack ) @ nil_state_a )
= ( take_state_a @ one_one_nat @ ( stack_list_state_a @ Stack ) ) ) ) ).
% first_take
thf(fact_397_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_398_take__hd,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( take_list_a @ ( suc @ zero_zero_nat ) @ Xs2 )
= ( cons_list_a @ ( hd_list_a @ Xs2 ) @ nil_list_a ) ) ) ).
% take_hd
thf(fact_399_take__hd,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( take_state_a @ ( suc @ zero_zero_nat ) @ Xs2 )
= ( cons_state_a @ ( hd_state_a @ Xs2 ) @ nil_state_a ) ) ) ).
% take_hd
thf(fact_400_take__hd,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( take_a @ ( suc @ zero_zero_nat ) @ Xs2 )
= ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) ) ) ).
% take_hd
thf(fact_401_lex__take__index,axiom,
! [Xs2: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys ) @ ( lex_list_a @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( take_list_a @ I @ Xs2 )
= ( take_list_a @ I @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs2 @ I ) @ ( nth_list_a @ Ys @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_402_lex__take__index,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lex_a @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I @ Xs2 )
= ( take_a @ I @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs2 @ I ) @ ( nth_a @ Ys @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_403_lex__take__index,axiom,
! [Xs2: list_state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( lex_state_a @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ( ( take_state_a @ I @ Xs2 )
= ( take_state_a @ I @ Ys ) )
=> ~ ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ ( nth_state_a @ Xs2 @ I ) @ ( nth_state_a @ Ys @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_404_pop__list__length,axiom,
! [Stack: stack_state_a] :
( ~ ( type_i6544059586428101441tate_a @ Stack )
=> ( ( suc @ ( size_s4663437186682225508tate_a @ ( stack_list_state_a @ ( pop_state_a4 @ Stack ) ) ) )
= ( size_s4663437186682225508tate_a @ ( stack_list_state_a @ Stack ) ) ) ) ).
% pop_list_length
thf(fact_405_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_406_take__hd_H,axiom,
! [Ys: list_list_a,X: list_a,Xs2: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( ( take_list_a @ ( size_s349497388124573686list_a @ Ys ) @ ( cons_list_a @ X @ Xs2 ) )
= ( take_list_a @ ( suc @ ( size_s349497388124573686list_a @ Xs2 ) ) @ Ys ) )
=> ( ( hd_list_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_407_take__hd_H,axiom,
! [Ys: list_state_a,X: state_a,Xs2: list_state_a] :
( ( Ys != nil_state_a )
=> ( ( ( take_state_a @ ( size_s4663437186682225508tate_a @ Ys ) @ ( cons_state_a @ X @ Xs2 ) )
= ( take_state_a @ ( suc @ ( size_s4663437186682225508tate_a @ Xs2 ) ) @ Ys ) )
=> ( ( hd_state_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_408_take__hd_H,axiom,
! [Ys: list_a,X: a,Xs2: list_a] :
( ( Ys != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys ) @ ( cons_a @ X @ Xs2 ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs2 ) ) @ Ys ) )
=> ( ( hd_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_409_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_410_length__code,axiom,
( size_s4663437186682225508tate_a
= ( gen_length_state_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_411_listrel__iff__nth,axiom,
! [Xs2: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys ) @ ( listre6772471554020304241list_a @ R ) )
= ( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs2 @ N3 ) @ ( nth_list_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_412_listrel__iff__nth,axiom,
! [Xs2: list_a_a_o,Ys: list_list_a,R: set_Pr7667830121170965149list_a] :
( ( member476489107781553766list_a @ ( produc106582675883131181list_a @ Xs2 @ Ys ) @ ( listrel_a_a_o_list_a @ R ) )
= ( ( ( size_size_list_a_a_o @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a_a_o @ Xs2 ) )
=> ( member5490092443863680486list_a @ ( produc8111569692950616493list_a @ ( nth_a_a_o @ Xs2 @ N3 ) @ ( nth_list_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_413_listrel__iff__nth,axiom,
! [Xs2: list_a,Ys: list_current_a,R: set_Pr3922190185655878585rent_a] :
( ( member1785988321452022018rent_a @ ( produc5645532383025311945rent_a @ Xs2 @ Ys ) @ ( listrel_a_current_a @ R ) )
= ( ( ( size_size_list_a @ Xs2 )
= ( size_s190638326304920994rent_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member8183269688883217154rent_a @ ( produc8503237746132909001rent_a @ ( nth_a @ Xs2 @ N3 ) @ ( nth_current_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_414_listrel__iff__nth,axiom,
! [Xs2: list_a,Ys: list_idle_a,R: set_Pr7225968907286947725idle_a] :
( ( member9073500730917845590idle_a @ ( produc4587228114721326877idle_a @ Xs2 @ Ys ) @ ( listrel_a_idle_a @ R ) )
= ( ( ( size_size_list_a @ Xs2 )
= ( size_s8477453896275132790idle_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member3794253416837758678idle_a @ ( produc1265230069547855005idle_a @ ( nth_a @ Xs2 @ N3 ) @ ( nth_idle_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_415_listrel__iff__nth,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) )
= ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs2 @ N3 ) @ ( nth_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_416_listrel__iff__nth,axiom,
! [Xs2: list_state_a,Ys: list_a,R: set_Pr1095656866282543721te_a_a] :
( ( member491016310653805376list_a @ ( produc69382289565035227list_a @ Xs2 @ Ys ) @ ( listrel_state_a_a @ R ) )
= ( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( member196721918138178250te_a_a @ ( produc6693812758571628389te_a_a @ ( nth_state_a @ Xs2 @ N3 ) @ ( nth_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_417_listrel__iff__nth,axiom,
! [Xs2: list_state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( listre8205798609706990835tate_a @ R ) )
= ( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( member8645189796577445642tate_a @ ( produc4227292171933100441tate_a @ ( nth_state_a @ Xs2 @ N3 ) @ ( nth_state_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_418_listrel__iff__nth,axiom,
! [Xs2: list_a,Ys: list_state_a,R: set_Pr324718442235990179tate_a] :
( ( member7736023580690228378tate_a @ ( produc4137809223961881341tate_a @ Xs2 @ Ys ) @ ( listrel_a_state_a @ R ) )
= ( ( ( size_size_list_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ ( nth_a @ Xs2 @ N3 ) @ ( nth_state_a @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_419_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_a @ 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_420_first__take__pop,axiom,
! [Stack: stack_state_a,X: nat] :
( ~ ( type_i6544059586428101441tate_a @ Stack )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( cons_state_a @ ( first_state_a @ Stack ) @ ( take_state_a @ ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) ) @ ( stack_list_state_a @ ( pop_state_a4 @ Stack ) ) ) )
= ( take_state_a @ X @ ( stack_list_state_a @ Stack ) ) ) ) ) ).
% first_take_pop
thf(fact_421_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_422_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_423_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_424_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_425_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_426_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_427_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_428_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_429_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_430_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_431_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_432_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_433_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_434_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_435_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_436_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_437_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_438_nth__Cons__Suc,axiom,
! [X: list_a,Xs2: list_list_a,N: nat] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_list_a @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_439_nth__Cons__Suc,axiom,
! [X: state_a,Xs2: list_state_a,N: nat] :
( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_state_a @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_440_nth__Cons__Suc,axiom,
! [X: a,Xs2: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_a @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_441_nth__Cons__0,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_442_nth__Cons__0,axiom,
! [X: state_a,Xs2: list_state_a] :
( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_443_nth__Cons__0,axiom,
! [X: a,Xs2: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_444_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_445_take__Suc__Cons,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( take_list_a @ ( suc @ N ) @ ( cons_list_a @ X @ Xs2 ) )
= ( cons_list_a @ X @ ( take_list_a @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_446_take__Suc__Cons,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( take_state_a @ ( suc @ N ) @ ( cons_state_a @ X @ Xs2 ) )
= ( cons_state_a @ X @ ( take_state_a @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_447_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs2 ) )
= ( cons_a @ X @ ( take_a @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_448_nth__take,axiom,
! [I3: nat,N: nat,Xs2: list_state_a] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_state_a @ ( take_state_a @ N @ Xs2 ) @ I3 )
= ( nth_state_a @ Xs2 @ I3 ) ) ) ).
% nth_take
thf(fact_449_nth__take,axiom,
! [I3: nat,N: nat,Xs2: list_a] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_a @ ( take_a @ N @ Xs2 ) @ I3 )
= ( nth_a @ Xs2 @ I3 ) ) ) ).
% nth_take
thf(fact_450_Stack__Proof_Osize__push,axiom,
! [X: a,Stack: stack_a] :
( ( size_size_stack_a @ ( push_a4 @ X @ Stack ) )
= ( suc @ ( size_size_stack_a @ Stack ) ) ) ).
% Stack_Proof.size_push
thf(fact_451_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_452_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_453_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_454_nth__Cons__pos,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ N )
= ( nth_list_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_455_nth__Cons__pos,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ N )
= ( nth_state_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_456_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_457_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_458_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_459_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_460_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_461_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_462_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_463_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_464_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_465_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_466_nth__Cons_H,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ N )
= ( nth_list_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_467_nth__Cons_H,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ N )
= ( nth_state_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_468_nth__Cons_H,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_469_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_470_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_471_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_472_Suc__sub,axiom,
! [N: nat,M2: nat] :
( ( ( suc @ N )
= M2 )
=> ( N
= ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_473_gen__length__code_I2_J,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( gen_length_list_a @ N @ ( cons_list_a @ X @ Xs2 ) )
= ( gen_length_list_a @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_474_gen__length__code_I2_J,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( gen_length_state_a @ N @ ( cons_state_a @ X @ Xs2 ) )
= ( gen_length_state_a @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_475_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs2 ) )
= ( gen_length_a @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_476_nth__non__equal__first__eq,axiom,
! [X: list_a,Y: list_a,Xs2: list_list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_a @ ( cons_list_a @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_list_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_477_nth__non__equal__first__eq,axiom,
! [X: state_a,Y: state_a,Xs2: list_state_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_state_a @ ( cons_state_a @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_state_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_478_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs2: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_a @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_479_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_480_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_481_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_482_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_483_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_484_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_485_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_486_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_487_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_488_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_489_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_490_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_491_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X5: nat,Y5: nat] :
( ( P @ X5 @ Y5 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y5 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_492_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_493_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_494_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_495_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_496_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_497_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_498_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_499_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_500_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_501_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_502_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_503_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_504_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_505_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_506_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_507_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_508_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_509_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_510_less__trans__Suc,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_511_less__Suc__induct,axiom,
! [I3: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_512_strict__inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_513_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_514_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_515_list__eq__iff__nth__eq,axiom,
( ( ^ [Y7: list_a,Z2: list_a] : ( Y7 = Z2 ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I4 )
= ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_516_list__eq__iff__nth__eq,axiom,
( ( ^ [Y7: list_state_a,Z2: list_state_a] : ( Y7 = Z2 ) )
= ( ^ [Xs3: list_state_a,Ys3: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs3 )
= ( size_s4663437186682225508tate_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s4663437186682225508tate_a @ Xs3 ) )
=> ( ( nth_state_a @ Xs3 @ I4 )
= ( nth_state_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_517_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X7: a] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_518_Skolem__list__nth,axiom,
! [K: nat,P: nat > state_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X7: state_a] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs3: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_state_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_519_nth__equalityI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I )
= ( nth_a @ Ys @ I ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_520_nth__equalityI,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( nth_state_a @ Xs2 @ I )
= ( nth_state_a @ Ys @ I ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_521_take__Cons_H,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_list_a @ N @ ( cons_list_a @ X @ Xs2 ) )
= nil_list_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_list_a @ N @ ( cons_list_a @ X @ Xs2 ) )
= ( cons_list_a @ X @ ( take_list_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_522_take__Cons_H,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_state_a @ N @ ( cons_state_a @ X @ Xs2 ) )
= nil_state_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_state_a @ N @ ( cons_state_a @ X @ Xs2 ) )
= ( cons_state_a @ X @ ( take_state_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_523_take__Cons_H,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs2 ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs2 ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_524_lift__Suc__mono__less,axiom,
! [F2: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_525_lift__Suc__mono__less,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_526_lift__Suc__mono__less__iff,axiom,
! [F2: nat > num,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F2 @ N ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_527_lift__Suc__mono__less__iff,axiom,
! [F2: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_528_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_529_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_530_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_531_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_532_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_533_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_534_Suc__length__conv,axiom,
! [N: nat,Xs2: list_list_a] :
( ( ( suc @ N )
= ( size_s349497388124573686list_a @ Xs2 ) )
= ( ? [Y6: list_a,Ys3: list_list_a] :
( ( Xs2
= ( cons_list_a @ Y6 @ Ys3 ) )
& ( ( size_s349497388124573686list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_535_Suc__length__conv,axiom,
! [N: nat,Xs2: list_state_a] :
( ( ( suc @ N )
= ( size_s4663437186682225508tate_a @ Xs2 ) )
= ( ? [Y6: state_a,Ys3: list_state_a] :
( ( Xs2
= ( cons_state_a @ Y6 @ Ys3 ) )
& ( ( size_s4663437186682225508tate_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_536_Suc__length__conv,axiom,
! [N: nat,Xs2: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs2 ) )
= ( ? [Y6: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ Y6 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_537_length__Suc__conv,axiom,
! [Xs2: list_list_a,N: nat] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: list_a,Ys3: list_list_a] :
( ( Xs2
= ( cons_list_a @ Y6 @ Ys3 ) )
& ( ( size_s349497388124573686list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_538_length__Suc__conv,axiom,
! [Xs2: list_state_a,N: nat] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: state_a,Ys3: list_state_a] :
( ( Xs2
= ( cons_state_a @ Y6 @ Ys3 ) )
& ( ( size_s4663437186682225508tate_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_539_length__Suc__conv,axiom,
! [Xs2: list_a,N: nat] :
( ( ( size_size_list_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ Y6 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_540_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_541_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_a @ N @ nil_list_a )
= N ) ).
% gen_length_code(1)
thf(fact_542_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_state_a @ N @ nil_state_a )
= N ) ).
% gen_length_code(1)
thf(fact_543_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_544_hd__conv__nth,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( hd_list_a @ Xs2 )
= ( nth_list_a @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_545_hd__conv__nth,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( hd_state_a @ Xs2 )
= ( nth_state_a @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_546_hd__conv__nth,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ Xs2 )
= ( nth_a @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_547_take__1,axiom,
! [X: nat,Y: nat,Xs2: list_a,Ys: list_a] :
( ( ( ord_less_nat @ zero_zero_nat @ X )
& ( ord_less_nat @ zero_zero_nat @ Y ) )
=> ( ( ( take_a @ X @ Xs2 )
= ( take_a @ Y @ Ys ) )
=> ( ( take_a @ one_one_nat @ Xs2 )
= ( take_a @ one_one_nat @ Ys ) ) ) ) ).
% take_1
thf(fact_548_length__one__hd,axiom,
! [Xs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= one_one_nat )
=> ( Xs2
= ( cons_list_a @ ( hd_list_a @ Xs2 ) @ nil_list_a ) ) ) ).
