TPTP Problem File: SWW473_2.p
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%------------------------------------------------------------------------------
% File : SWW473_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 383, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_tff_l383 [Bla11]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.20 v8.1.0, 0.09 v7.5.0, 0.12 v7.4.0, 0.00 v6.0.0, 0.20 v5.5.0, 0.00 v5.4.0, 0.33 v5.3.0
% Syntax : Number of formulae : 1168 ( 238 unt; 354 typ; 0 def)
% Number of atoms : 1912 ( 506 equ)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 1307 ( 209 ~; 54 |; 130 &)
% ( 137 <=>; 777 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 87 ( 86 usr)
% Number of type conns : 363 ( 210 >; 153 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 267 ( 267 usr; 58 con; 0-2 aty)
% Number of variables : 1942 (1877 !; 65 ?;1942 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:41:28
%------------------------------------------------------------------------------
%----Should-be-implicit typings (86)
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tff(sy_c_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__f_018,type,
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tff(sy_c_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__f_019,type,
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tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL,type,
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tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL_020,type,
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tff(sy_c_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL_021,type,
cOMBB_1823939024ol_nat: ( fun_fu1471507361l_bool * fun_nat_fun_a_bool ) > fun_na1632405922l_bool ).
tff(sy_c_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc,type,
cOMBB_1137537805bool_a: ( fun_fu802393907l_bool * fun_a_fun_pname_bool ) > fun_a_968466022l_bool ).
tff(sy_c_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc_022,type,
cOMBB_408569982_pname: ( fun_fu802393907l_bool * fun_pn800050071e_bool ) > fun_pn422929397l_bool ).
tff(sy_c_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc_023,type,
cOMBB_2117322052ol_nat: ( fun_fu802393907l_bool * fun_na936072029e_bool ) > fun_na2122364079l_bool ).
tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__,type,
cOMBB_743407885bool_a: ( fun_fu1217155507l_bool * fun_a_fun_nat_bool ) > fun_a_762926252l_bool ).
tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc___024,type,
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tff(sy_c_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc___025,type,
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tff(sy_c_COMBC_000t__a_000t__a_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000t__a_000tc__Com__Opname_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000t__a_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obo,type,
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tff(sy_c_COMBC_000t__a_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Com__Opname_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obo,type,
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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_fu410713561e_bool ).
tff(sy_c_COMBC_000tc__Com__Opname_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc_,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000t__a_000tc__HOL__Obool,type,
cOMBC_nat_a_bool: fun_nat_fun_a_bool > fun_a_fun_nat_bool ).
tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Com__Opname_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc_,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__H,type,
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tff(sy_c_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname_000tc__HOL__Obo,type,
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tff(sy_c_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
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tff(sy_c_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL,type,
cOMBC_1880041174l_bool: fun_fu386216885l_bool > fun_fu911136611l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opname_000t,type,
cOMBC_1004116266e_bool: fun_fu410713561e_bool > fun_pn422929397l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Nat__Onat_000tc_,type,
cOMBC_1666426608t_bool: fun_fu900310687t_bool > fun_na2122364079l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Op,type,
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tff(sy_c_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc,type,
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tff(sy_c_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Com__Opname_000tc_,type,
cOMBC_615407716e_bool: fun_fu1492573599e_bool > fun_pn1165013435l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Onat_000tc__H,type,
cOMBC_385542954t_bool: fun_fu821463397t_bool > fun_na1436237685l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat,type,
cOMBC_1693257480l_bool: fun_fu1217155507l_bool > fun_fu1217155507l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__,type,
cOMBC_1245412066l_bool: fun_fu1436348701l_bool > fun_fu616551101l_bool ).
tff(sy_c_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__,type,
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tff(sy_c_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo,type,
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tff(sy_c_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_,type,
cOMBC_595898202l_bool: fun_fu140186515l_bool > fun_fu140186515l_bool ).
tff(sy_c_COMBK_000t__a_000tc__Com__Opname,type,
cOMBK_a_pname: x_a > fun_pname_a ).
tff(sy_c_COMBK_000tc__HOL__Obool_000t__a,type,
cOMBK_bool_a: bool > fun_a_bool ).
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__Nat__Onat,type,
cOMBK_bool_nat: bool > fun_nat_bool ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
cOMBK_324466864a_bool: bool > fun_fun_a_bool_bool ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
cOMBK_1857069011e_bool: bool > fun_fu1430349052l_bool ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
cOMBK_1994329625t_bool: bool > fun_fu425979586l_bool ).
tff(sy_c_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool,type,
cOMBS_a_bool_bool: ( fun_a_fun_bool_bool * fun_a_bool ) > fun_a_bool ).
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__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
cOMBS_nat_bool_bool: ( fun_na1469252690l_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__HOL__Oboo,type,
cOMBS_1035972772l_bool: ( fun_fu554186387l_bool * fun_fun_a_bool_bool ) > fun_fun_a_bool_bool ).
tff(sy_c_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc,type,
cOMBS_350070575l_bool: ( fun_fu31783638l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).
tff(sy_c_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__,type,
cOMBS_1187019125l_bool: ( fun_fu1016514960l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).
tff(sy_c_Com_Obody,type,
body: pname > option_com ).
tff(sy_c_Finite__Set_Ocard_000t__a,type,
finite_card_a: fun_fun_a_bool_nat ).
tff(sy_c_Finite__Set_Ocard_000tc__Com__Opname,type,
finite_card_pname: fun_fu1668467777ol_nat ).
tff(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: fun_fun_nat_bool_nat ).
tff(sy_c_Finite__Set_Ocard_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
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tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
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tff(sy_c_Finite__Set_Ocard_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
finite346522414t_bool: fun_fu411113733ol_nat ).
tff(sy_c_Finite__Set_Ofinite_000t__a,type,
finite_finite_a: fun_fun_a_bool_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
finite_finite_pname: fun_fu1430349052l_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: fun_fu425979586l_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
finite347923420a_bool: fun_fu885608257l_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
finite595471783e_bool: fun_fu399576434l_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
finite2012431853t_bool: fun_fu814369080l_bool ).
tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__O,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_,type,
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tff(sy_c_Finite__Set_Ofinite_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mt,type,
finite1701474069l_bool: fun_fu61768826l_bool ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > fun_nat_nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_Nat_OSuc,type,
suc: fun_nat_nat ).
tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
nat_case_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: option_com > com ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
bot_bot_bool: bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
bot_bot_fun_a_bool: fun_a_bool ).
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__Nat__Onat_Mtc__HOL__Obool_J,type,
bot_bot_fun_nat_bool: fun_nat_bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc_,type,
bot_bo1389914601l_bool: fun_fun_a_bool_bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__O,type,
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tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obo,type,
bot_bo1701429464l_bool: fun_fu425979586l_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__HOL__Obool,type,
ord_less_eq_bool: fun_bo1549164019l_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__H,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL,type,
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tff(sy_c_Set_OCollect_000t__a,type,
collect_a: fun_a_bool > fun_a_bool ).
tff(sy_c_Set_OCollect_000tc__Com__Opname,type,
collect_pname: fun_pname_bool > fun_pname_bool ).
tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: fun_nat_bool > fun_nat_bool ).
tff(sy_c_Set_OCollect_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
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tff(sy_c_Set_OCollect_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
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tff(sy_c_Set_OCollect_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
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tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HO,type,
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tff(sy_c_Set_OCollect_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL_,type,
collec1015864663l_bool: fun_fu814369080l_bool > fun_fu814369080l_bool ).
tff(sy_c_Set_Oimage_000t__a_000t__a,type,
image_a_a: ( fun_a_a * fun_a_bool ) > fun_a_bool ).
tff(sy_c_Set_Oimage_000t__a_000tc__Com__Opname,type,
image_a_pname: ( fun_a_pname * fun_a_bool ) > fun_pname_bool ).
tff(sy_c_Set_Oimage_000t__a_000tc__Nat__Onat,type,
image_a_nat: ( fun_a_nat * fun_a_bool ) > fun_nat_bool ).
tff(sy_c_Set_Oimage_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
image_a_fun_a_bool: ( fun_a_fun_a_bool * fun_a_bool ) > fun_fun_a_bool_bool ).
tff(sy_c_Set_Oimage_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
image_819518260e_bool: ( fun_a_fun_pname_bool * fun_a_bool ) > fun_fu1430349052l_bool ).
