TPTP Problem File: SWW471_1.p
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%------------------------------------------------------------------------------
% File : SWW471_1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 100 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_100_tff_l269 [Bla11]
% Status : CounterSatisfiable
% Rating : 0.33 v8.2.0, 0.67 v6.3.0, 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 350 ( 101 unt; 119 typ; 0 def)
% Number of atoms : 446 ( 182 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 278 ( 63 ~; 29 |; 23 &)
% ( 54 <=>; 109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 118 ( 64 >; 54 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 85 ( 85 usr; 22 con; 0-3 aty)
% Number of variables : 586 ( 576 !; 10 ?; 586 :)
% SPC : TF0_CSA_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:28:47
%------------------------------------------------------------------------------
%----Should-be-implicit typings (33)
tff(ty_ty_tc__Com__Ocom,type,
com: $tType ).
tff(ty_ty_tc__Com__Opname,type,
pname: $tType ).
tff(ty_ty_tc__Com__Ostate,type,
state: $tType ).
tff(ty_ty_tc__HOL__Obool,type,
bool: $tType ).
tff(ty_ty_tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J,type,
hoare_1927711152iple_a: $tType ).
tff(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
tff(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
option_com: $tType ).
tff(ty_ty_tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J,type,
fun_a_fun_state_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Ocom_Mtc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate_M,type,
fun_co1155576772iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__Com__Ocom_J,type,
fun_pname_com: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__Com__Opname_J,type,
fun_pname_pname: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
fun_pname_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__Hoare____Mirabelle____yiemogtkbg__Otriple_It,type,
fun_pn708290217iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__Option__Ooption_Itc__Com__Ocom_J_J,type,
fun_pname_option_com: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL,type,
fun_pn1683930517e_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_Itc__Com__Ocom_Mtc__fun_Itc__fun_It__a_M,type,
fun_pn308211645iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_J,type,
fun_pn800050071e_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_pn250273176l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate,type,
fun_pn579076298iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__Com__Opname_Mtc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,type,
fun_pn422929397l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
fun_bool_bool: $tType ).
tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_bo1549164019l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__Com__O,type,
fun_Ho842746065_pname: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__HOL__O,type,
fun_Ho1877127206a_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__Hoare_,type,
fun_Ho843200573iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__fun_It,type,
fun_Ho957066028l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__fun_It_001,type,
fun_Ho440810351a_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc__fun_It_002,type,
fun_Ho525994229l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Option__Ooption_Itc__Com__Ocom_J_Mtc__Com__Ocom_J,type,
fun_option_com_com: $tType ).
tff(ty_ty_tc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J_Mtc__,type,
fun_fu1344872529iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J_Mtc___003,type,
fun_fu90068325iple_a: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
fun_fu1430349052l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mt,type,
fun_fu832487784l_bool: $tType ).
%----Explicit typings (86)
tff(sy_c_COMBB_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_I,type,
cOMBB_1110279240iple_a: ( fun_pn708290217iple_a * fun_Ho842746065_pname ) > fun_Ho843200573iple_a ).
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname,type,
cOMBB_647938656_pname: ( fun_bool_bool * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabelle____yiem,type,
cOMBB_213049548iple_a: ( fun_bool_bool * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc_,type,
cOMBB_675860798_pname: ( fun_bo1549164019l_bool * fun_pname_bool ) > fun_pn250273176l_bool ).
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__004,type,
cOMBB_196465322iple_a: ( fun_bo1549164019l_bool * fun_Ho1877127206a_bool ) > fun_Ho957066028l_bool ).
tff(sy_c_COMBB_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Com__,type,
cOMBB_1433562676_pname: ( fun_Ho842746065_pname * fun_pn708290217iple_a ) > fun_pname_pname ).
tff(sy_c_COMBB_000tc__Option__Ooption_Itc__Com__Ocom_J_000tc__Com__Ocom_000tc__Com__,type,
cOMBB_923936821_pname: ( fun_option_com_com * fun_pname_option_com ) > fun_pname_com ).
tff(sy_c_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J_000tc_,type,
cOMBB_1515136928_pname: ( fun_fu90068325iple_a * fun_pn1683930517e_bool ) > fun_pn308211645iple_a ).
tff(sy_c_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool,type,
cOMBC_1149511130e_bool: ( fun_pn800050071e_bool * pname ) > fun_pname_bool ).
tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t,type,
cOMBC_1058051404l_bool: ( fun_pn422929397l_bool * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_COMBC_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Hoare,type,
cOMBC_671859290a_bool: ( fun_Ho440810351a_bool * hoare_1927711152iple_a ) > fun_Ho1877127206a_bool ).
tff(sy_c_COMBC_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__fun_I,type,
cOMBC_862840740l_bool: ( fun_Ho525994229l_bool * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_COMBK_000tc__Com__Opname_000tc__Com__Opname,type,
cOMBK_pname_pname: pname > fun_pname_pname ).
tff(sy_c_COMBK_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_I,type,
cOMBK_1495131898iple_a: pname > fun_Ho842746065_pname ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Com__Opname,type,
cOMBK_bool_pname: bool > fun_pname_bool ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It,type,
cOMBK_712844119iple_a: bool > fun_Ho1877127206a_bool ).
tff(sy_c_COMBK_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Com__,type,
cOMBK_669226658_pname: hoare_1927711152iple_a > fun_pn708290217iple_a ).
tff(sy_c_COMBK_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Hoare,type,
cOMBK_2109678094iple_a: hoare_1927711152iple_a > fun_Ho843200573iple_a ).
tff(sy_c_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_It__a_Mtc__fu,type,
cOMBS_1125763966iple_a: ( fun_pn308211645iple_a * fun_pname_com ) > fun_pn579076298iple_a ).
tff(sy_c_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool,type,
cOMBS_568398431l_bool: ( fun_pn250273176l_bool * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_COMBS_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HO,type,
cOMBS_821474699iple_a: ( fun_pn579076298iple_a * fun_pn1683930517e_bool ) > fun_pn708290217iple_a ).
tff(sy_c_COMBS_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__HOL__,type,
cOMBS_2061548107l_bool: ( fun_Ho957066028l_bool * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_Com_Obody,type,
body_1: fun_pname_option_com ).
tff(sy_c_Com_Ocom_OBODY,type,
body: fun_pname_com ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_Hoare__Mirabelle__yiemogtkbg_Ohoare__derivs_000t__a,type,
hoare_1617968510rivs_a: ( fun_Ho1877127206a_bool * fun_Ho1877127206a_bool ) > bool ).
tff(sy_c_Hoare__Mirabelle__yiemogtkbg_Ohoare__valids_000t__a,type,
hoare_1955801856lids_a: ( fun_Ho1877127206a_bool * fun_Ho1877127206a_bool ) > bool ).
tff(sy_c_Hoare__Mirabelle__yiemogtkbg_Otriple_Otriple_000t__a,type,
hoare_1652181356iple_a: fun_fu90068325iple_a ).
tff(sy_c_Hoare__Mirabelle__yiemogtkbg_Otriple__valid_000t__a,type,
hoare_1572001082alid_a: nat > fun_Ho1877127206a_bool ).
tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__HOL__Obool,type,
semila1168014441p_bool: ( bool * bool ) > bool ).
tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Com__Opname_Mtc__HOL,type,
semila278973382e_bool: ( fun_pname_bool * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__fun_Itc__Hoare____Mirabelle__,type,
semila1525949746a_bool: ( fun_Ho1877127206a_bool * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Natural_Oevalc,type,
evalc: ( com * state * state ) > bool ).
tff(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: fun_option_com_com ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
bot_bo844097828e_bool: fun_pname_bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg,type,
bot_bo1208640912a_bool: fun_Ho1877127206a_bool ).
tff(sy_c_Set_OCollect_000tc__Com__Opname,type,
collect_pname: fun_pname_bool > fun_pname_bool ).
tff(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J,type,
collec829051333iple_a: fun_Ho1877127206a_bool > fun_Ho1877127206a_bool ).
tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
image_pname_pname: ( fun_pname_pname * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg__Otri,type,
image_68284913iple_a: ( fun_pn708290217iple_a * fun_pname_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__,type,
image_1389863321_pname: ( fun_Ho842746065_pname * fun_Ho1877127206a_bool ) > fun_pname_bool ).
tff(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc___005,type,
image_590713477iple_a: ( fun_Ho843200573iple_a * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_Set_Oinsert_000tc__Com__Opname,type,
insert_pname: ( pname * fun_pname_bool ) > fun_pname_bool ).
tff(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J,type,
insert1434104874iple_a: ( hoare_1927711152iple_a * fun_Ho1877127206a_bool ) > fun_Ho1877127206a_bool ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun_bool_bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun_bo1549164019l_bool ).