% length_one_hd
thf(fact_549_length__one__hd,axiom,
! [Xs2: list_state_a] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= one_one_nat )
=> ( Xs2
= ( cons_state_a @ ( hd_state_a @ Xs2 ) @ nil_state_a ) ) ) ).
% length_one_hd
thf(fact_550_length__one__hd,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= one_one_nat )
=> ( Xs2
= ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) ) ) ).
% length_one_hd
thf(fact_551_Current_Opop_Oelims,axiom,
! [X: current_list_a,Y: produc603899091129270873list_a] :
( ( ( pop_list_a2 @ 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_a @ Old ) @ ( current_list_a2 @ nil_list_a @ Added @ ( pop_list_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ ( cons_list_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc1356073977093405001list_a @ X5 @ ( current_list_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_552_Current_Opop_Oelims,axiom,
! [X: current_state_a,Y: produc1507450493377391557tate_a] :
( ( ( pop_state_a2 @ X )
= Y )
=> ( ! [Added: nat,Old: stack_state_a,Remained: nat] :
( ( X
= ( current_state_a2 @ nil_state_a @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc8606655205717325309tate_a @ ( first_state_a @ Old ) @ ( current_state_a2 @ nil_state_a @ Added @ ( pop_state_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Added: nat,Old: stack_state_a,Remained: nat] :
( ( X
= ( current_state_a2 @ ( cons_state_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc8606655205717325309tate_a @ X5 @ ( current_state_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_553_Current_Opop_Oelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a2 @ 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 ) ) ) ) )
=> ~ ! [X5: a,Xs: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( Y
!= ( produc8503237746132909001rent_a @ X5 @ ( current_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) ) ) ) ) ).
% Current.pop.elims
thf(fact_554_Idle_Opop_Oelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a3 @ X )
= Y )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( Y
!= ( produc1265230069547855005idle_a @ ( first_a2 @ Stack2 ) @ ( idle_a3 @ ( pop_a4 @ Stack2 ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) ) ) ) ).
% Idle.pop.elims
thf(fact_555_Idle_Opop_Osimps,axiom,
! [Stack: stack_a,StackSize2: nat] :
( ( pop_a3 @ ( idle_a3 @ Stack @ StackSize2 ) )
= ( produc1265230069547855005idle_a @ ( first_a2 @ Stack ) @ ( idle_a3 @ ( pop_a4 @ Stack ) @ ( minus_minus_nat @ StackSize2 @ one_one_nat ) ) ) ) ).
% Idle.pop.simps
thf(fact_556_Current_Opop_Osimps_I1_J,axiom,
! [Added2: nat,Old2: stack_list_a,Remained2: nat] :
( ( pop_list_a2 @ ( current_list_a2 @ nil_list_a @ Added2 @ Old2 @ Remained2 ) )
= ( produc1356073977093405001list_a @ ( first_list_a @ 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_557_Current_Opop_Osimps_I1_J,axiom,
! [Added2: nat,Old2: stack_state_a,Remained2: nat] :
( ( pop_state_a2 @ ( current_state_a2 @ nil_state_a @ Added2 @ Old2 @ Remained2 ) )
= ( produc8606655205717325309tate_a @ ( first_state_a @ Old2 ) @ ( current_state_a2 @ nil_state_a @ Added2 @ ( pop_state_a4 @ Old2 ) @ ( minus_minus_nat @ Remained2 @ one_one_nat ) ) ) ) ).
% Current.pop.simps(1)
thf(fact_558_Current_Opop_Osimps_I1_J,axiom,
! [Added2: nat,Old2: stack_a,Remained2: nat] :
( ( pop_a2 @ ( 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_559_length__Cons,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ ( cons_list_a @ X @ Xs2 ) )
= ( suc @ ( size_s349497388124573686list_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_560_length__Cons,axiom,
! [X: state_a,Xs2: list_state_a] :
( ( size_s4663437186682225508tate_a @ ( cons_state_a @ X @ Xs2 ) )
= ( suc @ ( size_s4663437186682225508tate_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_561_length__Cons,axiom,
! [X: a,Xs2: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs2 ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_562_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_563_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_564_first__take__tl,axiom,
! [Big: stack_list_a,Count: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6256786526647004292list_a @ Big ) )
=> ( ( cons_list_a @ ( first_list_a @ Big ) @ ( take_list_a @ Count @ ( tl_list_a @ ( stack_list_list_a @ Big ) ) ) )
= ( take_list_a @ ( suc @ Count ) @ ( stack_list_list_a @ Big ) ) ) ) ).
% first_take_tl
thf(fact_565_first__take__tl,axiom,
! [Big: stack_state_a,Count: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s2616604874200068438tate_a @ Big ) )
=> ( ( cons_state_a @ ( first_state_a @ Big ) @ ( take_state_a @ Count @ ( tl_state_a @ ( stack_list_state_a @ Big ) ) ) )
= ( take_state_a @ ( suc @ Count ) @ ( stack_list_state_a @ Big ) ) ) ) ).
% first_take_tl
thf(fact_566_first__take__tl,axiom,
! [Big: stack_a,Count: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_stack_a @ Big ) )
=> ( ( cons_a @ ( first_a2 @ Big ) @ ( take_a @ Count @ ( tl_a @ ( stack_list_a2 @ Big ) ) ) )
= ( take_a @ ( suc @ Count ) @ ( stack_list_a2 @ Big ) ) ) ) ).
% first_take_tl
thf(fact_567_pop__tl,axiom,
! [Stack: stack_a] :
( ( stack_list_a2 @ ( pop_a4 @ Stack ) )
= ( tl_a @ ( stack_list_a2 @ Stack ) ) ) ).
% pop_tl
thf(fact_568_length__tl,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_569_length__tl,axiom,
! [Xs2: list_state_a] :
( ( size_s4663437186682225508tate_a @ ( tl_state_a @ Xs2 ) )
= ( minus_minus_nat @ ( size_s4663437186682225508tate_a @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_570_hd__Cons__tl,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs2 ) @ ( tl_list_a @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_571_hd__Cons__tl,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( cons_state_a @ ( hd_state_a @ Xs2 ) @ ( tl_state_a @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_572_hd__Cons__tl,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs2 ) @ ( tl_a @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_573_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_574_list_Ocollapse,axiom,
! [List: list_state_a] :
( ( List != nil_state_a )
=> ( ( cons_state_a @ ( hd_state_a @ List ) @ ( tl_state_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_575_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_576_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_577_list_Osel_I3_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( tl_list_a @ ( cons_list_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_578_list_Osel_I3_J,axiom,
! [X21: state_a,X22: list_state_a] :
( ( tl_state_a @ ( cons_state_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_579_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_580_cons__tl,axiom,
! [X: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( ( cons_list_a @ X @ Xs2 )
= Ys )
=> ( Xs2
= ( tl_list_a @ Ys ) ) ) ).
% cons_tl
thf(fact_581_cons__tl,axiom,
! [X: state_a,Xs2: list_state_a,Ys: list_state_a] :
( ( ( cons_state_a @ X @ Xs2 )
= Ys )
=> ( Xs2
= ( tl_state_a @ Ys ) ) ) ).
% cons_tl
thf(fact_582_cons__tl,axiom,
! [X: a,Xs2: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs2 )
= Ys )
=> ( Xs2
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_583_list_Osel_I2_J,axiom,
( ( tl_list_a @ nil_list_a )
= nil_list_a ) ).
% list.sel(2)
thf(fact_584_list_Osel_I2_J,axiom,
( ( tl_state_a @ nil_state_a )
= nil_state_a ) ).
% list.sel(2)
thf(fact_585_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_586_tl__Nil,axiom,
! [Xs2: list_list_a] :
( ( ( tl_list_a @ Xs2 )
= nil_list_a )
= ( ( Xs2 = nil_list_a )
| ? [X6: list_a] :
( Xs2
= ( cons_list_a @ X6 @ nil_list_a ) ) ) ) ).
% tl_Nil
thf(fact_587_tl__Nil,axiom,
! [Xs2: list_state_a] :
( ( ( tl_state_a @ Xs2 )
= nil_state_a )
= ( ( Xs2 = nil_state_a )
| ? [X6: state_a] :
( Xs2
= ( cons_state_a @ X6 @ nil_state_a ) ) ) ) ).
% tl_Nil
thf(fact_588_tl__Nil,axiom,
! [Xs2: list_a] :
( ( ( tl_a @ Xs2 )
= nil_a )
= ( ( Xs2 = nil_a )
| ? [X6: a] :
( Xs2
= ( cons_a @ X6 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_589_Nil__tl,axiom,
! [Xs2: list_list_a] :
( ( nil_list_a
= ( tl_list_a @ Xs2 ) )
= ( ( Xs2 = nil_list_a )
| ? [X6: list_a] :
( Xs2
= ( cons_list_a @ X6 @ nil_list_a ) ) ) ) ).
% Nil_tl
thf(fact_590_Nil__tl,axiom,
! [Xs2: list_state_a] :
( ( nil_state_a
= ( tl_state_a @ Xs2 ) )
= ( ( Xs2 = nil_state_a )
| ? [X6: state_a] :
( Xs2
= ( cons_state_a @ X6 @ nil_state_a ) ) ) ) ).
% Nil_tl
thf(fact_591_Nil__tl,axiom,
! [Xs2: list_a] :
( ( nil_a
= ( tl_a @ Xs2 ) )
= ( ( Xs2 = nil_a )
| ? [X6: a] :
( Xs2
= ( cons_a @ X6 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_592_take__tl,axiom,
! [N: nat,Xs2: list_a] :
( ( take_a @ N @ ( tl_a @ Xs2 ) )
= ( tl_a @ ( take_a @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_tl
thf(fact_593_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_594_list_Oexpand,axiom,
! [List: list_state_a,List2: list_state_a] :
( ( ( List = nil_state_a )
= ( List2 = nil_state_a ) )
=> ( ( ( List != nil_state_a )
=> ( ( List2 != nil_state_a )
=> ( ( ( hd_state_a @ List )
= ( hd_state_a @ List2 ) )
& ( ( tl_state_a @ List )
= ( tl_state_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_595_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_596_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_597_list_Oexhaust__sel,axiom,
! [List: list_state_a] :
( ( List != nil_state_a )
=> ( List
= ( cons_state_a @ ( hd_state_a @ List ) @ ( tl_state_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_598_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_599_tl__take,axiom,
! [N: nat,Xs2: list_a] :
( ( tl_a @ ( take_a @ N @ Xs2 ) )
= ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_a @ Xs2 ) ) ) ).
% tl_take
thf(fact_600_nth__tl,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( tl_a @ Xs2 ) ) )
=> ( ( nth_a @ ( tl_a @ Xs2 ) @ N )
= ( nth_a @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_601_nth__tl,axiom,
! [N: nat,Xs2: list_state_a] :
( ( ord_less_nat @ N @ ( size_s4663437186682225508tate_a @ ( tl_state_a @ Xs2 ) ) )
=> ( ( nth_state_a @ ( tl_state_a @ Xs2 ) @ N )
= ( nth_state_a @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_602_take__Suc,axiom,
! [Xs2: list_list_a,N: nat] :
( ( Xs2 != nil_list_a )
=> ( ( take_list_a @ ( suc @ N ) @ Xs2 )
= ( cons_list_a @ ( hd_list_a @ Xs2 ) @ ( take_list_a @ N @ ( tl_list_a @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_603_take__Suc,axiom,
! [Xs2: list_state_a,N: nat] :
( ( Xs2 != nil_state_a )
=> ( ( take_state_a @ ( suc @ N ) @ Xs2 )
= ( cons_state_a @ ( hd_state_a @ Xs2 ) @ ( take_state_a @ N @ ( tl_state_a @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_604_take__Suc,axiom,
! [Xs2: list_a,N: nat] :
( ( Xs2 != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs2 )
= ( cons_a @ ( hd_a @ Xs2 ) @ ( take_a @ N @ ( tl_a @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_605_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_606_Current_Opop_Osimps_I2_J,axiom,
! [X: list_a,Xs2: list_list_a,Added2: nat,Old2: stack_list_a,Remained2: nat] :
( ( pop_list_a2 @ ( current_list_a2 @ ( cons_list_a @ X @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) )
= ( produc1356073977093405001list_a @ X @ ( current_list_a2 @ Xs2 @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ).
% Current.pop.simps(2)
thf(fact_607_Current_Opop_Osimps_I2_J,axiom,
! [X: state_a,Xs2: list_state_a,Added2: nat,Old2: stack_state_a,Remained2: nat] :
( ( pop_state_a2 @ ( current_state_a2 @ ( cons_state_a @ X @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) )
= ( produc8606655205717325309tate_a @ X @ ( current_state_a2 @ Xs2 @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ).
% Current.pop.simps(2)
thf(fact_608_Current_Opop_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a,Added2: nat,Old2: stack_a,Remained2: nat] :
( ( pop_a2 @ ( current_a2 @ ( cons_a @ X @ Xs2 ) @ Added2 @ Old2 @ Remained2 ) )
= ( produc8503237746132909001rent_a @ X @ ( current_a2 @ Xs2 @ ( minus_minus_nat @ Added2 @ one_one_nat ) @ Old2 @ Remained2 ) ) ) ).
% Current.pop.simps(2)
thf(fact_609_Idle__Proof_Osize__pop__sub,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ( ( pop_a3 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( size_size_idle_a @ Idle2 )
= ( minus_minus_nat @ ( size_size_idle_a @ Idle ) @ one_one_nat ) ) ) ).
% Idle_Proof.size_pop_sub
thf(fact_610_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s349497388124573686list_a
= ( ^ [Xs3: list_list_a] : ( if_nat @ ( Xs3 = nil_list_a ) @ zero_zero_nat @ ( suc @ ( size_s349497388124573686list_a @ ( tl_list_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_611_Nitpick_Osize__list__simp_I2_J,axiom,
( size_s4663437186682225508tate_a
= ( ^ [Xs3: list_state_a] : ( if_nat @ ( Xs3 = nil_state_a ) @ zero_zero_nat @ ( suc @ ( size_s4663437186682225508tate_a @ ( tl_state_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_612_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_613_Common_Opush_Osimps_I1_J,axiom,
! [X: a,Current: current_a,Stack: stack_a,StackSize2: nat] :
( ( push_a @ X @ ( idle_a2 @ Current @ ( idle_a3 @ Stack @ StackSize2 ) ) )
= ( idle_a2 @ ( push_a2 @ X @ Current ) @ ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize2 ) ) ) ) ).
% Common.push.simps(1)
thf(fact_614_Idle_Opush_Osimps,axiom,
! [X: a,Stack: stack_a,StackSize2: nat] :
( ( push_a3 @ X @ ( idle_a3 @ Stack @ StackSize2 ) )
= ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize2 ) ) ) ).
% Idle.push.simps
thf(fact_615_Idle_Opush_Oelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( Y
!= ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize ) ) ) ) ) ).