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tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_028,type,
hAPP_f55526627ol_nat: ( fun_fu1730389579ol_nat * fun_fu1430349052l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_029,type,
hAPP_f559147733l_bool: ( fun_fu2087345469l_bool * fun_fu1430349052l_bool ) > fun_fu1430349052l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Obool_030,type,
hAPP_f510955609l_bool: ( fun_fu1911931399l_bool * fun_fu1430349052l_bool ) > fun_fu399576434l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
hAPP_f1637334154l_bool: ( fun_fu814369080l_bool * fun_fu425979586l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_031,type,
hAPP_f696928925ol_nat: ( fun_fu411113733ol_nat * fun_fu425979586l_bool ) > nat ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_032,type,
hAPP_f1246832597l_bool: ( fun_fu616551101l_bool * fun_fu425979586l_bool ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_033,type,
hAPP_f1772781669l_bool: ( fun_fu140186515l_bool * fun_fu425979586l_bool ) > fun_fu814369080l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
hAPP_f292226953l_bool: ( fun_fu255076663l_bool * fun_fu885608257l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__H,type,
hAPP_f389811538l_bool: ( fun_fu1438281908l_bool * fun_fu399576434l_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL,type,
hAPP_f937997336l_bool: ( fun_fu61768826l_bool * fun_fu814369080l_bool ) > bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000t__a,type,
member_a: fun_a_1255737515l_bool ).
tff(sy_c_member_000tc__Com__Opname,type,
member_pname: fun_pn422929397l_bool ).
tff(sy_c_member_000tc__Nat__Onat,type,
member_nat: fun_na1436237685l_bool ).
tff(sy_c_member_000tc__fun_It__a_Mtc__HOL__Obool_J,type,
member_fun_a_bool: fun_fu386216885l_bool ).
tff(sy_c_member_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J,type,
member799430823e_bool: fun_fu931343505l_bool ).
tff(sy_c_member_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
member_fun_nat_bool: fun_fu1436348701l_bool ).
tff(sy_v_G,type,
g: fun_a_bool ).
tff(sy_v_P,type,
p: fun_a_bool > fun_fun_a_bool_bool ).
tff(sy_v_U,type,
u: fun_pname_bool ).
tff(sy_v_mgt,type,
mgt: com > x_a ).
tff(sy_v_mgt__call,type,
mgt_call: fun_pname_a ).
tff(sy_v_na,type,
na: nat ).
tff(sy_v_pn,type,
pn: pname ).
tff(sy_v_wt,type,
wt: com > bool ).
%----Relevant facts (700)
tff(fact_0_assms_I1_J,axiom,
! [Ts: fun_a_bool,G: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Ts),G))
=> hBOOL(hAPP_fun_a_bool_bool(p(G),Ts)) ) ).
tff(fact_1_finite__Collect__subsets,axiom,
! [A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,collec1015864663l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(ord_le1454342156l_bool),A_1)))) ) ).
tff(fact_2_finite__Collect__subsets,axiom,
! [A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,collec1613912337l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(ord_le675606854l_bool),A_1)))) ) ).
tff(fact_3_finite__Collect__subsets,axiom,
! [A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,collec707592106l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(ord_le1375614389l_bool),A_1)))) ) ).
tff(fact_4_finite__Collect__subsets,axiom,
! [A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(ord_le1311769555a_bool),A_1)))) ) ).
tff(fact_5_finite__Collect__subsets,axiom,
! [A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(ord_le313189616e_bool),A_1)))) ) ).
tff(fact_6_finite__Collect__subsets,axiom,
! [A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(ord_le1568362934t_bool),A_1)))) ) ).
tff(fact_7_finite__imageI,axiom,
! [H: fun_pn406123357t_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(H,F_1))) ) ).
tff(fact_8_finite__imageI,axiom,
! [H: fun_pn800050071e_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(H,F_1))) ) ).
tff(fact_9_finite__imageI,axiom,
! [H: fun_pname_fun_a_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(H,F_1))) ) ).
tff(fact_10_finite__imageI,axiom,
! [H: fun_nat_a,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(H,F_1))) ) ).
tff(fact_11_finite__imageI,axiom,
! [H: fun_nat_fun_nat_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(H,F_1))) ) ).
tff(fact_12_finite__imageI,axiom,
! [H: fun_na936072029e_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(H,F_1))) ) ).
tff(fact_13_finite__imageI,axiom,
! [H: fun_nat_fun_a_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(H,F_1))) ) ).
tff(fact_14_finite__imageI,axiom,
! [H: fun_a_pname,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(H,F_1))) ) ).
tff(fact_15_finite__imageI,axiom,
! [H: fun_fu665170229_pname,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(H,F_1))) ) ).
tff(fact_16_finite__imageI,axiom,
! [H: fun_fu1499449723_pname,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(H,F_1))) ) ).
tff(fact_17_finite__imageI,axiom,
! [H: fun_fun_a_bool_pname,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(H,F_1))) ) ).
tff(fact_18_finite__imageI,axiom,
! [H: fun_a_nat,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(H,F_1))) ) ).
tff(fact_19_finite__imageI,axiom,
! [H: fun_fun_nat_bool_nat,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(H,F_1))) ) ).
tff(fact_20_finite__imageI,axiom,
! [H: fun_fu1668467777ol_nat,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(H,F_1))) ) ).
tff(fact_21_finite__imageI,axiom,
! [H: fun_fun_a_bool_nat,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(H,F_1))) ) ).
tff(fact_22_finite__imageI,axiom,
! [H: fun_pname_a,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(H,F_1))) ) ).
tff(fact_23_finite_OinsertI,axiom,
! [A_3: fun_nat_bool,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_3,A_1))) ) ).
tff(fact_24_finite_OinsertI,axiom,
! [A_3: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_3,A_1))) ) ).
tff(fact_25_finite_OinsertI,axiom,
! [A_3: fun_a_bool,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_3,A_1))) ) ).
tff(fact_26_finite_OinsertI,axiom,
! [A_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_3,A_1))) ) ).
tff(fact_27_finite_OinsertI,axiom,
! [A_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_3,A_1))) ) ).
tff(fact_28_finite_OinsertI,axiom,
! [A_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_3,A_1))) ) ).
tff(fact_29_card__image__le,axiom,
! [F: fun_pname_pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_pname_pname(F,A_1))),hAPP_f921600141ol_nat(finite_card_pname,A_1))) ) ).
tff(fact_30_card__image__le,axiom,
! [F: fun_a_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_a_a(F,A_1))),hAPP_fun_a_bool_nat(finite_card_a,A_1))) ) ).
tff(fact_31_card__image__le,axiom,
! [F: fun_fun_nat_bool_a,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_nat_bool_a(F,A_1))),hAPP_f696928925ol_nat(finite346522414t_bool,A_1))) ) ).
tff(fact_32_card__image__le,axiom,
! [F: fun_fun_pname_bool_a,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_876012084bool_a(F,A_1))),hAPP_f55526627ol_nat(finite1340463720e_bool,A_1))) ) ).
tff(fact_33_card__image__le,axiom,
! [F: fun_fun_a_bool_a,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_a_bool_a(F,A_1))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1))) ) ).
tff(fact_34_card__image__le,axiom,
! [F: fun_pname_nat,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_pname_nat(F,A_1))),hAPP_f921600141ol_nat(finite_card_pname,A_1))) ) ).
tff(fact_35_card__image__le,axiom,
! [F: fun_a_nat,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_a_nat(F,A_1))),hAPP_fun_a_bool_nat(finite_card_a,A_1))) ) ).
tff(fact_36_card__image__le,axiom,
! [F: fun_fun_nat_bool_nat,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_496248727ol_nat(F,A_1))),hAPP_f696928925ol_nat(finite346522414t_bool,A_1))) ) ).
tff(fact_37_card__image__le,axiom,
! [F: fun_fu1668467777ol_nat,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1551609309ol_nat(F,A_1))),hAPP_f55526627ol_nat(finite1340463720e_bool,A_1))) ) ).
tff(fact_38_card__image__le,axiom,
! [F: fun_fun_a_bool_nat,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_fun_a_bool_nat(F,A_1))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1))) ) ).
tff(fact_39_card__image__le,axiom,
! [F: fun_a_pname,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_a_pname(F,A_1))),hAPP_fun_a_bool_nat(finite_card_a,A_1))) ) ).
tff(fact_40_card__image__le,axiom,
! [F: fun_nat_pname,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_nat_pname(F,A_1))),hAPP_f22106695ol_nat(finite_card_nat,A_1))) ) ).
tff(fact_41_card__image__le,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(F,A_1))),hAPP_f921600141ol_nat(finite_card_pname,A_1))) ) ).
tff(fact_42_card__mono,axiom,
! [A_1: fun_fu425979586l_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A_1)),hAPP_f696928925ol_nat(finite346522414t_bool,B_6))) ) ) ).