tff(sy_c_fdisj,type,
fdisj: fun_bo1549164019l_bool ).
tff(sy_c_fequal_000tc__Com__Opname,type,
fequal_pname: fun_pn800050071e_bool ).
tff(sy_c_fequal_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J,type,
fequal1440857775iple_a: fun_Ho440810351a_bool ).
tff(sy_c_fimplies,type,
fimplies: fun_bo1549164019l_bool ).
tff(sy_c_hAPP_000tc__Com__Ocom_000tc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate_M,type,
hAPP_c429049308iple_a: ( fun_co1155576772iple_a * com ) > fun_fu1344872529iple_a ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__Com__Ocom,type,
hAPP_pname_com: ( fun_pname_com * pname ) > com ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__Com__Opname,type,
hAPP_pname_pname: ( fun_pname_pname * pname ) > pname ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__HOL__Obool,type,
hAPP_pname_bool: ( fun_pname_bool * pname ) > bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It,type,
hAPP_p824302401iple_a: ( fun_pn708290217iple_a * pname ) > hoare_1927711152iple_a ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__Option__Ooption_Itc__Com__Ocom_J,type,
hAPP_p799580910on_com: ( fun_pname_option_com * pname ) > option_com ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL,type,
hAPP_p635540397e_bool: ( fun_pn1683930517e_bool * pname ) > fun_a_fun_state_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Ocom_Mtc__fun_Itc__fun_It__a_M,type,
hAPP_p1788720341iple_a: ( fun_pn308211645iple_a * pname ) > fun_co1155576772iple_a ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
hAPP_p61793385e_bool: ( fun_pn800050071e_bool * pname ) > fun_pname_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_p393069232l_bool: ( fun_pn250273176l_bool * pname ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_It__a_Mtc__fun_Itc__Com__Ostate,type,
hAPP_p1513881570iple_a: ( fun_pn579076298iple_a * pname ) > fun_fu1344872529iple_a ).
tff(sy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,type,
hAPP_p338031245l_bool: ( fun_pn422929397l_bool * pname ) > fun_fu1430349052l_bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Com__O,type,
hAPP_H2145880809_pname: ( fun_Ho842746065_pname * hoare_1927711152iple_a ) > pname ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__HOL__O,type,
hAPP_H1448631928a_bool: ( fun_Ho1877127206a_bool * hoare_1927711152iple_a ) > bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__Hoare_,type,
hAPP_H963118037iple_a: ( fun_Ho843200573iple_a * hoare_1927711152iple_a ) > hoare_1927711152iple_a ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__fun_It,type,
hAPP_H1487873860l_bool: ( fun_Ho957066028l_bool * hoare_1927711152iple_a ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__fun_It_006,type,
hAPP_H1027145665a_bool: ( fun_Ho440810351a_bool * hoare_1927711152iple_a ) > fun_Ho1877127206a_bool ).
tff(sy_c_hAPP_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_000tc__fun_It_007,type,
hAPP_H694056973l_bool: ( fun_Ho525994229l_bool * hoare_1927711152iple_a ) > fun_fu832487784l_bool ).
tff(sy_c_hAPP_000tc__Option__Ooption_Itc__Com__Ocom_J_000tc__Com__Ocom,type,
hAPP_option_com_com: ( fun_option_com_com * option_com ) > com ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J_000tc__,type,
hAPP_f711275241iple_a: ( fun_fu1344872529iple_a * fun_a_fun_state_bool ) > hoare_1927711152iple_a ).
tff(sy_c_hAPP_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_J_000tc___008,type,
hAPP_f185596029iple_a: ( fun_fu90068325iple_a * fun_a_fun_state_bool ) > fun_co1155576772iple_a ).
tff(sy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f1664156314l_bool: ( fun_fu1430349052l_bool * fun_pname_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_Mtc_,type,
hAPP_f1454306822l_bool: ( fun_fu832487784l_bool * fun_Ho1877127206a_bool ) > bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000tc__Com__Opname,type,
member_pname: fun_pn422929397l_bool ).
tff(sy_c_member_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J,type,
member127332739iple_a: fun_Ho525994229l_bool ).
tff(sy_v_G,type,
g: fun_Ho1877127206a_bool ).
tff(sy_v_P,type,
p: fun_pn1683930517e_bool ).
tff(sy_v_Procs,type,
procs: fun_pname_bool ).
tff(sy_v_Q,type,
q: fun_pn1683930517e_bool ).
tff(sy_v_n,type,
n: nat ).
%----Relevant facts (190)
tff(fact_0_triple_Oinject,axiom,
! [Fun1_2: fun_a_fun_state_bool,Com_2: com,Fun2_2: fun_a_fun_state_bool,Fun1_1: fun_a_fun_state_bool,Com_1: com,Fun2_1: fun_a_fun_state_bool] :
( ( hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Fun1_2),Com_2),Fun2_2) = hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Fun1_1),Com_1),Fun2_1) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_2 = Com_1 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
tff(fact_1_hoare__valids__def,axiom,
! [Ga: fun_Ho1877127206a_bool,Ts: fun_Ho1877127206a_bool] :
( hBOOL(hoare_1955801856lids_a(Ga,Ts))
<=> ! [N: nat] :
( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),Ga))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(N),X)) )
=> ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),Ts))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(N),X)) ) ) ) ).
tff(fact_2_hoare__derivs_OBody,axiom,
! [Ga: fun_Ho1877127206a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
( hBOOL(hoare_1617968510rivs_a(semila1525949746a_bool(Ga,image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,Pa),body),Qa),Procsa)),image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,Pa),cOMBB_923936821_pname(the_com,body_1)),Qa),Procsa)))
=> hBOOL(hoare_1617968510rivs_a(Ga,image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,Pa),body),Qa),Procsa))) ) ).
tff(fact_3_UnE,axiom,
! [C: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),semila1525949746a_bool(A,B_1)))
=> ( ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),A))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),B_1)) ) ) ).
tff(fact_4_UnE,axiom,
! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),semila278973382e_bool(A,B_1)))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).
tff(fact_5_sup1E,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(A,B_1),X_2))
=> ( ~ hBOOL(hAPP_H1448631928a_bool(A,X_2))
=> hBOOL(hAPP_H1448631928a_bool(B_1,X_2)) ) ) ).
tff(fact_6_sup1E,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(semila278973382e_bool(A,B_1),X_2))
=> ( ~ hBOOL(hAPP_pname_bool(A,X_2))
=> hBOOL(hAPP_pname_bool(B_1,X_2)) ) ) ).
tff(fact_7_sup1CI,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( ( ~ hBOOL(hAPP_H1448631928a_bool(B_1,X_2))
=> hBOOL(hAPP_H1448631928a_bool(A,X_2)) )
=> hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(A,B_1),X_2)) ) ).
tff(fact_8_sup1CI,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
( ( ~ hBOOL(hAPP_pname_bool(B_1,X_2))
=> hBOOL(hAPP_pname_bool(A,X_2)) )
=> hBOOL(hAPP_pname_bool(semila278973382e_bool(A,B_1),X_2)) ) ).
tff(fact_9_UnCI,axiom,
! [A: fun_Ho1877127206a_bool,C: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] :
( ( ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),B_1))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),A)) )
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),semila1525949746a_bool(A,B_1))) ) ).
tff(fact_10_UnCI,axiom,
! [A: fun_pname_bool,C: pname,B_1: fun_pname_bool] :
( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),semila278973382e_bool(A,B_1))) ) ).
tff(fact_11_image__eqI,axiom,
! [A: fun_pname_bool,B: hoare_1927711152iple_a,F: fun_pn708290217iple_a,X_2: pname] :
( ( B = hAPP_p824302401iple_a(F,X_2) )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,B),image_68284913iple_a(F,A))) ) ) ).
tff(fact_12_image__eqI,axiom,
! [A: fun_Ho1877127206a_bool,B: pname,F: fun_Ho842746065_pname,X_2: hoare_1927711152iple_a] :
( ( B = hAPP_H2145880809_pname(F,X_2) )
=> ( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),image_1389863321_pname(F,A))) ) ) ).
tff(fact_13_image__Un,axiom,
! [F: fun_pn708290217iple_a,A: fun_pname_bool,B_1: fun_pname_bool] : image_68284913iple_a(F,semila278973382e_bool(A,B_1)) = semila1525949746a_bool(image_68284913iple_a(F,A),image_68284913iple_a(F,B_1)) ).
tff(fact_14_image__Un,axiom,
! [F: fun_Ho842746065_pname,A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] : image_1389863321_pname(F,semila1525949746a_bool(A,B_1)) = semila278973382e_bool(image_1389863321_pname(F,A),image_1389863321_pname(F,B_1)) ).