% Idle.push.elims
thf(fact_616_Idle__Proof_Osize__push,axiom,
! [X: a,Idle: idle_a] :
( ( size_size_idle_a @ ( push_a3 @ X @ Idle ) )
= ( suc @ ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.size_push
thf(fact_617_Current__Proof_Osize__push,axiom,
! [X: a,Current: current_a] :
( ( size_size_current_a @ ( push_a2 @ X @ Current ) )
= ( suc @ ( size_size_current_a @ Current ) ) ) ).
% Current_Proof.size_push
thf(fact_618_Idle__Proof_Osize__pop,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( ( ( pop_a3 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( suc @ ( size_size_idle_a @ Idle2 ) )
= ( size_size_idle_a @ Idle ) ) ) ) ).
% Idle_Proof.size_pop
thf(fact_619_Current_Opop_Opelims,axiom,
! [X: current_list_a,Y: produc603899091129270873list_a] :
( ( ( pop_list_a2 @ 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_a @ 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 ) ) ) )
=> ~ ! [X5: list_a,Xs: list_list_a,Added: nat,Old: stack_list_a,Remained: nat] :
( ( X
= ( current_list_a2 @ ( cons_list_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc1356073977093405001list_a @ X5 @ ( current_list_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) )
=> ~ ( accp_current_list_a @ pop_rel_list_a @ ( current_list_a2 @ ( cons_list_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_620_Current_Opop_Opelims,axiom,
! [X: current_state_a,Y: produc1507450493377391557tate_a] :
( ( ( pop_state_a2 @ X )
= Y )
=> ( ( accp_current_state_a @ pop_rel_state_a @ X )
=> ( ! [Added: nat,Old: stack_state_a,Remained: nat] :
( ( X
= ( current_state_a2 @ nil_state_a @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc8606655205717325309tate_a @ ( first_state_a @ Old ) @ ( current_state_a2 @ nil_state_a @ Added @ ( pop_state_a4 @ Old ) @ ( minus_minus_nat @ Remained @ one_one_nat ) ) ) )
=> ~ ( accp_current_state_a @ pop_rel_state_a @ ( current_state_a2 @ nil_state_a @ Added @ Old @ Remained ) ) ) )
=> ~ ! [X5: state_a,Xs: list_state_a,Added: nat,Old: stack_state_a,Remained: nat] :
( ( X
= ( current_state_a2 @ ( cons_state_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc8606655205717325309tate_a @ X5 @ ( current_state_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) )
=> ~ ( accp_current_state_a @ pop_rel_state_a @ ( current_state_a2 @ ( cons_state_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_621_Current_Opop_Opelims,axiom,
! [X: current_a,Y: produc7805042584321970905rent_a] :
( ( ( pop_a2 @ 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 ) ) ) )
=> ~ ! [X5: a,Xs: list_a,Added: nat,Old: stack_a,Remained: nat] :
( ( X
= ( current_a2 @ ( cons_a @ X5 @ Xs ) @ Added @ Old @ Remained ) )
=> ( ( Y
= ( produc8503237746132909001rent_a @ X5 @ ( current_a2 @ Xs @ ( minus_minus_nat @ Added @ one_one_nat ) @ Old @ Remained ) ) )
=> ~ ( accp_current_a @ pop_rel_a @ ( current_a2 @ ( cons_a @ X5 @ Xs ) @ Added @ Old @ Remained ) ) ) ) ) ) ) ).
% Current.pop.pelims
thf(fact_622_Common_Opush_Oelims,axiom,
! [X: a,Xa: state_a,Y: state_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ! [Current2: current_a,Stack2: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Current2 @ ( idle_a3 @ Stack2 @ StackSize ) ) )
=> ( Y
!= ( idle_a2 @ ( push_a2 @ X @ Current2 ) @ ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize ) ) ) ) )
=> ~ ! [Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( Xa
= ( copy_a @ Current2 @ Aux @ New @ Moved ) )
=> ( Y
!= ( copy_a @ ( push_a2 @ X @ Current2 ) @ Aux @ New @ Moved ) ) ) ) ) ).
% Common.push.elims
thf(fact_623_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs2: list_list_a] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( ( take_list_a @ ( suc @ I3 ) @ Xs2 )
= ( append_list_a @ ( take_list_a @ I3 @ Xs2 ) @ ( cons_list_a @ ( nth_list_a @ Xs2 @ I3 ) @ nil_list_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_624_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs2: list_state_a] :
( ( ord_less_nat @ I3 @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( take_state_a @ ( suc @ I3 ) @ Xs2 )
= ( append_state_a @ ( take_state_a @ I3 @ Xs2 ) @ ( cons_state_a @ ( nth_state_a @ Xs2 @ I3 ) @ nil_state_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_625_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs2: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( take_a @ ( suc @ I3 ) @ Xs2 )
= ( append_a @ ( take_a @ I3 @ Xs2 ) @ ( cons_a @ ( nth_a @ Xs2 @ I3 ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_626_append_Oassoc,axiom,
! [A: list_a,B: list_a,C2: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C2 )
= ( append_a @ A @ ( append_a @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_627_append__assoc,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_628_append__same__eq,axiom,
! [Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_629_same__append__eq,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_630_state_Oinject_I1_J,axiom,
! [X11: current_a,X122: list_a,X13: list_a,X14: nat,Y11: current_a,Y12: list_a,Y13: list_a,Y14: nat] :
( ( ( copy_a @ X11 @ X122 @ X13 @ X14 )
= ( copy_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% state.inject(1)
thf(fact_631_append__is__Nil__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= nil_list_a )
= ( ( Xs2 = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_632_append__is__Nil__conv,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( append_state_a @ Xs2 @ Ys )
= nil_state_a )
= ( ( Xs2 = nil_state_a )
& ( Ys = nil_state_a ) ) ) ).
% append_is_Nil_conv
thf(fact_633_append__is__Nil__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= nil_a )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_634_Nil__is__append__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_635_Nil__is__append__conv,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( nil_state_a
= ( append_state_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_state_a )
& ( Ys = nil_state_a ) ) ) ).
% Nil_is_append_conv
thf(fact_636_Nil__is__append__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_637_self__append__conv2,axiom,
! [Y: list_list_a,Xs2: list_list_a] :
( ( Y
= ( append_list_a @ Xs2 @ Y ) )
= ( Xs2 = nil_list_a ) ) ).
% self_append_conv2
thf(fact_638_self__append__conv2,axiom,
! [Y: list_state_a,Xs2: list_state_a] :
( ( Y
= ( append_state_a @ Xs2 @ Y ) )
= ( Xs2 = nil_state_a ) ) ).
% self_append_conv2
thf(fact_639_self__append__conv2,axiom,
! [Y: list_a,Xs2: list_a] :
( ( Y
= ( append_a @ Xs2 @ Y ) )
= ( Xs2 = nil_a ) ) ).
% self_append_conv2
thf(fact_640_append__self__conv2,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_list_a ) ) ).
% append_self_conv2
thf(fact_641_append__self__conv2,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( append_state_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_state_a ) ) ).
% append_self_conv2
thf(fact_642_append__self__conv2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_a ) ) ).
% append_self_conv2
thf(fact_643_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_644_self__append__conv,axiom,
! [Y: list_state_a,Ys: list_state_a] :
( ( Y
= ( append_state_a @ Y @ Ys ) )
= ( Ys = nil_state_a ) ) ).
% self_append_conv
thf(fact_645_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_646_append__self__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_647_append__self__conv,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( append_state_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_state_a ) ) ).
% append_self_conv
thf(fact_648_append__self__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_649_append__Nil2,axiom,
! [Xs2: list_list_a] :
( ( append_list_a @ Xs2 @ nil_list_a )
= Xs2 ) ).
% append_Nil2
thf(fact_650_append__Nil2,axiom,
! [Xs2: list_state_a] :
( ( append_state_a @ Xs2 @ nil_state_a )
= Xs2 ) ).
% append_Nil2
thf(fact_651_append__Nil2,axiom,
! [Xs2: list_a] :
( ( append_a @ Xs2 @ nil_a )
= Xs2 ) ).
% append_Nil2
thf(fact_652_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_653_append_Oright__neutral,axiom,
! [A: list_state_a] :
( ( append_state_a @ A @ nil_state_a )
= A ) ).
% append.right_neutral
thf(fact_654_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_655_append__eq__append__conv,axiom,
! [Xs2: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs2 @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_656_append__eq__append__conv,axiom,
! [Xs2: list_state_a,Ys: list_state_a,Us: list_state_a,Vs: list_state_a] :
( ( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
| ( ( size_s4663437186682225508tate_a @ Us )
= ( size_s4663437186682225508tate_a @ Vs ) ) )
=> ( ( ( append_state_a @ Xs2 @ Us )
= ( append_state_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_657_append1__eq__conv,axiom,
! [Xs2: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_658_append1__eq__conv,axiom,
! [Xs2: list_state_a,X: state_a,Ys: list_state_a,Y: state_a] :
( ( ( append_state_a @ Xs2 @ ( cons_state_a @ X @ nil_state_a ) )
= ( append_state_a @ Ys @ ( cons_state_a @ Y @ nil_state_a ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_659_append1__eq__conv,axiom,
! [Xs2: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_660_hd__append2,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( hd_list_a @ Xs2 ) ) ) ).
% hd_append2
thf(fact_661_hd__append2,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( hd_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( hd_state_a @ Xs2 ) ) ) ).
% hd_append2
thf(fact_662_hd__append2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ).
% hd_append2
thf(fact_663_tl__append2,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_664_tl__append2,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( tl_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( append_state_a @ ( tl_state_a @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_665_tl__append2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( tl_a @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_666_nth__append__length,axiom,
! [Xs2: list_list_a,X: list_a,Ys: list_list_a] :
( ( nth_list_a @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ Ys ) ) @ ( size_s349497388124573686list_a @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_667_nth__append__length,axiom,
! [Xs2: list_state_a,X: state_a,Ys: list_state_a] :
( ( nth_state_a @ ( append_state_a @ Xs2 @ ( cons_state_a @ X @ Ys ) ) @ ( size_s4663437186682225508tate_a @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_668_nth__append__length,axiom,
! [Xs2: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs2 @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_669_take__append,axiom,
! [N: nat,Xs2: list_a,Ys: list_a] :
( ( take_a @ N @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( take_a @ N @ Xs2 ) @ ( take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).
% take_append
thf(fact_670_take__append,axiom,
! [N: nat,Xs2: list_state_a,Ys: list_state_a] :
( ( take_state_a @ N @ ( append_state_a @ Xs2 @ Ys ) )
= ( append_state_a @ ( take_state_a @ N @ Xs2 ) @ ( take_state_a @ ( minus_minus_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) ) @ Ys ) ) ) ).
% take_append
thf(fact_671_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_672_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_673_append__Cons,axiom,
! [X: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs2 ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_674_append__Cons,axiom,
! [X: state_a,Xs2: list_state_a,Ys: list_state_a] :
( ( append_state_a @ ( cons_state_a @ X @ Xs2 ) @ Ys )
= ( cons_state_a @ X @ ( append_state_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_675_append__Cons,axiom,
! [X: a,Xs2: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs2 ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_676_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs2: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X @ Xs2 )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_677_Cons__eq__appendI,axiom,
! [X: state_a,Xs1: list_state_a,Ys: list_state_a,Xs2: list_state_a,Zs: list_state_a] :
( ( ( cons_state_a @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_state_a @ Xs1 @ Zs ) )
=> ( ( cons_state_a @ X @ Xs2 )
= ( append_state_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_678_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs2 )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_679_eq__Nil__appendI,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_680_eq__Nil__appendI,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_state_a @ nil_state_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_681_eq__Nil__appendI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_682_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_683_append_Oleft__neutral,axiom,
! [A: list_state_a] :
( ( append_state_a @ nil_state_a @ A )
= A ) ).
% append.left_neutral
thf(fact_684_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_685_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_686_append__Nil,axiom,
! [Ys: list_state_a] :
( ( append_state_a @ nil_state_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_687_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_688_state_Odistinct_I1_J,axiom,
! [X11: current_a,X122: list_a,X13: list_a,X14: nat,X21: current_a,X22: idle_a] :
( ( copy_a @ X11 @ X122 @ X13 @ X14 )
!= ( idle_a2 @ X21 @ X22 ) ) ).
% state.distinct(1)
thf(fact_689_state_Oexhaust,axiom,
! [Y: state_a] :
( ! [X112: current_a,X123: list_a,X132: list_a,X142: nat] :
( Y
!= ( copy_a @ X112 @ X123 @ X132 @ X142 ) )
=> ~ ! [X212: current_a,X222: idle_a] :
( Y
!= ( idle_a2 @ X212 @ X222 ) ) ) ).
% state.exhaust
thf(fact_690_normalize_Ocases,axiom,
! [X: state_a] :
( ! [Current2: current_a,Old: list_a,New: list_a,Moved: nat] :
( X
!= ( copy_a @ Current2 @ Old @ New @ Moved ) )
=> ~ ! [V: current_a,Va: idle_a] :
( X
!= ( idle_a2 @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_691_rev__nonempty__induct,axiom,
! [Xs2: list_list_a,P: list_list_a > $o] :
( ( Xs2 != nil_list_a )
=> ( ! [X5: list_a] : ( P @ ( cons_list_a @ X5 @ nil_list_a ) )
=> ( ! [X5: list_a,Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( P @ Xs )
=> ( P @ ( append_list_a @ Xs @ ( cons_list_a @ X5 @ nil_list_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_692_rev__nonempty__induct,axiom,
! [Xs2: list_state_a,P: list_state_a > $o] :
( ( Xs2 != nil_state_a )
=> ( ! [X5: state_a] : ( P @ ( cons_state_a @ X5 @ nil_state_a ) )
=> ( ! [X5: state_a,Xs: list_state_a] :
( ( Xs != nil_state_a )
=> ( ( P @ Xs )
=> ( P @ ( append_state_a @ Xs @ ( cons_state_a @ X5 @ nil_state_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_693_rev__nonempty__induct,axiom,
! [Xs2: list_a,P: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X5: a] : ( P @ ( cons_a @ X5 @ nil_a ) )
=> ( ! [X5: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P @ Xs )
=> ( P @ ( append_a @ Xs @ ( cons_a @ X5 @ nil_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_694_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X: list_a,Xs2: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X @ Xs2 ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X @ Xs2 ) ) )
| ? [Ys5: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Ys5 ) )
& ( ( append_list_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_695_append__eq__Cons__conv,axiom,
! [Ys: list_state_a,Zs: list_state_a,X: state_a,Xs2: list_state_a] :
( ( ( append_state_a @ Ys @ Zs )
= ( cons_state_a @ X @ Xs2 ) )
= ( ( ( Ys = nil_state_a )
& ( Zs
= ( cons_state_a @ X @ Xs2 ) ) )
| ? [Ys5: list_state_a] :
( ( Ys
= ( cons_state_a @ X @ Ys5 ) )
& ( ( append_state_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_696_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs2: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs2 ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs2 ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_697_Cons__eq__append__conv,axiom,
! [X: list_a,Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X @ Xs2 )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_list_a] :
( ( ( cons_list_a @ X @ Ys5 )
= Ys )
& ( Xs2
= ( append_list_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_698_Cons__eq__append__conv,axiom,
! [X: state_a,Xs2: list_state_a,Ys: list_state_a,Zs: list_state_a] :
( ( ( cons_state_a @ X @ Xs2 )
= ( append_state_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_state_a )
& ( ( cons_state_a @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_state_a] :
( ( ( cons_state_a @ X @ Ys5 )
= Ys )
& ( Xs2
= ( append_state_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_699_Cons__eq__append__conv,axiom,
! [X: a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys )
& ( Xs2
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_700_rev__exhaust,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ~ ! [Ys2: list_list_a,Y5: list_a] :
( Xs2
!= ( append_list_a @ Ys2 @ ( cons_list_a @ Y5 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_701_rev__exhaust,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
=> ~ ! [Ys2: list_state_a,Y5: state_a] :
( Xs2
!= ( append_state_a @ Ys2 @ ( cons_state_a @ Y5 @ nil_state_a ) ) ) ) ).