tff(fact_43_card__mono,axiom,
! [A_1: fun_fu1430349052l_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A_1)),hAPP_f55526627ol_nat(finite1340463720e_bool,B_6))) ) ) ).
tff(fact_44_card__mono,axiom,
! [A_1: fun_fun_a_bool_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1)),hAPP_f2009550088ol_nat(finite1306199131a_bool,B_6))) ) ) ).
tff(fact_45_card__mono,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A_1)),hAPP_f921600141ol_nat(finite_card_pname,B_6))) ) ) ).
tff(fact_46_card__mono,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A_1)),hAPP_fun_a_bool_nat(finite_card_a,B_6))) ) ) ).
tff(fact_47_card__mono,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A_1)),hAPP_f22106695ol_nat(finite_card_nat,B_6))) ) ) ).
tff(fact_48_card__seteq,axiom,
! [A_1: fun_fu425979586l_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,B_6)),hAPP_f696928925ol_nat(finite346522414t_bool,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_49_card__seteq,axiom,
! [A_1: fun_fu1430349052l_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,B_6)),hAPP_f55526627ol_nat(finite1340463720e_bool,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_50_card__seteq,axiom,
! [A_1: fun_fun_a_bool_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,B_6)),hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_51_card__seteq,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,B_6)),hAPP_f921600141ol_nat(finite_card_pname,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_52_card__seteq,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,B_6)),hAPP_fun_a_bool_nat(finite_card_a,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_53_card__seteq,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B_6)),hAPP_f22106695ol_nat(finite_card_nat,A_1)))
=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_54_card__insert__le,axiom,
! [X_3: fun_nat_bool,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A_1)),hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_3,A_1)))) ) ).
tff(fact_55_card__insert__le,axiom,
! [X_3: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A_1)),hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_3,A_1)))) ) ).
tff(fact_56_card__insert__le,axiom,
! [X_3: fun_a_bool,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1)),hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_3,A_1)))) ) ).
tff(fact_57_card__insert__le,axiom,
! [X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A_1)),hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_3,A_1)))) ) ).
tff(fact_58_card__insert__le,axiom,
! [X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A_1)),hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_3,A_1)))) ) ).
tff(fact_59_card__insert__le,axiom,
! [X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A_1)),hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_3,A_1)))) ) ).
tff(fact_60_card__insert__if,axiom,
! [X_3: fun_nat_bool,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_3),A_1))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_3,A_1)) = hAPP_f696928925ol_nat(finite346522414t_bool,A_1) ) )
& ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_3),A_1))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A_1)) ) ) ) ) ).
tff(fact_61_card__insert__if,axiom,
! [X_3: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_3),A_1))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_3,A_1)) = hAPP_f55526627ol_nat(finite1340463720e_bool,A_1) ) )
& ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_3),A_1))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A_1)) ) ) ) ) ).
tff(fact_62_card__insert__if,axiom,
! [X_3: fun_a_bool,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( ( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_3),A_1))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_3,A_1)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1) ) )
& ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_3),A_1))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1)) ) ) ) ) ).
tff(fact_63_card__insert__if,axiom,
! [X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_3,A_1)) = hAPP_f22106695ol_nat(finite_card_nat,A_1) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A_1)) ) ) ) ) ).
tff(fact_64_card__insert__if,axiom,
! [X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_3,A_1)) = hAPP_f921600141ol_nat(finite_card_pname,A_1) ) )
& ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A_1)) ) ) ) ) ).
tff(fact_65_card__insert__if,axiom,
! [X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_3,A_1)) = hAPP_fun_a_bool_nat(finite_card_a,A_1) ) )
& ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A_1)) ) ) ) ) ).
tff(fact_66_card__insert__disjoint,axiom,
! [X_3: fun_nat_bool,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_3),A_1))
=> ( hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A_1)) ) ) ) ).
tff(fact_67_card__insert__disjoint,axiom,
! [X_3: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_3),A_1))
=> ( hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A_1)) ) ) ) ).
tff(fact_68_card__insert__disjoint,axiom,
! [X_3: fun_a_bool,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_3),A_1))
=> ( hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A_1)) ) ) ) ).
tff(fact_69_card__insert__disjoint,axiom,
! [X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A_1)) ) ) ) ).
tff(fact_70_card__insert__disjoint,axiom,
! [X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A_1)) ) ) ) ).
tff(fact_71_card__insert__disjoint,axiom,
! [X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_3,A_1)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A_1)) ) ) ) ).
tff(fact_72_finite__Collect__conjI,axiom,
! [Q_1: fun_a_bool,Pa: fun_a_bool] :
( ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
| hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))) )
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,Pa),Q_1)))) ) ).
tff(fact_73_finite__Collect__conjI,axiom,
! [Q_1: fun_fu425979586l_bool,Pa: fun_fu425979586l_bool] :
( ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
| hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) )
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_74_finite__Collect__conjI,axiom,
! [Q_1: fun_fu1430349052l_bool,Pa: fun_fu1430349052l_bool] :
( ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
| hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) )
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_75_finite__Collect__conjI,axiom,
! [Q_1: fun_fun_a_bool_bool,Pa: fun_fun_a_bool_bool] :
( ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
| hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) )
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,Pa),Q_1)))) ) ).
tff(fact_76_finite__Collect__conjI,axiom,
! [Q_1: fun_pname_bool,Pa: fun_pname_bool] :
( ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
| hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) )
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,Pa),Q_1)))) ) ).
tff(fact_77_finite__Collect__conjI,axiom,
! [Q_1: fun_nat_bool,Pa: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
| hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),Q_1)))) ) ).
tff(fact_78_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(M),N)) ) ) ).
tff(fact_79_finite__Collect__le__nat,axiom,
! [K_2: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),K_2)))) ).
tff(fact_80_card__Collect__le__nat,axiom,
! [Na: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),Na))) = hAPP_nat_nat(suc,Na) ).
tff(fact_81_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_82_nat_Oinject,axiom,
! [Nat_3: nat,Nat_2: nat] :
( ( hAPP_nat_nat(suc,Nat_3) = hAPP_nat_nat(suc,Nat_2) )
<=> ( Nat_3 = Nat_2 ) ) ).
tff(fact_83_Suc__n__not__n,axiom,
! [N: nat] : hAPP_nat_nat(suc,N) != N ).
tff(fact_84_n__not__Suc__n,axiom,
! [N: nat] : N != hAPP_nat_nat(suc,N) ).
tff(fact_85_le__antisym,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
tff(fact_86_le__trans,axiom,
! [K: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).
tff(fact_87_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_88_nat__le__linear,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
tff(fact_89_le__refl,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
tff(fact_90_diff__commute,axiom,
! [I: nat,J_2: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J_2)),K) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),K)),J_2) ).
tff(fact_91_finite__Collect__disjI,axiom,
! [Pa: fun_a_bool,Q_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa)))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))) ) ) ).
tff(fact_92_finite__Collect__disjI,axiom,
! [Pa: fun_fu425979586l_bool,Q_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa)))
& hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) ) ) ).
tff(fact_93_finite__Collect__disjI,axiom,
! [Pa: fun_fu1430349052l_bool,Q_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa)))
& hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) ) ) ).
tff(fact_94_finite__Collect__disjI,axiom,
! [Pa: fun_fun_a_bool_bool,Q_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa)))
& hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) ) ) ).
tff(fact_95_finite__Collect__disjI,axiom,
! [Pa: fun_pname_bool,Q_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa)))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) ) ) ).
tff(fact_96_finite__Collect__disjI,axiom,
! [Pa: fun_nat_bool,Q_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,Pa),Q_1))))
<=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa)))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) ) ) ).
tff(fact_97_finite__insert,axiom,
! [A_3: fun_nat_bool,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_3,A_1)))
<=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1)) ) ).
tff(fact_98_finite__insert,axiom,
! [A_3: fun_pname_bool,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_3,A_1)))
<=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1)) ) ).
tff(fact_99_finite__insert,axiom,
! [A_3: fun_a_bool,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_3,A_1)))
<=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1)) ) ).
tff(fact_100_finite__insert,axiom,
! [A_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_3,A_1)))
<=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1)) ) ).
tff(fact_101_finite__insert,axiom,
! [A_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_3,A_1)))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) ) ).
tff(fact_102_finite__insert,axiom,
! [A_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_3,A_1)))
<=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1)) ) ).
tff(fact_103_finite__subset,axiom,
! [A_1: fun_fu425979586l_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1)) ) ) ).
tff(fact_104_finite__subset,axiom,
! [A_1: fun_fu1430349052l_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1)) ) ) ).
