TPTP Problem File: ITP145^1.p
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%------------------------------------------------------------------------------
% File : ITP145^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer PHoareTotal problem prob_632__3264706_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : PHoareTotal/prob_632__3264706_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 468 ( 133 unt; 113 typ; 0 def)
% Number of atoms : 1002 ( 421 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 5196 ( 215 ~; 35 |; 108 &;4218 @)
% ( 0 <=>; 620 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 10 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 299 ( 299 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 84 usr; 10 con; 0-3 aty)
% Number of variables : 1551 ( 8 ^;1465 !; 78 ?;1551 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:29.764
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J_J,type,
set_Pr541394119_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
produc2046512871_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J_J,type,
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produc1586269031_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
set_Pr2096893127_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
produc186437863_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
set_Pr1142642759_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
produc1225889383_state: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
list_P890094829_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
set_Pr1165141447_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
produc410756839_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J_J,type,
set_Pr391631483_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J,type,
produc1393453339_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__PLang__Ocom_M_062_It__PLang__Ocom_M_Eo_J_J_Mt__List__Olist_It__PLang__Ocom_J_J,type,
produc2058209240st_com: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J_J,type,
set_Pr602326075st_com: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
list_P1040014977_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
set_Pr849496667_state: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J,type,
produc258871771st_com: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
produc2041926651_state: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PLang__Ocom_Mt__PLang__Ocom_J_J,type,
set_Pr1822651163om_com: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__PLang__Ocom_Mt__PLang__Ocom_J,type,
product_prod_com_com: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__PLang__Ocom_J_J,type,
list_list_com: $tType ).
thf(ty_n_t__List__Olist_It__PLang__Ostate_J,type,
list_state: $tType ).
thf(ty_n_t__Set__Oset_It__PLang__Ostate_J,type,
set_state: $tType ).
thf(ty_n_t__List__Olist_It__PLang__Ocom_J,type,
list_com: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__PLang__Ostate,type,
state: $tType ).
thf(ty_n_t__PLang__Ocom,type,
com: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (84)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__PLang__Ocom_J,type,
append_list_com: list_list_com > list_list_com > list_list_com ).
thf(sy_c_List_Oappend_001t__PLang__Ocom,type,
append_com: list_com > list_com > list_com ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
append982487830_state: list_P1040014977_state > list_P1040014977_state > list_P1040014977_state ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
append221843330_state: list_P890094829_state > list_P890094829_state > list_P890094829_state ).
thf(sy_c_List_Obind_001t__PLang__Ocom_001t__PLang__Ocom,type,
bind_com_com: list_com > ( com > list_com ) > list_com ).
thf(sy_c_List_Oconcat_001t__PLang__Ocom,type,
concat_com: list_list_com > list_com ).
thf(sy_c_List_Ogen__length_001t__PLang__Ocom,type,
gen_length_com: nat > list_com > nat ).
thf(sy_c_List_Oinsert_001t__PLang__Ocom,type,
insert_com: com > list_com > list_com ).
thf(sy_c_List_Olexord_001t__PLang__Ocom,type,
lexord_com: set_Pr1822651163om_com > set_Pr602326075st_com ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
lexord1062073966_state: set_Pr1165141447_state > set_Pr1142642759_state ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
lexord1893203674_state: set_Pr2096893127_state > set_Pr1325664839_state ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__PLang__Ocom_J,type,
cons_list_com: list_com > list_list_com > list_list_com ).
thf(sy_c_List_Olist_OCons_001t__PLang__Ocom,type,
cons_com: com > list_com > list_com ).
thf(sy_c_List_Olist_OCons_001t__PLang__Ostate,type,
cons_state: state > list_state > list_state ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
cons_P465369265_state: produc2041926651_state > list_P1040014977_state > list_P1040014977_state ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
cons_P438230685_state: produc410756839_state > list_P890094829_state > list_P890094829_state ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__PLang__Ocom_J,type,
nil_list_com: list_list_com ).
thf(sy_c_List_Olist_ONil_001t__PLang__Ocom,type,
nil_com: list_com ).
thf(sy_c_List_Olist_ONil_001t__PLang__Ostate,type,
nil_state: list_state ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
nil_Pr1844458753_state: list_P1040014977_state ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
nil_Pr1347322605_state: list_P890094829_state ).
thf(sy_c_List_Olistrel1_001t__PLang__Ocom,type,
listrel1_com: set_Pr1822651163om_com > set_Pr602326075st_com ).
thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
listre1738592858_state: set_Pr1165141447_state > set_Pr1142642759_state ).
thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
listre1725776070_state: set_Pr2096893127_state > set_Pr1325664839_state ).
thf(sy_c_List_Olistrel_001t__List__Olist_It__PLang__Ocom_J_001t__PLang__Ostate,type,
listre1268245261_state: set_Pr849496667_state > set_Pr391631483_state ).
thf(sy_c_List_Olistrel_001t__PLang__Ocom_001t__PLang__Ocom,type,
listrel_com_com: set_Pr1822651163om_com > set_Pr602326075st_com ).
thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
listre194873521_state: set_Pr1165141447_state > set_Pr1142642759_state ).
thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
listre1244407729_state: set_Pr2096893127_state > set_Pr1325664839_state ).
thf(sy_c_List_Omaps_001t__PLang__Ocom_001t__PLang__Ocom,type,
maps_com_com: ( com > list_com ) > list_com > list_com ).
thf(sy_c_List_On__lists_001t__PLang__Ocom,type,
n_lists_com: nat > list_com > list_list_com ).
thf(sy_c_List_Onths_001t__PLang__Ocom,type,
nths_com: list_com > set_nat > list_com ).
thf(sy_c_List_Oproduct__lists_001t__PLang__Ocom,type,
product_lists_com: list_list_com > list_list_com ).
thf(sy_c_List_Osubseqs_001t__PLang__Ocom,type,
subseqs_com: list_com > list_list_com ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_PHoareTotal__Mirabelle__ynemugpbor_Oexec1,type,
pHoare259243666_exec1: set_Pr1165141447_state ).
thf(sy_c_PHoareTotal__Mirabelle__ynemugpbor_Oexecs,type,
pHoare259243732_execs: state > list_com > state > $o ).
thf(sy_c_PHoareTotal__Mirabelle__ynemugpbor_Oinf,type,
pHoare2126160787le_inf: list_com > state > $o ).
thf(sy_c_PHoareTotal__Mirabelle__ynemugpbor_Otermis,type,
pHoare806997636termis: list_com > state > $o ).
thf(sy_c_PLang_Ocom_OCond,type,
cond: ( state > $o ) > com > com > com ).
thf(sy_c_PLang_Ocom_ODo,type,
do: ( state > set_state ) > com ).
thf(sy_c_PLang_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_PLang_Ocom_OWhile,type,
while: ( state > $o ) > com > com ).
thf(sy_c_PLang_Oexec,type,
exec: state > com > state > $o ).
thf(sy_c_PTermi_Otermi,type,
termi: com > state > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__PLang__Ocom_M_062_It__PLang__Ocom_M_Eo_J_J_001t__List__Olist_It__PLang__Ocom_J,type,
produc1914379986st_com: ( com > com > $o ) > list_com > produc2058209240st_com ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_001t__List__Olist_It__PLang__Ostate_J,type,
produc1742567891_state: list_list_com > list_state > produc1393453339_state ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__PLang__Ocom_J_001t__List__Olist_It__PLang__Ocom_J,type,
produc1307105171st_com: list_com > list_com > produc258871771st_com ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__PLang__Ocom_J_001t__PLang__Ostate,type,
produc1204172211_state: list_com > state > produc2041926651_state ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
produc1298457559_state: list_P1040014977_state > list_P1040014977_state > produc1225889383_state ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
produc1354769367_state: list_P890094829_state > list_P890094829_state > produc1586269031_state ).
thf(sy_c_Product__Type_OPair_001t__PLang__Ocom_001t__PLang__Ocom,type,
product_Pair_com_com: com > com > product_prod_com_com ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
produc1909270103_state: produc2041926651_state > produc2041926651_state > produc410756839_state ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
produc2065881943_state: produc410756839_state > produc410756839_state > produc186437863_state ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
produc2049704791_state: produc186437863_state > produc186437863_state > produc2046512871_state ).
thf(sy_c_Relation_Oirrefl_001t__PLang__Ocom,type,
irrefl_com: set_Pr1822651163om_com > $o ).
thf(sy_c_Set_OCollect_001t__PLang__Ostate,type,
collect_state: ( state > $o ) > set_state ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
collec300063826_state: ( produc410756839_state > $o ) > set_Pr1165141447_state ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
collec489924946_state: ( produc186437863_state > $o ) > set_Pr2096893127_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__PLang__Ocom_J,type,
transi436345274st_com: set_Pr602326075st_com > set_Pr602326075st_com ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
transi339813252_state: set_Pr1142642759_state > set_Pr1142642759_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
transi1965148016_state: set_Pr1325664839_state > set_Pr1325664839_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__PLang__Ocom,type,
transi863150634cl_com: set_Pr1822651163om_com > set_Pr1822651163om_com ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
transi1302705790_state: set_Pr1165141447_state > set_Pr1165141447_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
transi463013610_state: set_Pr2096893127_state > set_Pr2096893127_state ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
transi152726506_state: set_Pr541394119_state > set_Pr541394119_state ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
transi1726587420_state: set_Pr1165141447_state > set_Pr1165141447_state ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
transi1666849160_state: set_Pr2096893127_state > set_Pr2096893127_state ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
transi1995721864_state: set_Pr541394119_state > set_Pr541394119_state ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__PLang__Ostate,type,
member_state: state > set_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__PLang__Ocom_J_J_Mt__List__Olist_It__PLang__Ostate_J_J,type,
member1442949316_state: produc1393453339_state > set_Pr391631483_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__List__Olist_It__PLang__Ocom_J_J,type,
member32358020st_com: produc258871771st_com > set_Pr602326075st_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J,type,
member64234660_state: produc2041926651_state > set_Pr849496667_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
member1267359376_state: produc1225889383_state > set_Pr1142642759_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
member423306384_state: produc1586269031_state > set_Pr1325664839_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__PLang__Ocom_Mt__PLang__Ocom_J,type,
member1333916004om_com: product_prod_com_com > set_Pr1822651163om_com > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J,type,
member1069318160_state: produc410756839_state > set_Pr1165141447_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J,type,
member512523024_state: produc186437863_state > set_Pr2096893127_state > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_Mt__Product____Type__Oprod_It__List__Olist_It__PLang__Ocom_J_Mt__PLang__Ostate_J_J_J_J,type,
member284400400_state: produc2046512871_state > set_Pr541394119_state > $o ).
thf(sy_v_c,type,
c: com ).
thf(sy_v_f,type,
f: nat > produc2041926651_state ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_s,type,
s: state ).
% Relevant facts (350)
thf(fact_0_exec1E_I1_J,axiom,
! [S: state,Cs: list_com,S2: state] :
~ ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ nil_com @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 ) ).
% exec1E(1)
thf(fact_1_renumber,axiom,
! [A: produc410756839_state,F: nat > produc410756839_state,R: set_Pr2096893127_state] :
( ! [I: nat] :
( ( member512523024_state @ ( produc2065881943_state @ A @ ( F @ I ) ) @ ( transi463013610_state @ R ) )
& ( member512523024_state @ ( produc2065881943_state @ ( F @ I ) @ ( F @ ( suc @ I ) ) ) @ R ) )
=> ? [F2: nat > produc410756839_state] :
( ( ( F2 @ zero_zero_nat )
= A )
& ! [I2: nat] : ( member512523024_state @ ( produc2065881943_state @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ).
% renumber
thf(fact_2_renumber,axiom,
! [A: produc2041926651_state,F: nat > produc2041926651_state,R: set_Pr1165141447_state] :
( ! [I: nat] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ ( F @ I ) ) @ ( transi1302705790_state @ R ) )
& ( member1069318160_state @ ( produc1909270103_state @ ( F @ I ) @ ( F @ ( suc @ I ) ) ) @ R ) )
=> ? [F2: nat > produc2041926651_state] :
( ( ( F2 @ zero_zero_nat )
= A )
& ! [I2: nat] : ( member1069318160_state @ ( produc1909270103_state @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ).
% renumber
thf(fact_3_renumber__aux,axiom,
! [A: produc410756839_state,F: nat > produc410756839_state,R: set_Pr2096893127_state,B: produc410756839_state] :
( ! [I: nat] :
( ( member512523024_state @ ( produc2065881943_state @ A @ ( F @ I ) ) @ ( transi463013610_state @ R ) )
& ( member512523024_state @ ( produc2065881943_state @ ( F @ I ) @ ( F @ ( suc @ I ) ) ) @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( B
= ( F @ zero_zero_nat ) )
=> ? [F2: nat > produc410756839_state] :
( ( ( F2 @ zero_zero_nat )
= A )
& ! [I2: nat] : ( member512523024_state @ ( produc2065881943_state @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ) ) ).
% renumber_aux
thf(fact_4_renumber__aux,axiom,
! [A: produc2041926651_state,F: nat > produc2041926651_state,R: set_Pr1165141447_state,B: produc2041926651_state] :
( ! [I: nat] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ ( F @ I ) ) @ ( transi1302705790_state @ R ) )
& ( member1069318160_state @ ( produc1909270103_state @ ( F @ I ) @ ( F @ ( suc @ I ) ) ) @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( B
= ( F @ zero_zero_nat ) )
=> ? [F2: nat > produc2041926651_state] :
( ( ( F2 @ zero_zero_nat )
= A )
& ! [I2: nat] : ( member1069318160_state @ ( produc1909270103_state @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ) ) ).
% renumber_aux
thf(fact_5_execs__pres__termi,axiom,
! [C: com,S: state,C2: com,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ nil_com ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C2 @ Cs ) @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( ( termi @ C @ S )
=> ( termi @ C2 @ S2 ) ) ) ).
% execs_pres_termi
thf(fact_6_rtrancl__trancl__absorb,axiom,
! [R2: set_Pr2096893127_state] :
( ( transi1666849160_state @ ( transi463013610_state @ R2 ) )
= ( transi463013610_state @ R2 ) ) ).
% rtrancl_trancl_absorb
thf(fact_7_rtrancl__trancl__absorb,axiom,
! [R2: set_Pr1165141447_state] :
( ( transi1726587420_state @ ( transi1302705790_state @ R2 ) )
= ( transi1302705790_state @ R2 ) ) ).