% rev_exhaust
thf(fact_702_rev__exhaust,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ~ ! [Ys2: list_a,Y5: a] :
( Xs2
!= ( append_a @ Ys2 @ ( cons_a @ Y5 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_703_rev__induct,axiom,
! [P: list_list_a > $o,Xs2: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X5: list_a,Xs: list_list_a] :
( ( P @ Xs )
=> ( P @ ( append_list_a @ Xs @ ( cons_list_a @ X5 @ nil_list_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_704_rev__induct,axiom,
! [P: list_state_a > $o,Xs2: list_state_a] :
( ( P @ nil_state_a )
=> ( ! [X5: state_a,Xs: list_state_a] :
( ( P @ Xs )
=> ( P @ ( append_state_a @ Xs @ ( cons_state_a @ X5 @ nil_state_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_705_rev__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ( P @ nil_a )
=> ( ! [X5: a,Xs: list_a] :
( ( P @ Xs )
=> ( P @ ( append_a @ Xs @ ( cons_a @ X5 @ nil_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_706_longest__common__prefix,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
? [Ps: list_list_a,Xs4: list_list_a,Ys6: list_list_a] :
( ( Xs2
= ( append_list_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_list_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_list_a )
| ( Ys6 = nil_list_a )
| ( ( hd_list_a @ Xs4 )
!= ( hd_list_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_707_longest__common__prefix,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
? [Ps: list_state_a,Xs4: list_state_a,Ys6: list_state_a] :
( ( Xs2
= ( append_state_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_state_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_state_a )
| ( Ys6 = nil_state_a )
| ( ( hd_state_a @ Xs4 )
!= ( hd_state_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_708_longest__common__prefix,axiom,
! [Xs2: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs2
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_709_hd__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( Xs2 = nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( hd_list_a @ Ys ) ) )
& ( ( Xs2 != nil_list_a )
=> ( ( hd_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( hd_list_a @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_710_hd__append,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( Xs2 = nil_state_a )
=> ( ( hd_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( hd_state_a @ Ys ) ) )
& ( ( Xs2 != nil_state_a )
=> ( ( hd_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( hd_state_a @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_711_hd__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( Xs2 = nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_712_tl__append__if,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( Xs2 = nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( tl_list_a @ Ys ) ) )
& ( ( Xs2 != nil_list_a )
=> ( ( tl_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_list_a @ ( tl_list_a @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_713_tl__append__if,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( ( Xs2 = nil_state_a )
=> ( ( tl_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( tl_state_a @ Ys ) ) )
& ( ( Xs2 != nil_state_a )
=> ( ( tl_state_a @ ( append_state_a @ Xs2 @ Ys ) )
= ( append_state_a @ ( tl_state_a @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_714_tl__append__if,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( Xs2 = nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs2 != nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( tl_a @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_715_lex__append__leftI,axiom,
! [Ys: list_a,Zs: list_a,R: set_Product_prod_a_a,Xs2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lex_a @ R ) ) ) ).
% lex_append_leftI
thf(fact_716_Common_Opush_Osimps_I2_J,axiom,
! [X: a,Current: current_a,Aux2: list_a,New2: list_a,Moved2: nat] :
( ( push_a @ X @ ( copy_a @ Current @ Aux2 @ New2 @ Moved2 ) )
= ( copy_a @ ( push_a2 @ X @ Current ) @ Aux2 @ New2 @ Moved2 ) ) ).
% Common.push.simps(2)
thf(fact_717_same__length__different,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( Xs2 != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X5: list_a,Xs4: list_list_a,Y5: list_a,Ys6: list_list_a] :
( ( X5 != Y5 )
& ( Xs2
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X5 @ nil_list_a ) @ Xs4 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y5 @ nil_list_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_718_same__length__different,axiom,
! [Xs2: list_state_a,Ys: list_state_a] :
( ( Xs2 != Ys )
=> ( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ? [Pre: list_state_a,X5: state_a,Xs4: list_state_a,Y5: state_a,Ys6: list_state_a] :
( ( X5 != Y5 )
& ( Xs2
= ( append_state_a @ Pre @ ( append_state_a @ ( cons_state_a @ X5 @ nil_state_a ) @ Xs4 ) ) )
& ( Ys
= ( append_state_a @ Pre @ ( append_state_a @ ( cons_state_a @ Y5 @ nil_state_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_719_same__length__different,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X5: a,Xs4: list_a,Y5: a,Ys6: list_a] :
( ( X5 != Y5 )
& ( Xs2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X5 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y5 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_720_lex__append__left__iff,axiom,
! [R: set_Pr4048851178543822343list_a,Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs2 @ Ys ) @ ( append_list_a @ Xs2 @ Zs ) ) @ ( lex_list_a @ R ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_721_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lex_a @ R ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_722_lex__append__leftD,axiom,
! [R: set_Pr4048851178543822343list_a,Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs2 @ Ys ) @ ( append_list_a @ Xs2 @ Zs ) ) @ ( lex_list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_723_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Ys ) @ ( append_a @ Xs2 @ Zs ) ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_724_lex__append__rightI,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lex_a @ R ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_725_lex__append__rightI,axiom,
! [Xs2: list_state_a,Ys: list_state_a,R: set_Pr7022286013061942721tate_a,Vs: list_state_a,Us: list_state_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Xs2 @ Ys ) @ ( lex_state_a @ R ) )
=> ( ( ( size_s4663437186682225508tate_a @ Vs )
= ( size_s4663437186682225508tate_a @ Us ) )
=> ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( append_state_a @ Xs2 @ Us ) @ ( append_state_a @ Ys @ Vs ) ) @ ( lex_state_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_726_lenlex__append1,axiom,
! [Us: list_a,Xs2: list_a,R2: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs2 ) @ ( lenlex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs2 @ Ys ) ) @ ( lenlex_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_727_lenlex__append1,axiom,
! [Us: list_state_a,Xs2: list_state_a,R2: set_Pr7022286013061942721tate_a,Vs: list_state_a,Ys: list_state_a] :
( ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ Us @ Xs2 ) @ ( lenlex_state_a @ R2 ) )
=> ( ( ( size_s4663437186682225508tate_a @ Vs )
= ( size_s4663437186682225508tate_a @ Ys ) )
=> ( member1625279973161930154tate_a @ ( produc5982907372945109049tate_a @ ( append_state_a @ Us @ Vs ) @ ( append_state_a @ Xs2 @ Ys ) ) @ ( lenlex_state_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_728_Common_Opush_Ocases,axiom,
! [X: produc3409137331138395373tate_a] :
( ! [X5: a,Current2: current_a,Stack2: stack_a,StackSize: nat] :
( X
!= ( produc8263595898873874535tate_a @ X5 @ ( idle_a2 @ Current2 @ ( idle_a3 @ Stack2 @ StackSize ) ) ) )
=> ~ ! [X5: a,Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( X
!= ( produc8263595898873874535tate_a @ X5 @ ( copy_a @ Current2 @ Aux @ New @ Moved ) ) ) ) ).
% Common.push.cases
thf(fact_729_Idle__Proof_Osize__empty__2,axiom,
! [Idle: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( zero_zero_nat
!= ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.size_empty_2
thf(fact_730_Idle__Proof_Osize__empty,axiom,
! [Idle: idle_a] :
( ( ( size_size_idle_a @ Idle )
= zero_zero_nat )
= ( type_i7304311975391125061idle_a @ Idle ) ) ).
% Idle_Proof.size_empty
thf(fact_731_is__empty__idle_Oelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ X )
=> ~ ! [Stack2: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ).
% is_empty_idle.elims(3)
thf(fact_732_is__empty__idle_Oelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i7304311975391125061idle_a @ X )
=> ~ ! [Stack2: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ~ ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ).
% is_empty_idle.elims(2)
thf(fact_733_is__empty__idle_Oelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i7304311975391125061idle_a @ X )
= Y )
=> ~ ! [Stack2: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( Y
= ( ~ ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ) ) ).
% is_empty_idle.elims(1)
thf(fact_734_is__empty__idle_Osimps,axiom,
! [Stack: stack_a,Uu2: nat] :
( ( type_i7304311975391125061idle_a @ ( idle_a3 @ Stack @ Uu2 ) )
= ( type_i3216275384938974675tack_a @ Stack ) ) ).
% is_empty_idle.simps
thf(fact_735_length__append__singleton,axiom,
! [Xs2: list_list_a,X: list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( suc @ ( size_s349497388124573686list_a @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_736_length__append__singleton,axiom,
! [Xs2: list_state_a,X: state_a] :
( ( size_s4663437186682225508tate_a @ ( append_state_a @ Xs2 @ ( cons_state_a @ X @ nil_state_a ) ) )
= ( suc @ ( size_s4663437186682225508tate_a @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_737_length__append__singleton,axiom,
! [Xs2: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_738_length__Suc__conv__rev,axiom,
! [Xs2: list_list_a,N: nat] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: list_a,Ys3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ Y6 @ nil_list_a ) ) )
& ( ( size_s349497388124573686list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_739_length__Suc__conv__rev,axiom,
! [Xs2: list_state_a,N: nat] :
( ( ( size_s4663437186682225508tate_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: state_a,Ys3: list_state_a] :
( ( Xs2
= ( append_state_a @ Ys3 @ ( cons_state_a @ Y6 @ nil_state_a ) ) )
& ( ( size_s4663437186682225508tate_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_740_length__Suc__conv__rev,axiom,
! [Xs2: list_a,N: nat] :
( ( ( size_size_list_a @ Xs2 )
= ( suc @ N ) )
= ( ? [Y6: a,Ys3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ Y6 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_741_nth__append,axiom,
! [N: nat,Xs2: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( append_a @ Xs2 @ Ys ) @ N )
= ( nth_a @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( append_a @ Xs2 @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_742_nth__append,axiom,
! [N: nat,Xs2: list_state_a,Ys: list_state_a] :
( ( ( ord_less_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( nth_state_a @ ( append_state_a @ Xs2 @ Ys ) @ N )
= ( nth_state_a @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( nth_state_a @ ( append_state_a @ Xs2 @ Ys ) @ N )
= ( nth_state_a @ Ys @ ( minus_minus_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_743_Idle__Proof_Osize__not__empty__2,axiom,
! [Idle: idle_a] :
( ( type_i7304311975391125061idle_a @ Idle )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.size_not_empty_2
thf(fact_744_Idle__Proof_Osize__not__empty,axiom,
! [Idle: idle_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle ) )
= ( ~ ( type_i7304311975391125061idle_a @ Idle ) ) ) ).
% Idle_Proof.size_not_empty
thf(fact_745_take__rev__tl__hd,axiom,
! [N: nat,Xs2: list_list_a,Ys: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs2 != nil_list_a )
=> ( ( append_list_a @ ( common3049970819459547540list_a @ N @ Xs2 ) @ Ys )
= ( append_list_a @ ( common3049970819459547540list_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_list_a @ Xs2 ) ) @ ( cons_list_a @ ( hd_list_a @ Xs2 ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_746_take__rev__tl__hd,axiom,
! [N: nat,Xs2: list_state_a,Ys: list_state_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs2 != nil_state_a )
=> ( ( append_state_a @ ( common2372828473477909296tate_a @ N @ Xs2 ) @ Ys )
= ( append_state_a @ ( common2372828473477909296tate_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_state_a @ Xs2 ) ) @ ( cons_state_a @ ( hd_state_a @ Xs2 ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_747_take__rev__tl__hd,axiom,
! [N: nat,Xs2: list_a,Ys: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs2 != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs2 ) @ Ys )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs2 ) ) @ ( cons_a @ ( hd_a @ Xs2 ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_748_remaining__steps__state_Ocases,axiom,
! [X: state_a] :
( ! [Uu: current_a,Uv: idle_a] :
( X
!= ( idle_a2 @ Uu @ Uv ) )
=> ~ ! [Uw: list_a,Ux: nat,Uy: stack_a,Remained: nat,Aux: list_a,New: list_a,Moved: nat] :
( X
!= ( copy_a @ ( current_a2 @ Uw @ Ux @ Uy @ Remained ) @ Aux @ New @ Moved ) ) ) ).
% remaining_steps_state.cases
thf(fact_749_size__pop__suc,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( suc @ ( size_size_current_a @ Current3 ) )
= ( size_size_current_a @ Current ) ) ) ) ) ).
% size_pop_suc
thf(fact_750_pop__list__tl,axiom,
! [Idle: idle_list_a,X: list_a,Idle2: idle_list_a] :
( ~ ( type_i3138959845546662347list_a @ Idle )
=> ( ( ( pop_list_a3 @ Idle )
= ( produc1061802603431610141list_a @ X @ Idle2 ) )
=> ( ( cons_list_a @ X @ ( tl_list_a @ ( idle_list_list_a @ Idle ) ) )
= ( idle_list_list_a @ Idle ) ) ) ) ).
% pop_list_tl
thf(fact_751_pop__list__tl,axiom,
! [Idle: idle_state_a,X: state_a,Idle2: idle_state_a] :
( ~ ( type_i4947487625591023951tate_a @ Idle )
=> ( ( ( pop_state_a3 @ Idle )
= ( produc5721854270093297961tate_a @ X @ Idle2 ) )
=> ( ( cons_state_a @ X @ ( tl_state_a @ ( idle_list_state_a @ Idle ) ) )
= ( idle_list_state_a @ Idle ) ) ) ) ).
% pop_list_tl
thf(fact_752_pop__list__tl,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( ( ( pop_a3 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( cons_a @ X @ ( tl_a @ ( idle_list_a2 @ Idle ) ) )
= ( idle_list_a2 @ Idle ) ) ) ) ).
% pop_list_tl
thf(fact_753_take__rev__step,axiom,
! [Xs2: list_list_a,N: nat,Acc: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( append_list_a @ ( common3049970819459547540list_a @ N @ ( tl_list_a @ Xs2 ) ) @ ( cons_list_a @ ( hd_list_a @ Xs2 ) @ Acc ) )
= ( append_list_a @ ( common3049970819459547540list_a @ ( suc @ N ) @ Xs2 ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_754_take__rev__step,axiom,
! [Xs2: list_state_a,N: nat,Acc: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( append_state_a @ ( common2372828473477909296tate_a @ N @ ( tl_state_a @ Xs2 ) ) @ ( cons_state_a @ ( hd_state_a @ Xs2 ) @ Acc ) )
= ( append_state_a @ ( common2372828473477909296tate_a @ ( suc @ N ) @ Xs2 ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_755_take__rev__step,axiom,
! [Xs2: list_a,N: nat,Acc: list_a] :
( ( Xs2 != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ ( tl_a @ Xs2 ) ) @ ( cons_a @ ( hd_a @ Xs2 ) @ Acc ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs2 ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_756_take__rev__empty,axiom,
! [N: nat] :
( ( common3049970819459547540list_a @ N @ nil_list_a )
= nil_list_a ) ).