tff(fact_105_finite__subset,axiom,
! [A_1: fun_fun_a_bool_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A_1),B_6))
=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1)) ) ) ).
tff(fact_106_finite__subset,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1)) ) ) ).
tff(fact_107_finite__subset,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1)) ) ) ).
tff(fact_108_finite__subset,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) ) ) ).
tff(fact_109_rev__finite__subset,axiom,
! [A_1: fun_fu425979586l_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A_1),B_6))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1)) ) ) ).
tff(fact_110_rev__finite__subset,axiom,
! [A_1: fun_fu1430349052l_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A_1),B_6))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1)) ) ) ).
tff(fact_111_rev__finite__subset,axiom,
! [A_1: fun_fun_a_bool_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A_1),B_6))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1)) ) ) ).
tff(fact_112_rev__finite__subset,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1)) ) ) ).
tff(fact_113_rev__finite__subset,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1)) ) ) ).
tff(fact_114_rev__finite__subset,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) ) ) ).
tff(fact_115_Suc__leD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_116_le__SucE,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( M = hAPP_nat_nat(suc,N) ) ) ) ).
tff(fact_117_le__SucI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_118_Suc__le__mono,axiom,
! [Na: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),M_1)) ) ).
tff(fact_119_le__Suc__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
| ( M_1 = hAPP_nat_nat(suc,Na) ) ) ) ).
tff(fact_120_not__less__eq__eq,axiom,
! [M_1: nat,Na: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M_1)) ) ).
tff(fact_121_Suc__n__not__le__n,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).
tff(fact_122_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(suc,K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),N)),K) ).
tff(fact_123_diff__Suc__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(minus_minus_nat(M),N) ).
tff(fact_124_le__diff__iff,axiom,
! [Na: nat,K_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_2)),hAPP_nat_nat(minus_minus_nat(Na),K_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)) ) ) ) ).
tff(fact_125_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),K)),hAPP_nat_nat(minus_minus_nat(N),K)) = hAPP_nat_nat(minus_minus_nat(M),N) ) ) ) ).
tff(fact_126_eq__diff__iff,axiom,
! [Na: nat,K_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),Na))
=> ( ( hAPP_nat_nat(minus_minus_nat(M_1),K_2) = hAPP_nat_nat(minus_minus_nat(Na),K_2) )
<=> ( M_1 = Na ) ) ) ) ).
tff(fact_127_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> ( hAPP_nat_nat(minus_minus_nat(N),hAPP_nat_nat(minus_minus_nat(N),I)) = I ) ) ).
tff(fact_128_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),L)),hAPP_nat_nat(minus_minus_nat(N),L))) ) ).
tff(fact_129_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L),N)),hAPP_nat_nat(minus_minus_nat(L),M))) ) ).
tff(fact_130_diff__le__self,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),N)),M)) ).
tff(fact_131_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_a_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_a_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_132_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_fun_nat_bool_a,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_fun_nat_bool_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_133_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_fun_pname_bool_a,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_876012084bool_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_134_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_fun_a_bool_a,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_fun_a_bool_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_135_finite__surj,axiom,
! [B_6: fun_fu425979586l_bool,F: fun_pn406123357t_bool,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B_6),image_2129980159t_bool(F,A_1)))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6)) ) ) ).
tff(fact_136_finite__surj,axiom,
! [B_6: fun_fu1430349052l_bool,F: fun_pn800050071e_bool,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B_6),image_47868345e_bool(F,A_1)))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6)) ) ) ).
tff(fact_137_finite__surj,axiom,
! [B_6: fun_fun_a_bool_bool,F: fun_pname_fun_a_bool,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B_6),image_112932426a_bool(F,A_1)))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6)) ) ) ).
tff(fact_138_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_pname_pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_pname_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_139_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_pname_nat,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_pname_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_140_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_nat_a,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_nat_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_141_finite__surj,axiom,
! [B_6: fun_fu425979586l_bool,F: fun_nat_fun_nat_bool,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B_6),image_26036933t_bool(F,A_1)))
=> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6)) ) ) ).
tff(fact_142_finite__surj,axiom,
! [B_6: fun_fu1430349052l_bool,F: fun_na936072029e_bool,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B_6),image_1655916159e_bool(F,A_1)))
=> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6)) ) ) ).
tff(fact_143_finite__surj,axiom,
! [B_6: fun_fun_a_bool_bool,F: fun_nat_fun_a_bool,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B_6),image_nat_fun_a_bool(F,A_1)))
=> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6)) ) ) ).
tff(fact_144_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_nat_pname,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_nat_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_145_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_nat_nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_nat_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_146_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_a_pname,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_a_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_147_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_fu665170229_pname,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1921560913_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_148_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_fu1499449723_pname,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1283814551_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_149_finite__surj,axiom,
! [B_6: fun_pname_bool,F: fun_fun_a_bool_pname,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1854862208_pname(F,A_1)))
=> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6)) ) ) ).
tff(fact_150_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_a_nat,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_a_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_151_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_fun_nat_bool_nat,A_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_496248727ol_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_152_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_fu1668467777ol_nat,A_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_1551609309ol_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_153_finite__surj,axiom,
! [B_6: fun_nat_bool,F: fun_fun_a_bool_nat,A_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_fun_a_bool_nat(F,A_1)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6)) ) ) ).
tff(fact_154_finite__surj,axiom,
! [B_6: fun_a_bool,F: fun_pname_a,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_pname_a(F,A_1)))
=> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6)) ) ) ).
tff(fact_155_finite__subset__image,axiom,
! [F: fun_fun_nat_bool_a,A_1: fun_fu425979586l_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_fun_nat_bool_a(F,A_1)))
=> ? [C_7: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_7),A_1))
& hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_7))
& ( B_6 = image_fun_nat_bool_a(F,C_7) ) ) ) ) ).
tff(fact_156_finite__subset__image,axiom,
! [F: fun_fun_pname_bool_a,A_1: fun_fu1430349052l_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_876012084bool_a(F,A_1)))
=> ? [C_7: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_7),A_1))
& hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_7))
& ( B_6 = image_876012084bool_a(F,C_7) ) ) ) ) ).
tff(fact_157_finite__subset__image,axiom,
! [F: fun_fun_a_bool_a,A_1: fun_fun_a_bool_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_fun_a_bool_a(F,A_1)))
=> ? [C_7: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_7),A_1))
& hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_7))
& ( B_6 = image_fun_a_bool_a(F,C_7) ) ) ) ) ).
tff(fact_158_finite__subset__image,axiom,
! [F: fun_a_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_a_a(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_a_a(F,C_7) ) ) ) ) ).
tff(fact_159_finite__subset__image,axiom,
! [F: fun_a_fun_nat_bool,A_1: fun_a_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B_6),image_a_fun_nat_bool(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_a_fun_nat_bool(F,C_7) ) ) ) ) ).
tff(fact_160_finite__subset__image,axiom,
! [F: fun_a_fun_pname_bool,A_1: fun_a_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B_6),image_819518260e_bool(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_819518260e_bool(F,C_7) ) ) ) ) ).
tff(fact_161_finite__subset__image,axiom,
! [F: fun_a_fun_a_bool,A_1: fun_a_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B_6),image_a_fun_a_bool(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_a_fun_a_bool(F,C_7) ) ) ) ) ).
tff(fact_162_finite__subset__image,axiom,
! [F: fun_a_pname,A_1: fun_a_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_a_pname(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_a_pname(F,C_7) ) ) ) ) ).
tff(fact_163_finite__subset__image,axiom,
! [F: fun_fu665170229_pname,A_1: fun_fu425979586l_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1921560913_pname(F,A_1)))
=> ? [C_7: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_7),A_1))
& hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_7))
& ( B_6 = image_1921560913_pname(F,C_7) ) ) ) ) ).
tff(fact_164_finite__subset__image,axiom,
! [F: fun_fu1499449723_pname,A_1: fun_fu1430349052l_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1283814551_pname(F,A_1)))
=> ? [C_7: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_7),A_1))
& hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_7))
& ( B_6 = image_1283814551_pname(F,C_7) ) ) ) ) ).
tff(fact_165_finite__subset__image,axiom,
! [F: fun_fun_a_bool_pname,A_1: fun_fun_a_bool_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_1854862208_pname(F,A_1)))
=> ? [C_7: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_7),A_1))
& hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_7))
& ( B_6 = image_1854862208_pname(F,C_7) ) ) ) ) ).
tff(fact_166_finite__subset__image,axiom,
! [F: fun_a_nat,A_1: fun_a_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_a_nat(F,A_1)))
=> ? [C_7: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_7),A_1))
& hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,C_7))
& ( B_6 = image_a_nat(F,C_7) ) ) ) ) ).