tff(fact_15_sup__fun__def,axiom,
! [F: fun_Ho1877127206a_bool,G: fun_Ho1877127206a_bool,X: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(F,G),X))
<=> hBOOL(semila1168014441p_bool(hAPP_H1448631928a_bool(F,X),hAPP_H1448631928a_bool(G,X))) ) ).
tff(fact_16_sup__fun__def,axiom,
! [F: fun_pname_bool,G: fun_pname_bool,X: pname] :
( hBOOL(hAPP_pname_bool(semila278973382e_bool(F,G),X))
<=> hBOOL(semila1168014441p_bool(hAPP_pname_bool(F,X),hAPP_pname_bool(G,X))) ) ).
tff(fact_17_sup__apply,axiom,
! [F: fun_Ho1877127206a_bool,G: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(F,G),X_2))
<=> hBOOL(semila1168014441p_bool(hAPP_H1448631928a_bool(F,X_2),hAPP_H1448631928a_bool(G,X_2))) ) ).
tff(fact_18_sup__apply,axiom,
! [F: fun_pname_bool,G: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(semila278973382e_bool(F,G),X_2))
<=> hBOOL(semila1168014441p_bool(hAPP_pname_bool(F,X_2),hAPP_pname_bool(G,X_2))) ) ).
tff(fact_19_cut,axiom,
! [Ga: fun_Ho1877127206a_bool,G_1: fun_Ho1877127206a_bool,Ts: fun_Ho1877127206a_bool] :
( hBOOL(hoare_1617968510rivs_a(G_1,Ts))
=> ( hBOOL(hoare_1617968510rivs_a(Ga,G_1))
=> hBOOL(hoare_1617968510rivs_a(Ga,Ts)) ) ) ).
tff(fact_20_sup__assoc,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool,Z: fun_Ho1877127206a_bool] : semila1525949746a_bool(semila1525949746a_bool(X_2,Y_1),Z) = semila1525949746a_bool(X_2,semila1525949746a_bool(Y_1,Z)) ).
tff(fact_21_sup__assoc,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z: fun_pname_bool] : semila278973382e_bool(semila278973382e_bool(X_2,Y_1),Z) = semila278973382e_bool(X_2,semila278973382e_bool(Y_1,Z)) ).
tff(fact_22_sup__assoc,axiom,
! [X_2: bool,Y_1: bool,Z: bool] :
( hBOOL(semila1168014441p_bool(semila1168014441p_bool(X_2,Y_1),Z))
<=> hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(Y_1,Z))) ) ).
tff(fact_23_inf__sup__aci_I6_J,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool,Z: fun_Ho1877127206a_bool] : semila1525949746a_bool(semila1525949746a_bool(X_2,Y_1),Z) = semila1525949746a_bool(X_2,semila1525949746a_bool(Y_1,Z)) ).
tff(fact_24_inf__sup__aci_I6_J,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z: fun_pname_bool] : semila278973382e_bool(semila278973382e_bool(X_2,Y_1),Z) = semila278973382e_bool(X_2,semila278973382e_bool(Y_1,Z)) ).
tff(fact_25_inf__sup__aci_I6_J,axiom,
! [X_2: bool,Y_1: bool,Z: bool] :
( hBOOL(semila1168014441p_bool(semila1168014441p_bool(X_2,Y_1),Z))
<=> hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(Y_1,Z))) ) ).
tff(fact_26_sup_Oassoc,axiom,
! [A_1: fun_Ho1877127206a_bool,B: fun_Ho1877127206a_bool,C: fun_Ho1877127206a_bool] : semila1525949746a_bool(semila1525949746a_bool(A_1,B),C) = semila1525949746a_bool(A_1,semila1525949746a_bool(B,C)) ).
tff(fact_27_sup_Oassoc,axiom,
! [A_1: fun_pname_bool,B: fun_pname_bool,C: fun_pname_bool] : semila278973382e_bool(semila278973382e_bool(A_1,B),C) = semila278973382e_bool(A_1,semila278973382e_bool(B,C)) ).
tff(fact_28_sup_Oassoc,axiom,
! [A_1: bool,B: bool,C: bool] :
( hBOOL(semila1168014441p_bool(semila1168014441p_bool(A_1,B),C))
<=> hBOOL(semila1168014441p_bool(A_1,semila1168014441p_bool(B,C))) ) ).
tff(fact_29_sup__left__commute,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool,Z: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,semila1525949746a_bool(Y_1,Z)) = semila1525949746a_bool(Y_1,semila1525949746a_bool(X_2,Z)) ).
tff(fact_30_sup__left__commute,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z: fun_pname_bool] : semila278973382e_bool(X_2,semila278973382e_bool(Y_1,Z)) = semila278973382e_bool(Y_1,semila278973382e_bool(X_2,Z)) ).
tff(fact_31_sup__left__commute,axiom,
! [X_2: bool,Y_1: bool,Z: bool] :
( hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(Y_1,Z)))
<=> hBOOL(semila1168014441p_bool(Y_1,semila1168014441p_bool(X_2,Z))) ) ).
tff(fact_32_inf__sup__aci_I7_J,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool,Z: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,semila1525949746a_bool(Y_1,Z)) = semila1525949746a_bool(Y_1,semila1525949746a_bool(X_2,Z)) ).
tff(fact_33_inf__sup__aci_I7_J,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool,Z: fun_pname_bool] : semila278973382e_bool(X_2,semila278973382e_bool(Y_1,Z)) = semila278973382e_bool(Y_1,semila278973382e_bool(X_2,Z)) ).
tff(fact_34_inf__sup__aci_I7_J,axiom,
! [X_2: bool,Y_1: bool,Z: bool] :
( hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(Y_1,Z)))
<=> hBOOL(semila1168014441p_bool(Y_1,semila1168014441p_bool(X_2,Z))) ) ).
tff(fact_35_sup_Oleft__commute,axiom,
! [B: fun_Ho1877127206a_bool,A_1: fun_Ho1877127206a_bool,C: fun_Ho1877127206a_bool] : semila1525949746a_bool(B,semila1525949746a_bool(A_1,C)) = semila1525949746a_bool(A_1,semila1525949746a_bool(B,C)) ).
tff(fact_36_sup_Oleft__commute,axiom,
! [B: fun_pname_bool,A_1: fun_pname_bool,C: fun_pname_bool] : semila278973382e_bool(B,semila278973382e_bool(A_1,C)) = semila278973382e_bool(A_1,semila278973382e_bool(B,C)) ).
tff(fact_37_sup_Oleft__commute,axiom,
! [B: bool,A_1: bool,C: bool] :
( hBOOL(semila1168014441p_bool(B,semila1168014441p_bool(A_1,C)))
<=> hBOOL(semila1168014441p_bool(A_1,semila1168014441p_bool(B,C))) ) ).
tff(fact_38_sup__left__idem,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,semila1525949746a_bool(X_2,Y_1)) = semila1525949746a_bool(X_2,Y_1) ).
tff(fact_39_sup__left__idem,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : semila278973382e_bool(X_2,semila278973382e_bool(X_2,Y_1)) = semila278973382e_bool(X_2,Y_1) ).
tff(fact_40_sup__left__idem,axiom,
! [X_2: bool,Y_1: bool] :
( hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(X_2,Y_1)))
<=> hBOOL(semila1168014441p_bool(X_2,Y_1)) ) ).
tff(fact_41_inf__sup__aci_I8_J,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,semila1525949746a_bool(X_2,Y_1)) = semila1525949746a_bool(X_2,Y_1) ).
tff(fact_42_inf__sup__aci_I8_J,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : semila278973382e_bool(X_2,semila278973382e_bool(X_2,Y_1)) = semila278973382e_bool(X_2,Y_1) ).
tff(fact_43_inf__sup__aci_I8_J,axiom,
! [X_2: bool,Y_1: bool] :
( hBOOL(semila1168014441p_bool(X_2,semila1168014441p_bool(X_2,Y_1)))
<=> hBOOL(semila1168014441p_bool(X_2,Y_1)) ) ).
tff(fact_44_sup_Oleft__idem,axiom,
! [A_1: fun_Ho1877127206a_bool,B: fun_Ho1877127206a_bool] : semila1525949746a_bool(A_1,semila1525949746a_bool(A_1,B)) = semila1525949746a_bool(A_1,B) ).
tff(fact_45_sup_Oleft__idem,axiom,
! [A_1: fun_pname_bool,B: fun_pname_bool] : semila278973382e_bool(A_1,semila278973382e_bool(A_1,B)) = semila278973382e_bool(A_1,B) ).
tff(fact_46_sup_Oleft__idem,axiom,
! [A_1: bool,B: bool] :
( hBOOL(semila1168014441p_bool(A_1,semila1168014441p_bool(A_1,B)))
<=> hBOOL(semila1168014441p_bool(A_1,B)) ) ).