% take_rev_empty
thf(fact_757_take__rev__empty,axiom,
! [N: nat] :
( ( common2372828473477909296tate_a @ N @ nil_state_a )
= nil_state_a ) ).
% take_rev_empty
thf(fact_758_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_a @ N @ nil_a )
= nil_a ) ).
% take_rev_empty
thf(fact_759_Idle__Proof_Opush__list,axiom,
! [X: list_a,Idle: idle_list_a] :
( ( idle_list_list_a @ ( push_list_a3 @ X @ Idle ) )
= ( cons_list_a @ X @ ( idle_list_list_a @ Idle ) ) ) ).
% Idle_Proof.push_list
thf(fact_760_Idle__Proof_Opush__list,axiom,
! [X: state_a,Idle: idle_state_a] :
( ( idle_list_state_a @ ( push_state_a3 @ X @ Idle ) )
= ( cons_state_a @ X @ ( idle_list_state_a @ Idle ) ) ) ).
% Idle_Proof.push_list
thf(fact_761_Idle__Proof_Opush__list,axiom,
! [X: a,Idle: idle_a] :
( ( idle_list_a2 @ ( push_a3 @ X @ Idle ) )
= ( cons_a @ X @ ( idle_list_a2 @ Idle ) ) ) ).
% Idle_Proof.push_list
thf(fact_762_pop__list__tl_H,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ( ( pop_a3 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( idle_list_a2 @ Idle2 )
= ( tl_a @ ( idle_list_a2 @ Idle ) ) ) ) ).
% pop_list_tl'
thf(fact_763_Idle__Proof_Opop__list,axiom,
! [Idle: idle_list_a,X: list_a,Idle2: idle_list_a] :
( ~ ( type_i3138959845546662347list_a @ Idle )
=> ( ( ( pop_list_a3 @ Idle )
= ( produc1061802603431610141list_a @ X @ Idle2 ) )
=> ( ( cons_list_a @ X @ ( idle_list_list_a @ Idle2 ) )
= ( idle_list_list_a @ Idle ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_764_Idle__Proof_Opop__list,axiom,
! [Idle: idle_state_a,X: state_a,Idle2: idle_state_a] :
( ~ ( type_i4947487625591023951tate_a @ Idle )
=> ( ( ( pop_state_a3 @ Idle )
= ( produc5721854270093297961tate_a @ X @ Idle2 ) )
=> ( ( cons_state_a @ X @ ( idle_list_state_a @ Idle2 ) )
= ( idle_list_state_a @ Idle ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_765_Idle__Proof_Opop__list,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( ( ( pop_a3 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( cons_a @ X @ ( idle_list_a2 @ Idle2 ) )
= ( idle_list_a2 @ Idle ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_766_Common__Aux_Olist_Osimps_I1_J,axiom,
! [Uu2: current_a,Idle: idle_a] :
( ( common_list_a @ ( idle_a2 @ Uu2 @ Idle ) )
= ( idle_list_a2 @ Idle ) ) ).
% Common_Aux.list.simps(1)
thf(fact_767_Idle__Proof_Olist__empty,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
= nil_list_a )
= ( type_i3138959845546662347list_a @ Idle ) ) ).
% Idle_Proof.list_empty
thf(fact_768_Idle__Proof_Olist__empty,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
= nil_state_a )
= ( type_i4947487625591023951tate_a @ Idle ) ) ).
% Idle_Proof.list_empty
thf(fact_769_Idle__Proof_Olist__empty,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
= nil_a )
= ( type_i7304311975391125061idle_a @ Idle ) ) ).
% Idle_Proof.list_empty
thf(fact_770_Idle__Proof_Olist__empty__2,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
= nil_list_a )
=> ( type_i3138959845546662347list_a @ Idle ) ) ).
% Idle_Proof.list_empty_2
thf(fact_771_Idle__Proof_Olist__empty__2,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
= nil_state_a )
=> ( type_i4947487625591023951tate_a @ Idle ) ) ).
% Idle_Proof.list_empty_2
thf(fact_772_Idle__Proof_Olist__empty__2,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
= nil_a )
=> ( type_i7304311975391125061idle_a @ Idle ) ) ).
% Idle_Proof.list_empty_2
thf(fact_773_Idle__Proof_Olist__not__empty,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
!= nil_list_a )
= ( ~ ( type_i3138959845546662347list_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty
thf(fact_774_Idle__Proof_Olist__not__empty,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
!= nil_state_a )
= ( ~ ( type_i4947487625591023951tate_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty
thf(fact_775_Idle__Proof_Olist__not__empty,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
!= nil_a )
= ( ~ ( type_i7304311975391125061idle_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty
thf(fact_776_Idle__Proof_Olist__not__empty__2,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
!= nil_list_a )
=> ~ ( type_i3138959845546662347list_a @ Idle ) ) ).
% Idle_Proof.list_not_empty_2
thf(fact_777_Idle__Proof_Olist__not__empty__2,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
!= nil_state_a )
=> ~ ( type_i4947487625591023951tate_a @ Idle ) ) ).
% Idle_Proof.list_not_empty_2
thf(fact_778_Idle__Proof_Olist__not__empty__2,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
!= nil_a )
=> ~ ( type_i7304311975391125061idle_a @ Idle ) ) ).
% Idle_Proof.list_not_empty_2
thf(fact_779_Idle__Aux_Olist_Osimps,axiom,
! [Stack: stack_a,Uu2: nat] :
( ( idle_list_a2 @ ( idle_a3 @ Stack @ Uu2 ) )
= ( stack_list_a2 @ Stack ) ) ).
% Idle_Aux.list.simps
thf(fact_780_Idle__Aux_Olist_Oelims,axiom,
! [X: idle_a,Y: list_a] :
( ( ( idle_list_a2 @ X )
= Y )
=> ~ ! [Stack2: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( Y
!= ( stack_list_a2 @ Stack2 ) ) ) ) ).
% Idle_Aux.list.elims
thf(fact_781_Common__Aux_Olist_Oelims,axiom,
! [X: state_a,Y: list_a] :
( ( ( common_list_a @ X )
= Y )
=> ( ! [Uu: current_a,Idle3: idle_a] :
( ( X
= ( idle_a2 @ Uu @ Idle3 ) )
=> ( Y
!= ( idle_list_a2 @ Idle3 ) ) )
=> ~ ! [Extra: list_a,Uv: nat,Uw: stack_a,Remained: nat,Old: list_a,New: list_a,Moved: nat] :
( ( X
= ( copy_a @ ( current_a2 @ Extra @ Uv @ Uw @ Remained ) @ Old @ New @ Moved ) )
=> ( Y
!= ( append_a @ Extra @ ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ Remained @ Moved ) @ Old ) @ New ) ) ) ) ) ) ).
% Common_Aux.list.elims
thf(fact_782_Idle__Proof_Olist__empty__size,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
= nil_list_a )
= ( zero_zero_nat
= ( size_s4952028384985416054list_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size
thf(fact_783_Idle__Proof_Olist__empty__size,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
= nil_state_a )
= ( zero_zero_nat
= ( size_s905445271783748068tate_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size
thf(fact_784_Idle__Proof_Olist__empty__size,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
= nil_a )
= ( zero_zero_nat
= ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size
thf(fact_785_Idle__Proof_Olist__empty__size__2,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
!= nil_list_a )
=> ( zero_zero_nat
!= ( size_s4952028384985416054list_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size_2
thf(fact_786_Idle__Proof_Olist__empty__size__2,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
!= nil_state_a )
=> ( zero_zero_nat
!= ( size_s905445271783748068tate_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size_2
thf(fact_787_Idle__Proof_Olist__empty__size__2,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
!= nil_a )
=> ( zero_zero_nat
!= ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.list_empty_size_2
thf(fact_788_Idle__Proof_Olist__not__empty__size,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
!= nil_list_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s4952028384985416054list_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size
thf(fact_789_Idle__Proof_Olist__not__empty__size,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
!= nil_state_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s905445271783748068tate_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size
thf(fact_790_Idle__Proof_Olist__not__empty__size,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
!= nil_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size
thf(fact_791_Idle__Proof_Olist__not__empty__size__2,axiom,
! [Idle: idle_list_a] :
( ( ( idle_list_list_a @ Idle )
= nil_list_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s4952028384985416054list_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size_2
thf(fact_792_Idle__Proof_Olist__not__empty__size__2,axiom,
! [Idle: idle_state_a] :
( ( ( idle_list_state_a @ Idle )
= nil_state_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s905445271783748068tate_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size_2
thf(fact_793_Idle__Proof_Olist__not__empty__size__2,axiom,
! [Idle: idle_a] :
( ( ( idle_list_a2 @ Idle )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_idle_a @ Idle ) ) ) ).
% Idle_Proof.list_not_empty_size_2
thf(fact_794_Common__Aux_Olist_Osimps_I2_J,axiom,
! [Extra2: list_a,Uv2: nat,Uw2: stack_a,Remained2: nat,Old2: list_a,New2: list_a,Moved2: nat] :
( ( common_list_a @ ( copy_a @ ( current_a2 @ Extra2 @ Uv2 @ Uw2 @ Remained2 ) @ Old2 @ New2 @ Moved2 ) )
= ( append_a @ Extra2 @ ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ Remained2 @ Moved2 ) @ Old2 ) @ New2 ) ) ) ).
% Common_Aux.list.simps(2)
thf(fact_795_size__new__state_Ocases,axiom,
! [X: state_a] :
( ! [Current2: current_a,Uu: idle_a] :
( X
!= ( idle_a2 @ Current2 @ Uu ) )
=> ~ ! [Current2: current_a,Uv: list_a,Uw: list_a,Ux: nat] :
( X
!= ( copy_a @ Current2 @ Uv @ Uw @ Ux ) ) ) ).
% size_new_state.cases
thf(fact_796_Current__Proof_Oinvar__pop,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( type_i6141643110573041459rent_a @ Current3 ) ) ) ) ).
% Current_Proof.invar_pop
thf(fact_797_take__rev__nth,axiom,
! [N: nat,Xs2: list_list_a,X: list_a,Ys: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( ( X
= ( nth_list_a @ Xs2 @ N ) )
=> ( ( cons_list_a @ X @ ( append_list_a @ ( common3049970819459547540list_a @ N @ Xs2 ) @ Ys ) )
= ( append_list_a @ ( common3049970819459547540list_a @ ( suc @ N ) @ Xs2 ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_798_take__rev__nth,axiom,
! [N: nat,Xs2: list_state_a,X: state_a,Ys: list_state_a] :
( ( ord_less_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( X
= ( nth_state_a @ Xs2 @ N ) )
=> ( ( cons_state_a @ X @ ( append_state_a @ ( common2372828473477909296tate_a @ N @ Xs2 ) @ Ys ) )
= ( append_state_a @ ( common2372828473477909296tate_a @ ( suc @ N ) @ Xs2 ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_799_take__rev__nth,axiom,
! [N: nat,Xs2: list_a,X: a,Ys: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( X
= ( nth_a @ Xs2 @ N ) )
=> ( ( cons_a @ X @ ( append_a @ ( common_take_rev_a @ N @ Xs2 ) @ Ys ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs2 ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_800_Current__Proof_Osize__pop__sub,axiom,
! [Current: current_a,X: a,Current3: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current3 ) )
=> ( ( size_size_current_a @ Current3 )
= ( minus_minus_nat @ ( size_size_current_a @ Current ) @ one_one_nat ) ) ) ) ) ).
% Current_Proof.size_pop_sub
thf(fact_801_remaining__steps__state_Oelims,axiom,
! [X: state_a,Y: nat] :
( ( ( type_r2212416260012024137tate_a @ X )
= Y )
=> ( ( ? [Uu: current_a,Uv: idle_a] :
( X
= ( idle_a2 @ Uu @ Uv ) )
=> ( Y != zero_zero_nat ) )
=> ~ ! [Uw: list_a,Ux: nat,Uy: stack_a,Remained: nat,Aux: list_a,New: list_a,Moved: nat] :
( ( X
= ( copy_a @ ( current_a2 @ Uw @ Ux @ Uy @ Remained ) @ Aux @ New @ Moved ) )
=> ( Y
!= ( minus_minus_nat @ Remained @ Moved ) ) ) ) ) ).
% remaining_steps_state.elims
thf(fact_802_invar__current_Oelims_I3_J,axiom,
! [X: current_state_a] :
( ~ ( type_i729744499658654177tate_a @ X )
=> ~ ! [Extra: list_state_a,Added: nat] :
( ? [Uu: stack_state_a,Uv: nat] :
( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_s4663437186682225508tate_a @ Extra )
= Added ) ) ) ).
% invar_current.elims(3)
thf(fact_803_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_804_invar__current_Oelims_I2_J,axiom,
! [X: current_state_a] :
( ( type_i729744499658654177tate_a @ X )
=> ~ ! [Extra: list_state_a,Added: nat] :
( ? [Uu: stack_state_a,Uv: nat] :
( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_s4663437186682225508tate_a @ Extra )
!= Added ) ) ) ).
% invar_current.elims(2)
thf(fact_805_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_806_invar__current_Oelims_I1_J,axiom,
! [X: current_state_a,Y: $o] :
( ( ( type_i729744499658654177tate_a @ X )
= Y )
=> ~ ! [Extra: list_state_a,Added: nat] :
( ? [Uu: stack_state_a,Uv: nat] :
( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( Y
= ( ( size_s4663437186682225508tate_a @ Extra )
!= Added ) ) ) ) ).
% invar_current.elims(1)
thf(fact_807_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_808_invar__current_Osimps,axiom,
! [Extra2: list_state_a,Added2: nat,Uu2: stack_state_a,Uv2: nat] :
( ( type_i729744499658654177tate_a @ ( current_state_a2 @ Extra2 @ Added2 @ Uu2 @ Uv2 ) )
= ( ( size_s4663437186682225508tate_a @ Extra2 )
= Added2 ) ) ).
% invar_current.simps
thf(fact_809_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_810_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_811_remaining__steps__state_Osimps_I1_J,axiom,
! [Uu2: current_a,Uv2: idle_a] :
( ( type_r2212416260012024137tate_a @ ( idle_a2 @ Uu2 @ Uv2 ) )
= zero_zero_nat ) ).
% remaining_steps_state.simps(1)
thf(fact_812_remaining__steps__state_Osimps_I2_J,axiom,
! [Uw2: list_a,Ux2: nat,Uy2: stack_a,Remained2: nat,Aux2: list_a,New2: list_a,Moved2: nat] :
( ( type_r2212416260012024137tate_a @ ( copy_a @ ( current_a2 @ Uw2 @ Ux2 @ Uy2 @ Remained2 ) @ Aux2 @ New2 @ Moved2 ) )
= ( minus_minus_nat @ Remained2 @ Moved2 ) ) ).