tff(fact_167_finite__subset__image,axiom,
! [F: fun_fun_nat_bool_nat,A_1: fun_fu425979586l_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_496248727ol_nat(F,A_1)))
=> ? [C_7: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,C_7),A_1))
& hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,C_7))
& ( B_6 = image_496248727ol_nat(F,C_7) ) ) ) ) ).
tff(fact_168_finite__subset__image,axiom,
! [F: fun_fu1668467777ol_nat,A_1: fun_fu1430349052l_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_1551609309ol_nat(F,A_1)))
=> ? [C_7: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,C_7),A_1))
& hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,C_7))
& ( B_6 = image_1551609309ol_nat(F,C_7) ) ) ) ) ).
tff(fact_169_finite__subset__image,axiom,
! [F: fun_fun_a_bool_nat,A_1: fun_fun_a_bool_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_fun_a_bool_nat(F,A_1)))
=> ? [C_7: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,C_7),A_1))
& hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,C_7))
& ( B_6 = image_fun_a_bool_nat(F,C_7) ) ) ) ) ).
tff(fact_170_finite__subset__image,axiom,
! [F: fun_pn406123357t_bool,A_1: fun_pname_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B_6),image_2129980159t_bool(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_2129980159t_bool(F,C_7) ) ) ) ) ).
tff(fact_171_finite__subset__image,axiom,
! [F: fun_pn800050071e_bool,A_1: fun_pname_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B_6),image_47868345e_bool(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_47868345e_bool(F,C_7) ) ) ) ) ).
tff(fact_172_finite__subset__image,axiom,
! [F: fun_pname_fun_a_bool,A_1: fun_pname_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B_6),image_112932426a_bool(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_112932426a_bool(F,C_7) ) ) ) ) ).
tff(fact_173_finite__subset__image,axiom,
! [F: fun_pname_pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_pname_pname(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_pname_pname(F,C_7) ) ) ) ) ).
tff(fact_174_finite__subset__image,axiom,
! [F: fun_pname_nat,A_1: fun_pname_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_pname_nat(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_pname_nat(F,C_7) ) ) ) ) ).
tff(fact_175_finite__subset__image,axiom,
! [F: fun_nat_a,A_1: fun_nat_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_nat_a(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_nat_a(F,C_7) ) ) ) ) ).
tff(fact_176_finite__subset__image,axiom,
! [F: fun_nat_fun_nat_bool,A_1: fun_nat_bool,B_6: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B_6))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,B_6),image_26036933t_bool(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_26036933t_bool(F,C_7) ) ) ) ) ).
tff(fact_177_finite__subset__image,axiom,
! [F: fun_na936072029e_bool,A_1: fun_nat_bool,B_6: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B_6))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,B_6),image_1655916159e_bool(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_1655916159e_bool(F,C_7) ) ) ) ) ).
tff(fact_178_finite__subset__image,axiom,
! [F: fun_nat_fun_a_bool,A_1: fun_nat_bool,B_6: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B_6))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,B_6),image_nat_fun_a_bool(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_nat_fun_a_bool(F,C_7) ) ) ) ) ).
tff(fact_179_finite__subset__image,axiom,
! [F: fun_nat_pname,A_1: fun_nat_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_nat_pname(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_nat_pname(F,C_7) ) ) ) ) ).
tff(fact_180_finite__subset__image,axiom,
! [F: fun_nat_nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_nat_nat(F,A_1)))
=> ? [C_7: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_7),A_1))
& hBOOL(hAPP_f54304608l_bool(finite_finite_nat,C_7))
& ( B_6 = image_nat_nat(F,C_7) ) ) ) ) ).
tff(fact_181_finite__subset__image,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_pname_a(F,A_1)))
=> ? [C_7: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_7),A_1))
& hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_7))
& ( B_6 = image_pname_a(F,C_7) ) ) ) ) ).
tff(fact_182_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_bool] :
( ! [N_2: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,N_2)),hAPP_nat_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Na)),hAPP_nat_bool(F,N_3))) ) ) ).
tff(fact_183_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_na936072029e_bool] :
( ! [N_2: nat] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,N_2)),hAPP_n1025906991e_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,Na)),hAPP_n1025906991e_bool(F,N_3))) ) ) ).
tff(fact_184_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_fun_nat_bool] :
( ! [N_2: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,N_2)),hAPP_n1699378549t_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,Na)),hAPP_n1699378549t_bool(F,N_3))) ) ) ).
tff(fact_185_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_fun_a_bool] :
( ! [N_2: nat] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,N_2)),hAPP_nat_fun_a_bool(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,Na)),hAPP_nat_fun_a_bool(F,N_3))) ) ) ).
tff(fact_186_lift__Suc__mono__le,axiom,
! [Na: nat,N_3: nat,F: fun_nat_nat] :
( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_3))) ) ) ).
tff(fact_187_pigeonhole__infinite,axiom,
! [F: fun_nat_pname,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_nat_pname(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_p1499970991t_bool(cOMBC_nat_pname_bool(cOMBB_1212655066ol_nat(fequal_pname,F)),hAPP_nat_pname(F,X_1)))))) ) ) ) ).
tff(fact_188_pigeonhole__infinite,axiom,
! [F: fun_a_pname,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_p1534023578a_bool(cOMBC_a_pname_bool(cOMBB_610033911bool_a(fequal_pname,F)),hAPP_a_pname(F,X_1)))))) ) ) ) ).
tff(fact_189_pigeonhole__infinite,axiom,
! [F: fun_pname_pname,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_pname_pname(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_p61793385e_bool(cOMBC_1149511130e_bool(cOMBB_542850580_pname(fequal_pname,F)),hAPP_pname_pname(F,X_1)))))) ) ) ) ).
tff(fact_190_pigeonhole__infinite,axiom,
! [F: fun_fu665170229_pname,A_1: fun_fu425979586l_bool] :
( ~ hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(F,A_1)))
=> ? [X_1: fun_nat_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A_1))
& ~ hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),A_1)),hAPP_p130839763l_bool(cOMBC_615407716e_bool(cOMBB_141086128t_bool(fequal_pname,F)),hAPP_f1291551745_pname(F,X_1)))))) ) ) ) ).
tff(fact_191_pigeonhole__infinite,axiom,
! [F: fun_fu1499449723_pname,A_1: fun_fu1430349052l_bool] :
( ~ hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(F,A_1)))
=> ? [X_1: fun_pname_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A_1))
& ~ hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),A_1)),hAPP_p338031245l_bool(cOMBC_1004116266e_bool(cOMBB_598082538e_bool(fequal_pname,F)),hAPP_f1297739591_pname(F,X_1)))))) ) ) ) ).
tff(fact_192_pigeonhole__infinite,axiom,
! [F: fun_fun_a_bool_pname,A_1: fun_fun_a_bool_bool] :
( ~ hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(F,A_1)))
=> ? [X_1: fun_a_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A_1))
& ~ hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),A_1)),hAPP_p1824510254l_bool(cOMBC_1738168533e_bool(cOMBB_1693087065a_bool(fequal_pname,F)),hAPP_f1128469712_pname(F,X_1)))))) ) ) ) ).
tff(fact_193_pigeonhole__infinite,axiom,
! [F: fun_nat_nat,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_nat_nat(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(cOMBB_800536526ol_nat(fequal_nat,F)),hAPP_nat_nat(F,X_1)))))) ) ) ) ).
tff(fact_194_pigeonhole__infinite,axiom,
! [F: fun_a_nat,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_nat_fun_a_bool(cOMBC_a_nat_bool(cOMBB_1321347587bool_a(fequal_nat,F)),hAPP_a_nat(F,X_1)))))) ) ) ) ).
tff(fact_195_pigeonhole__infinite,axiom,
! [F: fun_pname_nat,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_pname_nat(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_n1025906991e_bool(cOMBC_pname_nat_bool(cOMBB_523834888_pname(fequal_nat,F)),hAPP_pname_nat(F,X_1)))))) ) ) ) ).
tff(fact_196_pigeonhole__infinite,axiom,
! [F: fun_fun_nat_bool_nat,A_1: fun_fu425979586l_bool] :
( ~ hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(F,A_1)))
=> ? [X_1: fun_nat_bool] :
( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),A_1))
& ~ hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),A_1)),hAPP_n215258509l_bool(cOMBC_385542954t_bool(cOMBB_1250273980t_bool(fequal_nat,F)),hAPP_f22106695ol_nat(F,X_1)))))) ) ) ) ).