% rtrancl_trancl_absorb
thf(fact_8_trancl__rtrancl__absorb,axiom,
! [R2: set_Pr2096893127_state] :
( ( transi463013610_state @ ( transi1666849160_state @ R2 ) )
= ( transi463013610_state @ R2 ) ) ).
% trancl_rtrancl_absorb
thf(fact_9_trancl__rtrancl__absorb,axiom,
! [R2: set_Pr1165141447_state] :
( ( transi1302705790_state @ ( transi1726587420_state @ R2 ) )
= ( transi1302705790_state @ R2 ) ) ).
% trancl_rtrancl_absorb
thf(fact_10_trancl__induct2,axiom,
! [Ax: produc410756839_state,Ay: produc410756839_state,Bx: produc410756839_state,By: produc410756839_state,R: set_Pr541394119_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi1995721864_state @ R ) )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ A2 @ B2 ) ) @ R )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state,Aa: produc410756839_state,Ba: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ A2 @ B2 ) ) @ ( transi1995721864_state @ R ) )
=> ( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( produc2065881943_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_11_trancl__induct2,axiom,
! [Ax: produc2041926651_state,Ay: produc2041926651_state,Bx: produc2041926651_state,By: produc2041926651_state,R: set_Pr2096893127_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi1666849160_state @ R ) )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ A2 @ B2 ) ) @ R )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state,Aa: produc2041926651_state,Ba: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ A2 @ B2 ) ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( produc1909270103_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_12_trancl__induct2,axiom,
! [Ax: list_com,Ay: state,Bx: list_com,By: state,R: set_Pr1165141447_state,P: list_com > state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1726587420_state @ R ) )
=> ( ! [A2: list_com,B2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ A2 @ B2 ) ) @ R )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: list_com,B2: state,Aa: list_com,Ba: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ A2 @ B2 ) ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A2 @ B2 ) @ ( produc1204172211_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_13_inf__def,axiom,
( pHoare2126160787le_inf
= ( ^ [Cs2: list_com,S3: state] :
? [F3: nat > produc2041926651_state] :
( ( ( F3 @ zero_zero_nat )
= ( produc1204172211_state @ Cs2 @ S3 ) )
& ! [I3: nat] : ( member1069318160_state @ ( produc1909270103_state @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ pHoare259243666_exec1 ) ) ) ) ).
% inf_def
thf(fact_14_rtrancl__induct2,axiom,
! [Ax: produc410756839_state,Ay: produc410756839_state,Bx: produc410756839_state,By: produc410756839_state,R: set_Pr541394119_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state,Aa: produc410756839_state,Ba: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ A2 @ B2 ) ) @ ( transi152726506_state @ R ) )
=> ( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( produc2065881943_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_15_rtrancl__induct2,axiom,
! [Ax: produc2041926651_state,Ay: produc2041926651_state,Bx: produc2041926651_state,By: produc2041926651_state,R: set_Pr2096893127_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state,Aa: produc2041926651_state,Ba: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ A2 @ B2 ) ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( produc1909270103_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_16_rtrancl__induct2,axiom,
! [Ax: list_com,Ay: state,Bx: list_com,By: state,R: set_Pr1165141447_state,P: list_com > state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A2: list_com,B2: state,Aa: list_com,Ba: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ A2 @ B2 ) ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A2 @ B2 ) @ ( produc1204172211_state @ Aa @ Ba ) ) @ R )
=> ( ( P @ A2 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_17_converse__rtranclE2,axiom,
! [Xa: produc410756839_state,Xb: produc410756839_state,Za: produc410756839_state,Zb: produc410756839_state,R: set_Pr541394119_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Xa @ Xb ) @ ( produc2065881943_state @ Za @ Zb ) ) @ ( transi152726506_state @ R ) )
=> ( ( ( produc2065881943_state @ Xa @ Xb )
!= ( produc2065881943_state @ Za @ Zb ) )
=> ~ ! [A2: produc410756839_state,B2: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Xa @ Xb ) @ ( produc2065881943_state @ A2 @ B2 ) ) @ R )
=> ~ ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( produc2065881943_state @ Za @ Zb ) ) @ ( transi152726506_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_18_converse__rtranclE2,axiom,
! [Xa: produc2041926651_state,Xb: produc2041926651_state,Za: produc2041926651_state,Zb: produc2041926651_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Xa @ Xb ) @ ( produc1909270103_state @ Za @ Zb ) ) @ ( transi463013610_state @ R ) )
=> ( ( ( produc1909270103_state @ Xa @ Xb )
!= ( produc1909270103_state @ Za @ Zb ) )
=> ~ ! [A2: produc2041926651_state,B2: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Xa @ Xb ) @ ( produc1909270103_state @ A2 @ B2 ) ) @ R )
=> ~ ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( produc1909270103_state @ Za @ Zb ) ) @ ( transi463013610_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_19_converse__rtranclE2,axiom,
! [Xa: list_com,Xb: state,Za: list_com,Zb: state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Xa @ Xb ) @ ( produc1204172211_state @ Za @ Zb ) ) @ ( transi1302705790_state @ R ) )
=> ( ( ( produc1204172211_state @ Xa @ Xb )
!= ( produc1204172211_state @ Za @ Zb ) )
=> ~ ! [A2: list_com,B2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Xa @ Xb ) @ ( produc1204172211_state @ A2 @ B2 ) ) @ R )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A2 @ B2 ) @ ( produc1204172211_state @ Za @ Zb ) ) @ ( transi1302705790_state @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_20_converse__rtrancl__induct2,axiom,
! [Ax: produc410756839_state,Ay: produc410756839_state,Bx: produc410756839_state,By: produc410756839_state,R: set_Pr541394119_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Ax @ Ay ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state,Aa: produc410756839_state,Ba: produc410756839_state] :
( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( produc2065881943_state @ Aa @ Ba ) ) @ R )
=> ( ( member284400400_state @ ( produc2049704791_state @ ( produc2065881943_state @ Aa @ Ba ) @ ( produc2065881943_state @ Bx @ By ) ) @ ( transi152726506_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A2 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_21_converse__rtrancl__induct2,axiom,
! [Ax: list_com,Ay: state,Bx: list_com,By: state,R: set_Pr1165141447_state,P: list_com > state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ax @ Ay ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A2: list_com,B2: state,Aa: list_com,Ba: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ A2 @ B2 ) @ ( produc1204172211_state @ Aa @ Ba ) ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Aa @ Ba ) @ ( produc1204172211_state @ Bx @ By ) ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A2 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_22_converse__rtrancl__induct2,axiom,
! [Ax: produc2041926651_state,Ay: produc2041926651_state,Bx: produc2041926651_state,By: produc2041926651_state,R: set_Pr2096893127_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Ax @ Ay ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state,Aa: produc2041926651_state,Ba: produc2041926651_state] :
( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( produc1909270103_state @ Aa @ Ba ) ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ ( produc1909270103_state @ Aa @ Ba ) @ ( produc1909270103_state @ Bx @ By ) ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A2 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_23_tranclD,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R2: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1726587420_state @ R2 ) )
=> ? [Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Z ) @ R2 )
& ( member1069318160_state @ ( produc1909270103_state @ Z @ Y ) @ ( transi1302705790_state @ R2 ) ) ) ) ).
% tranclD
thf(fact_24_tranclD,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R2: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi1666849160_state @ R2 ) )
=> ? [Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Z ) @ R2 )
& ( member512523024_state @ ( produc2065881943_state @ Z @ Y ) @ ( transi463013610_state @ R2 ) ) ) ) ).
% tranclD
thf(fact_25_rtranclD,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R2: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R2 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R2 ) ) ) ) ) ).
% rtranclD
thf(fact_26_rtranclD,axiom,
! [A: produc410756839_state,B: produc410756839_state,R2: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R2 ) )
=> ( ( A = B )
| ( ( A != B )
& ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R2 ) ) ) ) ) ).
% rtranclD
thf(fact_27_tranclD2,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R2: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1726587420_state @ R2 ) )
=> ? [Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Z ) @ ( transi1302705790_state @ R2 ) )
& ( member1069318160_state @ ( produc1909270103_state @ Z @ Y ) @ R2 ) ) ) ).
% tranclD2
thf(fact_28_tranclD2,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R2: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi1666849160_state @ R2 ) )
=> ? [Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Z ) @ ( transi463013610_state @ R2 ) )
& ( member512523024_state @ ( produc2065881943_state @ Z @ Y ) @ R2 ) ) ) ).
% tranclD2
thf(fact_29_trancl__into__rtrancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_30_trancl__into__rtrancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_31_rtrancl__eq__or__trancl,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R2: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R2 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1726587420_state @ R2 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_32_rtrancl__eq__or__trancl,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R2: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R2 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi1666849160_state @ R2 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_33_r__into__rtrancl,axiom,
! [P2: produc410756839_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ P2 @ R )
=> ( member1069318160_state @ P2 @ ( transi1302705790_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_34_r__into__rtrancl,axiom,
! [P2: produc186437863_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ P2 @ R )
=> ( member512523024_state @ P2 @ ( transi463013610_state @ R ) ) ) ).
% r_into_rtrancl
thf(fact_35_rtrancl__idemp,axiom,
! [R: set_Pr1165141447_state] :
( ( transi1302705790_state @ ( transi1302705790_state @ R ) )
= ( transi1302705790_state @ R ) ) ).
% rtrancl_idemp
thf(fact_36_rtrancl__idemp,axiom,
! [R: set_Pr2096893127_state] :
( ( transi463013610_state @ ( transi463013610_state @ R ) )
= ( transi463013610_state @ R ) ) ).
% rtrancl_idemp
thf(fact_37_r__into__trancl_H,axiom,
! [P2: produc410756839_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ P2 @ R )
=> ( member1069318160_state @ P2 @ ( transi1726587420_state @ R ) ) ) ).
% r_into_trancl'
thf(fact_38_r__into__trancl_H,axiom,
! [P2: produc186437863_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ P2 @ R )
=> ( member512523024_state @ P2 @ ( transi1666849160_state @ R ) ) ) ).
% r_into_trancl'
thf(fact_39_converse__rtrancl__into__rtrancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1302705790_state @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_40_converse__rtrancl__into__rtrancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi463013610_state @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_41_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ R )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1302705790_state @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_42_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ R )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi463013610_state @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_43_converse__rtrancl__induct,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ B )
=> ( ! [Y2: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Z @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y2 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_44_converse__rtrancl__induct,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( P @ B )
=> ( ! [Y2: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ Z @ B ) @ ( transi463013610_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y2 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_45_rtrancl_Ortrancl__refl,axiom,
! [A: produc2041926651_state,R: set_Pr1165141447_state] : ( member1069318160_state @ ( produc1909270103_state @ A @ A ) @ ( transi1302705790_state @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_46_rtrancl_Ortrancl__refl,axiom,
! [A: produc410756839_state,R: set_Pr2096893127_state] : ( member512523024_state @ ( produc2065881943_state @ A @ A ) @ ( transi463013610_state @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_47_converse__rtranclE,axiom,
! [X: produc2041926651_state,Z2: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1302705790_state @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y2 ) @ R )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z2 ) @ ( transi1302705790_state @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_48_converse__rtranclE,axiom,
! [X: produc410756839_state,Z2: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi463013610_state @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y2 ) @ R )
=> ~ ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z2 ) @ ( transi463013610_state @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_49_rtrancl_Oinducts,axiom,
! [X1: produc2041926651_state,X2: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ X1 @ X2 ) @ ( transi1302705790_state @ R ) )
=> ( ! [A2: produc2041926651_state] : ( P @ A2 @ A2 )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state,C3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ A2 @ B2 )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B2 @ C3 ) @ R )
=> ( P @ A2 @ C3 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% rtrancl.inducts
thf(fact_50_rtrancl_Oinducts,axiom,
! [X1: produc410756839_state,X2: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ X1 @ X2 ) @ ( transi463013610_state @ R ) )
=> ( ! [A2: produc410756839_state] : ( P @ A2 @ A2 )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state,C3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi463013610_state @ R ) )
=> ( ( P @ A2 @ B2 )
=> ( ( member512523024_state @ ( produc2065881943_state @ B2 @ C3 ) @ R )
=> ( P @ A2 @ C3 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% rtrancl.inducts
thf(fact_51_rtrancl__induct,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( P @ A )
=> ( ! [Y2: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ Y2 ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z ) @ R )
=> ( ( P @ Y2 )
=> ( P @ Z ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_52_rtrancl__induct,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( P @ A )
=> ( ! [Y2: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ Y2 ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z ) @ R )
=> ( ( P @ Y2 )
=> ( P @ Z ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_53_rtrancl__trans,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Z2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Z2 ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1302705790_state @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_54_rtrancl__trans,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Z2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Z2 ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi463013610_state @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_55_rtrancl_Osimps,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1302705790_state @ R ) )
= ( ? [A3: produc2041926651_state] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: produc2041926651_state,B3: produc2041926651_state,C4: produc2041926651_state] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( member1069318160_state @ ( produc1909270103_state @ A3 @ B3 ) @ ( transi1302705790_state @ R ) )
& ( member1069318160_state @ ( produc1909270103_state @ B3 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_56_rtrancl_Osimps,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi463013610_state @ R ) )
= ( ? [A3: produc410756839_state] :
( ( A1 = A3 )
& ( A22 = A3 ) )
| ? [A3: produc410756839_state,B3: produc410756839_state,C4: produc410756839_state] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( member512523024_state @ ( produc2065881943_state @ A3 @ B3 ) @ ( transi463013610_state @ R ) )
& ( member512523024_state @ ( produc2065881943_state @ B3 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_57_rtrancl_Ocases,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1302705790_state @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ B2 ) @ ( transi1302705790_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ B2 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_58_rtrancl_Ocases,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi463013610_state @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ B2 ) @ ( transi463013610_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ B2 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_59_rtranclE,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( A != B )
=> ~ ! [Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ Y2 ) @ ( transi1302705790_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ Y2 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_60_rtranclE,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( A != B )
=> ~ ! [Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ Y2 ) @ ( transi463013610_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ Y2 @ B ) @ R ) ) ) ) ).
% rtranclE
thf(fact_61_trancl_Otrancl__into__trancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ R )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R ) ) ) ) ).
% trancl.trancl_into_trancl
thf(fact_62_trancl_Otrancl__into__trancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ R )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R ) ) ) ) ).