% remaining_steps_state.simps(2)
thf(fact_813_invar__current_Opelims_I1_J,axiom,
! [X: current_state_a,Y: $o] :
( ( ( type_i729744499658654177tate_a @ X )
= Y )
=> ( ( accp_current_state_a @ curren1694177707811796940tate_a @ X )
=> ~ ! [Extra: list_state_a,Added: nat,Uu: stack_state_a,Uv: nat] :
( ( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( Y
= ( ( size_s4663437186682225508tate_a @ Extra )
= Added ) )
=> ~ ( accp_current_state_a @ curren1694177707811796940tate_a @ ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) ) ) ) ) ) ).
% invar_current.pelims(1)
thf(fact_814_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_815_invar__current_Opelims_I2_J,axiom,
! [X: current_state_a] :
( ( type_i729744499658654177tate_a @ X )
=> ( ( accp_current_state_a @ curren1694177707811796940tate_a @ X )
=> ~ ! [Extra: list_state_a,Added: nat,Uu: stack_state_a,Uv: nat] :
( ( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( accp_current_state_a @ curren1694177707811796940tate_a @ ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_s4663437186682225508tate_a @ Extra )
!= Added ) ) ) ) ) ).
% invar_current.pelims(2)
thf(fact_816_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_817_invar__current_Opelims_I3_J,axiom,
! [X: current_state_a] :
( ~ ( type_i729744499658654177tate_a @ X )
=> ( ( accp_current_state_a @ curren1694177707811796940tate_a @ X )
=> ~ ! [Extra: list_state_a,Added: nat,Uu: stack_state_a,Uv: nat] :
( ( X
= ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( accp_current_state_a @ curren1694177707811796940tate_a @ ( current_state_a2 @ Extra @ Added @ Uu @ Uv ) )
=> ( ( size_s4663437186682225508tate_a @ Extra )
= Added ) ) ) ) ) ).
% invar_current.pelims(3)
thf(fact_818_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_819_Common_Opush_Opelims,axiom,
! [X: a,Xa: state_a,Y: state_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ Xa ) )
=> ( ! [Current2: current_a,Stack2: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Current2 @ ( idle_a3 @ Stack2 @ StackSize ) ) )
=> ( ( Y
= ( idle_a2 @ ( push_a2 @ X @ Current2 ) @ ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize ) ) ) )
=> ~ ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ ( idle_a2 @ Current2 @ ( idle_a3 @ Stack2 @ StackSize ) ) ) ) ) )
=> ~ ! [Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( Xa
= ( copy_a @ Current2 @ Aux @ New @ Moved ) )
=> ( ( Y
= ( copy_a @ ( push_a2 @ X @ Current2 ) @ Aux @ New @ Moved ) )
=> ~ ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ ( copy_a @ Current2 @ Aux @ New @ Moved ) ) ) ) ) ) ) ) ).
% Common.push.pelims
thf(fact_820_list__size,axiom,
! [Current: current_list_a] :
( ( type_i8419033571676804025list_a @ Current )
=> ( ( ( current_list_list_a @ Current )
= nil_list_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s5124551957227789858list_a @ Current ) ) ) ) ).
% list_size
thf(fact_821_list__size,axiom,
! [Current: current_state_a] :
( ( type_i729744499658654177tate_a @ Current )
=> ( ( ( current_list_state_a @ Current )
= nil_state_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s1806206149518693944tate_a @ Current ) ) ) ) ).
% list_size
thf(fact_822_list__size,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( current_list_a3 @ Current )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) ) ) ) ).
% list_size
thf(fact_823_Current__Proof_Opush__list,axiom,
! [X: list_a,Current: current_list_a] :
( ( current_list_list_a @ ( push_list_a2 @ X @ Current ) )
= ( cons_list_a @ X @ ( current_list_list_a @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_824_Current__Proof_Opush__list,axiom,
! [X: state_a,Current: current_state_a] :
( ( current_list_state_a @ ( push_state_a2 @ X @ Current ) )
= ( cons_state_a @ X @ ( current_list_state_a @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_825_Current__Proof_Opush__list,axiom,
! [X: a,Current: current_a] :
( ( current_list_a3 @ ( push_a2 @ X @ Current ) )
= ( cons_a @ X @ ( current_list_a3 @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_826_list__current_Osimps_I1_J,axiom,
! [Current: current_a,Uu2: idle_a] :
( ( common1102728217005306191rent_a @ ( idle_a2 @ Current @ Uu2 ) )
= ( current_list_a3 @ Current ) ) ).
% list_current.simps(1)
thf(fact_827_list__current_Osimps_I2_J,axiom,
! [Current: current_a,Uv2: list_a,Uw2: list_a,Ux2: nat] :
( ( common1102728217005306191rent_a @ ( copy_a @ Current @ Uv2 @ Uw2 @ Ux2 ) )
= ( current_list_a3 @ Current ) ) ).
% list_current.simps(2)
thf(fact_828_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_829_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_830_list__current_Oelims,axiom,
! [X: state_a,Y: list_a] :
( ( ( common1102728217005306191rent_a @ X )
= Y )
=> ( ! [Current2: current_a] :
( ? [Uu: idle_a] :
( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( Y
!= ( current_list_a3 @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uv: list_a,Uw: list_a,Ux: nat] :
( X
= ( copy_a @ Current2 @ Uv @ Uw @ Ux ) )
=> ( Y
!= ( current_list_a3 @ Current2 ) ) ) ) ) ).
% list_current.elims
thf(fact_831_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_832_Current__Proof_Opop__list,axiom,
! [Current: current_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5124551957227789858list_a @ Current ) )
=> ( ( type_i8419033571676804025list_a @ Current )
=> ( ( cons_list_a @ ( produc8258313757495307061list_a @ ( pop_list_a2 @ Current ) ) @ ( tl_list_a @ ( current_list_list_a @ Current ) ) )
= ( current_list_list_a @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_833_Current__Proof_Opop__list,axiom,
! [Current: current_state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1806206149518693944tate_a @ Current ) )
=> ( ( type_i729744499658654177tate_a @ Current )
=> ( ( cons_state_a @ ( produc2307077730481702377tate_a @ ( pop_state_a2 @ Current ) ) @ ( tl_state_a @ ( current_list_state_a @ Current ) ) )
= ( current_list_state_a @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_834_Current__Proof_Opop__list,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( cons_a @ ( produc4952273589686483381rent_a @ ( pop_a2 @ Current ) ) @ ( tl_a @ ( current_list_a3 @ Current ) ) )
= ( current_list_a3 @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_835_drop__first__list,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( current_list_a3 @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) )
= ( tl_a @ ( current_list_a3 @ Current ) ) ) ) ) ).
% drop_first_list
thf(fact_836_take__hd__drop,axiom,
! [N: nat,Xs2: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( ( append_list_a @ ( take_list_a @ N @ Xs2 ) @ ( cons_list_a @ ( hd_list_a @ ( drop_list_a @ N @ Xs2 ) ) @ nil_list_a ) )
= ( take_list_a @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_837_take__hd__drop,axiom,
! [N: nat,Xs2: list_state_a] :
( ( ord_less_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) )
=> ( ( append_state_a @ ( take_state_a @ N @ Xs2 ) @ ( cons_state_a @ ( hd_state_a @ ( drop_state_a @ N @ Xs2 ) ) @ nil_state_a ) )
= ( take_state_a @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_838_take__hd__drop,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( append_a @ ( take_a @ N @ Xs2 ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs2 ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_839_Current__Proof_Osize__pop,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( suc @ ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) )
= ( size_size_current_a @ Current ) ) ) ) ).
% Current_Proof.size_pop
thf(fact_840_drop0,axiom,
( ( drop_a @ zero_zero_nat )
= ( ^ [X6: list_a] : X6 ) ) ).
% drop0
thf(fact_841_prod_Ocollapse,axiom,
! [Prod: produc9164743771328383783list_a] :
( ( produc6837034575241423639list_a @ ( produc3698117735987127555list_a @ Prod ) @ ( produc8617614985401127493list_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_842_prod_Ocollapse,axiom,
! [Prod: produc5032551385658279741list_a] :
( ( produc8111569692950616493list_a @ ( produc6311120620833064345list_a @ Prod ) @ ( produc3976258275648695771list_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_843_prod_Ocollapse,axiom,
! [Prod: produc7590564867095724333idle_a] :
( ( produc1265230069547855005idle_a @ ( product_fst_a_idle_a @ Prod ) @ ( product_snd_a_idle_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_844_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_845_prod_Ocollapse,axiom,
! [Prod: produc7805042584321970905rent_a] :
( ( produc8503237746132909001rent_a @ ( produc4952273589686483381rent_a @ Prod ) @ ( produc4695312889421393143rent_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_846_prod_Ocollapse,axiom,
! [Prod: produc3409137331138395373tate_a] :
( ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_847_drop__Suc__Cons,axiom,
! [N: nat,X: list_a,Xs2: list_list_a] :
( ( drop_list_a @ ( suc @ N ) @ ( cons_list_a @ X @ Xs2 ) )
= ( drop_list_a @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_848_drop__Suc__Cons,axiom,
! [N: nat,X: state_a,Xs2: list_state_a] :
( ( drop_state_a @ ( suc @ N ) @ ( cons_state_a @ X @ Xs2 ) )
= ( drop_state_a @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_849_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs2 ) )
= ( drop_a @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_850_length__drop,axiom,
! [N: nat,Xs2: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_851_length__drop,axiom,
! [N: nat,Xs2: list_state_a] :
( ( size_s4663437186682225508tate_a @ ( drop_state_a @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_s4663437186682225508tate_a @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_852_append__take__drop__id,axiom,
! [N: nat,Xs2: list_a] :
( ( append_a @ ( take_a @ N @ Xs2 ) @ ( drop_a @ N @ Xs2 ) )
= Xs2 ) ).
% append_take_drop_id
thf(fact_853_drop__append,axiom,
! [N: nat,Xs2: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs2 ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_854_drop__append,axiom,
! [N: nat,Xs2: list_state_a,Ys: list_state_a] :
( ( drop_state_a @ N @ ( append_state_a @ Xs2 @ Ys ) )
= ( append_state_a @ ( drop_state_a @ N @ Xs2 ) @ ( drop_state_a @ ( minus_minus_nat @ N @ ( size_s4663437186682225508tate_a @ Xs2 ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_855_hd__drop__1,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( cons_list_a @ ( hd_list_a @ Xs2 ) @ ( drop_list_a @ ( suc @ zero_zero_nat ) @ Xs2 ) )
= Xs2 ) ) ).
% hd_drop_1
thf(fact_856_hd__drop__1,axiom,
! [Xs2: list_state_a] :
( ( Xs2 != nil_state_a )
=> ( ( cons_state_a @ ( hd_state_a @ Xs2 ) @ ( drop_state_a @ ( suc @ zero_zero_nat ) @ Xs2 ) )
= Xs2 ) ) ).
% hd_drop_1
thf(fact_857_hd__drop__1,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs2 ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Xs2 ) )
= Xs2 ) ) ).
% hd_drop_1
thf(fact_858_hd__drop,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs2 ) ) @ ( drop_a @ ( suc @ N ) @ Xs2 ) )
= ( drop_a @ N @ Xs2 ) ) ) ).
% hd_drop
thf(fact_859_snd__eqD,axiom,
! [X: a,Y: state_a,A: state_a] :
( ( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_860_snd__conv,axiom,
! [X1: a,X2: state_a] :
( ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_861_fst__eqD,axiom,
! [X: a,Y: state_a,A: a] :
( ( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_862_fst__conv,axiom,
! [X1: a,X2: state_a] :
( ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_863_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_864_prod_Oexhaust__sel,axiom,
! [Prod: produc3409137331138395373tate_a] :
( Prod
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ Prod ) @ ( produc681690970763031737tate_a @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_865_surjective__pairing,axiom,
! [T: produc3409137331138395373tate_a] :
( T
= ( produc8263595898873874535tate_a @ ( produc3154331710141225339tate_a @ T ) @ ( produc681690970763031737tate_a @ T ) ) ) ).
% surjective_pairing
thf(fact_866_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_867_nth__via__drop,axiom,
! [N: nat,Xs2: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs2 )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_868_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_869_drop__Cons_H,axiom,
! [N: nat,X: a,Xs2: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs2 ) )
= ( cons_a @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs2 ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_870_hd__drop__conv__nth,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( hd_a @ ( drop_a @ N @ Xs2 ) )
= ( nth_a @ Xs2 @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_871_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs2: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( cons_a @ ( nth_a @ Xs2 @ I3 ) @ ( drop_a @ ( suc @ I3 ) @ Xs2 ) )
= ( drop_a @ I3 @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_872_id__take__nth__drop,axiom,
! [I3: nat,Xs2: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( Xs2
= ( append_a @ ( take_a @ I3 @ Xs2 ) @ ( cons_a @ ( nth_a @ Xs2 @ I3 ) @ ( drop_a @ ( suc @ I3 ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_873_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_874_conjI__realizer,axiom,
! [P: a > $o,P3: a,Q: state_a > $o,Q2: state_a] :
( ( P @ P3 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc3154331710141225339tate_a @ ( produc8263595898873874535tate_a @ P3 @ Q2 ) ) )
& ( Q @ ( produc681690970763031737tate_a @ ( produc8263595898873874535tate_a @ P3 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_875_invar__state_Osimps_I1_J,axiom,
! [Current: current_a,Idle: idle_a] :
( ( type_i4669920168676019581tate_a @ ( idle_a2 @ Current @ Idle ) )
= ( ( type_i8151583586401621767idle_a @ Idle )
& ( type_i6141643110573041459rent_a @ Current )
& ( ( type_s933026853152659577rent_a @ Current )
= ( size_size_idle_a @ Idle ) )
& ( ( take_a @ ( size_size_idle_a @ Idle ) @ ( current_list_a3 @ Current ) )
= ( take_a @ ( size_size_current_a @ Current ) @ ( idle_list_a2 @ Idle ) ) ) ) ) ).
% invar_state.simps(1)
thf(fact_876_invar__idle_Osimps,axiom,
! [Stack: stack_a,StackSize2: nat] :
( ( type_i8151583586401621767idle_a @ ( idle_a3 @ Stack @ StackSize2 ) )
= ( ( size_size_stack_a @ Stack )
= StackSize2 ) ) ).
% invar_idle.simps
thf(fact_877_invar__idle_Oelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( Y
= ( ( size_size_stack_a @ Stack2 )
!= StackSize ) ) ) ) ).
% invar_idle.elims(1)
thf(fact_878_invar__idle_Oelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( size_size_stack_a @ Stack2 )
!= StackSize ) ) ) ).
% invar_idle.elims(2)
thf(fact_879_invar__idle_Oelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( size_size_stack_a @ Stack2 )
= StackSize ) ) ) ).