tff(fact_47_sup__commute,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,Y_1) = semila1525949746a_bool(Y_1,X_2) ).
tff(fact_48_sup__commute,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : semila278973382e_bool(X_2,Y_1) = semila278973382e_bool(Y_1,X_2) ).
tff(fact_49_sup__commute,axiom,
! [X_2: bool,Y_1: bool] :
( hBOOL(semila1168014441p_bool(X_2,Y_1))
<=> hBOOL(semila1168014441p_bool(Y_1,X_2)) ) ).
tff(fact_50_inf__sup__aci_I5_J,axiom,
! [X_2: fun_Ho1877127206a_bool,Y_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,Y_1) = semila1525949746a_bool(Y_1,X_2) ).
tff(fact_51_inf__sup__aci_I5_J,axiom,
! [X_2: fun_pname_bool,Y_1: fun_pname_bool] : semila278973382e_bool(X_2,Y_1) = semila278973382e_bool(Y_1,X_2) ).
tff(fact_52_inf__sup__aci_I5_J,axiom,
! [X_2: bool,Y_1: bool] :
( hBOOL(semila1168014441p_bool(X_2,Y_1))
<=> hBOOL(semila1168014441p_bool(Y_1,X_2)) ) ).
tff(fact_53_sup_Ocommute,axiom,
! [A_1: fun_Ho1877127206a_bool,B: fun_Ho1877127206a_bool] : semila1525949746a_bool(A_1,B) = semila1525949746a_bool(B,A_1) ).
tff(fact_54_sup_Ocommute,axiom,
! [A_1: fun_pname_bool,B: fun_pname_bool] : semila278973382e_bool(A_1,B) = semila278973382e_bool(B,A_1) ).
tff(fact_55_sup_Ocommute,axiom,
! [A_1: bool,B: bool] :
( hBOOL(semila1168014441p_bool(A_1,B))
<=> hBOOL(semila1168014441p_bool(B,A_1)) ) ).
tff(fact_56_sup__idem,axiom,
! [X_2: fun_Ho1877127206a_bool] : semila1525949746a_bool(X_2,X_2) = X_2 ).
tff(fact_57_sup__idem,axiom,
! [X_2: fun_pname_bool] : semila278973382e_bool(X_2,X_2) = X_2 ).
tff(fact_58_sup__idem,axiom,
! [X_2: bool] :
( hBOOL(semila1168014441p_bool(X_2,X_2))
<=> hBOOL(X_2) ) ).
tff(fact_59_sup_Oidem,axiom,
! [A_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(A_1,A_1) = A_1 ).
tff(fact_60_sup_Oidem,axiom,
! [A_1: fun_pname_bool] : semila278973382e_bool(A_1,A_1) = A_1 ).
tff(fact_61_sup_Oidem,axiom,
! [A_1: bool] :
( hBOOL(semila1168014441p_bool(A_1,A_1))
<=> hBOOL(A_1) ) ).
tff(fact_62_rev__image__eqI,axiom,
! [B: pname,F: fun_Ho842746065_pname,X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> ( ( B = hAPP_H2145880809_pname(F,X_2) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),image_1389863321_pname(F,A))) ) ) ).
tff(fact_63_rev__image__eqI,axiom,
! [B: hoare_1927711152iple_a,F: fun_pn708290217iple_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( ( B = hAPP_p824302401iple_a(F,X_2) )
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,B),image_68284913iple_a(F,A))) ) ) ).
tff(fact_64_imageI,axiom,
! [F: fun_Ho842746065_pname,X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_H2145880809_pname(F,X_2)),image_1389863321_pname(F,A))) ) ).
tff(fact_65_imageI,axiom,
! [F: fun_pn708290217iple_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,hAPP_p824302401iple_a(F,X_2)),image_68284913iple_a(F,A))) ) ).
tff(fact_66_image__iff,axiom,
! [Z: pname,F: fun_Ho842746065_pname,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Z),image_1389863321_pname(F,A)))
<=> ? [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A))
& ( Z = hAPP_H2145880809_pname(F,X) ) ) ) ).
tff(fact_67_image__iff,axiom,
! [Z: hoare_1927711152iple_a,F: fun_pn708290217iple_a,A: fun_pname_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,Z),image_68284913iple_a(F,A)))
<=> ? [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
& ( Z = hAPP_p824302401iple_a(F,X) ) ) ) ).
tff(fact_68_UnI2,axiom,
! [A: fun_Ho1877127206a_bool,C: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),B_1))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),semila1525949746a_bool(A,B_1))) ) ).
tff(fact_69_UnI2,axiom,
! [A: fun_pname_bool,C: pname,B_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),semila278973382e_bool(A,B_1))) ) ).
tff(fact_70_UnI1,axiom,
! [B_1: fun_Ho1877127206a_bool,C: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),A))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),semila1525949746a_bool(A,B_1))) ) ).
tff(fact_71_UnI1,axiom,
! [B_1: fun_pname_bool,C: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),semila278973382e_bool(A,B_1))) ) ).
tff(fact_72_sup1I2,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(B_1,X_2))
=> hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(A,B_1),X_2)) ) ).
tff(fact_73_sup1I2,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(B_1,X_2))
=> hBOOL(hAPP_pname_bool(semila278973382e_bool(A,B_1),X_2)) ) ).
tff(fact_74_sup1I1,axiom,
! [B_1: fun_Ho1877127206a_bool,A: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(A,X_2))
=> hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(A,B_1),X_2)) ) ).
tff(fact_75_sup1I1,axiom,
! [B_1: fun_pname_bool,A: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(A,X_2))
=> hBOOL(hAPP_pname_bool(semila278973382e_bool(A,B_1),X_2)) ) ).
tff(fact_76_ball__Un,axiom,
! [Pa: fun_Ho1877127206a_bool,A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] :
( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),semila1525949746a_bool(A,B_1)))
=> hBOOL(hAPP_H1448631928a_bool(Pa,X)) )
<=> ( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A))
=> hBOOL(hAPP_H1448631928a_bool(Pa,X)) )
& ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),B_1))
=> hBOOL(hAPP_H1448631928a_bool(Pa,X)) ) ) ) ).
tff(fact_77_ball__Un,axiom,
! [Pa: fun_pname_bool,A: fun_pname_bool,B_1: fun_pname_bool] :
( ! [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),semila278973382e_bool(A,B_1)))
=> hBOOL(hAPP_pname_bool(Pa,X)) )
<=> ( ! [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
=> hBOOL(hAPP_pname_bool(Pa,X)) )
& ! [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_1))
=> hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).
tff(fact_78_bex__Un,axiom,
! [Pa: fun_Ho1877127206a_bool,A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] :
( ? [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),semila1525949746a_bool(A,B_1)))
& hBOOL(hAPP_H1448631928a_bool(Pa,X)) )
<=> ( ? [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A))
& hBOOL(hAPP_H1448631928a_bool(Pa,X)) )
| ? [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),B_1))
& hBOOL(hAPP_H1448631928a_bool(Pa,X)) ) ) ) ).
tff(fact_79_bex__Un,axiom,
! [Pa: fun_pname_bool,A: fun_pname_bool,B_1: fun_pname_bool] :
( ? [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),semila278973382e_bool(A,B_1)))
& hBOOL(hAPP_pname_bool(Pa,X)) )
<=> ( ? [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
& hBOOL(hAPP_pname_bool(Pa,X)) )
| ? [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),B_1))
& hBOOL(hAPP_pname_bool(Pa,X)) ) ) ) ).
tff(fact_80_Un__assoc,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool,C_2: fun_Ho1877127206a_bool] : semila1525949746a_bool(semila1525949746a_bool(A,B_1),C_2) = semila1525949746a_bool(A,semila1525949746a_bool(B_1,C_2)) ).
tff(fact_81_Un__assoc,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool,C_2: fun_pname_bool] : semila278973382e_bool(semila278973382e_bool(A,B_1),C_2) = semila278973382e_bool(A,semila278973382e_bool(B_1,C_2)) ).
tff(fact_82_Un__iff,axiom,
! [C: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),semila1525949746a_bool(A,B_1)))
<=> ( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),A))
| hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),B_1)) ) ) ).
tff(fact_83_Un__iff,axiom,
! [C: pname,A: fun_pname_bool,B_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),semila278973382e_bool(A,B_1)))
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),A))
| hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),B_1)) ) ) ).
tff(fact_84_Un__left__commute,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool,C_2: fun_Ho1877127206a_bool] : semila1525949746a_bool(A,semila1525949746a_bool(B_1,C_2)) = semila1525949746a_bool(B_1,semila1525949746a_bool(A,C_2)) ).
tff(fact_85_Un__left__commute,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool,C_2: fun_pname_bool] : semila278973382e_bool(A,semila278973382e_bool(B_1,C_2)) = semila278973382e_bool(B_1,semila278973382e_bool(A,C_2)) ).