% trancl.trancl_into_trancl
thf(fact_63_converse__trancl__induct,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( ! [Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ B ) @ R )
=> ( P @ Y2 ) )
=> ( ! [Y2: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Z @ B ) @ ( transi1726587420_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y2 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_64_converse__trancl__induct,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( ! [Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ Y2 @ B ) @ R )
=> ( P @ Y2 ) )
=> ( ! [Y2: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ Z @ B ) @ ( transi1666849160_state @ R ) )
=> ( ( P @ Z )
=> ( P @ Y2 ) ) ) )
=> ( P @ A ) ) ) ) ).
% converse_trancl_induct
thf(fact_65_trancl_Or__into__trancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_66_trancl_Or__into__trancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_67_trancl__trans__induct,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1726587420_state @ R ) )
=> ( ! [X3: produc2041926651_state,Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y2 ) @ R )
=> ( P @ X3 @ Y2 ) )
=> ( ! [X3: produc2041926651_state,Y2: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y2 ) @ ( transi1726587420_state @ R ) )
=> ( ( P @ X3 @ Y2 )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z ) @ ( transi1726587420_state @ R ) )
=> ( ( P @ Y2 @ Z )
=> ( P @ X3 @ Z ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_68_trancl__trans__induct,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi1666849160_state @ R ) )
=> ( ! [X3: produc410756839_state,Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y2 ) @ R )
=> ( P @ X3 @ Y2 ) )
=> ( ! [X3: produc410756839_state,Y2: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y2 ) @ ( transi1666849160_state @ R ) )
=> ( ( P @ X3 @ Y2 )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z ) @ ( transi1666849160_state @ R ) )
=> ( ( P @ Y2 @ Z )
=> ( P @ X3 @ Z ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_69_trancl__into__trancl2,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ ( transi1726587420_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_70_trancl__into__trancl2,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ ( transi1666849160_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_71_irrefl__trancl__rD,axiom,
! [R: set_Pr1165141447_state,X: produc2041926651_state,Y: produc2041926651_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_72_irrefl__trancl__rD,axiom,
! [R: set_Pr2096893127_state,X: produc410756839_state,Y: produc410756839_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_73_converse__tranclE,axiom,
! [X: produc2041926651_state,Z2: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1726587420_state @ R ) )
=> ( ~ ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ R )
=> ~ ! [Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y2 ) @ R )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z2 ) @ ( transi1726587420_state @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_74_converse__tranclE,axiom,
! [X: produc410756839_state,Z2: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi1666849160_state @ R ) )
=> ( ~ ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ R )
=> ~ ! [Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y2 ) @ R )
=> ~ ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z2 ) @ ( transi1666849160_state @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_75_r__r__into__trancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R2: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R2 )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ R2 )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_76_r__r__into__trancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R2: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R2 )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ R2 )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_77_trancl_Oinducts,axiom,
! [X1: produc2041926651_state,X2: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ X1 @ X2 ) @ ( transi1726587420_state @ R ) )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ R )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: produc2041926651_state,B2: produc2041926651_state,C3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ ( transi1726587420_state @ R ) )
=> ( ( P @ A2 @ B2 )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B2 @ C3 ) @ R )
=> ( P @ A2 @ C3 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% trancl.inducts
thf(fact_78_trancl_Oinducts,axiom,
! [X1: produc410756839_state,X2: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ X1 @ X2 ) @ ( transi1666849160_state @ R ) )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ R )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: produc410756839_state,B2: produc410756839_state,C3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ ( transi1666849160_state @ R ) )
=> ( ( P @ A2 @ B2 )
=> ( ( member512523024_state @ ( produc2065881943_state @ B2 @ C3 ) @ R )
=> ( P @ A2 @ C3 ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% trancl.inducts
thf(fact_79_trancl__induct,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,P: produc2041926651_state > $o] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( ! [Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ Y2 ) @ R )
=> ( P @ Y2 ) )
=> ( ! [Y2: produc2041926651_state,Z: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ Y2 ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y2 @ Z ) @ R )
=> ( ( P @ Y2 )
=> ( P @ Z ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_80_trancl__induct,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,P: produc410756839_state > $o] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( ! [Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ Y2 ) @ R )
=> ( P @ Y2 ) )
=> ( ! [Y2: produc410756839_state,Z: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ Y2 ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y2 @ Z ) @ R )
=> ( ( P @ Y2 )
=> ( P @ Z ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_81_trancl__trans,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Z2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Z2 ) @ ( transi1726587420_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1726587420_state @ R ) ) ) ) ).
% trancl_trans
thf(fact_82_trancl__trans,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Z2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Z2 ) @ ( transi1666849160_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi1666849160_state @ R ) ) ) ) ).
% trancl_trans
thf(fact_83_trancl_Osimps,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1726587420_state @ R ) )
= ( ? [A3: produc2041926651_state,B3: produc2041926651_state] :
( ( A1 = A3 )
& ( A22 = B3 )
& ( member1069318160_state @ ( produc1909270103_state @ A3 @ B3 ) @ R ) )
| ? [A3: produc2041926651_state,B3: produc2041926651_state,C4: produc2041926651_state] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( member1069318160_state @ ( produc1909270103_state @ A3 @ B3 ) @ ( transi1726587420_state @ R ) )
& ( member1069318160_state @ ( produc1909270103_state @ B3 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_84_trancl_Osimps,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi1666849160_state @ R ) )
= ( ? [A3: produc410756839_state,B3: produc410756839_state] :
( ( A1 = A3 )
& ( A22 = B3 )
& ( member512523024_state @ ( produc2065881943_state @ A3 @ B3 ) @ R ) )
| ? [A3: produc410756839_state,B3: produc410756839_state,C4: produc410756839_state] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( member512523024_state @ ( produc2065881943_state @ A3 @ B3 ) @ ( transi1666849160_state @ R ) )
& ( member512523024_state @ ( produc2065881943_state @ B3 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_85_trancl_Ocases,axiom,
! [A1: produc2041926651_state,A22: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ ( transi1726587420_state @ R ) )
=> ( ~ ( member1069318160_state @ ( produc1909270103_state @ A1 @ A22 ) @ R )
=> ~ ! [B2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A1 @ B2 ) @ ( transi1726587420_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ B2 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_86_trancl_Ocases,axiom,
! [A1: produc410756839_state,A22: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ ( transi1666849160_state @ R ) )
=> ( ~ ( member512523024_state @ ( produc2065881943_state @ A1 @ A22 ) @ R )
=> ~ ! [B2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A1 @ B2 ) @ ( transi1666849160_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ B2 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_87_tranclE,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( ~ ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ~ ! [C3: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ C3 ) @ ( transi1726587420_state @ R ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ C3 @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_88_tranclE,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( ~ ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ~ ! [C3: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ C3 ) @ ( transi1666849160_state @ R ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ C3 @ B ) @ R ) ) ) ) ).
% tranclE
thf(fact_89_termi__impl__not__inf,axiom,
! [C: com,S: state] :
( ( termi @ C @ S )
=> ~ ( pHoare2126160787le_inf @ ( cons_com @ C @ nil_com ) @ S ) ) ).
% termi_impl_not_inf
thf(fact_90_mem__Collect__eq,axiom,
! [A: produc410756839_state,P: produc410756839_state > $o] :
( ( member1069318160_state @ A @ ( collec300063826_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A: state,P: state > $o] :
( ( member_state @ A @ ( collect_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A: produc186437863_state,P: produc186437863_state > $o] :
( ( member512523024_state @ A @ ( collec489924946_state @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_93_Collect__mem__eq,axiom,
! [A4: set_Pr1165141447_state] :
( ( collec300063826_state
@ ^ [X4: produc410756839_state] : ( member1069318160_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A4: set_state] :
( ( collect_state
@ ^ [X4: state] : ( member_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A4: set_Pr2096893127_state] :
( ( collec489924946_state
@ ^ [X4: produc186437863_state] : ( member512523024_state @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_96_trancl__rtrancl__trancl,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1726587420_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_97_trancl__rtrancl__trancl,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi1666849160_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_98_rtrancl__trancl__trancl,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Z2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Z2 ) @ ( transi1726587420_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ X @ Z2 ) @ ( transi1726587420_state @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_99_rtrancl__trancl__trancl,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Z2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Z2 ) @ ( transi1666849160_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ X @ Z2 ) @ ( transi1666849160_state @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_100_rtrancl__into__trancl2,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ ( transi1302705790_state @ R ) )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_101_rtrancl__into__trancl2,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ ( transi463013610_state @ R ) )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_102_rtrancl__into__trancl1,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,C: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ ( transi1302705790_state @ R ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ B @ C ) @ R )
=> ( member1069318160_state @ ( produc1909270103_state @ A @ C ) @ ( transi1726587420_state @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_103_rtrancl__into__trancl1,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,C: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ ( transi463013610_state @ R ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ B @ C ) @ R )
=> ( member512523024_state @ ( produc2065881943_state @ A @ C ) @ ( transi1666849160_state @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_104_exec1s__impl__execs,axiom,
! [Cs3: list_com,S: state,T: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs3 @ S ) @ ( produc1204172211_state @ nil_com @ T ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( pHoare259243732_execs @ S @ Cs3 @ T ) ) ).
% exec1s_impl_execs
thf(fact_105_execs__pres__termis,axiom,
! [Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs3 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( ( pHoare806997636termis @ Cs3 @ S )
=> ( pHoare806997636termis @ Cs @ S2 ) ) ) ).
% execs_pres_termis
thf(fact_106_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_107_nat_Oinject,axiom,
! [X2: nat,Y22: nat] :
( ( ( suc @ X2 )
= ( suc @ Y22 ) )
= ( X2 = Y22 ) ) ).
% nat.inject
thf(fact_108_list_Oinject,axiom,
! [X21: com,X22: list_com,Y21: com,Y222: list_com] :
( ( ( cons_com @ X21 @ X22 )
= ( cons_com @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X22 = Y222 ) ) ) ).
% list.inject
thf(fact_109_exec1s__impl__exec,axiom,
! [C: com,S: state,T: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ nil_com ) @ S ) @ ( produc1204172211_state @ nil_com @ T ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( exec @ S @ C @ T ) ) ).
% exec1s_impl_exec
thf(fact_110_old_Oprod_Oinject,axiom,
! [A: list_com,B: state,A5: list_com,B4: state] :
( ( ( produc1204172211_state @ A @ B )
= ( produc1204172211_state @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_111_old_Oprod_Oinject,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,A5: produc2041926651_state,B4: produc2041926651_state] :
( ( ( produc1909270103_state @ A @ B )
= ( produc1909270103_state @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_112_old_Oprod_Oinject,axiom,
! [A: produc410756839_state,B: produc410756839_state,A5: produc410756839_state,B4: produc410756839_state] :
( ( ( produc2065881943_state @ A @ B )
= ( produc2065881943_state @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_113_prod_Oinject,axiom,
! [X1: list_com,X2: state,Y1: list_com,Y22: state] :
( ( ( produc1204172211_state @ X1 @ X2 )
= ( produc1204172211_state @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_114_prod_Oinject,axiom,
! [X1: produc2041926651_state,X2: produc2041926651_state,Y1: produc2041926651_state,Y22: produc2041926651_state] :
( ( ( produc1909270103_state @ X1 @ X2 )
= ( produc1909270103_state @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_115_prod_Oinject,axiom,
! [X1: produc410756839_state,X2: produc410756839_state,Y1: produc410756839_state,Y22: produc410756839_state] :
( ( ( produc2065881943_state @ X1 @ X2 )
= ( produc2065881943_state @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_116_execs_Ointros_I2_J,axiom,
! [S: state,C: com,T: state,Cs3: list_com,U: state] :
( ( exec @ S @ C @ T )
=> ( ( pHoare259243732_execs @ T @ Cs3 @ U )
=> ( pHoare259243732_execs @ S @ ( cons_com @ C @ Cs3 ) @ U ) ) ) ).
% execs.intros(2)
thf(fact_117_execs_Ocases,axiom,
! [A1: state,A22: list_com,A32: state] :
( ( pHoare259243732_execs @ A1 @ A22 @ A32 )
=> ( ( ( A22 = nil_com )
=> ( A32 != A1 ) )
=> ~ ! [C3: com,T2: state,Cs4: list_com] :
( ( A22
= ( cons_com @ C3 @ Cs4 ) )
=> ( ( exec @ A1 @ C3 @ T2 )
=> ~ ( pHoare259243732_execs @ T2 @ Cs4 @ A32 ) ) ) ) ) ).
% execs.cases
thf(fact_118_execs_Osimps,axiom,
( pHoare259243732_execs
= ( ^ [A12: state,A23: list_com,A33: state] :
( ? [S3: state] :
( ( A12 = S3 )
& ( A23 = nil_com )
& ( A33 = S3 ) )
| ? [S3: state,C4: com,T3: state,Cs2: list_com,U2: state] :
( ( A12 = S3 )
& ( A23
= ( cons_com @ C4 @ Cs2 ) )
& ( A33 = U2 )
& ( exec @ S3 @ C4 @ T3 )
& ( pHoare259243732_execs @ T3 @ Cs2 @ U2 ) ) ) ) ) ).
% execs.simps
thf(fact_119_execs_Oinducts,axiom,
! [X1: state,X2: list_com,X32: state,P: state > list_com > state > $o] :
( ( pHoare259243732_execs @ X1 @ X2 @ X32 )
=> ( ! [S4: state] : ( P @ S4 @ nil_com @ S4 )
=> ( ! [S4: state,C3: com,T2: state,Cs4: list_com,U3: state] :
( ( exec @ S4 @ C3 @ T2 )
=> ( ( pHoare259243732_execs @ T2 @ Cs4 @ U3 )
=> ( ( P @ T2 @ Cs4 @ U3 )
=> ( P @ S4 @ ( cons_com @ C3 @ Cs4 ) @ U3 ) ) ) )
=> ( P @ X1 @ X2 @ X32 ) ) ) ) ).
% execs.inducts
thf(fact_120_termis_Osimps_I2_J,axiom,
! [C: com,Cs3: list_com,S: state] :
( ( pHoare806997636termis @ ( cons_com @ C @ Cs3 ) @ S )
= ( ( termi @ C @ S )
& ! [T3: state] :
( ( exec @ S @ C @ T3 )
=> ( pHoare806997636termis @ Cs3 @ T3 ) ) ) ) ).