% invar_idle.elims(3)
thf(fact_880_size__new__state_Oelims,axiom,
! [X: state_a,Y: nat] :
( ( ( type_s8424385952999958455tate_a @ X )
= Y )
=> ( ! [Current2: current_a] :
( ? [Uu: idle_a] :
( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uv: list_a,Uw: list_a,Ux: nat] :
( X
= ( copy_a @ Current2 @ Uv @ Uw @ Ux ) )
=> ( Y
!= ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ).
% size_new_state.elims
thf(fact_881_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs2: list_a,A: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( list_update_a @ Xs2 @ I3 @ A )
= ( append_a @ ( take_a @ I3 @ Xs2 ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I3 ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_882_rotate1__hd__tl,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( rotate1_a @ Xs2 )
= ( append_a @ ( tl_a @ Xs2 ) @ ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_883_eq__snd__iff,axiom,
! [B: state_a,P3: produc3409137331138395373tate_a] :
( ( B
= ( produc681690970763031737tate_a @ P3 ) )
= ( ? [A5: a] :
( P3
= ( produc8263595898873874535tate_a @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_884_list__update__nonempty,axiom,
! [Xs2: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs2 @ K @ X )
= nil_a )
= ( Xs2 = nil_a ) ) ).
% list_update_nonempty
thf(fact_885_rotate1__is__Nil__conv,axiom,
! [Xs2: list_a] :
( ( ( rotate1_a @ Xs2 )
= nil_a )
= ( Xs2 = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_886_list__update__length,axiom,
! [Xs2: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs2 @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs2 ) @ Y )
= ( append_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_887_list__update__code_I1_J,axiom,
! [I3: nat,Y: a] :
( ( list_update_a @ nil_a @ I3 @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_888_list__update_Osimps_I1_J,axiom,
! [I3: nat,V2: a] :
( ( list_update_a @ nil_a @ I3 @ V2 )
= nil_a ) ).
% list_update.simps(1)
thf(fact_889_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_890_list__update__code_I3_J,axiom,
! [X: a,Xs2: list_a,I3: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs2 ) @ ( suc @ I3 ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs2 @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_891_list__update__code_I2_J,axiom,
! [X: a,Xs2: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs2 ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_892_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs2 ) )
= ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_893_size__new__state_Osimps_I1_J,axiom,
! [Current: current_a,Uu2: idle_a] :
( ( type_s8424385952999958455tate_a @ ( idle_a2 @ Current @ Uu2 ) )
= ( type_s933026853152659577rent_a @ Current ) ) ).
% size_new_state.simps(1)
thf(fact_894_eq__fst__iff,axiom,
! [A: a,P3: produc3409137331138395373tate_a] :
( ( A
= ( produc3154331710141225339tate_a @ P3 ) )
= ( ? [B4: state_a] :
( P3
= ( produc8263595898873874535tate_a @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_895_sndI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z3: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z3 ) )
=> ( ( produc681690970763031737tate_a @ X )
= Z3 ) ) ).
% sndI
thf(fact_896_size__state_Oelims,axiom,
! [X: state_a,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ! [Current2: current_a,Idle3: idle_a] :
( ( X
= ( idle_a2 @ Current2 @ Idle3 ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( size_size_idle_a @ Idle3 ) ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uu: list_a,Uv: list_a,Uw: nat] :
( X
= ( copy_a @ Current2 @ Uu @ Uv @ Uw ) )
=> ( Y
!= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) ) ) ) ) ).
% size_state.elims
thf(fact_897_fstI,axiom,
! [X: produc3409137331138395373tate_a,Y: a,Z3: state_a] :
( ( X
= ( produc8263595898873874535tate_a @ Y @ Z3 ) )
=> ( ( produc3154331710141225339tate_a @ X )
= Y ) ) ).
% fstI
thf(fact_898_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_899_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_900_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_901_min__less__iff__conj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z3 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z3 @ X )
& ( ord_less_nat @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_902_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_903_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_904_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_905_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_906_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_907_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_908_min__diff,axiom,
! [M2: nat,I3: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I3 ) @ ( minus_minus_nat @ N @ I3 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I3 ) ) ).
% min_diff
thf(fact_909_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_nat @ X @ Z3 )
| ( ord_less_nat @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_910_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C2 ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_911_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_912_min_Ostrict__coboundedI1,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ A @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_913_min_Ostrict__coboundedI2,axiom,
! [B: nat,C2: nat,A: nat] :
( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_914_size__state_Osimps_I1_J,axiom,
! [Current: current_a,Idle: idle_a] :
( ( size_size_state_a @ ( idle_a2 @ Current @ Idle ) )
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( size_size_idle_a @ Idle ) ) ) ).
% size_state.simps(1)
thf(fact_915_size__state_Osimps_I2_J,axiom,
! [Current: current_a,Uu2: list_a,Uv2: list_a,Uw2: nat] :
( ( size_size_state_a @ ( copy_a @ Current @ Uu2 @ Uv2 @ Uw2 ) )
= ( ord_min_nat @ ( size_size_current_a @ Current ) @ ( type_s933026853152659577rent_a @ Current ) ) ) ).
% size_state.simps(2)
thf(fact_916_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_917_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_918_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_919_Cons__listrel1__Cons,axiom,
! [X: a,Xs2: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( Xs2 = Ys ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_920_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_921_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,X6: list_a] :
( Y
= ( cons_a @ A5 @ X6 ) ) ) ) ).
% lexord_Nil_left
thf(fact_922_listrel1I2,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel1_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I2
thf(fact_923_not__listrel1__Nil,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ nil_a ) @ ( listrel1_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_924_not__Nil__listrel1,axiom,
! [Xs2: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs2 ) @ ( listrel1_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_925_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_926_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_927_Cons__listrel1E2,axiom,
! [Xs2: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
=> ( ! [X5: a] :
( ( Xs2
= ( cons_a @ X5 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Xs2
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_928_Cons__listrel1E1,axiom,
! [X: a,Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ Ys ) @ ( listrel1_a @ R ) )
=> ( ! [Y5: a] :
( ( Ys
= ( cons_a @ Y5 @ Xs2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_929_listrel1I1,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y @ Xs2 ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I1
thf(fact_930_lexord__append__rightI,axiom,
! [Y: list_a,X: list_a,R: set_Product_prod_a_a] :
( ? [B5: a,Z4: list_a] :
( Y
= ( cons_a @ B5 @ Z4 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ ( append_a @ X @ Y ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_rightI
thf(fact_931_listrel1I,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs2: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_932_listrel1E,axiom,
! [Xs2: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel1_a @ R ) )
=> ~ ! [X5: a,Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y5 ) @ R )
=> ! [Us3: list_a,Vs2: list_a] :
( ( Xs2
= ( append_a @ Us3 @ ( cons_a @ X5 @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us3 @ ( cons_a @ Y5 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_933_lexord__append__left__rightI,axiom,
! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_934_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_a,X: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_935_Idle_Opop_Opelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a3 @ X )
= Y )
=> ( ( accp_idle_a @ pop_rel_a2 @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( Y
= ( produc1265230069547855005idle_a @ ( first_a2 @ Stack2 ) @ ( idle_a3 @ ( pop_a4 @ Stack2 ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) )
=> ~ ( accp_idle_a @ pop_rel_a2 @ ( idle_a3 @ Stack2 @ StackSize ) ) ) ) ) ) ).
% Idle.pop.pelims
thf(fact_936_size__idle_Opelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_size_idle_rel_a @ X )
=> ~ ! [Stack2: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( ( Y
= ( size_size_stack_a @ Stack2 ) )
=> ~ ( accp_idle_a @ idle_size_idle_rel_a @ ( idle_a3 @ Stack2 @ Uu ) ) ) ) ) ) ).
% size_idle.pelims
thf(fact_937_Idle__Aux_Olist_Opelims,axiom,
! [X: idle_a,Y: list_a] :
( ( ( idle_list_a2 @ X )
= Y )
=> ( ( accp_idle_a @ idle_list_rel_a @ X )
=> ~ ! [Stack2: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( ( Y
= ( stack_list_a2 @ Stack2 ) )
=> ~ ( accp_idle_a @ idle_list_rel_a @ ( idle_a3 @ Stack2 @ Uu ) ) ) ) ) ) ).
% Idle_Aux.list.pelims
thf(fact_938_Common_Opop_Osimps_I2_J,axiom,
! [Current: current_a,Aux2: list_a,New2: list_a,Moved2: nat] :
( ( pop_a @ ( copy_a @ Current @ Aux2 @ New2 @ Moved2 ) )
= ( produc8263595898873874535tate_a @ ( first_a @ Current ) @ ( normalize_a @ ( copy_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) @ Aux2 @ New2 @ Moved2 ) ) ) ) ).
% Common.pop.simps(2)
thf(fact_939_invar__idle_Opelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( size_size_stack_a @ Stack2 )
= StackSize ) ) ) ) ) ).
% invar_idle.pelims(3)
thf(fact_940_invar__idle_Opelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( size_size_stack_a @ Stack2 )
!= StackSize ) ) ) ) ) ).
% invar_idle.pelims(2)
thf(fact_941_invar__idle_Opelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( Y
= ( ( size_size_stack_a @ Stack2 )
= StackSize ) )
=> ~ ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack2 @ StackSize ) ) ) ) ) ) ).
% invar_idle.pelims(1)
thf(fact_942_is__empty__idle_Opelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ X )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack2: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a3 @ Stack2 @ Uu ) )
=> ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ) ) ).
% is_empty_idle.pelims(3)
thf(fact_943_is__empty__idle_Opelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i7304311975391125061idle_a @ X )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack2: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a3 @ Stack2 @ Uu ) )
=> ~ ( type_i3216275384938974675tack_a @ Stack2 ) ) ) ) ) ).
% is_empty_idle.pelims(2)
thf(fact_944_is__empty__idle_Opelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i7304311975391125061idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_i5927890838958520983_rel_a @ X )
=> ~ ! [Stack2: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack2 @ Uu ) )
=> ( ( Y
= ( type_i3216275384938974675tack_a @ Stack2 ) )
=> ~ ( accp_idle_a @ idle_i5927890838958520983_rel_a @ ( idle_a3 @ Stack2 @ Uu ) ) ) ) ) ) ).
% is_empty_idle.pelims(1)
thf(fact_945_size__state_Opelims,axiom,
! [X: state_a,Y: nat] :
( ( ( size_size_state_a @ X )
= Y )
=> ( ( accp_state_a @ common288305775107427143_rel_a @ X )
=> ( ! [Current2: current_a,Idle3: idle_a] :
( ( X
= ( idle_a2 @ Current2 @ Idle3 ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( size_size_idle_a @ Idle3 ) ) )
=> ~ ( accp_state_a @ common288305775107427143_rel_a @ ( idle_a2 @ Current2 @ Idle3 ) ) ) )
=> ~ ! [Current2: current_a,Uu: list_a,Uv: list_a,Uw: nat] :
( ( X
= ( copy_a @ Current2 @ Uu @ Uv @ Uw ) )
=> ( ( Y
= ( ord_min_nat @ ( size_size_current_a @ Current2 ) @ ( type_s933026853152659577rent_a @ Current2 ) ) )
=> ~ ( accp_state_a @ common288305775107427143_rel_a @ ( copy_a @ Current2 @ Uu @ Uv @ Uw ) ) ) ) ) ) ) ).
% size_state.pelims
thf(fact_946_remaining__steps__state_Opelims,axiom,
! [X: state_a,Y: nat] :
( ( ( type_r2212416260012024137tate_a @ X )
= Y )
=> ( ( accp_state_a @ common2258151948077950263_rel_a @ X )
=> ( ! [Uu: current_a,Uv: idle_a] :
( ( X
= ( idle_a2 @ Uu @ Uv ) )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_state_a @ common2258151948077950263_rel_a @ ( idle_a2 @ Uu @ Uv ) ) ) )
=> ~ ! [Uw: list_a,Ux: nat,Uy: stack_a,Remained: nat,Aux: list_a,New: list_a,Moved: nat] :
( ( X
= ( copy_a @ ( current_a2 @ Uw @ Ux @ Uy @ Remained ) @ Aux @ New @ Moved ) )
=> ( ( Y
= ( minus_minus_nat @ Remained @ Moved ) )
=> ~ ( accp_state_a @ common2258151948077950263_rel_a @ ( copy_a @ ( current_a2 @ Uw @ Ux @ Uy @ Remained ) @ Aux @ New @ Moved ) ) ) ) ) ) ) ).
% remaining_steps_state.pelims
thf(fact_947_size__new__state_Opelims,axiom,
! [X: state_a,Y: nat] :
( ( ( type_s8424385952999958455tate_a @ X )
= Y )
=> ( ( accp_state_a @ common8464561214029693731_rel_a @ X )
=> ( ! [Current2: current_a,Uu: idle_a] :
( ( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ common8464561214029693731_rel_a @ ( idle_a2 @ Current2 @ Uu ) ) ) )
=> ~ ! [Current2: current_a,Uv: list_a,Uw: list_a,Ux: nat] :
( ( X
= ( copy_a @ Current2 @ Uv @ Uw @ Ux ) )
=> ( ( Y
= ( type_s933026853152659577rent_a @ Current2 ) )
=> ~ ( accp_state_a @ common8464561214029693731_rel_a @ ( copy_a @ Current2 @ Uv @ Uw @ Ux ) ) ) ) ) ) ) ).
% size_new_state.pelims
thf(fact_948_list__current_Opelims,axiom,
! [X: state_a,Y: list_a] :
( ( ( common1102728217005306191rent_a @ X )
= Y )
=> ( ( accp_state_a @ common1269017819723767218_rel_a @ X )
=> ( ! [Current2: current_a,Uu: idle_a] :
( ( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( ( Y
= ( current_list_a3 @ Current2 ) )
=> ~ ( accp_state_a @ common1269017819723767218_rel_a @ ( idle_a2 @ Current2 @ Uu ) ) ) )
=> ~ ! [Current2: current_a,Uv: list_a,Uw: list_a,Ux: nat] :
( ( X
= ( copy_a @ Current2 @ Uv @ Uw @ Ux ) )
=> ( ( Y
= ( current_list_a3 @ Current2 ) )
=> ~ ( accp_state_a @ common1269017819723767218_rel_a @ ( copy_a @ Current2 @ Uv @ Uw @ Ux ) ) ) ) ) ) ) ).
% list_current.pelims
thf(fact_949_last__list__update,axiom,
! [Xs2: list_a,K: nat,X: a] :
( ( Xs2 != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K @ X ) )
= ( last_a @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_950_bind__simps_I1_J,axiom,
! [F2: a > list_a] :
( ( bind_a_a @ nil_a @ F2 )
= nil_a ) ).
% bind_simps(1)
thf(fact_951_last__appendL,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Xs2 ) ) ) ).