tff(fact_197_pigeonhole__infinite,axiom,
! [F: fun_fu1668467777ol_nat,A_1: fun_fu1430349052l_bool] :
( ~ hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(F,A_1)))
=> ? [X_1: fun_pname_bool] :
( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),A_1))
& ~ hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),A_1)),hAPP_n850744903l_bool(cOMBC_1666426608t_bool(cOMBB_1896684278e_bool(fequal_nat,F)),hAPP_f921600141ol_nat(F,X_1)))))) ) ) ) ).
tff(fact_198_pigeonhole__infinite,axiom,
! [F: fun_fun_a_bool_nat,A_1: fun_fun_a_bool_bool] :
( ~ hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A_1))
=> ( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(F,A_1)))
=> ? [X_1: fun_a_bool] :
( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),A_1))
& ~ hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),A_1)),hAPP_n1414589940l_bool(cOMBC_619334683t_bool(cOMBB_164527437a_bool(fequal_nat,F)),hAPP_fun_a_bool_nat(F,X_1)))))) ) ) ) ).
tff(fact_199_pigeonhole__infinite,axiom,
! [F: fun_nat_a,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_a_fun_nat_bool(cOMBC_nat_a_bool(cOMBB_1885489796ol_nat(fequal_a,F)),hAPP_nat_a(F,X_1)))))) ) ) ) ).
tff(fact_200_pigeonhole__infinite,axiom,
! [F: fun_nat_fun_nat_bool,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_2026977092ol_nat(fequal_fun_nat_bool,F)),hAPP_n1699378549t_bool(F,X_1)))))) ) ) ) ).
tff(fact_201_pigeonhole__infinite,axiom,
! [F: fun_na936072029e_bool,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_f1066163005t_bool(cOMBC_386238098l_bool(cOMBB_2117322052ol_nat(fequal533582459e_bool,F)),hAPP_n1025906991e_bool(F,X_1)))))) ) ) ) ).
tff(fact_202_pigeonhole__infinite,axiom,
! [F: fun_nat_fun_a_bool,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1))
=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(F,A_1)))
=> ? [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
& ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A_1)),hAPP_f282463732t_bool(cOMBC_1928494297l_bool(cOMBB_1823939024ol_nat(fequal_fun_a_bool,F)),hAPP_nat_fun_a_bool(F,X_1)))))) ) ) ) ).
tff(fact_203_pigeonhole__infinite,axiom,
! [F: fun_pn406123357t_bool,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_f654413245e_bool(cOMBC_267053842l_bool(cOMBB_928955006_pname(fequal_fun_nat_bool,F)),hAPP_p1499970991t_bool(F,X_1)))))) ) ) ) ).
tff(fact_204_pigeonhole__infinite,axiom,
! [F: fun_pn800050071e_bool,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(cOMBB_408569982_pname(fequal533582459e_bool,F)),hAPP_p61793385e_bool(F,X_1)))))) ) ) ) ).
tff(fact_205_pigeonhole__infinite,axiom,
! [F: fun_pname_fun_a_bool,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_f1794073134e_bool(cOMBC_445755039l_bool(cOMBB_392435466_pname(fequal_fun_a_bool,F)),hAPP_p1534023578a_bool(F,X_1)))))) ) ) ) ).
tff(fact_206_pigeonhole__infinite,axiom,
! [F: fun_a_a,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_a_a(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_a_fun_a_bool(cOMBC_a_a_bool(cOMBB_a_fun_a_bool_a(fequal_a,F)),hAPP_a_a(F,X_1)))))) ) ) ) ).
tff(fact_207_pigeonhole__infinite,axiom,
! [F: fun_a_fun_nat_bool,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_a_fun_nat_bool(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_f1549043526a_bool(cOMBC_777206479l_bool(cOMBB_743407885bool_a(fequal_fun_nat_bool,F)),hAPP_a_fun_nat_bool(F,X_1)))))) ) ) ) ).
tff(fact_208_pigeonhole__infinite,axiom,
! [F: fun_a_fun_pname_bool,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_819518260e_bool(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_f1051908748a_bool(cOMBC_1834145417l_bool(cOMBB_1137537805bool_a(fequal533582459e_bool,F)),hAPP_a93125764e_bool(F,X_1)))))) ) ) ) ).
tff(fact_209_pigeonhole__infinite,axiom,
! [F: fun_a_fun_a_bool,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A_1))
=> ( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_a_fun_a_bool(F,A_1)))
=> ? [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
& ~ hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),A_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(cOMBB_472261505bool_a(fequal_fun_a_bool,F)),hAPP_a_fun_a_bool(F,X_1)))))) ) ) ) ).
tff(fact_210_pigeonhole__infinite,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(F,A_1)))
=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ~ hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A_1)),hAPP_a93125764e_bool(cOMBC_pname_a_bool(cOMBB_1897541054_pname(fequal_a,F)),hAPP_pname_a(F,X_1)))))) ) ) ) ).
tff(fact_211_image__eqI,axiom,
! [A_1: fun_pname_bool,B_1: nat,F: fun_pname_nat,X_3: pname] :
( ( B_1 = hAPP_pname_nat(F,X_3) )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_pname_nat(F,A_1))) ) ) ).
tff(fact_212_image__eqI,axiom,
! [A_1: fun_a_bool,B_1: nat,F: fun_a_nat,X_3: x_a] :
( ( B_1 = hAPP_a_nat(F,X_3) )
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_a_nat(F,A_1))) ) ) ).
tff(fact_213_image__eqI,axiom,
! [A_1: fun_nat_bool,B_1: pname,F: fun_nat_pname,X_3: nat] :
( ( B_1 = hAPP_nat_pname(F,X_3) )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),image_nat_pname(F,A_1))) ) ) ).
tff(fact_214_image__eqI,axiom,
! [A_1: fun_nat_bool,B_1: x_a,F: fun_nat_a,X_3: nat] :
( ( B_1 = hAPP_nat_a(F,X_3) )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_nat_a(F,A_1))) ) ) ).
tff(fact_215_image__eqI,axiom,
! [A_1: fun_pname_bool,B_1: x_a,F: fun_pname_a,X_3: pname] :
( ( B_1 = hAPP_pname_a(F,X_3) )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A_1))) ) ) ).
tff(fact_216_equalityI,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),A_1))
=> ( A_1 = B_6 ) ) ) ).
tff(fact_217_equalityI,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),A_1))
=> ( A_1 = B_6 ) ) ) ).
tff(fact_218_equalityI,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),A_1))
=> ( A_1 = B_6 ) ) ) ).
tff(fact_219_subsetD,axiom,
! [C_1: nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B_6)) ) ) ).
tff(fact_220_subsetD,axiom,
! [C_1: pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),B_6)) ) ) ).
tff(fact_221_subsetD,axiom,
! [C_1: x_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),B_6)) ) ) ).
tff(fact_222_insertCI,axiom,
! [B_1: nat,A_3: nat,B_6: fun_nat_bool] :
( ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),B_6))
=> ( A_3 = B_1 ) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),insert_nat(B_1,B_6))) ) ).
tff(fact_223_insertCI,axiom,
! [B_1: pname,A_3: pname,B_6: fun_pname_bool] :
( ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),B_6))
=> ( A_3 = B_1 ) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),insert_pname(B_1,B_6))) ) ).
tff(fact_224_insertCI,axiom,
! [B_1: x_a,A_3: x_a,B_6: fun_a_bool] :
( ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),B_6))
=> ( A_3 = B_1 ) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),insert_a(B_1,B_6))) ) ).
tff(fact_225_insertE,axiom,
! [A_3: nat,B_1: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),insert_nat(B_1,A_1)))
=> ( ( A_3 != B_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),A_1)) ) ) ).
tff(fact_226_insertE,axiom,
! [A_3: pname,B_1: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),insert_pname(B_1,A_1)))
=> ( ( A_3 != B_1 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),A_1)) ) ) ).
tff(fact_227_insertE,axiom,
! [A_3: x_a,B_1: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),insert_a(B_1,A_1)))
=> ( ( A_3 != B_1 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),A_1)) ) ) ).
tff(fact_228_zero__induct__lemma,axiom,
! [I_1: nat,Pa: fun_nat_bool,K_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,K_2))
=> ( ! [N_2: nat] :
( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2)))
=> hBOOL(hAPP_nat_bool(Pa,N_2)) )
=> hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(minus_minus_nat(K_2),I_1))) ) ) ).
tff(fact_229_Suc__le__D,axiom,
! [N: nat,M_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_3))
=> ? [M_2: nat] : M_3 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_230_insertI1,axiom,
! [A_3: nat,B_6: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),insert_nat(A_3,B_6))) ).
tff(fact_231_insertI1,axiom,
! [A_3: pname,B_6: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),insert_pname(A_3,B_6))) ).