% termis.simps(2)
thf(fact_121_execs_Ointros_I1_J,axiom,
! [S: state] : ( pHoare259243732_execs @ S @ nil_com @ S ) ).
% execs.intros(1)
thf(fact_122_termis_Osimps_I1_J,axiom,
! [S: state] : ( pHoare806997636termis @ nil_com @ S ) ).
% termis.simps(1)
thf(fact_123_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_124_surj__pair,axiom,
! [P2: produc2041926651_state] :
? [X3: list_com,Y2: state] :
( P2
= ( produc1204172211_state @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_125_surj__pair,axiom,
! [P2: produc410756839_state] :
? [X3: produc2041926651_state,Y2: produc2041926651_state] :
( P2
= ( produc1909270103_state @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_126_surj__pair,axiom,
! [P2: produc186437863_state] :
? [X3: produc410756839_state,Y2: produc410756839_state] :
( P2
= ( produc2065881943_state @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_127_prod__cases,axiom,
! [P: produc2041926651_state > $o,P2: produc2041926651_state] :
( ! [A2: list_com,B2: state] : ( P @ ( produc1204172211_state @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_128_prod__cases,axiom,
! [P: produc410756839_state > $o,P2: produc410756839_state] :
( ! [A2: produc2041926651_state,B2: produc2041926651_state] : ( P @ ( produc1909270103_state @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_129_prod__cases,axiom,
! [P: produc186437863_state > $o,P2: produc186437863_state] :
( ! [A2: produc410756839_state,B2: produc410756839_state] : ( P @ ( produc2065881943_state @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_130_Pair__inject,axiom,
! [A: list_com,B: state,A5: list_com,B4: state] :
( ( ( produc1204172211_state @ A @ B )
= ( produc1204172211_state @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_131_Pair__inject,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,A5: produc2041926651_state,B4: produc2041926651_state] :
( ( ( produc1909270103_state @ A @ B )
= ( produc1909270103_state @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_132_Pair__inject,axiom,
! [A: produc410756839_state,B: produc410756839_state,A5: produc410756839_state,B4: produc410756839_state] :
( ( ( produc2065881943_state @ A @ B )
= ( produc2065881943_state @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_133_old_Oprod_Oexhaust,axiom,
! [Y: produc2041926651_state] :
~ ! [A2: list_com,B2: state] :
( Y
!= ( produc1204172211_state @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_134_old_Oprod_Oexhaust,axiom,
! [Y: produc410756839_state] :
~ ! [A2: produc2041926651_state,B2: produc2041926651_state] :
( Y
!= ( produc1909270103_state @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_135_old_Oprod_Oexhaust,axiom,
! [Y: produc186437863_state] :
~ ! [A2: produc410756839_state,B2: produc410756839_state] :
( Y
!= ( produc2065881943_state @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_136_old_Oprod_Oinducts,axiom,
! [P: produc2041926651_state > $o,Prod: produc2041926651_state] :
( ! [A2: list_com,B2: state] : ( P @ ( produc1204172211_state @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_137_old_Oprod_Oinducts,axiom,
! [P: produc410756839_state > $o,Prod: produc410756839_state] :
( ! [A2: produc2041926651_state,B2: produc2041926651_state] : ( P @ ( produc1909270103_state @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_138_old_Oprod_Oinducts,axiom,
! [P: produc186437863_state > $o,Prod: produc186437863_state] :
( ! [A2: produc410756839_state,B2: produc410756839_state] : ( P @ ( produc2065881943_state @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_139_not__Cons__self2,axiom,
! [X: com,Xs: list_com] :
( ( cons_com @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_140_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_141_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_142_inf__cases,axiom,
! [C: com,Cs3: list_com,S: state] :
( ( pHoare2126160787le_inf @ ( cons_com @ C @ Cs3 ) @ S )
=> ( ( pHoare2126160787le_inf @ ( cons_com @ C @ nil_com ) @ S )
| ? [T2: state] :
( ( exec @ S @ C @ T2 )
& ( pHoare2126160787le_inf @ Cs3 @ T2 ) ) ) ) ).
% inf_cases
thf(fact_143_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_144_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_145_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_146_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_147_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_148_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_149_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_150_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_151_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_152_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_153_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_154_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_155_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_156_list_Odistinct_I1_J,axiom,
! [X21: com,X22: list_com] :
( nil_com
!= ( cons_com @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_157_list_OdiscI,axiom,
! [List: list_com,X21: com,X22: list_com] :
( ( List
= ( cons_com @ X21 @ X22 ) )
=> ( List != nil_com ) ) ).
% list.discI
thf(fact_158_list_Oexhaust,axiom,
! [Y: list_com] :
( ( Y != nil_com )
=> ~ ! [X212: com,X222: list_com] :
( Y
!= ( cons_com @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_159_list_Oinducts,axiom,
! [P: list_com > $o,List: list_com] :
( ( P @ nil_com )
=> ( ! [X12: com,X23: list_com] :
( ( P @ X23 )
=> ( P @ ( cons_com @ X12 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_160_neq__Nil__conv,axiom,
! [Xs: list_com] :
( ( Xs != nil_com )
= ( ? [Y3: com,Ys: list_com] :
( Xs
= ( cons_com @ Y3 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_161_list__induct2_H,axiom,
! [P: list_com > list_com > $o,Xs: list_com,Ys2: list_com] :
( ( P @ nil_com @ nil_com )
=> ( ! [X3: com,Xs2: list_com] : ( P @ ( cons_com @ X3 @ Xs2 ) @ nil_com )
=> ( ! [Y2: com,Ys3: list_com] : ( P @ nil_com @ ( cons_com @ Y2 @ Ys3 ) )
=> ( ! [X3: com,Xs2: list_com,Y2: com,Ys3: list_com] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) @ ( cons_com @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_162_splice_Oinduct,axiom,
! [P: list_com > list_com > $o,A0: list_com,A1: list_com] :
( ! [X_1: list_com] : ( P @ nil_com @ X_1 )
=> ( ! [X3: com,Xs2: list_com,Ys3: list_com] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_163_induct__list012,axiom,
! [P: list_com > $o,Xs: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Y2: com,Zs: list_com] :
( ( P @ Zs )
=> ( ( P @ ( cons_com @ Y2 @ Zs ) )
=> ( P @ ( cons_com @ X3 @ ( cons_com @ Y2 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_164_shuffles_Oinduct,axiom,
! [P: list_com > list_com > $o,A0: list_com,A1: list_com] :
( ! [X_1: list_com] : ( P @ nil_com @ X_1 )
=> ( ! [Xs2: list_com] : ( P @ Xs2 @ nil_com )
=> ( ! [X3: com,Xs2: list_com,Y2: com,Ys3: list_com] :
( ( P @ Xs2 @ ( cons_com @ Y2 @ Ys3 ) )
=> ( ( P @ ( cons_com @ X3 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) @ ( cons_com @ Y2 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_165_transpose_Ocases,axiom,
! [X: list_list_com] :
( ( X != nil_list_com )
=> ( ! [Xss: list_list_com] :
( X
!= ( cons_list_com @ nil_com @ Xss ) )
=> ~ ! [X3: com,Xs2: list_com,Xss: list_list_com] :
( X
!= ( cons_list_com @ ( cons_com @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_166_remdups__adj_Ocases,axiom,
! [X: list_com] :
( ( X != nil_com )
=> ( ! [X3: com] :
( X
!= ( cons_com @ X3 @ nil_com ) )
=> ~ ! [X3: com,Y2: com,Xs2: list_com] :
( X
!= ( cons_com @ X3 @ ( cons_com @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_167_sorted__wrt_Oinduct,axiom,
! [P: ( com > com > $o ) > list_com > $o,A0: com > com > $o,A1: list_com] :
( ! [P3: com > com > $o] : ( P @ P3 @ nil_com )
=> ( ! [P3: com > com > $o,X3: com,Ys3: list_com] :
( ( P @ P3 @ Ys3 )
=> ( P @ P3 @ ( cons_com @ X3 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_168_remdups__adj_Oinduct,axiom,
! [P: list_com > $o,A0: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Y2: com,Xs2: list_com] :
( ( ( X3 = Y2 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y2 )
=> ( P @ ( cons_com @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons_com @ X3 @ ( cons_com @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_169_successively_Oinduct,axiom,
! [P: ( com > com > $o ) > list_com > $o,A0: com > com > $o,A1: list_com] :
( ! [P3: com > com > $o] : ( P @ P3 @ nil_com )
=> ( ! [P3: com > com > $o,X3: com] : ( P @ P3 @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [P3: com > com > $o,X3: com,Y2: com,Xs2: list_com] :
( ( P @ P3 @ ( cons_com @ Y2 @ Xs2 ) )
=> ( P @ P3 @ ( cons_com @ X3 @ ( cons_com @ Y2 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_170_list__nonempty__induct,axiom,
! [Xs: list_com,P: list_com > $o] :
( ( Xs != nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Xs2: list_com] :
( ( Xs2 != nil_com )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_171_map__tailrec__rev_Oinduct,axiom,
! [P: ( com > com ) > list_com > list_com > $o,A0: com > com,A1: list_com,A22: list_com] :
( ! [F2: com > com,X_1: list_com] : ( P @ F2 @ nil_com @ X_1 )
=> ( ! [F2: com > com,A2: com,As: list_com,Bs: list_com] :
( ( P @ F2 @ As @ ( cons_com @ ( F2 @ A2 ) @ Bs ) )
=> ( P @ F2 @ ( cons_com @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_172_splice_Ocases,axiom,
! [X: produc258871771st_com] :
( ! [Ys3: list_com] :
( X
!= ( produc1307105171st_com @ nil_com @ Ys3 ) )
=> ~ ! [X3: com,Xs2: list_com,Ys3: list_com] :
( X
!= ( produc1307105171st_com @ ( cons_com @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_173_shuffles_Ocases,axiom,
! [X: produc258871771st_com] :
( ! [Ys3: list_com] :
( X
!= ( produc1307105171st_com @ nil_com @ Ys3 ) )
=> ( ! [Xs2: list_com] :
( X
!= ( produc1307105171st_com @ Xs2 @ nil_com ) )
=> ~ ! [X3: com,Xs2: list_com,Y2: com,Ys3: list_com] :
( X
!= ( produc1307105171st_com @ ( cons_com @ X3 @ Xs2 ) @ ( cons_com @ Y2 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_174_sorted__wrt_Ocases,axiom,
! [X: produc2058209240st_com] :
( ! [P3: com > com > $o] :
( X
!= ( produc1914379986st_com @ P3 @ nil_com ) )
=> ~ ! [P3: com > com > $o,X3: com,Ys3: list_com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_175_successively_Ocases,axiom,
! [X: produc2058209240st_com] :
( ! [P3: com > com > $o] :
( X
!= ( produc1914379986st_com @ P3 @ nil_com ) )
=> ( ! [P3: com > com > $o,X3: com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ nil_com ) ) )
=> ~ ! [P3: com > com > $o,X3: com,Y2: com,Xs2: list_com] :
( X
!= ( produc1914379986st_com @ P3 @ ( cons_com @ X3 @ ( cons_com @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_176_exec__impl__execs,axiom,
! [S: state,C: com,S2: state,Cs3: list_com] :
( ( exec @ S @ C @ S2 )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs3 @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ) ) ).
% exec_impl_execs
thf(fact_177_prod__cases3,axiom,
! [Y: produc410756839_state] :
~ ! [A2: produc2041926651_state,B2: list_com,C3: state] :
( Y
!= ( produc1909270103_state @ A2 @ ( produc1204172211_state @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_178_prod__cases3,axiom,
! [Y: produc186437863_state] :
~ ! [A2: produc410756839_state,B2: produc2041926651_state,C3: produc2041926651_state] :
( Y
!= ( produc2065881943_state @ A2 @ ( produc1909270103_state @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_179_prod__cases4,axiom,
! [Y: produc186437863_state] :
~ ! [A2: produc410756839_state,B2: produc2041926651_state,C3: list_com,D: state] :
( Y
!= ( produc2065881943_state @ A2 @ ( produc1909270103_state @ B2 @ ( produc1204172211_state @ C3 @ D ) ) ) ) ).
% prod_cases4
thf(fact_180_prod__induct3,axiom,
! [P: produc410756839_state > $o,X: produc410756839_state] :
( ! [A2: produc2041926651_state,B2: list_com,C3: state] : ( P @ ( produc1909270103_state @ A2 @ ( produc1204172211_state @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_181_prod__induct3,axiom,
! [P: produc186437863_state > $o,X: produc186437863_state] :
( ! [A2: produc410756839_state,B2: produc2041926651_state,C3: produc2041926651_state] : ( P @ ( produc2065881943_state @ A2 @ ( produc1909270103_state @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_182_prod__induct4,axiom,
! [P: produc186437863_state > $o,X: produc186437863_state] :
( ! [A2: produc410756839_state,B2: produc2041926651_state,C3: list_com,D: state] : ( P @ ( produc2065881943_state @ A2 @ ( produc1909270103_state @ B2 @ ( produc1204172211_state @ C3 @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_183_exec1__pres__termis,axiom,
! [Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs3 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( pHoare806997636termis @ Cs3 @ S )
=> ( pHoare806997636termis @ Cs @ S2 ) ) ) ).
% exec1_pres_termis
thf(fact_184_exec1E_I3_J,axiom,
! [C1: com,C22: com,Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( semi @ C1 @ C22 ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ~ ( ( Cs
= ( cons_com @ C1 @ ( cons_com @ C22 @ Cs3 ) ) )
=> ( S2 != S ) ) ) ).
% exec1E(3)
thf(fact_185_exec1_OSemi,axiom,
! [C1: com,C22: com,Cs3: list_com,S: state] : ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( semi @ C1 @ C22 ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C1 @ ( cons_com @ C22 @ Cs3 ) ) @ S ) ) @ pHoare259243666_exec1 ) ).
% exec1.Semi
thf(fact_186_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_com @ N @ nil_com )
= ( cons_list_com @ nil_com @ nil_list_com ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_com @ N @ nil_com )
= nil_list_com ) ) ) ).
% n_lists_Nil
thf(fact_187_exec1E_I2_J,axiom,
! [F: state > set_state,Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( do @ F ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ~ ( ( Cs = Cs3 )
=> ~ ( member_state @ S2 @ ( F @ S ) ) ) ) ).