tff(fact_86_Un__left__absorb,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(A,semila1525949746a_bool(A,B_1)) = semila1525949746a_bool(A,B_1) ).
tff(fact_87_Un__left__absorb,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool] : semila278973382e_bool(A,semila278973382e_bool(A,B_1)) = semila278973382e_bool(A,B_1) ).
tff(fact_88_Un__commute,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(A,B_1) = semila1525949746a_bool(B_1,A) ).
tff(fact_89_Un__commute,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool] : semila278973382e_bool(A,B_1) = semila278973382e_bool(B_1,A) ).
tff(fact_90_Un__def,axiom,
! [A: fun_Ho1877127206a_bool,B_1: fun_Ho1877127206a_bool] : semila1525949746a_bool(A,B_1) = collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fdisj,cOMBC_862840740l_bool(member127332739iple_a,A)),cOMBC_862840740l_bool(member127332739iple_a,B_1))) ).
tff(fact_91_Un__def,axiom,
! [A: fun_pname_bool,B_1: fun_pname_bool] : semila278973382e_bool(A,B_1) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,cOMBC_1058051404l_bool(member_pname,A)),cOMBC_1058051404l_bool(member_pname,B_1))) ).
tff(fact_92_Un__absorb,axiom,
! [A: fun_Ho1877127206a_bool] : semila1525949746a_bool(A,A) = A ).
tff(fact_93_Un__absorb,axiom,
! [A: fun_pname_bool] : semila278973382e_bool(A,A) = A ).
tff(fact_94_image__image,axiom,
! [F: fun_pn708290217iple_a,G: fun_Ho842746065_pname,A: fun_Ho1877127206a_bool] : image_68284913iple_a(F,image_1389863321_pname(G,A)) = image_590713477iple_a(cOMBB_1110279240iple_a(F,G),A) ).
tff(fact_95_image__image,axiom,
! [F: fun_Ho842746065_pname,G: fun_pn708290217iple_a,A: fun_pname_bool] : image_1389863321_pname(F,image_68284913iple_a(G,A)) = image_pname_pname(cOMBB_1433562676_pname(F,G),A) ).
tff(fact_96_sup__Un__eq,axiom,
! [R_1: fun_Ho1877127206a_bool,S_1: fun_Ho1877127206a_bool,X: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(semila1525949746a_bool(cOMBC_862840740l_bool(member127332739iple_a,R_1),cOMBC_862840740l_bool(member127332739iple_a,S_1)),X))
<=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),semila1525949746a_bool(R_1,S_1))) ) ).
tff(fact_97_sup__Un__eq,axiom,
! [R_1: fun_pname_bool,S_1: fun_pname_bool,X: pname] :
( hBOOL(hAPP_pname_bool(semila278973382e_bool(cOMBC_1058051404l_bool(member_pname,R_1),cOMBC_1058051404l_bool(member_pname,S_1)),X))
<=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),semila278973382e_bool(R_1,S_1))) ) ).
tff(fact_98_Collect__disj__eq,axiom,
! [Pa: fun_Ho1877127206a_bool,Qa: fun_Ho1877127206a_bool] : collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fdisj,Pa),Qa)) = semila1525949746a_bool(collec829051333iple_a(Pa),collec829051333iple_a(Qa)) ).
tff(fact_99_Collect__disj__eq,axiom,
! [Pa: fun_pname_bool,Qa: fun_pname_bool] : collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Qa)) = semila278973382e_bool(collect_pname(Pa),collect_pname(Qa)) ).
tff(fact_100_imageE,axiom,
! [B: pname,F: fun_Ho842746065_pname,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),image_1389863321_pname(F,A)))
=> ~ ! [X: hoare_1927711152iple_a] :
( ( B = hAPP_H2145880809_pname(F,X) )
=> ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A)) ) ) ).
tff(fact_101_imageE,axiom,
! [B: hoare_1927711152iple_a,F: fun_pn708290217iple_a,A: fun_pname_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,B),image_68284913iple_a(F,A)))
=> ~ ! [X: pname] :
( ( B = hAPP_p824302401iple_a(F,X) )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A)) ) ) ).
tff(fact_102_Body__triple__valid__Suc,axiom,
! [N_2: nat,Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] :
( hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(N_2),hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Pa),hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn_1))),Qa)))
<=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(suc(N_2)),hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ) ).
tff(fact_103_triple_Oexhaust,axiom,
! [Y_1: hoare_1927711152iple_a] :
~ ! [Fun1: fun_a_fun_state_bool,Com: com,Fun2: fun_a_fun_state_bool] : Y_1 != hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Fun1),Com),Fun2) ).
tff(fact_104_Body1,axiom,
! [Pn_1: pname,Ga: fun_Ho1877127206a_bool,Pa: fun_pn1683930517e_bool,Qa: fun_pn1683930517e_bool,Procsa: fun_pname_bool] :
( hBOOL(hoare_1617968510rivs_a(semila1525949746a_bool(Ga,image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,Pa),body),Qa),Procsa)),image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,Pa),cOMBB_923936821_pname(the_com,body_1)),Qa),Procsa)))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn_1),Procsa))
=> hBOOL(hoare_1617968510rivs_a(Ga,insert1434104874iple_a(hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,hAPP_p635540397e_bool(Pa,Pn_1)),hAPP_pname_com(body,Pn_1)),hAPP_p635540397e_bool(Qa,Pn_1)),bot_bo1208640912a_bool))) ) ) ).
tff(fact_105_image__cong,axiom,
! [F: fun_Ho842746065_pname,G: fun_Ho842746065_pname,M: fun_Ho1877127206a_bool,N_1: fun_Ho1877127206a_bool] :
( ( M = N_1 )
=> ( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),N_1))
=> ( hAPP_H2145880809_pname(F,X) = hAPP_H2145880809_pname(G,X) ) )
=> ( image_1389863321_pname(F,M) = image_1389863321_pname(G,N_1) ) ) ) ).
tff(fact_106_image__cong,axiom,
! [F: fun_pn708290217iple_a,G: fun_pn708290217iple_a,M: fun_pname_bool,N_1: fun_pname_bool] :
( ( M = N_1 )
=> ( ! [X: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),N_1))
=> ( hAPP_p824302401iple_a(F,X) = hAPP_p824302401iple_a(G,X) ) )
=> ( image_68284913iple_a(F,M) = image_68284913iple_a(G,N_1) ) ) ) ).
tff(fact_107_Body__triple__valid__0,axiom,
! [Pa: fun_a_fun_state_bool,Pn_1: pname,Qa: fun_a_fun_state_bool] : hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(zero_zero_nat),hAPP_f711275241iple_a(hAPP_c429049308iple_a(hAPP_f185596029iple_a(hoare_1652181356iple_a,Pa),hAPP_pname_com(body,Pn_1)),Qa))) ).
tff(fact_108_com_Osimps_I6_J,axiom,
! [Pname_1: pname,Pname: pname] :
( ( hAPP_pname_com(body,Pname_1) = hAPP_pname_com(body,Pname) )
<=> ( Pname_1 = Pname ) ) ).
tff(fact_109_evalc_OBody,axiom,
! [Pn: pname,S0: state,S1: state] :
( hBOOL(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,Pn)),S0,S1))
=> hBOOL(evalc(hAPP_pname_com(body,Pn),S0,S1)) ) ).
tff(fact_110_emptyE,axiom,
! [A_1: hoare_1927711152iple_a] : ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),bot_bo1208640912a_bool)) ).
tff(fact_111_emptyE,axiom,
! [A_1: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),bot_bo844097828e_bool)) ).
tff(fact_112_insertE,axiom,
! [A_1: hoare_1927711152iple_a,B: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),insert1434104874iple_a(B,A)))
=> ( ( A_1 != B )
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),A)) ) ) ).
tff(fact_113_insertE,axiom,
! [A_1: pname,B: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B,A)))
=> ( ( A_1 != B )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).
tff(fact_114_insertCI,axiom,
! [B: hoare_1927711152iple_a,A_1: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] :
( ( ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),B_1))
=> ( A_1 = B ) )
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),insert1434104874iple_a(B,B_1))) ) ).
tff(fact_115_insertCI,axiom,
! [B: pname,A_1: pname,B_1: fun_pname_bool] :
( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B_1))
=> ( A_1 = B ) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B,B_1))) ) ).
tff(fact_116_evalc__elim__cases_I6_J,axiom,
! [P: pname,S: state,S1: state] :
( hBOOL(evalc(hAPP_pname_com(body,P),S,S1))
=> hBOOL(evalc(hAPP_option_com_com(the_com,hAPP_p799580910on_com(body_1,P)),S,S1)) ) ).