tff(fact_232_insertI1,axiom,
! [A_3: x_a,B_6: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),insert_a(A_3,B_6))) ).
tff(fact_233_insert__compr,axiom,
! [A_3: fun_nat_bool,B_6: fun_fu425979586l_bool] : insert_fun_nat_bool(A_3,B_6) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_3)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),B_6))) ).
tff(fact_234_insert__compr,axiom,
! [A_3: fun_pname_bool,B_6: fun_fu1430349052l_bool] : insert1325755072e_bool(A_3,B_6) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_3)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),B_6))) ).
tff(fact_235_insert__compr,axiom,
! [A_3: fun_a_bool,B_6: fun_fun_a_bool_bool] : insert_fun_a_bool(A_3,B_6) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_3)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),B_6))) ).
tff(fact_236_insert__compr,axiom,
! [A_3: nat,B_6: fun_nat_bool] : insert_nat(A_3,B_6) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_3)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B_6))) ).
tff(fact_237_insert__compr,axiom,
! [A_3: pname,B_6: fun_pname_bool] : insert_pname(A_3,B_6) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_3)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B_6))) ).
tff(fact_238_insert__compr,axiom,
! [A_3: x_a,B_6: fun_a_bool] : insert_a(A_3,B_6) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_3)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B_6))) ).
tff(fact_239_insert__Collect,axiom,
! [A_3: pname,Pa: fun_pname_bool] : insert_pname(A_3,collect_pname(Pa)) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fimplies,cOMBB_647938656_pname(fNot,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_3))),Pa)) ).
tff(fact_240_insert__Collect,axiom,
! [A_3: fun_nat_bool,Pa: fun_fu425979586l_bool] : insert_fun_nat_bool(A_3,collect_fun_nat_bool(Pa)) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fimplies,cOMBB_238756964t_bool(fNot,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_3))),Pa)) ).
tff(fact_241_insert__Collect,axiom,
! [A_3: fun_pname_bool,Pa: fun_fu1430349052l_bool] : insert1325755072e_bool(A_3,collec1974731493e_bool(Pa)) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fimplies,cOMBB_307249310e_bool(fNot,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_3))),Pa)) ).
tff(fact_242_insert__Collect,axiom,
! [A_3: fun_a_bool,Pa: fun_fun_a_bool_bool] : insert_fun_a_bool(A_3,collect_fun_a_bool(Pa)) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fimplies,cOMBB_2140588453a_bool(fNot,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_3))),Pa)) ).
tff(fact_243_insert__Collect,axiom,
! [A_3: nat,Pa: fun_nat_bool] : insert_nat(A_3,collect_nat(Pa)) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fimplies,cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_3))),Pa)) ).
tff(fact_244_insert__Collect,axiom,
! [A_3: x_a,Pa: fun_a_bool] : insert_a(A_3,collect_a(Pa)) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fimplies,cOMBB_bool_bool_a(fNot,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_3))),Pa)) ).
tff(fact_245_insert__absorb2,axiom,
! [X_3: pname,A_1: fun_pname_bool] : insert_pname(X_3,insert_pname(X_3,A_1)) = insert_pname(X_3,A_1) ).
tff(fact_246_insert__absorb2,axiom,
! [X_3: nat,A_1: fun_nat_bool] : insert_nat(X_3,insert_nat(X_3,A_1)) = insert_nat(X_3,A_1) ).
tff(fact_247_insert__absorb2,axiom,
! [X_3: x_a,A_1: fun_a_bool] : insert_a(X_3,insert_a(X_3,A_1)) = insert_a(X_3,A_1) ).
tff(fact_248_insert__commute,axiom,
! [X_3: pname,Y_2: pname,A_1: fun_pname_bool] : insert_pname(X_3,insert_pname(Y_2,A_1)) = insert_pname(Y_2,insert_pname(X_3,A_1)) ).
tff(fact_249_insert__commute,axiom,
! [X_3: nat,Y_2: nat,A_1: fun_nat_bool] : insert_nat(X_3,insert_nat(Y_2,A_1)) = insert_nat(Y_2,insert_nat(X_3,A_1)) ).
tff(fact_250_insert__commute,axiom,
! [X_3: x_a,Y_2: x_a,A_1: fun_a_bool] : insert_a(X_3,insert_a(Y_2,A_1)) = insert_a(Y_2,insert_a(X_3,A_1)) ).
tff(fact_251_insert__iff,axiom,
! [A_3: nat,B_1: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),insert_nat(B_1,A_1)))
<=> ( ( A_3 = B_1 )
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),A_1)) ) ) ).
tff(fact_252_insert__iff,axiom,
! [A_3: pname,B_1: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),insert_pname(B_1,A_1)))
<=> ( ( A_3 = B_1 )
| hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),A_1)) ) ) ).
tff(fact_253_insert__iff,axiom,
! [A_3: x_a,B_1: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),insert_a(B_1,A_1)))
<=> ( ( A_3 = B_1 )
| hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),A_1)) ) ) ).
tff(fact_254_insert__code,axiom,
! [Y_2: pname,A_1: fun_pname_bool,X_3: pname] :
( hBOOL(hAPP_pname_bool(insert_pname(Y_2,A_1),X_3))
<=> ( ( Y_2 = X_3 )
| hBOOL(hAPP_pname_bool(A_1,X_3)) ) ) ).
tff(fact_255_insert__code,axiom,
! [Y_2: nat,A_1: fun_nat_bool,X_3: nat] :
( hBOOL(hAPP_nat_bool(insert_nat(Y_2,A_1),X_3))
<=> ( ( Y_2 = X_3 )
| hBOOL(hAPP_nat_bool(A_1,X_3)) ) ) ).
tff(fact_256_insert__code,axiom,
! [Y_2: x_a,A_1: fun_a_bool,X_3: x_a] :
( hBOOL(hAPP_a_bool(insert_a(Y_2,A_1),X_3))
<=> ( ( Y_2 = X_3 )
| hBOOL(hAPP_a_bool(A_1,X_3)) ) ) ).
tff(fact_257_insert__ident,axiom,
! [B_6: fun_nat_bool,X_3: nat,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),B_6))
=> ( ( insert_nat(X_3,A_1) = insert_nat(X_3,B_6) )
<=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_258_insert__ident,axiom,
! [B_6: fun_pname_bool,X_3: pname,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),B_6))
=> ( ( insert_pname(X_3,A_1) = insert_pname(X_3,B_6) )
<=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_259_insert__ident,axiom,
! [B_6: fun_a_bool,X_3: x_a,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),B_6))
=> ( ( insert_a(X_3,A_1) = insert_a(X_3,B_6) )
<=> ( A_1 = B_6 ) ) ) ) ).
tff(fact_260_insertI2,axiom,
! [B_1: nat,A_3: nat,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),B_6))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),insert_nat(B_1,B_6))) ) ).
tff(fact_261_insertI2,axiom,
! [B_1: pname,A_3: pname,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),B_6))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),insert_pname(B_1,B_6))) ) ).
tff(fact_262_insertI2,axiom,
! [B_1: x_a,A_3: x_a,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),insert_a(B_1,B_6))) ) ).
tff(fact_263_insert__absorb,axiom,
! [A_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),A_1))
=> ( insert_nat(A_3,A_1) = A_1 ) ) ).
tff(fact_264_insert__absorb,axiom,
! [A_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),A_1))
=> ( insert_pname(A_3,A_1) = A_1 ) ) ).
tff(fact_265_insert__absorb,axiom,
! [A_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),A_1))
=> ( insert_a(A_3,A_1) = A_1 ) ) ).
tff(fact_266_subset__refl,axiom,
! [A_1: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),A_1)) ).
tff(fact_267_subset__refl,axiom,
! [A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),A_1)) ).
tff(fact_268_subset__refl,axiom,
! [A_1: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),A_1)) ).
tff(fact_269_set__eq__subset,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( ( A_1 = B_6 )
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
& hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),A_1)) ) ) ).
tff(fact_270_set__eq__subset,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( ( A_1 = B_6 )
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),A_1)) ) ) ).
tff(fact_271_set__eq__subset,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( ( A_1 = B_6 )
<=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
& hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),A_1)) ) ) ).
tff(fact_272_equalityD1,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6)) ) ).
tff(fact_273_equalityD1,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6)) ) ).
tff(fact_274_equalityD1,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6)) ) ).
tff(fact_275_equalityD2,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),A_1)) ) ).
tff(fact_276_equalityD2,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),A_1)) ) ).
tff(fact_277_equalityD2,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( ( A_1 = B_6 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),A_1)) ) ).
tff(fact_278_in__mono,axiom,
! [X_3: nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),B_6)) ) ) ).