% exec1E(2)
thf(fact_188_exec1_ODo,axiom,
! [T: state,F: state > set_state,S: state,Cs3: list_com] :
( ( member_state @ T @ ( F @ S ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( do @ F ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs3 @ T ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.Do
thf(fact_189_n__lists_Osimps_I1_J,axiom,
! [Xs: list_com] :
( ( n_lists_com @ zero_zero_nat @ Xs )
= ( cons_list_com @ nil_com @ nil_list_com ) ) ).
% n_lists.simps(1)
thf(fact_190_termi_OSemi,axiom,
! [C1: com,S0: state,C22: com] :
( ( termi @ C1 @ S0 )
=> ( ! [S1: state] :
( ( exec @ S0 @ C1 @ S1 )
=> ( termi @ C22 @ S1 ) )
=> ( termi @ ( semi @ C1 @ C22 ) @ S0 ) ) ) ).
% termi.Semi
thf(fact_191_com_Oinject_I2_J,axiom,
! [X21: com,X22: com,Y21: com,Y222: com] :
( ( ( semi @ X21 @ X22 )
= ( semi @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X22 = Y222 ) ) ) ).
% com.inject(2)
thf(fact_192_com_Oinject_I1_J,axiom,
! [X1: state > set_state,Y1: state > set_state] :
( ( ( do @ X1 )
= ( do @ Y1 ) )
= ( X1 = Y1 ) ) ).
% com.inject(1)
thf(fact_193_com_Odistinct_I1_J,axiom,
! [X1: state > set_state,X21: com,X22: com] :
( ( do @ X1 )
!= ( semi @ X21 @ X22 ) ) ).
% com.distinct(1)
thf(fact_194_exec_ODo,axiom,
! [T: state,F: state > set_state,S: state] :
( ( member_state @ T @ ( F @ S ) )
=> ( exec @ S @ ( do @ F ) @ T ) ) ).
% exec.Do
thf(fact_195_exec_OSemi,axiom,
! [S0: state,C1: com,S12: state,C22: com,S22: state] :
( ( exec @ S0 @ C1 @ S12 )
=> ( ( exec @ S12 @ C22 @ S22 )
=> ( exec @ S0 @ ( semi @ C1 @ C22 ) @ S22 ) ) ) ).
% exec.Semi
thf(fact_196_product__lists_Osimps_I1_J,axiom,
( ( product_lists_com @ nil_list_com )
= ( cons_list_com @ nil_com @ nil_list_com ) ) ).
% product_lists.simps(1)
thf(fact_197_subseqs_Osimps_I1_J,axiom,
( ( subseqs_com @ nil_com )
= ( cons_list_com @ nil_com @ nil_list_com ) ) ).
% subseqs.simps(1)
thf(fact_198_exec1E_I4_J,axiom,
! [B: state > $o,C1: com,C22: com,Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B @ C1 @ C22 ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( ( Cs
= ( cons_com @ C1 @ Cs3 ) )
=> ( ( S2 = S )
=> ~ ( B @ S ) ) )
=> ~ ( ( Cs
= ( cons_com @ C22 @ Cs3 ) )
=> ( ( S2 = S )
=> ( B @ S ) ) ) ) ) ).
% exec1E(4)
thf(fact_199_com_Oinject_I3_J,axiom,
! [X31: state > $o,X322: com,X33: com,Y31: state > $o,Y32: com,Y33: com] :
( ( ( cond @ X31 @ X322 @ X33 )
= ( cond @ Y31 @ Y32 @ Y33 ) )
= ( ( X31 = Y31 )
& ( X322 = Y32 )
& ( X33 = Y33 ) ) ) ).
% com.inject(3)
thf(fact_200_exec_OIfTrue,axiom,
! [B: state > $o,S: state,C1: com,T: state,C22: com] :
( ( B @ S )
=> ( ( exec @ S @ C1 @ T )
=> ( exec @ S @ ( cond @ B @ C1 @ C22 ) @ T ) ) ) ).
% exec.IfTrue
thf(fact_201_exec_OIfFalse,axiom,
! [B: state > $o,S: state,C22: com,T: state,C1: com] :
( ~ ( B @ S )
=> ( ( exec @ S @ C22 @ T )
=> ( exec @ S @ ( cond @ B @ C1 @ C22 ) @ T ) ) ) ).
% exec.IfFalse
thf(fact_202_termi_OIfTrue,axiom,
! [B: state > $o,S: state,C1: com,C22: com] :
( ( B @ S )
=> ( ( termi @ C1 @ S )
=> ( termi @ ( cond @ B @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfTrue
thf(fact_203_termi_OIfFalse,axiom,
! [B: state > $o,S: state,C22: com,C1: com] :
( ~ ( B @ S )
=> ( ( termi @ C22 @ S )
=> ( termi @ ( cond @ B @ C1 @ C22 ) @ S ) ) ) ).
% termi.IfFalse
thf(fact_204_com_Odistinct_I3_J,axiom,
! [X1: state > set_state,X31: state > $o,X322: com,X33: com] :
( ( do @ X1 )
!= ( cond @ X31 @ X322 @ X33 ) ) ).
% com.distinct(3)
thf(fact_205_com_Odistinct_I11_J,axiom,
! [X21: com,X22: com,X31: state > $o,X322: com,X33: com] :
( ( semi @ X21 @ X22 )
!= ( cond @ X31 @ X322 @ X33 ) ) ).
% com.distinct(11)
thf(fact_206_exec1_OIfFalse,axiom,
! [B: state > $o,S: state,C1: com,C22: com,Cs3: list_com] :
( ~ ( B @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B @ C1 @ C22 ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C22 @ Cs3 ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.IfFalse
thf(fact_207_exec1_OIfTrue,axiom,
! [B: state > $o,S: state,C1: com,C22: com,Cs3: list_com] :
( ( B @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( cond @ B @ C1 @ C22 ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C1 @ Cs3 ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.IfTrue
thf(fact_208_gen__length__code_I2_J,axiom,
! [N: nat,X: com,Xs: list_com] :
( ( gen_length_com @ N @ ( cons_com @ X @ Xs ) )
= ( gen_length_com @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_209_insert__Nil,axiom,
! [X: com] :
( ( insert_com @ X @ nil_com )
= ( cons_com @ X @ nil_com ) ) ).
% insert_Nil
thf(fact_210_exec1_OWhileFalse,axiom,
! [B: state > $o,S: state,C: com,Cs3: list_com] :
( ~ ( B @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B @ C ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs3 @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.WhileFalse
thf(fact_211_com_Oinject_I4_J,axiom,
! [X41: state > $o,X42: com,Y41: state > $o,Y42: com] :
( ( ( while @ X41 @ X42 )
= ( while @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% com.inject(4)
thf(fact_212_com_Odistinct_I19_J,axiom,
! [X31: state > $o,X322: com,X33: com,X41: state > $o,X42: com] :
( ( cond @ X31 @ X322 @ X33 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(19)
thf(fact_213_while__termiE,axiom,
! [B: state > $o,C: com,S: state] :
( ( termi @ ( while @ B @ C ) @ S )
=> ( ( B @ S )
=> ( termi @ C @ S ) ) ) ).
% while_termiE
thf(fact_214_exec_OWhileFalse,axiom,
! [B: state > $o,S: state,C: com] :
( ~ ( B @ S )
=> ( exec @ S @ ( while @ B @ C ) @ S ) ) ).
% exec.WhileFalse
thf(fact_215_exec_OWhileTrue,axiom,
! [B: state > $o,S: state,C: com,T: state,U: state] :
( ( B @ S )
=> ( ( exec @ S @ C @ T )
=> ( ( exec @ T @ ( while @ B @ C ) @ U )
=> ( exec @ S @ ( while @ B @ C ) @ U ) ) ) ) ).
% exec.WhileTrue
thf(fact_216_termi_OWhileFalse,axiom,
! [B: state > $o,S: state,C: com] :
( ~ ( B @ S )
=> ( termi @ ( while @ B @ C ) @ S ) ) ).
% termi.WhileFalse
thf(fact_217_com_Odistinct_I5_J,axiom,
! [X1: state > set_state,X41: state > $o,X42: com] :
( ( do @ X1 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(5)
thf(fact_218_com_Odistinct_I13_J,axiom,
! [X21: com,X22: com,X41: state > $o,X42: com] :
( ( semi @ X21 @ X22 )
!= ( while @ X41 @ X42 ) ) ).
% com.distinct(13)
thf(fact_219_termi_OWhileTrue,axiom,
! [B: state > $o,S: state,C: com] :
( ( B @ S )
=> ( ( termi @ C @ S )
=> ( ! [T2: state] :
( ( exec @ S @ C @ T2 )
=> ( termi @ ( while @ B @ C ) @ T2 ) )
=> ( termi @ ( while @ B @ C ) @ S ) ) ) ) ).
% termi.WhileTrue
thf(fact_220_while__termiE2,axiom,
! [B: state > $o,C: com,S: state,T: state] :
( ( termi @ ( while @ B @ C ) @ S )
=> ( ( B @ S )
=> ( ( exec @ S @ C @ T )
=> ( termi @ ( while @ B @ C ) @ T ) ) ) ) ).
% while_termiE2
thf(fact_221_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_com @ N @ nil_com )
= N ) ).
% gen_length_code(1)
thf(fact_222_termi__while,axiom,
! [B: state > $o,C: com,F: nat > state,K: nat] :
( ( termi @ ( while @ B @ C ) @ ( F @ K ) )
=> ( ! [I: nat] : ( exec @ ( F @ I ) @ C @ ( F @ ( suc @ I ) ) )
=> ? [I: nat] :
~ ( B @ ( F @ I ) ) ) ) ).
% termi_while
thf(fact_223_termi__while__lemma,axiom,
! [W: com,Fk: state,F: nat > state,K: nat,B: state > $o,C: com] :
( ( termi @ W @ Fk )
=> ( ( ( Fk
= ( F @ K ) )
& ( W
= ( while @ B @ C ) )
& ! [I: nat] : ( exec @ ( F @ I ) @ C @ ( F @ ( suc @ I ) ) ) )
=> ? [I: nat] :
~ ( B @ ( F @ I ) ) ) ) ).
% termi_while_lemma
thf(fact_224_exec1E_I5_J,axiom,
! [B: state > $o,C: com,Cs3: list_com,S: state,Cs: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B @ C ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( ( Cs = Cs3 )
=> ( ( S2 = S )
=> ( B @ S ) ) )
=> ~ ( ( Cs
= ( cons_com @ C @ ( cons_com @ ( while @ B @ C ) @ Cs3 ) ) )
=> ( ( S2 = S )
=> ~ ( B @ S ) ) ) ) ) ).
% exec1E(5)
thf(fact_225_exec1_OWhileTrue,axiom,
! [B: state > $o,S: state,C: com,Cs3: list_com] :
( ( B @ S )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ ( while @ B @ C ) @ Cs3 ) @ S ) @ ( produc1204172211_state @ ( cons_com @ C @ ( cons_com @ ( while @ B @ C ) @ Cs3 ) ) @ S ) ) @ pHoare259243666_exec1 ) ) ).
% exec1.WhileTrue
thf(fact_226_nths__singleton,axiom,
! [A4: set_nat,X: com] :
( ( ( member_nat @ zero_zero_nat @ A4 )
=> ( ( nths_com @ ( cons_com @ X @ nil_com ) @ A4 )
= ( cons_com @ X @ nil_com ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A4 )
=> ( ( nths_com @ ( cons_com @ X @ nil_com ) @ A4 )
= nil_com ) ) ) ).
% nths_singleton
thf(fact_227_app__execs,axiom,
! [Cs3: list_com,S: state,Cs: list_com,S2: state,Cs22: list_com] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs3 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( append_com @ Cs3 @ Cs22 ) @ S ) @ ( produc1204172211_state @ ( append_com @ Cs @ Cs22 ) @ S2 ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ) ) ).
% app_execs
thf(fact_228_listrel_Ocases,axiom,
! [A1: list_com,A22: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ A1 @ A22 ) @ ( listrel_com_com @ R ) )
=> ( ( ( A1 = nil_com )
=> ( A22 != nil_com ) )
=> ~ ! [X3: com,Y2: com,Xs2: list_com] :
( ( A1
= ( cons_com @ X3 @ Xs2 ) )
=> ! [Ys3: list_com] :
( ( A22
= ( cons_com @ Y2 @ Ys3 ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y2 ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys3 ) @ ( listrel_com_com @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_229_listrel_Ocases,axiom,
! [A1: list_list_com,A22: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ A1 @ A22 ) @ ( listre1268245261_state @ R ) )
=> ( ( ( A1 = nil_list_com )
=> ( A22 != nil_state ) )
=> ~ ! [X3: list_com,Y2: state,Xs2: list_list_com] :
( ( A1
= ( cons_list_com @ X3 @ Xs2 ) )
=> ! [Ys3: list_state] :
( ( A22
= ( cons_state @ Y2 @ Ys3 ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y2 ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ Ys3 ) @ ( listre1268245261_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_230_listrel_Ocases,axiom,
! [A1: list_P1040014977_state,A22: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ A1 @ A22 ) @ ( listre194873521_state @ R ) )
=> ( ( ( A1 = nil_Pr1844458753_state )
=> ( A22 != nil_Pr1844458753_state ) )
=> ~ ! [X3: produc2041926651_state,Y2: produc2041926651_state,Xs2: list_P1040014977_state] :
( ( A1
= ( cons_P465369265_state @ X3 @ Xs2 ) )
=> ! [Ys3: list_P1040014977_state] :
( ( A22
= ( cons_P465369265_state @ Y2 @ Ys3 ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y2 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys3 ) @ ( listre194873521_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_231_listrel_Ocases,axiom,
! [A1: list_P890094829_state,A22: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ A1 @ A22 ) @ ( listre1244407729_state @ R ) )
=> ( ( ( A1 = nil_Pr1347322605_state )
=> ( A22 != nil_Pr1347322605_state ) )
=> ~ ! [X3: produc410756839_state,Y2: produc410756839_state,Xs2: list_P890094829_state] :
( ( A1
= ( cons_P438230685_state @ X3 @ Xs2 ) )
=> ! [Ys3: list_P890094829_state] :
( ( A22
= ( cons_P438230685_state @ Y2 @ Ys3 ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y2 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys3 ) @ ( listre1244407729_state @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_232_same__append__eq,axiom,
! [Xs: list_com,Ys2: list_com,Zs2: list_com] :
( ( ( append_com @ Xs @ Ys2 )
= ( append_com @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_233_append__same__eq,axiom,
! [Ys2: list_com,Xs: list_com,Zs2: list_com] :
( ( ( append_com @ Ys2 @ Xs )
= ( append_com @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_234_append__assoc,axiom,
! [Xs: list_com,Ys2: list_com,Zs2: list_com] :
( ( append_com @ ( append_com @ Xs @ Ys2 ) @ Zs2 )
= ( append_com @ Xs @ ( append_com @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_235_append_Oassoc,axiom,
! [A: list_com,B: list_com,C: list_com] :
( ( append_com @ ( append_com @ A @ B ) @ C )
= ( append_com @ A @ ( append_com @ B @ C ) ) ) ).