tff(fact_117_empty__not__insert,axiom,
! [A_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] : bot_bo1208640912a_bool != insert1434104874iple_a(A_1,A) ).
tff(fact_118_empty__not__insert,axiom,
! [A_1: pname,A: fun_pname_bool] : bot_bo844097828e_bool != insert_pname(A_1,A) ).
tff(fact_119_insert__not__empty,axiom,
! [A_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] : insert1434104874iple_a(A_1,A) != bot_bo1208640912a_bool ).
tff(fact_120_insert__not__empty,axiom,
! [A_1: pname,A: fun_pname_bool] : insert_pname(A_1,A) != bot_bo844097828e_bool ).
tff(fact_121_bot__empty__eq,axiom,
! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(bot_bo1208640912a_bool,X))
<=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),bot_bo1208640912a_bool)) ) ).
tff(fact_122_bot__empty__eq,axiom,
! [X: pname] :
( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X))
<=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),bot_bo844097828e_bool)) ) ).
tff(fact_123_empty__def,axiom,
bot_bo844097828e_bool = collect_pname(cOMBK_bool_pname(fFalse)) ).
tff(fact_124_empty__def,axiom,
bot_bo1208640912a_bool = collec829051333iple_a(cOMBK_712844119iple_a(fFalse)) ).
tff(fact_125_insertI1,axiom,
! [A_1: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] : hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),insert1434104874iple_a(A_1,B_1))) ).
tff(fact_126_insertI1,axiom,
! [A_1: pname,B_1: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(A_1,B_1))) ).
tff(fact_127_all__not__in__conv,axiom,
! [A: fun_Ho1877127206a_bool] :
( ! [X: hoare_1927711152iple_a] : ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A))
<=> ( A = bot_bo1208640912a_bool ) ) ).
tff(fact_128_all__not__in__conv,axiom,
! [A: fun_pname_bool] :
( ! [X: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
<=> ( A = bot_bo844097828e_bool ) ) ).
tff(fact_129_singleton__conv2,axiom,
! [A_1: hoare_1927711152iple_a] : collec829051333iple_a(hAPP_H1027145665a_bool(fequal1440857775iple_a,A_1)) = insert1434104874iple_a(A_1,bot_bo1208640912a_bool) ).
tff(fact_130_singleton__conv2,axiom,
! [A_1: pname] : collect_pname(hAPP_p61793385e_bool(fequal_pname,A_1)) = insert_pname(A_1,bot_bo844097828e_bool) ).
tff(fact_131_ex__in__conv,axiom,
! [A: fun_Ho1877127206a_bool] :
( ? [X: hoare_1927711152iple_a] : hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),A))
<=> ( A != bot_bo1208640912a_bool ) ) ).
tff(fact_132_ex__in__conv,axiom,
! [A: fun_pname_bool] :
( ? [X: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X),A))
<=> ( A != bot_bo844097828e_bool ) ) ).
tff(fact_133_singleton__conv,axiom,
! [A_1: hoare_1927711152iple_a] : collec829051333iple_a(cOMBC_671859290a_bool(fequal1440857775iple_a,A_1)) = insert1434104874iple_a(A_1,bot_bo1208640912a_bool) ).
tff(fact_134_singleton__conv,axiom,
! [A_1: pname] : collect_pname(cOMBC_1149511130e_bool(fequal_pname,A_1)) = insert_pname(A_1,bot_bo844097828e_bool) ).
tff(fact_135_Collect__conv__if2,axiom,
! [Pa: fun_Ho1877127206a_bool,A_1: hoare_1927711152iple_a] :
( ( hBOOL(hAPP_H1448631928a_bool(Pa,A_1))
=> ( collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fconj,hAPP_H1027145665a_bool(fequal1440857775iple_a,A_1)),Pa)) = insert1434104874iple_a(A_1,bot_bo1208640912a_bool) ) )
& ( ~ hBOOL(hAPP_H1448631928a_bool(Pa,A_1))
=> ( collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fconj,hAPP_H1027145665a_bool(fequal1440857775iple_a,A_1)),Pa)) = bot_bo1208640912a_bool ) ) ) ).
tff(fact_136_Collect__conv__if2,axiom,
! [Pa: fun_pname_bool,A_1: pname] :
( ( hBOOL(hAPP_pname_bool(Pa,A_1))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(fequal_pname,A_1)),Pa)) = insert_pname(A_1,bot_bo844097828e_bool) ) )
& ( ~ hBOOL(hAPP_pname_bool(Pa,A_1))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(fequal_pname,A_1)),Pa)) = bot_bo844097828e_bool ) ) ) ).
tff(fact_137_Collect__conv__if,axiom,
! [Pa: fun_Ho1877127206a_bool,A_1: hoare_1927711152iple_a] :
( ( hBOOL(hAPP_H1448631928a_bool(Pa,A_1))
=> ( collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fconj,cOMBC_671859290a_bool(fequal1440857775iple_a,A_1)),Pa)) = insert1434104874iple_a(A_1,bot_bo1208640912a_bool) ) )
& ( ~ hBOOL(hAPP_H1448631928a_bool(Pa,A_1))
=> ( collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fconj,cOMBC_671859290a_bool(fequal1440857775iple_a,A_1)),Pa)) = bot_bo1208640912a_bool ) ) ) ).
tff(fact_138_Collect__conv__if,axiom,
! [Pa: fun_pname_bool,A_1: pname] :
( ( hBOOL(hAPP_pname_bool(Pa,A_1))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,cOMBC_1149511130e_bool(fequal_pname,A_1)),Pa)) = insert_pname(A_1,bot_bo844097828e_bool) ) )
& ( ~ hBOOL(hAPP_pname_bool(Pa,A_1))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,cOMBC_1149511130e_bool(fequal_pname,A_1)),Pa)) = bot_bo844097828e_bool ) ) ) ).
tff(fact_139_empty__Collect__eq,axiom,
! [Pa: fun_pname_bool] :
( ( bot_bo844097828e_bool = collect_pname(Pa) )
<=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).
tff(fact_140_empty__Collect__eq,axiom,
! [Pa: fun_Ho1877127206a_bool] :
( ( bot_bo1208640912a_bool = collec829051333iple_a(Pa) )
<=> ! [X: hoare_1927711152iple_a] : ~ hBOOL(hAPP_H1448631928a_bool(Pa,X)) ) ).
tff(fact_141_empty__iff,axiom,
! [C: hoare_1927711152iple_a] : ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,C),bot_bo1208640912a_bool)) ).
tff(fact_142_empty__iff,axiom,
! [C: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C),bot_bo844097828e_bool)) ).
tff(fact_143_mem__def,axiom,
! [X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
<=> hBOOL(hAPP_H1448631928a_bool(A,X_2)) ) ).
tff(fact_144_mem__def,axiom,
! [X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
<=> hBOOL(hAPP_pname_bool(A,X_2)) ) ).
tff(fact_145_Collect__def,axiom,
! [Pa: fun_pname_bool] : collect_pname(Pa) = Pa ).
tff(fact_146_Collect__def,axiom,
! [Pa: fun_Ho1877127206a_bool] : collec829051333iple_a(Pa) = Pa ).
tff(fact_147_insert__compr,axiom,
! [A_1: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] : insert1434104874iple_a(A_1,B_1) = collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fdisj,cOMBC_671859290a_bool(fequal1440857775iple_a,A_1)),cOMBC_862840740l_bool(member127332739iple_a,B_1))) ).
tff(fact_148_insert__compr,axiom,
! [A_1: pname,B_1: fun_pname_bool] : insert_pname(A_1,B_1) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,cOMBC_1149511130e_bool(fequal_pname,A_1)),cOMBC_1058051404l_bool(member_pname,B_1))) ).
tff(fact_149_insert__is__Un,axiom,
! [A_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] : insert1434104874iple_a(A_1,A) = semila1525949746a_bool(insert1434104874iple_a(A_1,bot_bo1208640912a_bool),A) ).
tff(fact_150_insert__is__Un,axiom,
! [A_1: pname,A: fun_pname_bool] : insert_pname(A_1,A) = semila278973382e_bool(insert_pname(A_1,bot_bo844097828e_bool),A) ).
tff(fact_151_insert__Collect,axiom,
! [A_1: hoare_1927711152iple_a,Pa: fun_Ho1877127206a_bool] : insert1434104874iple_a(A_1,collec829051333iple_a(Pa)) = collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fimplies,cOMBB_213049548iple_a(fNot,cOMBC_671859290a_bool(fequal1440857775iple_a,A_1))),Pa)) ).
tff(fact_152_insert__Collect,axiom,
! [A_1: pname,Pa: fun_pname_bool] : insert_pname(A_1,collect_pname(Pa)) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fimplies,cOMBB_647938656_pname(fNot,cOMBC_1149511130e_bool(fequal_pname,A_1))),Pa)) ).