% last_appendL
thf(fact_952_last__appendR,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_953_last__snoc,axiom,
! [Xs2: list_a,X: a] :
( ( last_a @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_954_last_Osimps,axiom,
! [Xs2: list_a,X: a] :
( ( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_955_last__ConsL,axiom,
! [Xs2: list_a,X: a] :
( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_956_last__ConsR,axiom,
! [Xs2: list_a,X: a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_957_last__append,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Xs2 ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_958_longest__common__suffix,axiom,
! [Xs2: list_a,Ys: list_a] :
? [Ss: list_a,Xs4: list_a,Ys6: list_a] :
( ( Xs2
= ( append_a @ Xs4 @ Ss ) )
& ( Ys
= ( append_a @ Ys6 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys6 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_959_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_960_last__tl,axiom,
! [Xs2: list_a] :
( ( ( Xs2 = nil_a )
| ( ( tl_a @ Xs2 )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ).
% last_tl
thf(fact_961_last__conv__nth,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ Xs2 )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_962_append__butlast__last__id,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs2 ) @ ( cons_a @ ( last_a @ Xs2 ) @ nil_a ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_963_Idle_Opush_Opelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ( ( accp_P2118310398061403062idle_a @ push_rel_a2 @ ( produc1265230069547855005idle_a @ X @ Xa ) )
=> ~ ! [Stack2: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a3 @ Stack2 @ StackSize ) )
=> ( ( Y
= ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize ) ) )
=> ~ ( accp_P2118310398061403062idle_a @ push_rel_a2 @ ( produc1265230069547855005idle_a @ X @ ( idle_a3 @ Stack2 @ StackSize ) ) ) ) ) ) ) ).
% Idle.push.pelims
thf(fact_964_distinct__adj__append__iff,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs2 @ Ys ) )
= ( ( distinct_adj_a @ Xs2 )
& ( distinct_adj_a @ Ys )
& ( ( Xs2 = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs2 )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_965_distinct__adj__Cons__Cons,axiom,
! [X: a,Y: a,Xs2: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) )
= ( ( X != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs2 ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_966_butlast__snoc,axiom,
! [Xs2: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_967_distinct__adj__ConsD,axiom,
! [X: a,Xs2: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs2 ) )
=> ( distinct_adj_a @ Xs2 ) ) ).
% distinct_adj_ConsD
thf(fact_968_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_969_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_970_butlast_Osimps_I2_J,axiom,
! [Xs2: list_a,X: a] :
( ( ( Xs2 = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs2 ) )
= nil_a ) )
& ( ( Xs2 != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs2 ) )
= ( cons_a @ X @ ( butlast_a @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_971_butlast__append,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs2 @ Ys ) )
= ( butlast_a @ Xs2 ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ Xs2 @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_972_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_973_distinct__adj__Cons,axiom,
! [X: a,Xs2: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs2 ) )
= ( ( Xs2 = nil_a )
| ( ( X
!= ( hd_a @ Xs2 ) )
& ( distinct_adj_a @ Xs2 ) ) ) ) ).
% distinct_adj_Cons
thf(fact_974_snoc__eq__iff__butlast,axiom,
! [Xs2: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs2 )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_975_drop__Cons__numeral,axiom,
! [V2: num,X: a,Xs2: list_a] :
( ( drop_a @ ( numeral_numeral_nat @ V2 ) @ ( cons_a @ X @ Xs2 ) )
= ( drop_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_976_take__Cons__numeral,axiom,
! [V2: num,X: a,Xs2: list_a] :
( ( take_a @ ( numeral_numeral_nat @ V2 ) @ ( cons_a @ X @ Xs2 ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_977_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_978_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_979_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_980_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_981_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_982_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_983_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_984_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_985_min__number__of_I1_J,axiom,
! [U: num,V2: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
= ( numeral_numeral_nat @ U ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
= ( numeral_numeral_nat @ V2 ) ) ) ) ).
% min_number_of(1)
thf(fact_986_min__0__1_I3_J,axiom,
! [X: num] :
( ( ord_min_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= zero_zero_nat ) ).
% min_0_1(3)
thf(fact_987_min__0__1_I4_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(4)
thf(fact_988_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= one_one_nat ) ).
% min_0_1(5)
thf(fact_989_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= one_one_nat ) ).
% min_0_1(6)
thf(fact_990_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_991_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_992_drop__all,axiom,
! [Xs2: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N )
=> ( ( drop_a @ N @ Xs2 )
= nil_a ) ) ).
% drop_all
thf(fact_993_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_a] :
( ( ( drop_a @ N @ Xs2 )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_994_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_995_nth__Cons__numeral,axiom,
! [X: a,Xs2: list_a,V2: num] :
( ( nth_a @ ( cons_a @ X @ Xs2 ) @ ( numeral_numeral_nat @ V2 ) )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_996_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_997_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_998_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_999_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1000_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1001_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1002_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1003_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1004_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1005_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1006_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X5: nat] : ( R2 @ X5 @ X5 )
=> ( ! [X5: nat,Y5: nat,Z: nat] :
( ( R2 @ X5 @ Y5 )
=> ( ( R2 @ Y5 @ Z )
=> ( R2 @ X5 @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1007_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1008_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_1009_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_1010_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1011_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F2 @ I ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1012_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1013_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1014_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1015_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1016_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1017_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1018_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1019_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1020_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1021_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1022_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1023_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1024_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1025_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 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y8: nat] :
( ( P @ Y8 )
=> ( ord_less_eq_nat @ Y8 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1026_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1027_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1028_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1029_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1030_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1031_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1032_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1033_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1034_lift__Suc__mono__le,axiom,
! [F2: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1035_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1036_lift__Suc__antimono__le,axiom,
! [F2: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1037_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1038_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1039_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1040_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1041_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1042_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1043_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_1044_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_1045_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1046_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1047_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1048_dec__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1049_inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1050_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1051_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1052_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1053_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1054_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1055_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1056_impossible__Cons,axiom,
! [Xs2: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
=> ( Xs2
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1057_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1058_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1059_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1060_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1061_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs2 ) )
= ( ? [X6: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ X6 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1062_take__last__length,axiom,
! [Xs2: list_a] :
( ( ( take_a @ ( suc @ zero_zero_nat ) @ ( rev_a @ Xs2 ) )
= ( cons_a @ ( last_a @ Xs2 ) @ nil_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ Xs2 ) ) ) ).
% take_last_length
thf(fact_1063_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1064_Nil__is__rev__conv,axiom,
! [Xs2: list_a] :
( ( nil_a
= ( rev_a @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_1065_rev__is__Nil__conv,axiom,
! [Xs2: list_a] :
( ( ( rev_a @ Xs2 )
= nil_a )
= ( Xs2 = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_1066_singleton__rev__conv,axiom,
! [X: a,Xs2: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs2 ) )
= ( ( cons_a @ X @ nil_a )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_1067_rev__singleton__conv,axiom,
! [Xs2: list_a,X: a] :
( ( ( rev_a @ Xs2 )
= ( cons_a @ X @ nil_a ) )
= ( Xs2
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_1068_rev__eq__Cons__iff,axiom,
! [Xs2: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs2 )
= ( cons_a @ Y @ Ys ) )
= ( Xs2
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_1069_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_1070_hd__rev,axiom,
! [Xs2: list_a] :
( ( hd_a @ ( rev_a @ Xs2 ) )
= ( last_a @ Xs2 ) ) ).
% hd_rev
thf(fact_1071_last__rev,axiom,
! [Xs2: list_a] :
( ( last_a @ ( rev_a @ Xs2 ) )
= ( hd_a @ Xs2 ) ) ).
% last_rev
thf(fact_1072_rev__app__single,axiom,
! [Xs2: list_a,X: a] :
( ( append_a @ ( rev_a @ Xs2 ) @ ( cons_a @ X @ nil_a ) )
= ( rev_a @ ( cons_a @ X @ Xs2 ) ) ) ).
% rev_app_single
thf(fact_1073_rev__tl__hd,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( append_a @ ( rev_a @ ( tl_a @ Xs2 ) ) @ ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) )
= ( rev_a @ Xs2 ) ) ) ).
% rev_tl_hd
thf(fact_1074_take__last,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( take_a @ one_one_nat @ ( rev_a @ Xs2 ) )
= ( cons_a @ ( last_a @ Xs2 ) @ nil_a ) ) ) ).
% take_last
thf(fact_1075_last__drop__rev,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( cons_a @ ( last_a @ Xs2 ) @ ( drop_a @ one_one_nat @ ( rev_a @ Xs2 ) ) )
= ( rev_a @ Xs2 ) ) ) ).
% last_drop_rev
thf(fact_1076_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X8 ) ) ).
% minf(8)
thf(fact_1077_minf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X8: num] :
( ( ord_less_num @ X8 @ Z )
=> ~ ( ord_less_eq_num @ T @ X8 ) ) ).
% minf(8)
thf(fact_1078_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( ord_less_eq_nat @ X8 @ T ) ) ).
% minf(6)
thf(fact_1079_minf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X8: num] :
( ( ord_less_num @ X8 @ Z )
=> ( ord_less_eq_num @ X8 @ T ) ) ).
% minf(6)
thf(fact_1080_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( ord_less_eq_nat @ T @ X8 ) ) ).
% pinf(8)
thf(fact_1081_pinf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X8: num] :
( ( ord_less_num @ Z @ X8 )
=> ( ord_less_eq_num @ T @ X8 ) ) ).
% pinf(8)
thf(fact_1082_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ~ ( ord_less_nat @ T @ X8 ) ) ).
% minf(7)
thf(fact_1083_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( ord_less_nat @ X8 @ T ) ) ).
% minf(5)
thf(fact_1084_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( X8 != T ) ) ).
% minf(4)
thf(fact_1085_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( X8 != T ) ) ).
% minf(3)
thf(fact_1086_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q3 @ X8 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1087_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q3 @ X8 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1088_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( ord_less_nat @ T @ X8 ) ) ).
% pinf(7)
thf(fact_1089_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ~ ( ord_less_nat @ X8 @ T ) ) ).
% pinf(5)
thf(fact_1090_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( X8 != T ) ) ).
% pinf(4)
thf(fact_1091_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( X8 != T ) ) ).
% pinf(3)
thf(fact_1092_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P4 @ X8 )
| ( Q3 @ X8 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1093_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P4 @ X8 )
& ( Q3 @ X8 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1094_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z @ X8 )
=> ~ ( ord_less_eq_nat @ X8 @ T ) ) ).
% pinf(6)
thf(fact_1095_pinf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X8: num] :
( ( ord_less_num @ Z @ X8 )
=> ~ ( ord_less_eq_num @ X8 @ T ) ) ).
% pinf(6)
thf(fact_1096_subrelI,axiom,
! [R: set_Pr324718442235990179tate_a,S: set_Pr324718442235990179tate_a] :
( ! [X5: a,Y5: state_a] :
( ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X5 @ Y5 ) @ R )
=> ( member7740437332958137860tate_a @ ( produc8263595898873874535tate_a @ X5 @ Y5 ) @ S ) )
=> ( ord_le8856296118373343491tate_a @ R @ S ) ) ).
% subrelI
thf(fact_1097_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_1098_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1099_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_1100_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1101_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1102_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_1103_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_1104_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_1105_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1106_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1107_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1108_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1109_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1110_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1111_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1112_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_1113_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1114_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1115_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_1116_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1117_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1118_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_1119_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_1120_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_1121_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X9: nat] : ( P5 @ X9 ) )
= ( ^ [P6: nat > $o] :
? [N3: nat] :
( ( P6 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P6 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1122_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1123_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1124_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1125_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1126_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y8: nat] :
( ( ord_less_nat @ Y8 @ X5 )
=> ( P @ Y8 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1127_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_1128_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_1129_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1130_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1131_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1132_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1133_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_1134_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1135_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_1136_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1137_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1138_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1139_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1140_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1141_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1142_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y6: nat] :
( ( ord_less_eq_nat @ X6 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_1143_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X6: num,Y6: num] :
( ( ord_less_eq_num @ X6 @ Y6 )
& ~ ( ord_less_eq_num @ Y6 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_1144_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_1145_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1146_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_1147_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1148_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_1149_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1150_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1151_order_Ostrict__trans1,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1152_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1153_order_Ostrict__trans2,axiom,
! [A: num,B: num,C2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1154_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_1155_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
& ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1156_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_1157_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_num @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1158_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_1159_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1160_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1161_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1162_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1163_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C2: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C2 @ B )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1164_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_1165_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B4: num,A5: num] :
( ( ord_less_eq_num @ B4 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1166_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_1167_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1168_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_1169_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1170_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y6: nat] :
( ( ord_less_nat @ X6 @ Y6 )
| ( X6 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1171_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X6: num,Y6: num] :
( ( ord_less_num @ X6 @ Y6 )
| ( X6 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_1172_order__less__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y6: nat] :
( ( ord_less_eq_nat @ X6 @ Y6 )
& ( X6 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1173_order__less__le,axiom,
( ord_less_num
= ( ^ [X6: num,Y6: num] :
( ( ord_less_eq_num @ X6 @ Y6 )
& ( X6 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_1174_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1175_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1176_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1177_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1178_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_1179_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1180_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_1181_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1182_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_1183_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1184_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1185_order__le__less__trans,axiom,
! [X: num,Y: num,Z3: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z3 )
=> ( ord_less_num @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1186_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1187_order__less__le__trans,axiom,
! [X: num,Y: num,Z3: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z3 )
=> ( ord_less_num @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1188_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1189_order__le__less__subst1,axiom,
! [A: num,F2: nat > num,B: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1190_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1191_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F2 @ B ) @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1192_order__le__less__subst2,axiom,
! [A: num,B: num,F2: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1193_order__le__less__subst2,axiom,
! [A: num,B: num,F2: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F2 @ B ) @ C2 )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1194_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1195_order__less__le__subst1,axiom,
! [A: num,F2: nat > num,B: nat,C2: nat] :
( ( ord_less_num @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1196_order__less__le__subst1,axiom,
! [A: nat,F2: num > nat,B: num,C2: num] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1197_order__less__le__subst1,axiom,
! [A: num,F2: num > num,B: num,C2: num] :
( ( ord_less_num @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_num @ B @ C2 )
=> ( ! [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
=> ( ord_less_eq_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1198_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1199_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > num,C2: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F2 @ B ) @ C2 )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
=> ( ord_less_num @ ( F2 @ X5 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1200_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_1201_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1202_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_1203_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A3: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A3 ) )
= ( ord_less_num @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1204_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 )
& ! [X8: nat] :
( ( ( ord_less_eq_nat @ A @ X8 )
& ( ord_less_nat @ X8 @ C3 ) )
=> ( P @ X8 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1205_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1206_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1207_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1208_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1209_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1210_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1211_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1212_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1213_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1214_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1215_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_1216_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_1217_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1218_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1219_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1220_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1221_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1222_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1223_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1224_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1225_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1233_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1234_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1235_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1236_add__numeral__left,axiom,
! [V2: num,W: num,Z3: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W ) ) @ Z3 ) ) ).
% add_numeral_left
thf(fact_1237_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_1238_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1239_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_1240_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1241_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1242_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1243_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1244_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_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_1256_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
% 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,
( ( cons_a @ xa @ ( common1102728217005306191rent_a @ common ) )
= ( common1102728217005306191rent_a @ ( idle_a2 @ ( current_a2 @ ( cons_a @ xb @ xs ) @ added @ old @ remained ) @ ( idle_a3 @ stack @ stackSize ) ) ) ) ).
%------------------------------------------------------------------------------