% append.assoc
thf(fact_236_append__Nil2,axiom,
! [Xs: list_com] :
( ( append_com @ Xs @ nil_com )
= Xs ) ).
% append_Nil2
thf(fact_237_append__self__conv,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( ( append_com @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_com ) ) ).
% append_self_conv
thf(fact_238_self__append__conv,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( Xs
= ( append_com @ Xs @ Ys2 ) )
= ( Ys2 = nil_com ) ) ).
% self_append_conv
thf(fact_239_append__self__conv2,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( ( append_com @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_com ) ) ).
% append_self_conv2
thf(fact_240_self__append__conv2,axiom,
! [Ys2: list_com,Xs: list_com] :
( ( Ys2
= ( append_com @ Xs @ Ys2 ) )
= ( Xs = nil_com ) ) ).
% self_append_conv2
thf(fact_241_Nil__is__append__conv,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( nil_com
= ( append_com @ Xs @ Ys2 ) )
= ( ( Xs = nil_com )
& ( Ys2 = nil_com ) ) ) ).
% Nil_is_append_conv
thf(fact_242_append__is__Nil__conv,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( ( append_com @ Xs @ Ys2 )
= nil_com )
= ( ( Xs = nil_com )
& ( Ys2 = nil_com ) ) ) ).
% append_is_Nil_conv
thf(fact_243_append_Oright__neutral,axiom,
! [A: list_com] :
( ( append_com @ A @ nil_com )
= A ) ).
% append.right_neutral
thf(fact_244_nths__nil,axiom,
! [A4: set_nat] :
( ( nths_com @ nil_com @ A4 )
= nil_com ) ).
% nths_nil
thf(fact_245_append1__eq__conv,axiom,
! [Xs: list_com,X: com,Ys2: list_com,Y: com] :
( ( ( append_com @ Xs @ ( cons_com @ X @ nil_com ) )
= ( append_com @ Ys2 @ ( cons_com @ Y @ nil_com ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_246_listrel__rtrancl__refl,axiom,
! [Xs: list_P1040014977_state,R: set_Pr1165141447_state] : ( member1267359376_state @ ( produc1298457559_state @ Xs @ Xs ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_247_listrel__rtrancl__refl,axiom,
! [Xs: list_P890094829_state,R: set_Pr2096893127_state] : ( member423306384_state @ ( produc1354769367_state @ Xs @ Xs ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_248_append__eq__append__conv2,axiom,
! [Xs: list_com,Ys2: list_com,Zs2: list_com,Ts: list_com] :
( ( ( append_com @ Xs @ Ys2 )
= ( append_com @ Zs2 @ Ts ) )
= ( ? [Us: list_com] :
( ( ( Xs
= ( append_com @ Zs2 @ Us ) )
& ( ( append_com @ Us @ Ys2 )
= Ts ) )
| ( ( ( append_com @ Xs @ Us )
= Zs2 )
& ( Ys2
= ( append_com @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_249_append__eq__appendI,axiom,
! [Xs: list_com,Xs1: list_com,Zs2: list_com,Ys2: list_com,Us2: list_com] :
( ( ( append_com @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append_com @ Xs1 @ Us2 ) )
=> ( ( append_com @ Xs @ Ys2 )
= ( append_com @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_250_eq__Nil__appendI,axiom,
! [Xs: list_com,Ys2: list_com] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_com @ nil_com @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_251_append__Nil,axiom,
! [Ys2: list_com] :
( ( append_com @ nil_com @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_252_append_Oleft__neutral,axiom,
! [A: list_com] :
( ( append_com @ nil_com @ A )
= A ) ).
% append.left_neutral
thf(fact_253_Cons__eq__appendI,axiom,
! [X: com,Xs1: list_com,Ys2: list_com,Xs: list_com,Zs2: list_com] :
( ( ( cons_com @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_com @ Xs1 @ Zs2 ) )
=> ( ( cons_com @ X @ Xs )
= ( append_com @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_254_append__Cons,axiom,
! [X: com,Xs: list_com,Ys2: list_com] :
( ( append_com @ ( cons_com @ X @ Xs ) @ Ys2 )
= ( cons_com @ X @ ( append_com @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_255_rev__nonempty__induct,axiom,
! [Xs: list_com,P: list_com > $o] :
( ( Xs != nil_com )
=> ( ! [X3: com] : ( P @ ( cons_com @ X3 @ nil_com ) )
=> ( ! [X3: com,Xs2: list_com] :
( ( Xs2 != nil_com )
=> ( ( P @ Xs2 )
=> ( P @ ( append_com @ Xs2 @ ( cons_com @ X3 @ nil_com ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_256_append__eq__Cons__conv,axiom,
! [Ys2: list_com,Zs2: list_com,X: com,Xs: list_com] :
( ( ( append_com @ Ys2 @ Zs2 )
= ( cons_com @ X @ Xs ) )
= ( ( ( Ys2 = nil_com )
& ( Zs2
= ( cons_com @ X @ Xs ) ) )
| ? [Ys4: list_com] :
( ( Ys2
= ( cons_com @ X @ Ys4 ) )
& ( ( append_com @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_257_Cons__eq__append__conv,axiom,
! [X: com,Xs: list_com,Ys2: list_com,Zs2: list_com] :
( ( ( cons_com @ X @ Xs )
= ( append_com @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_com )
& ( ( cons_com @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list_com] :
( ( ( cons_com @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append_com @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_258_rev__exhaust,axiom,
! [Xs: list_com] :
( ( Xs != nil_com )
=> ~ ! [Ys3: list_com,Y2: com] :
( Xs
!= ( append_com @ Ys3 @ ( cons_com @ Y2 @ nil_com ) ) ) ) ).
% rev_exhaust
thf(fact_259_rev__induct,axiom,
! [P: list_com > $o,Xs: list_com] :
( ( P @ nil_com )
=> ( ! [X3: com,Xs2: list_com] :
( ( P @ Xs2 )
=> ( P @ ( append_com @ Xs2 @ ( cons_com @ X3 @ nil_com ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_260_listrel_ONil,axiom,
! [R: set_Pr1822651163om_com] : ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ nil_com ) @ ( listrel_com_com @ R ) ) ).
% listrel.Nil
thf(fact_261_listrel__Nil1,axiom,
! [Xs: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Xs ) @ ( listrel_com_com @ R ) )
=> ( Xs = nil_com ) ) ).
% listrel_Nil1
thf(fact_262_listrel__Nil2,axiom,
! [Xs: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ nil_com ) @ ( listrel_com_com @ R ) )
=> ( Xs = nil_com ) ) ).
% listrel_Nil2
thf(fact_263_listrel__rtrancl__trans,axiom,
! [Xs: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state,Zs2: list_P1040014977_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Zs2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs @ Zs2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_264_listrel__rtrancl__trans,axiom,
! [Xs: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state,Zs2: list_P890094829_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Zs2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs @ Zs2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_265_app__exec,axiom,
! [Cs3: list_com,S: state,Cs: list_com,S2: state,Cs22: list_com] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs3 @ S ) @ ( produc1204172211_state @ Cs @ S2 ) ) @ pHoare259243666_exec1 )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( append_com @ Cs3 @ Cs22 ) @ S ) @ ( produc1204172211_state @ ( append_com @ Cs @ Cs22 ) @ S2 ) ) @ pHoare259243666_exec1 ) ) ).
% app_exec
thf(fact_266_listrel__Cons2,axiom,
! [Xs: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel_com_com @ R ) )
=> ~ ! [X3: com,Xs2: list_com] :
( ( Xs
= ( cons_com @ X3 @ Xs2 ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys2 ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_267_listrel__Cons2,axiom,
! [Xs: list_list_com,Y: state,Ys2: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ Xs @ ( cons_state @ Y @ Ys2 ) ) @ ( listre1268245261_state @ R ) )
=> ~ ! [X3: list_com,Xs2: list_list_com] :
( ( Xs
= ( cons_list_com @ X3 @ Xs2 ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ Ys2 ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_268_listrel__Cons2,axiom,
! [Xs: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre194873521_state @ R ) )
=> ~ ! [X3: produc2041926651_state,Xs2: list_P1040014977_state] :
( ( Xs
= ( cons_P465369265_state @ X3 @ Xs2 ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys2 ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_269_listrel__Cons2,axiom,
! [Xs: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1244407729_state @ R ) )
=> ~ ! [X3: produc410756839_state,Xs2: list_P890094829_state] :
( ( Xs
= ( cons_P438230685_state @ X3 @ Xs2 ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys2 ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_270_listrel__Cons1,axiom,
! [Y: com,Ys2: list_com,Xs: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ Y @ Ys2 ) @ Xs ) @ ( listrel_com_com @ R ) )
=> ~ ! [Y2: com,Ys3: list_com] :
( ( Xs
= ( cons_com @ Y2 @ Ys3 ) )
=> ( ( member1333916004om_com @ ( product_Pair_com_com @ Y @ Y2 ) @ R )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Ys3 ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_271_listrel__Cons1,axiom,
! [Y: list_com,Ys2: list_list_com,Xs: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ ( cons_list_com @ Y @ Ys2 ) @ Xs ) @ ( listre1268245261_state @ R ) )
=> ~ ! [Y2: state,Ys3: list_state] :
( ( Xs
= ( cons_state @ Y2 @ Ys3 ) )
=> ( ( member64234660_state @ ( produc1204172211_state @ Y @ Y2 ) @ R )
=> ~ ( member1442949316_state @ ( produc1742567891_state @ Ys2 @ Ys3 ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_272_listrel__Cons1,axiom,
! [Y: produc2041926651_state,Ys2: list_P1040014977_state,Xs: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ Y @ Ys2 ) @ Xs ) @ ( listre194873521_state @ R ) )
=> ~ ! [Y2: produc2041926651_state,Ys3: list_P1040014977_state] :
( ( Xs
= ( cons_P465369265_state @ Y2 @ Ys3 ) )
=> ( ( member1069318160_state @ ( produc1909270103_state @ Y @ Y2 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Ys2 @ Ys3 ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_273_listrel__Cons1,axiom,
! [Y: produc410756839_state,Ys2: list_P890094829_state,Xs: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ Y @ Ys2 ) @ Xs ) @ ( listre1244407729_state @ R ) )
=> ~ ! [Y2: produc410756839_state,Ys3: list_P890094829_state] :
( ( Xs
= ( cons_P438230685_state @ Y2 @ Ys3 ) )
=> ( ( member512523024_state @ ( produc2065881943_state @ Y @ Y2 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Ys2 @ Ys3 ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_274_listrel_OCons,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel_com_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel_com_com @ R ) ) ) ) ).
% listrel.Cons
thf(fact_275_listrel_OCons,axiom,
! [X: list_com,Y: state,R: set_Pr849496667_state,Xs: list_list_com,Ys2: list_state] :
( ( member64234660_state @ ( produc1204172211_state @ X @ Y ) @ R )
=> ( ( member1442949316_state @ ( produc1742567891_state @ Xs @ Ys2 ) @ ( listre1268245261_state @ R ) )
=> ( member1442949316_state @ ( produc1742567891_state @ ( cons_list_com @ X @ Xs ) @ ( cons_state @ Y @ Ys2 ) ) @ ( listre1268245261_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_276_listrel_OCons,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre194873521_state @ R ) )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre194873521_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_277_listrel_OCons,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1244407729_state @ R ) )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1244407729_state @ R ) ) ) ) ).
% listrel.Cons
thf(fact_278_exec1__only1__aux,axiom,
! [Ccs: list_com,S: state,Cs: list_com,T: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Ccs @ S ) @ ( produc1204172211_state @ Cs @ T ) ) @ pHoare259243666_exec1 )
=> ! [C5: com,Cs5: list_com] :
( ( Ccs
= ( cons_com @ C5 @ Cs5 ) )
=> ? [Cs1: list_com] :
( Cs
= ( append_com @ Cs1 @ Cs5 ) ) ) ) ).
% exec1_only1_aux
thf(fact_279_exec1__only1,axiom,
! [C: com,Cs3: list_com,S: state,Cs: list_com,T: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ C @ Cs3 ) @ S ) @ ( produc1204172211_state @ Cs @ T ) ) @ pHoare259243666_exec1 )
=> ? [Cs1: list_com] :
( Cs
= ( append_com @ Cs1 @ Cs3 ) ) ) ).
% exec1_only1
thf(fact_280_exec1__drop__suffix__aux,axiom,
! [Cs12: list_com,S: state,Cs1_2: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs12 @ S ) @ ( produc1204172211_state @ Cs1_2 @ S2 ) ) @ pHoare259243666_exec1 )
=> ! [Cs13: list_com,Cs23: list_com,Cs14: list_com] :
( ( ( Cs12
= ( append_com @ Cs13 @ Cs23 ) )
& ( Cs1_2
= ( append_com @ Cs14 @ Cs23 ) )
& ( Cs13 != nil_com ) )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs13 @ S ) @ ( produc1204172211_state @ Cs14 @ S2 ) ) @ pHoare259243666_exec1 ) ) ) ).
% exec1_drop_suffix_aux
thf(fact_281_exec1__drop__suffix,axiom,
! [Cs15: list_com,Cs22: list_com,S: state,Cs16: list_com,S2: state] :
( ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( append_com @ Cs15 @ Cs22 ) @ S ) @ ( produc1204172211_state @ ( append_com @ Cs16 @ Cs22 ) @ S2 ) ) @ pHoare259243666_exec1 )
=> ( ( Cs15 != nil_com )
=> ( member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ Cs15 @ S ) @ ( produc1204172211_state @ Cs16 @ S2 ) ) @ pHoare259243666_exec1 ) ) ) ).