tff(fact_153_singleton__iff,axiom,
! [B: hoare_1927711152iple_a,A_1: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,B),insert1434104874iple_a(A_1,bot_bo1208640912a_bool)))
<=> ( B = A_1 ) ) ).
tff(fact_154_singleton__iff,axiom,
! [B: pname,A_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),insert_pname(A_1,bot_bo844097828e_bool)))
<=> ( B = A_1 ) ) ).
tff(fact_155_insert__absorb2,axiom,
! [X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] : insert1434104874iple_a(X_2,insert1434104874iple_a(X_2,A)) = insert1434104874iple_a(X_2,A) ).
tff(fact_156_insert__absorb2,axiom,
! [X_2: pname,A: fun_pname_bool] : insert_pname(X_2,insert_pname(X_2,A)) = insert_pname(X_2,A) ).
tff(fact_157_insert__commute,axiom,
! [X_2: hoare_1927711152iple_a,Y_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] : insert1434104874iple_a(X_2,insert1434104874iple_a(Y_1,A)) = insert1434104874iple_a(Y_1,insert1434104874iple_a(X_2,A)) ).
tff(fact_158_insert__commute,axiom,
! [X_2: pname,Y_1: pname,A: fun_pname_bool] : insert_pname(X_2,insert_pname(Y_1,A)) = insert_pname(Y_1,insert_pname(X_2,A)) ).
tff(fact_159_insert__iff,axiom,
! [A_1: hoare_1927711152iple_a,B: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),insert1434104874iple_a(B,A)))
<=> ( ( A_1 = B )
| hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),A)) ) ) ).
tff(fact_160_insert__iff,axiom,
! [A_1: pname,B: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B,A)))
<=> ( ( A_1 = B )
| hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ) ).
tff(fact_161_Collect__empty__eq,axiom,
! [Pa: fun_pname_bool] :
( ( collect_pname(Pa) = bot_bo844097828e_bool )
<=> ! [X: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X)) ) ).
tff(fact_162_Collect__empty__eq,axiom,
! [Pa: fun_Ho1877127206a_bool] :
( ( collec829051333iple_a(Pa) = bot_bo1208640912a_bool )
<=> ! [X: hoare_1927711152iple_a] : ~ hBOOL(hAPP_H1448631928a_bool(Pa,X)) ) ).
tff(fact_163_doubleton__eq__iff,axiom,
! [A_1: hoare_1927711152iple_a,B: hoare_1927711152iple_a,C: hoare_1927711152iple_a,D: hoare_1927711152iple_a] :
( ( insert1434104874iple_a(A_1,insert1434104874iple_a(B,bot_bo1208640912a_bool)) = insert1434104874iple_a(C,insert1434104874iple_a(D,bot_bo1208640912a_bool)) )
<=> ( ( ( A_1 = C )
& ( B = D ) )
| ( ( A_1 = D )
& ( B = C ) ) ) ) ).
tff(fact_164_doubleton__eq__iff,axiom,
! [A_1: pname,B: pname,C: pname,D: pname] :
( ( insert_pname(A_1,insert_pname(B,bot_bo844097828e_bool)) = insert_pname(C,insert_pname(D,bot_bo844097828e_bool)) )
<=> ( ( ( A_1 = C )
& ( B = D ) )
| ( ( A_1 = D )
& ( B = C ) ) ) ) ).
tff(fact_165_insert__code,axiom,
! [Y_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a] :
( hBOOL(hAPP_H1448631928a_bool(insert1434104874iple_a(Y_1,A),X_2))
<=> ( ( Y_1 = X_2 )
| hBOOL(hAPP_H1448631928a_bool(A,X_2)) ) ) ).
tff(fact_166_insert__code,axiom,
! [Y_1: pname,A: fun_pname_bool,X_2: pname] :
( hBOOL(hAPP_pname_bool(insert_pname(Y_1,A),X_2))
<=> ( ( Y_1 = X_2 )
| hBOOL(hAPP_pname_bool(A,X_2)) ) ) ).
tff(fact_167_insert__compr__raw,axiom,
! [X: hoare_1927711152iple_a,Xa: fun_Ho1877127206a_bool] : insert1434104874iple_a(X,Xa) = collec829051333iple_a(cOMBS_2061548107l_bool(cOMBB_196465322iple_a(fdisj,cOMBC_671859290a_bool(fequal1440857775iple_a,X)),cOMBC_862840740l_bool(member127332739iple_a,Xa))) ).
tff(fact_168_insert__compr__raw,axiom,
! [X: pname,Xa: fun_pname_bool] : insert_pname(X,Xa) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,cOMBC_1149511130e_bool(fequal_pname,X)),cOMBC_1058051404l_bool(member_pname,Xa))) ).
tff(fact_169_insert__ident,axiom,
! [B_1: fun_Ho1877127206a_bool,X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> ( ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),B_1))
=> ( ( insert1434104874iple_a(X_2,A) = insert1434104874iple_a(X_2,B_1) )
<=> ( A = B_1 ) ) ) ) ).
tff(fact_170_insert__ident,axiom,
! [B_1: fun_pname_bool,X_2: pname,A: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B_1))
=> ( ( insert_pname(X_2,A) = insert_pname(X_2,B_1) )
<=> ( A = B_1 ) ) ) ) ).
tff(fact_171_equals0D,axiom,
! [A_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( ( A = bot_bo1208640912a_bool )
=> ~ hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),A)) ) ).
tff(fact_172_equals0D,axiom,
! [A_1: pname,A: fun_pname_bool] :
( ( A = bot_bo844097828e_bool )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) ) ).
tff(fact_173_insertI2,axiom,
! [B: hoare_1927711152iple_a,A_1: hoare_1927711152iple_a,B_1: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),B_1))
=> hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),insert1434104874iple_a(B,B_1))) ) ).
tff(fact_174_insertI2,axiom,
! [B: pname,A_1: pname,B_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B,B_1))) ) ).
tff(fact_175_insert__absorb,axiom,
! [A_1: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,A_1),A))
=> ( insert1434104874iple_a(A_1,A) = A ) ) ).
tff(fact_176_insert__absorb,axiom,
! [A_1: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A))
=> ( insert_pname(A_1,A) = A ) ) ).
tff(fact_177_hoare__derivs_Oinsert,axiom,
! [Ts: fun_Ho1877127206a_bool,Ga: fun_Ho1877127206a_bool,T_1: hoare_1927711152iple_a] :
( hBOOL(hoare_1617968510rivs_a(Ga,insert1434104874iple_a(T_1,bot_bo1208640912a_bool)))
=> ( hBOOL(hoare_1617968510rivs_a(Ga,Ts))
=> hBOOL(hoare_1617968510rivs_a(Ga,insert1434104874iple_a(T_1,Ts))) ) ) ).
tff(fact_178_singletonE,axiom,
! [B: hoare_1927711152iple_a,A_1: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,B),insert1434104874iple_a(A_1,bot_bo1208640912a_bool)))
=> ( B = A_1 ) ) ).
tff(fact_179_singletonE,axiom,
! [B: pname,A_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B),insert_pname(A_1,bot_bo844097828e_bool)))
=> ( B = A_1 ) ) ).
tff(fact_180_derivs__insertD,axiom,
! [Ga: fun_Ho1877127206a_bool,T_1: hoare_1927711152iple_a,Ts: fun_Ho1877127206a_bool] :
( hBOOL(hoare_1617968510rivs_a(Ga,insert1434104874iple_a(T_1,Ts)))
=> ( hBOOL(hoare_1617968510rivs_a(Ga,insert1434104874iple_a(T_1,bot_bo1208640912a_bool)))
& hBOOL(hoare_1617968510rivs_a(Ga,Ts)) ) ) ).
tff(fact_181_singleton__inject,axiom,
! [A_1: hoare_1927711152iple_a,B: hoare_1927711152iple_a] :
( ( insert1434104874iple_a(A_1,bot_bo1208640912a_bool) = insert1434104874iple_a(B,bot_bo1208640912a_bool) )
=> ( A_1 = B ) ) ).
tff(fact_182_singleton__inject,axiom,
! [A_1: pname,B: pname] :
( ( insert_pname(A_1,bot_bo844097828e_bool) = insert_pname(B,bot_bo844097828e_bool) )
=> ( A_1 = B ) ) ).
tff(fact_183_com__det,axiom,
! [U: state,C_1: com,S: state,T: state] :
( hBOOL(evalc(C_1,S,T))
=> ( hBOOL(evalc(C_1,S,U))
=> ( U = T ) ) ) ).