% exec1_drop_suffix
thf(fact_282_listrel_Oinducts,axiom,
! [X1: list_com,X2: list_com,R: set_Pr1822651163om_com,P: list_com > list_com > $o] :
( ( member32358020st_com @ ( produc1307105171st_com @ X1 @ X2 ) @ ( listrel_com_com @ R ) )
=> ( ( P @ nil_com @ nil_com )
=> ( ! [X3: com,Y2: com,Xs2: list_com,Ys3: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y2 ) @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs2 @ Ys3 ) @ ( listrel_com_com @ R ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_com @ X3 @ Xs2 ) @ ( cons_com @ Y2 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_283_listrel_Oinducts,axiom,
! [X1: list_list_com,X2: list_state,R: set_Pr849496667_state,P: list_list_com > list_state > $o] :
( ( member1442949316_state @ ( produc1742567891_state @ X1 @ X2 ) @ ( listre1268245261_state @ R ) )
=> ( ( P @ nil_list_com @ nil_state )
=> ( ! [X3: list_com,Y2: state,Xs2: list_list_com,Ys3: list_state] :
( ( member64234660_state @ ( produc1204172211_state @ X3 @ Y2 ) @ R )
=> ( ( member1442949316_state @ ( produc1742567891_state @ Xs2 @ Ys3 ) @ ( listre1268245261_state @ R ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_com @ X3 @ Xs2 ) @ ( cons_state @ Y2 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_284_listrel_Oinducts,axiom,
! [X1: list_P1040014977_state,X2: list_P1040014977_state,R: set_Pr1165141447_state,P: list_P1040014977_state > list_P1040014977_state > $o] :
( ( member1267359376_state @ ( produc1298457559_state @ X1 @ X2 ) @ ( listre194873521_state @ R ) )
=> ( ( P @ nil_Pr1844458753_state @ nil_Pr1844458753_state )
=> ( ! [X3: produc2041926651_state,Y2: produc2041926651_state,Xs2: list_P1040014977_state,Ys3: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y2 ) @ R )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs2 @ Ys3 ) @ ( listre194873521_state @ R ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_P465369265_state @ X3 @ Xs2 ) @ ( cons_P465369265_state @ Y2 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_285_listrel_Oinducts,axiom,
! [X1: list_P890094829_state,X2: list_P890094829_state,R: set_Pr2096893127_state,P: list_P890094829_state > list_P890094829_state > $o] :
( ( member423306384_state @ ( produc1354769367_state @ X1 @ X2 ) @ ( listre1244407729_state @ R ) )
=> ( ( P @ nil_Pr1347322605_state @ nil_Pr1347322605_state )
=> ( ! [X3: produc410756839_state,Y2: produc410756839_state,Xs2: list_P890094829_state,Ys3: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y2 ) @ R )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs2 @ Ys3 ) @ ( listre1244407729_state @ R ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_P438230685_state @ X3 @ Xs2 ) @ ( cons_P438230685_state @ Y2 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_286_listrel_Osimps,axiom,
! [A1: list_com,A22: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ A1 @ A22 ) @ ( listrel_com_com @ R ) )
= ( ( ( A1 = nil_com )
& ( A22 = nil_com ) )
| ? [X4: com,Y3: com,Xs3: list_com,Ys: list_com] :
( ( A1
= ( cons_com @ X4 @ Xs3 ) )
& ( A22
= ( cons_com @ Y3 @ Ys ) )
& ( member1333916004om_com @ ( product_Pair_com_com @ X4 @ Y3 ) @ R )
& ( member32358020st_com @ ( produc1307105171st_com @ Xs3 @ Ys ) @ ( listrel_com_com @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_287_listrel_Osimps,axiom,
! [A1: list_list_com,A22: list_state,R: set_Pr849496667_state] :
( ( member1442949316_state @ ( produc1742567891_state @ A1 @ A22 ) @ ( listre1268245261_state @ R ) )
= ( ( ( A1 = nil_list_com )
& ( A22 = nil_state ) )
| ? [X4: list_com,Y3: state,Xs3: list_list_com,Ys: list_state] :
( ( A1
= ( cons_list_com @ X4 @ Xs3 ) )
& ( A22
= ( cons_state @ Y3 @ Ys ) )
& ( member64234660_state @ ( produc1204172211_state @ X4 @ Y3 ) @ R )
& ( member1442949316_state @ ( produc1742567891_state @ Xs3 @ Ys ) @ ( listre1268245261_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_288_listrel_Osimps,axiom,
! [A1: list_P1040014977_state,A22: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ A1 @ A22 ) @ ( listre194873521_state @ R ) )
= ( ( ( A1 = nil_Pr1844458753_state )
& ( A22 = nil_Pr1844458753_state ) )
| ? [X4: produc2041926651_state,Y3: produc2041926651_state,Xs3: list_P1040014977_state,Ys: list_P1040014977_state] :
( ( A1
= ( cons_P465369265_state @ X4 @ Xs3 ) )
& ( A22
= ( cons_P465369265_state @ Y3 @ Ys ) )
& ( member1069318160_state @ ( produc1909270103_state @ X4 @ Y3 ) @ R )
& ( member1267359376_state @ ( produc1298457559_state @ Xs3 @ Ys ) @ ( listre194873521_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_289_listrel_Osimps,axiom,
! [A1: list_P890094829_state,A22: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ A1 @ A22 ) @ ( listre1244407729_state @ R ) )
= ( ( ( A1 = nil_Pr1347322605_state )
& ( A22 = nil_Pr1347322605_state ) )
| ? [X4: produc410756839_state,Y3: produc410756839_state,Xs3: list_P890094829_state,Ys: list_P890094829_state] :
( ( A1
= ( cons_P438230685_state @ X4 @ Xs3 ) )
& ( A22
= ( cons_P438230685_state @ Y3 @ Ys ) )
& ( member512523024_state @ ( produc2065881943_state @ X4 @ Y3 ) @ R )
& ( member423306384_state @ ( produc1354769367_state @ Xs3 @ Ys ) @ ( listre1244407729_state @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_290_bind__simps_I2_J,axiom,
! [X: com,Xs: list_com,F: com > list_com] :
( ( bind_com_com @ ( cons_com @ X @ Xs ) @ F )
= ( append_com @ ( F @ X ) @ ( bind_com_com @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_291_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_com,X: com,Ys2: list_com,Y: com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs @ ( cons_com @ X @ nil_com ) ) @ ( append_com @ Ys2 @ ( cons_com @ Y @ nil_com ) ) ) @ ( listrel1_com @ R ) )
= ( ( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_292_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_P1040014977_state,X: produc2041926651_state,Ys2: list_P1040014977_state,Y: produc2041926651_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ Xs @ ( cons_P465369265_state @ X @ nil_Pr1844458753_state ) ) @ ( append982487830_state @ Ys2 @ ( cons_P465369265_state @ Y @ nil_Pr1844458753_state ) ) ) @ ( listre1738592858_state @ R ) )
= ( ( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre1738592858_state @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_293_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_P890094829_state,X: produc410756839_state,Ys2: list_P890094829_state,Y: produc410756839_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ Xs @ ( cons_P438230685_state @ X @ nil_Pr1347322605_state ) ) @ ( append221843330_state @ Ys2 @ ( cons_P438230685_state @ Y @ nil_Pr1347322605_state ) ) ) @ ( listre1725776070_state @ R ) )
= ( ( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1725776070_state @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_294_maps__simps_I1_J,axiom,
! [F: com > list_com,X: com,Xs: list_com] :
( ( maps_com_com @ F @ ( cons_com @ X @ Xs ) )
= ( append_com @ ( F @ X ) @ ( maps_com_com @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_295_bind__simps_I1_J,axiom,
! [F: com > list_com] :
( ( bind_com_com @ nil_com @ F )
= nil_com ) ).
% bind_simps(1)
thf(fact_296_Cons__listrel1__Cons,axiom,
! [X: com,Xs: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel1_com @ R ) )
= ( ( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_297_Cons__listrel1__Cons,axiom,
! [X: produc2041926651_state,Xs: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre1738592858_state @ R ) )
= ( ( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_298_Cons__listrel1__Cons,axiom,
! [X: produc410756839_state,Xs: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1725776070_state @ R ) )
= ( ( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_299_rtrancl__listrel1__ConsI1,axiom,
! [Xs: list_com,Ys2: list_com,R: set_Pr1822651163om_com,X: com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ X @ Ys2 ) ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_300_listrel1I2,axiom,
! [Xs: list_com,Ys2: list_com,R: set_Pr1822651163om_com,X: com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ X @ Ys2 ) ) @ ( listrel1_com @ R ) ) ) ).
% listrel1I2
thf(fact_301_not__listrel1__Nil,axiom,
! [Xs: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ Xs @ nil_com ) @ ( listrel1_com @ R ) ) ).
% not_listrel1_Nil
thf(fact_302_not__Nil__listrel1,axiom,
! [Xs: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Xs ) @ ( listrel1_com @ R ) ) ).
% not_Nil_listrel1
thf(fact_303_append__listrel1I,axiom,
! [Xs: list_com,Ys2: list_com,R: set_Pr1822651163om_com,Us2: list_com,Vs: list_com] :
( ( ( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys2 )
& ( member32358020st_com @ ( produc1307105171st_com @ Us2 @ Vs ) @ ( listrel1_com @ R ) ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs @ Us2 ) @ ( append_com @ Ys2 @ Vs ) ) @ ( listrel1_com @ R ) ) ) ).
% append_listrel1I
thf(fact_304_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R: set_Pr1165141447_state] :
( ( listre194873521_state @ ( transi1302705790_state @ R ) )
= ( transi339813252_state @ ( listre1738592858_state @ R ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_305_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R: set_Pr2096893127_state] :
( ( listre1244407729_state @ ( transi463013610_state @ R ) )
= ( transi1965148016_state @ ( listre1725776070_state @ R ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_306_maps__simps_I2_J,axiom,
! [F: com > list_com] :
( ( maps_com_com @ F @ nil_com )
= nil_com ) ).
% maps_simps(2)
thf(fact_307_listrel1I1,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ Y @ Xs ) ) @ ( listrel1_com @ R ) ) ) ).
% listrel1I1
thf(fact_308_listrel1I1,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs ) @ ( cons_P465369265_state @ Y @ Xs ) ) @ ( listre1738592858_state @ R ) ) ) ).
% listrel1I1
thf(fact_309_listrel1I1,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs ) @ ( cons_P438230685_state @ Y @ Xs ) ) @ ( listre1725776070_state @ R ) ) ) ).
% listrel1I1
thf(fact_310_Cons__listrel1E1,axiom,
! [X: com,Xs: list_com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ Ys2 ) @ ( listrel1_com @ R ) )
=> ( ! [Y2: com] :
( ( Ys2
= ( cons_com @ Y2 @ Xs ) )
=> ~ ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y2 ) @ R ) )
=> ~ ! [Zs: list_com] :
( ( Ys2
= ( cons_com @ X @ Zs ) )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Zs ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_311_Cons__listrel1E1,axiom,
! [X: produc2041926651_state,Xs: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs ) @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ( ! [Y2: produc2041926651_state] :
( ( Ys2
= ( cons_P465369265_state @ Y2 @ Xs ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ X @ Y2 ) @ R ) )
=> ~ ! [Zs: list_P1040014977_state] :
( ( Ys2
= ( cons_P465369265_state @ X @ Zs ) )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs @ Zs ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_312_Cons__listrel1E1,axiom,
! [X: produc410756839_state,Xs: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs ) @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ( ! [Y2: produc410756839_state] :
( ( Ys2
= ( cons_P438230685_state @ Y2 @ Xs ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ X @ Y2 ) @ R ) )
=> ~ ! [Zs: list_P890094829_state] :
( ( Ys2
= ( cons_P438230685_state @ X @ Zs ) )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs @ Zs ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_313_Cons__listrel1E2,axiom,
! [Xs: list_com,Y: com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ ( cons_com @ Y @ Ys2 ) ) @ ( listrel1_com @ R ) )
=> ( ! [X3: com] :
( ( Xs
= ( cons_com @ X3 @ Ys2 ) )
=> ~ ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_com] :
( ( Xs
= ( cons_com @ Y @ Zs ) )
=> ~ ( member32358020st_com @ ( produc1307105171st_com @ Zs @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_314_Cons__listrel1E2,axiom,
! [Xs: list_P1040014977_state,Y: produc2041926651_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( listre1738592858_state @ R ) )
=> ( ! [X3: produc2041926651_state] :
( ( Xs
= ( cons_P465369265_state @ X3 @ Ys2 ) )
=> ~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_P1040014977_state] :
( ( Xs
= ( cons_P465369265_state @ Y @ Zs ) )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Zs @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_315_Cons__listrel1E2,axiom,
! [Xs: list_P890094829_state,Y: produc410756839_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( listre1725776070_state @ R ) )
=> ( ! [X3: produc410756839_state] :
( ( Xs
= ( cons_P438230685_state @ X3 @ Ys2 ) )
=> ~ ( member512523024_state @ ( produc2065881943_state @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_P890094829_state] :
( ( Xs
= ( cons_P438230685_state @ Y @ Zs ) )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Zs @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_316_listrel__reflcl__if__listrel1,axiom,
! [Xs: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre194873521_state @ ( transi1302705790_state @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_317_listrel__reflcl__if__listrel1,axiom,
! [Xs: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1244407729_state @ ( transi463013610_state @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_318_listrel1I,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs: list_com,Us2: list_com,Vs: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ R )
=> ( ( Xs
= ( append_com @ Us2 @ ( cons_com @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_com @ Us2 @ ( cons_com @ Y @ Vs ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) ) ) ) ) ).
% listrel1I
thf(fact_319_listrel1I,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs: list_P1040014977_state,Us2: list_P1040014977_state,Vs: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ R )
=> ( ( Xs
= ( append982487830_state @ Us2 @ ( cons_P465369265_state @ X @ Vs ) ) )
=> ( ( Ys2
= ( append982487830_state @ Us2 @ ( cons_P465369265_state @ Y @ Vs ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre1738592858_state @ R ) ) ) ) ) ).
% listrel1I
thf(fact_320_listrel1I,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs: list_P890094829_state,Us2: list_P890094829_state,Vs: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ R )
=> ( ( Xs
= ( append221843330_state @ Us2 @ ( cons_P438230685_state @ X @ Vs ) ) )
=> ( ( Ys2
= ( append221843330_state @ Us2 @ ( cons_P438230685_state @ Y @ Vs ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1725776070_state @ R ) ) ) ) ) ).
% listrel1I
thf(fact_321_listrel1E,axiom,
! [Xs: list_com,Ys2: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( listrel1_com @ R ) )
=> ~ ! [X3: com,Y2: com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X3 @ Y2 ) @ R )
=> ! [Us3: list_com,Vs2: list_com] :
( ( Xs
= ( append_com @ Us3 @ ( cons_com @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append_com @ Us3 @ ( cons_com @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_322_listrel1E,axiom,
! [Xs: list_P1040014977_state,Ys2: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( listre1738592858_state @ R ) )
=> ~ ! [X3: produc2041926651_state,Y2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X3 @ Y2 ) @ R )
=> ! [Us3: list_P1040014977_state,Vs2: list_P1040014977_state] :
( ( Xs
= ( append982487830_state @ Us3 @ ( cons_P465369265_state @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append982487830_state @ Us3 @ ( cons_P465369265_state @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_323_listrel1E,axiom,
! [Xs: list_P890094829_state,Ys2: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( listre1725776070_state @ R ) )
=> ~ ! [X3: produc410756839_state,Y2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ X3 @ Y2 ) @ R )
=> ! [Us3: list_P890094829_state,Vs2: list_P890094829_state] :
( ( Xs
= ( append221843330_state @ Us3 @ ( cons_P438230685_state @ X3 @ Vs2 ) ) )
=> ( Ys2
!= ( append221843330_state @ Us3 @ ( cons_P438230685_state @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_324_rtrancl__listrel1__ConsI2,axiom,
! [X: com,Y: com,R: set_Pr1822651163om_com,Xs: list_com,Ys2: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ X @ Y ) @ ( transi863150634cl_com @ R ) )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ Xs @ Ys2 ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ X @ Xs ) @ ( cons_com @ Y @ Ys2 ) ) @ ( transi436345274st_com @ ( listrel1_com @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_325_rtrancl__listrel1__ConsI2,axiom,
! [X: produc2041926651_state,Y: produc2041926651_state,R: set_Pr1165141447_state,Xs: list_P1040014977_state,Ys2: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ X @ Y ) @ ( transi1302705790_state @ R ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ Xs @ Ys2 ) @ ( transi339813252_state @ ( listre1738592858_state @ R ) ) )
=> ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ X @ Xs ) @ ( cons_P465369265_state @ Y @ Ys2 ) ) @ ( transi339813252_state @ ( listre1738592858_state @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_326_rtrancl__listrel1__ConsI2,axiom,
! [X: produc410756839_state,Y: produc410756839_state,R: set_Pr2096893127_state,Xs: list_P890094829_state,Ys2: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ X @ Y ) @ ( transi463013610_state @ R ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ Xs @ Ys2 ) @ ( transi1965148016_state @ ( listre1725776070_state @ R ) ) )
=> ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ X @ Xs ) @ ( cons_P438230685_state @ Y @ Ys2 ) ) @ ( transi1965148016_state @ ( listre1725776070_state @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_327_concat__eq__append__conv,axiom,
! [Xss2: list_list_com,Ys2: list_com,Zs2: list_com] :
( ( ( concat_com @ Xss2 )
= ( append_com @ Ys2 @ Zs2 ) )
= ( ( ( Xss2 = nil_list_com )
=> ( ( Ys2 = nil_com )
& ( Zs2 = nil_com ) ) )
& ( ( Xss2 != nil_list_com )
=> ? [Xss1: list_list_com,Xs3: list_com,Xs4: list_com,Xss22: list_list_com] :
( ( Xss2
= ( append_list_com @ Xss1 @ ( cons_list_com @ ( append_com @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append_com @ ( concat_com @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_com @ Xs4 @ ( concat_com @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_328_lexord__append__left__rightI,axiom,
! [A: com,B: com,R: set_Pr1822651163om_com,U: list_com,X: list_com,Y: list_com] :
( ( member1333916004om_com @ ( product_Pair_com_com @ A @ B ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ U @ ( cons_com @ A @ X ) ) @ ( append_com @ U @ ( cons_com @ B @ Y ) ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_329_lexord__append__left__rightI,axiom,
! [A: produc2041926651_state,B: produc2041926651_state,R: set_Pr1165141447_state,U: list_P1040014977_state,X: list_P1040014977_state,Y: list_P1040014977_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ U @ ( cons_P465369265_state @ A @ X ) ) @ ( append982487830_state @ U @ ( cons_P465369265_state @ B @ Y ) ) ) @ ( lexord1062073966_state @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_330_lexord__append__left__rightI,axiom,
! [A: produc410756839_state,B: produc410756839_state,R: set_Pr2096893127_state,U: list_P890094829_state,X: list_P890094829_state,Y: list_P890094829_state] :
( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ U @ ( cons_P438230685_state @ A @ X ) ) @ ( append221843330_state @ U @ ( cons_P438230685_state @ B @ Y ) ) ) @ ( lexord1893203674_state @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_331_concat__append,axiom,
! [Xs: list_list_com,Ys2: list_list_com] :
( ( concat_com @ ( append_list_com @ Xs @ Ys2 ) )
= ( append_com @ ( concat_com @ Xs ) @ ( concat_com @ Ys2 ) ) ) ).
% concat_append
thf(fact_332_lexord__cons__cons,axiom,
! [A: com,X: list_com,B: com,Y: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( cons_com @ A @ X ) @ ( cons_com @ B @ Y ) ) @ ( lexord_com @ R ) )
= ( ( member1333916004om_com @ ( product_Pair_com_com @ A @ B ) @ R )
| ( ( A = B )
& ( member32358020st_com @ ( produc1307105171st_com @ X @ Y ) @ ( lexord_com @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_333_lexord__cons__cons,axiom,
! [A: produc2041926651_state,X: list_P1040014977_state,B: produc2041926651_state,Y: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( cons_P465369265_state @ A @ X ) @ ( cons_P465369265_state @ B @ Y ) ) @ ( lexord1062073966_state @ R ) )
= ( ( member1069318160_state @ ( produc1909270103_state @ A @ B ) @ R )
| ( ( A = B )
& ( member1267359376_state @ ( produc1298457559_state @ X @ Y ) @ ( lexord1062073966_state @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_334_lexord__cons__cons,axiom,
! [A: produc410756839_state,X: list_P890094829_state,B: produc410756839_state,Y: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( cons_P438230685_state @ A @ X ) @ ( cons_P438230685_state @ B @ Y ) ) @ ( lexord1893203674_state @ R ) )
= ( ( member512523024_state @ ( produc2065881943_state @ A @ B ) @ R )
| ( ( A = B )
& ( member423306384_state @ ( produc1354769367_state @ X @ Y ) @ ( lexord1893203674_state @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_335_lexord__Nil__left,axiom,
! [Y: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ nil_com @ Y ) @ ( lexord_com @ R ) )
= ( ? [A3: com,X4: list_com] :
( Y
= ( cons_com @ A3 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_336_lexord__linear,axiom,
! [R: set_Pr1165141447_state,X: list_P1040014977_state,Y: list_P1040014977_state] :
( ! [A2: produc2041926651_state,B2: produc2041926651_state] :
( ( member1069318160_state @ ( produc1909270103_state @ A2 @ B2 ) @ R )
| ( A2 = B2 )
| ( member1069318160_state @ ( produc1909270103_state @ B2 @ A2 ) @ R ) )
=> ( ( member1267359376_state @ ( produc1298457559_state @ X @ Y ) @ ( lexord1062073966_state @ R ) )
| ( X = Y )
| ( member1267359376_state @ ( produc1298457559_state @ Y @ X ) @ ( lexord1062073966_state @ R ) ) ) ) ).
% lexord_linear
thf(fact_337_lexord__linear,axiom,
! [R: set_Pr2096893127_state,X: list_P890094829_state,Y: list_P890094829_state] :
( ! [A2: produc410756839_state,B2: produc410756839_state] :
( ( member512523024_state @ ( produc2065881943_state @ A2 @ B2 ) @ R )
| ( A2 = B2 )
| ( member512523024_state @ ( produc2065881943_state @ B2 @ A2 ) @ R ) )
=> ( ( member423306384_state @ ( produc1354769367_state @ X @ Y ) @ ( lexord1893203674_state @ R ) )
| ( X = Y )
| ( member423306384_state @ ( produc1354769367_state @ Y @ X ) @ ( lexord1893203674_state @ R ) ) ) ) ).
% lexord_linear
thf(fact_338_lexord__irreflexive,axiom,
! [R: set_Pr1165141447_state,Xs: list_P1040014977_state] :
( ! [X3: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ X3 @ X3 ) @ R )
=> ~ ( member1267359376_state @ ( produc1298457559_state @ Xs @ Xs ) @ ( lexord1062073966_state @ R ) ) ) ).
% lexord_irreflexive
thf(fact_339_lexord__irreflexive,axiom,
! [R: set_Pr2096893127_state,Xs: list_P890094829_state] :
( ! [X3: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ X3 @ X3 ) @ R )
=> ~ ( member423306384_state @ ( produc1354769367_state @ Xs @ Xs ) @ ( lexord1893203674_state @ R ) ) ) ).
% lexord_irreflexive
thf(fact_340_concat_Osimps_I1_J,axiom,
( ( concat_com @ nil_list_com )
= nil_com ) ).
% concat.simps(1)
thf(fact_341_lexord__Nil__right,axiom,
! [X: list_com,R: set_Pr1822651163om_com] :
~ ( member32358020st_com @ ( produc1307105171st_com @ X @ nil_com ) @ ( lexord_com @ R ) ) ).
% lexord_Nil_right
thf(fact_342_concat_Osimps_I2_J,axiom,
! [X: list_com,Xs: list_list_com] :
( ( concat_com @ ( cons_list_com @ X @ Xs ) )
= ( append_com @ X @ ( concat_com @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_343_lexord__append__leftI,axiom,
! [U: list_com,V: list_com,R: set_Pr1822651163om_com,X: list_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ U @ V ) @ ( lexord_com @ R ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ X @ U ) @ ( append_com @ X @ V ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_leftI
thf(fact_344_lexord__append__leftD,axiom,
! [X: list_com,U: list_com,V: list_com,R: set_Pr1822651163om_com] :
( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ X @ U ) @ ( append_com @ X @ V ) ) @ ( lexord_com @ R ) )
=> ( ! [A2: com] :
~ ( member1333916004om_com @ ( product_Pair_com_com @ A2 @ A2 ) @ R )
=> ( member32358020st_com @ ( produc1307105171st_com @ U @ V ) @ ( lexord_com @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_345_lexord__append__leftD,axiom,
! [X: list_P1040014977_state,U: list_P1040014977_state,V: list_P1040014977_state,R: set_Pr1165141447_state] :
( ( member1267359376_state @ ( produc1298457559_state @ ( append982487830_state @ X @ U ) @ ( append982487830_state @ X @ V ) ) @ ( lexord1062073966_state @ R ) )
=> ( ! [A2: produc2041926651_state] :
~ ( member1069318160_state @ ( produc1909270103_state @ A2 @ A2 ) @ R )
=> ( member1267359376_state @ ( produc1298457559_state @ U @ V ) @ ( lexord1062073966_state @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_346_lexord__append__leftD,axiom,
! [X: list_P890094829_state,U: list_P890094829_state,V: list_P890094829_state,R: set_Pr2096893127_state] :
( ( member423306384_state @ ( produc1354769367_state @ ( append221843330_state @ X @ U ) @ ( append221843330_state @ X @ V ) ) @ ( lexord1893203674_state @ R ) )
=> ( ! [A2: produc410756839_state] :
~ ( member512523024_state @ ( produc2065881943_state @ A2 @ A2 ) @ R )
=> ( member423306384_state @ ( produc1354769367_state @ U @ V ) @ ( lexord1893203674_state @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_347_lexord__append__rightI,axiom,
! [Y: list_com,X: list_com,R: set_Pr1822651163om_com] :
( ? [B5: com,Z3: list_com] :
( Y
= ( cons_com @ B5 @ Z3 ) )
=> ( member32358020st_com @ ( produc1307105171st_com @ X @ ( append_com @ X @ Y ) ) @ ( lexord_com @ R ) ) ) ).
% lexord_append_rightI
thf(fact_348_concat__eq__appendD,axiom,
! [Xss2: list_list_com,Ys2: list_com,Zs2: list_com] :
( ( ( concat_com @ Xss2 )
= ( append_com @ Ys2 @ Zs2 ) )
=> ( ( Xss2 != nil_list_com )
=> ? [Xss12: list_list_com,Xs2: list_com,Xs5: list_com,Xss23: list_list_com] :
( ( Xss2
= ( append_list_com @ Xss12 @ ( cons_list_com @ ( append_com @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_com @ ( concat_com @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append_com @ Xs5 @ ( concat_com @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_349_lexord__same__pref__if__irrefl,axiom,
! [R: set_Pr1822651163om_com,Xs: list_com,Ys2: list_com,Zs2: list_com] :
( ( irrefl_com @ R )
=> ( ( member32358020st_com @ ( produc1307105171st_com @ ( append_com @ Xs @ Ys2 ) @ ( append_com @ Xs @ Zs2 ) ) @ ( lexord_com @ R ) )
= ( member32358020st_com @ ( produc1307105171st_com @ Ys2 @ Zs2 ) @ ( lexord_com @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
% Conjectures (5)
thf(conj_0,hypothesis,
termi @ c @ s ).
thf(conj_1,hypothesis,
( ( f @ zero_zero_nat )
= ( produc1204172211_state @ ( cons_com @ c @ nil_com ) @ s ) ) ).
thf(conj_2,hypothesis,
! [I2: nat] : ( member1069318160_state @ ( produc1909270103_state @ ( f @ I2 ) @ ( f @ ( suc @ I2 ) ) ) @ ( transi1726587420_state @ pHoare259243666_exec1 ) ) ).
thf(conj_3,hypothesis,
member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ c @ nil_com ) @ s ) @ ( f @ n ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ).
thf(conj_4,conjecture,
member1069318160_state @ ( produc1909270103_state @ ( produc1204172211_state @ ( cons_com @ c @ nil_com ) @ s ) @ ( f @ ( suc @ n ) ) ) @ ( transi1302705790_state @ pHoare259243666_exec1 ) ).
%------------------------------------------------------------------------------