tff(fact_184_image__constant__conv,axiom,
! [C: pname,A: fun_Ho1877127206a_bool] :
( ( ( A = bot_bo1208640912a_bool )
=> ( image_1389863321_pname(cOMBK_1495131898iple_a(C),A) = bot_bo844097828e_bool ) )
& ( ( A != bot_bo1208640912a_bool )
=> ( image_1389863321_pname(cOMBK_1495131898iple_a(C),A) = insert_pname(C,bot_bo844097828e_bool) ) ) ) ).
tff(fact_185_image__constant__conv,axiom,
! [C: hoare_1927711152iple_a,A: fun_pname_bool] :
( ( ( A = bot_bo844097828e_bool )
=> ( image_68284913iple_a(cOMBK_669226658_pname(C),A) = bot_bo1208640912a_bool ) )
& ( ( A != bot_bo844097828e_bool )
=> ( image_68284913iple_a(cOMBK_669226658_pname(C),A) = insert1434104874iple_a(C,bot_bo1208640912a_bool) ) ) ) ).
tff(fact_186_image__constant,axiom,
! [C: pname,X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> ( image_1389863321_pname(cOMBK_1495131898iple_a(C),A) = insert_pname(C,bot_bo844097828e_bool) ) ) ).
tff(fact_187_image__constant,axiom,
! [C: hoare_1927711152iple_a,X_2: hoare_1927711152iple_a,A: fun_Ho1877127206a_bool] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X_2),A))
=> ( image_590713477iple_a(cOMBK_2109678094iple_a(C),A) = insert1434104874iple_a(C,bot_bo1208640912a_bool) ) ) ).
tff(fact_188_image__constant,axiom,
! [C: pname,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( image_pname_pname(cOMBK_pname_pname(C),A) = insert_pname(C,bot_bo844097828e_bool) ) ) ).
tff(fact_189_image__constant,axiom,
! [C: hoare_1927711152iple_a,X_2: pname,A: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A))
=> ( image_68284913iple_a(cOMBK_669226658_pname(C),A) = insert1434104874iple_a(C,bot_bo1208640912a_bool) ) ) ).
%----Helper facts (39)
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ hBOOL(hAPP_bool_bool(fNot,P))
| ~ hBOOL(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fNot,P)) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
tff(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X_1),Y))
| ( X_1 = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ( X_1 != Y )
| hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X_1),Y)) ) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
! [P: bool,Q: pname] : hAPP_pname_bool(cOMBK_bool_pname(P),Q) = P ).
tff(help_COMBK_1_1_COMBK_000tc__Com__Opname_000tc__Com__Opname_U,axiom,
! [P: pname,Q: pname] : hAPP_pname_pname(cOMBK_pname_pname(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U,axiom,
! [P: fun_bool_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(cOMBB_647938656_pname(P,Q),R) = hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_pn250273176l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(cOMBS_568398431l_bool(P,Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
! [P: fun_pn800050071e_bool,Q: pname,R: pname] : hAPP_pname_bool(cOMBC_1149511130e_bool(P,Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ).
tff(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_,axiom,
! [X_1: hoare_1927711152iple_a,Y: hoare_1927711152iple_a] :
( ~ hBOOL(hAPP_H1448631928a_bool(hAPP_H1027145665a_bool(fequal1440857775iple_a,X_1),Y))
| ( X_1 = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_,axiom,
! [X_1: hoare_1927711152iple_a,Y: hoare_1927711152iple_a] :
( ( X_1 != Y )
| hBOOL(hAPP_H1448631928a_bool(hAPP_H1027145665a_bool(fequal1440857775iple_a,X_1),Y)) ) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____yiemogtkbg__,axiom,
! [P: bool,Q: hoare_1927711152iple_a] : hAPP_H1448631928a_bool(cOMBK_712844119iple_a(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__Option__Ooption_Itc__Com__Ocom_J_000tc__Com__Ocom_00,axiom,
! [P: fun_option_com_com,Q: fun_pname_option_com,R: pname] : hAPP_pname_com(cOMBB_923936821_pname(P,Q),R) = hAPP_option_com_com(P,hAPP_p799580910on_com(Q,R)) ).
tff(help_COMBK_1_1_COMBK_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg_,axiom,
! [P: pname,Q: hoare_1927711152iple_a] : hAPP_H2145880809_pname(cOMBK_1495131898iple_a(P),Q) = P ).
tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00,axiom,
! [P: hoare_1927711152iple_a,Q: pname] : hAPP_p824302401iple_a(cOMBK_669226658_pname(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_pname_bool,R: pname] : hAPP_p393069232l_bool(cOMBB_675860798_pname(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
! [P: fun_pn422929397l_bool,Q: fun_pname_bool,R: pname] : hAPP_pname_bool(cOMBC_1058051404l_bool(P,Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
! [P: fun_bool_bool,Q: fun_Ho1877127206a_bool,R: hoare_1927711152iple_a] : hAPP_H1448631928a_bool(cOMBB_213049548iple_a(P,Q),R) = hAPP_bool_bool(P,hAPP_H1448631928a_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00,axiom,
! [P: fun_Ho957066028l_bool,Q: fun_Ho1877127206a_bool,R: hoare_1927711152iple_a] : hAPP_H1448631928a_bool(cOMBS_2061548107l_bool(P,Q),R) = hAPP_bool_bool(hAPP_H1487873860l_bool(P,R),hAPP_H1448631928a_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00,axiom,
! [P: fun_Ho842746065_pname,Q: fun_pn708290217iple_a,R: pname] : hAPP_pname_pname(cOMBB_1433562676_pname(P,Q),R) = hAPP_H2145880809_pname(P,hAPP_p824302401iple_a(Q,R)) ).
tff(help_COMBK_1_1_COMBK_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00_009,axiom,
! [P: hoare_1927711152iple_a,Q: hoare_1927711152iple_a] : hAPP_H963118037iple_a(cOMBK_2109678094iple_a(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_010,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_Ho1877127206a_bool,R: hoare_1927711152iple_a] : hAPP_H1487873860l_bool(cOMBB_196465322iple_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_H1448631928a_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00,axiom,
! [P: fun_Ho440810351a_bool,Q: hoare_1927711152iple_a,R: hoare_1927711152iple_a] : hAPP_H1448631928a_bool(cOMBC_671859290a_bool(P,Q),R) = hAPP_H1448631928a_bool(hAPP_H1027145665a_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__Hoare____Mirabelle____yiemogtkbg_,axiom,
! [P: fun_pn708290217iple_a,Q: fun_Ho842746065_pname,R: hoare_1927711152iple_a] : hAPP_H963118037iple_a(cOMBB_1110279240iple_a(P,Q),R) = hAPP_p824302401iple_a(P,hAPP_H2145880809_pname(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__fun_It__a_Mtc__fun_Itc__Com__Osta,axiom,
! [P: fun_pn579076298iple_a,Q: fun_pn1683930517e_bool,R: pname] : hAPP_p824302401iple_a(cOMBS_821474699iple_a(P,Q),R) = hAPP_f711275241iple_a(hAPP_p1513881570iple_a(P,R),hAPP_p635540397e_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____yiemogtkbg__Otriple_It__a_J_00_011,axiom,
! [P: fun_Ho525994229l_bool,Q: fun_Ho1877127206a_bool,R: hoare_1927711152iple_a] : hAPP_H1448631928a_bool(cOMBC_862840740l_bool(P,Q),R) = hAPP_f1454306822l_bool(hAPP_H694056973l_bool(P,R),Q) ).
tff(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__Com__Ocom_000tc__fun_Itc__fun_It_,axiom,
! [P: fun_pn308211645iple_a,Q: fun_pname_com,R: pname] : hAPP_p1513881570iple_a(cOMBS_1125763966iple_a(P,Q),R) = hAPP_c429049308iple_a(hAPP_p1788720341iple_a(P,R),hAPP_pname_com(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_,axiom,
! [P: fun_fu90068325iple_a,Q: fun_pn1683930517e_bool,R: pname] : hAPP_p1788720341iple_a(cOMBB_1515136928_pname(P,Q),R) = hAPP_f185596029iple_a(P,hAPP_p635540397e_bool(Q,R)) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
! [N: nat] :
( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),semila1525949746a_bool(g,image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,p),body),q),procs))))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(N),X)) )
=> ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,p),cOMBB_923936821_pname(the_com,body_1)),q),procs)))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(N),X)) ) ) ).
tff(conj_1,conjecture,
( ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),g))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(n),X)) )
=> ! [X: hoare_1927711152iple_a] :
( hBOOL(hAPP_f1454306822l_bool(hAPP_H694056973l_bool(member127332739iple_a,X),image_68284913iple_a(cOMBS_821474699iple_a(cOMBS_1125763966iple_a(cOMBB_1515136928_pname(hoare_1652181356iple_a,p),body),q),procs)))
=> hBOOL(hAPP_H1448631928a_bool(hoare_1572001082alid_a(n),X)) ) ) ).
%------------------------------------------------------------------------------