tff(fact_279_in__mono,axiom,
! [X_3: pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),B_6)) ) ) ).
tff(fact_280_in__mono,axiom,
! [X_3: x_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),B_6)) ) ) ).
tff(fact_281_set__rev__mp,axiom,
! [B_6: fun_nat_bool,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),B_6)) ) ) ).
tff(fact_282_set__rev__mp,axiom,
! [B_6: fun_pname_bool,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),B_6)) ) ) ).
tff(fact_283_set__rev__mp,axiom,
! [B_6: fun_a_bool,X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),B_6)) ) ) ).
tff(fact_284_set__mp,axiom,
! [X_3: nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),B_6)) ) ) ).
tff(fact_285_set__mp,axiom,
! [X_3: pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),B_6)) ) ) ).
tff(fact_286_set__mp,axiom,
! [X_3: x_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),B_6)) ) ) ).
tff(fact_287_mem__def,axiom,
! [X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
<=> hBOOL(hAPP_nat_bool(A_1,X_3)) ) ).
tff(fact_288_mem__def,axiom,
! [X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
<=> hBOOL(hAPP_pname_bool(A_1,X_3)) ) ).
tff(fact_289_mem__def,axiom,
! [X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
<=> hBOOL(hAPP_a_bool(A_1,X_3)) ) ).
tff(fact_290_Collect__def,axiom,
! [Pa: fun_a_bool] : collect_a(Pa) = Pa ).
tff(fact_291_Collect__def,axiom,
! [Pa: fun_pname_bool] : collect_pname(Pa) = Pa ).
tff(fact_292_Collect__def,axiom,
! [Pa: fun_fu425979586l_bool] : collect_fun_nat_bool(Pa) = Pa ).
tff(fact_293_Collect__def,axiom,
! [Pa: fun_fu1430349052l_bool] : collec1974731493e_bool(Pa) = Pa ).
tff(fact_294_Collect__def,axiom,
! [Pa: fun_fun_a_bool_bool] : collect_fun_a_bool(Pa) = Pa ).
tff(fact_295_Collect__def,axiom,
! [Pa: fun_nat_bool] : collect_nat(Pa) = Pa ).
tff(fact_296_subset__trans,axiom,
! [C_6: fun_pname_bool,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),C_6))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),C_6)) ) ) ).
tff(fact_297_subset__trans,axiom,
! [C_6: fun_nat_bool,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),C_6))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),C_6)) ) ) ).
tff(fact_298_subset__trans,axiom,
! [C_6: fun_a_bool,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),C_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),C_6)) ) ) ).
tff(fact_299_equalityE,axiom,
! [A_1: fun_pname_bool,B_6: fun_pname_bool] :
( ( A_1 = B_6 )
=> ~ ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),A_1)) ) ) ).
tff(fact_300_equalityE,axiom,
! [A_1: fun_nat_bool,B_6: fun_nat_bool] :
( ( A_1 = B_6 )
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),A_1)) ) ) ).
tff(fact_301_equalityE,axiom,
! [A_1: fun_a_bool,B_6: fun_a_bool] :
( ( A_1 = B_6 )
=> ~ ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),A_1)) ) ) ).
tff(fact_302_image__iff,axiom,
! [Z_1: x_a,F: fun_pname_a,A_1: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,Z_1),image_pname_a(F,A_1)))
<=> ? [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
& ( Z_1 = hAPP_pname_a(F,X_1) ) ) ) ).
tff(fact_303_imageI,axiom,
! [F: fun_pname_nat,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_pname_nat(F,X_3)),image_pname_nat(F,A_1))) ) ).
tff(fact_304_imageI,axiom,
! [F: fun_a_nat,X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_a_nat(F,X_3)),image_a_nat(F,A_1))) ) ).
tff(fact_305_imageI,axiom,
! [F: fun_nat_pname,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_nat_pname(F,X_3)),image_nat_pname(F,A_1))) ) ).
tff(fact_306_imageI,axiom,
! [F: fun_nat_a,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_nat_a(F,X_3)),image_nat_a(F,A_1))) ) ).
tff(fact_307_imageI,axiom,
! [F: fun_pname_a,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_3)),image_pname_a(F,A_1))) ) ).
tff(fact_308_rev__image__eqI,axiom,
! [B_1: nat,F: fun_pname_nat,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( ( B_1 = hAPP_pname_nat(F,X_3) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_pname_nat(F,A_1))) ) ) ).
tff(fact_309_rev__image__eqI,axiom,
! [B_1: nat,F: fun_a_nat,X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( ( B_1 = hAPP_a_nat(F,X_3) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_a_nat(F,A_1))) ) ) ).
tff(fact_310_rev__image__eqI,axiom,
! [B_1: pname,F: fun_nat_pname,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( ( B_1 = hAPP_nat_pname(F,X_3) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),image_nat_pname(F,A_1))) ) ) ).
tff(fact_311_rev__image__eqI,axiom,
! [B_1: x_a,F: fun_nat_a,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( ( B_1 = hAPP_nat_a(F,X_3) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_nat_a(F,A_1))) ) ) ).
tff(fact_312_rev__image__eqI,axiom,
! [B_1: x_a,F: fun_pname_a,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( ( B_1 = hAPP_pname_a(F,X_3) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A_1))) ) ) ).
tff(fact_313_insert__compr__raw,axiom,
! [X_1: fun_nat_bool,Xa: fun_fu425979586l_bool] : insert_fun_nat_bool(X_1,Xa) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),X_1)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),Xa))) ).
tff(fact_314_insert__compr__raw,axiom,
! [X_1: fun_pname_bool,Xa: fun_fu1430349052l_bool] : insert1325755072e_bool(X_1,Xa) = collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),X_1)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),Xa))) ).
tff(fact_315_insert__compr__raw,axiom,
! [X_1: fun_a_bool,Xa: fun_fun_a_bool_bool] : insert_fun_a_bool(X_1,Xa) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),X_1)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),Xa))) ).
tff(fact_316_insert__compr__raw,axiom,
! [X_1: nat,Xa: fun_nat_bool] : insert_nat(X_1,Xa) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),Xa))) ).
tff(fact_317_insert__compr__raw,axiom,
! [X_1: pname,Xa: fun_pname_bool] : insert_pname(X_1,Xa) = collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),X_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),Xa))) ).
tff(fact_318_insert__compr__raw,axiom,
! [X_1: x_a,Xa: fun_a_bool] : insert_a(X_1,Xa) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),X_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),Xa))) ).
tff(fact_319_subset__insertI,axiom,
! [B_6: fun_pname_bool,A_3: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),insert_pname(A_3,B_6))) ).
tff(fact_320_subset__insertI,axiom,
! [B_6: fun_nat_bool,A_3: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),insert_nat(A_3,B_6))) ).
tff(fact_321_subset__insertI,axiom,
! [B_6: fun_a_bool,A_3: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),insert_a(A_3,B_6))) ).
tff(fact_322_insert__subset,axiom,
! [X_3: nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(X_3,A_1)),B_6))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),B_6))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6)) ) ) ).
tff(fact_323_insert__subset,axiom,
! [X_3: pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(X_3,A_1)),B_6))
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),B_6))
& hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6)) ) ) ).
tff(fact_324_insert__subset,axiom,
! [X_3: x_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(X_3,A_1)),B_6))
<=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),B_6))
& hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6)) ) ) ).
tff(fact_325_subset__insert,axiom,
! [B_6: fun_nat_bool,X_3: nat,A_1: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),insert_nat(X_3,B_6)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6)) ) ) ).
tff(fact_326_subset__insert,axiom,
! [B_6: fun_pname_bool,X_3: pname,A_1: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),insert_pname(X_3,B_6)))
<=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6)) ) ) ).
tff(fact_327_subset__insert,axiom,
! [B_6: fun_a_bool,X_3: x_a,A_1: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),insert_a(X_3,B_6)))
<=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6)) ) ) ).
tff(fact_328_subset__insertI2,axiom,
! [B_1: pname,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),insert_pname(B_1,B_6))) ) ).
tff(fact_329_subset__insertI2,axiom,
! [B_1: nat,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),insert_nat(B_1,B_6))) ) ).
tff(fact_330_subset__insertI2,axiom,
! [B_1: x_a,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),insert_a(B_1,B_6))) ) ).
tff(fact_331_insert__mono,axiom,
! [A_3: pname,C_6: fun_pname_bool,D_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_6),D_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(A_3,C_6)),insert_pname(A_3,D_1))) ) ).
tff(fact_332_insert__mono,axiom,
! [A_3: nat,C_6: fun_nat_bool,D_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_6),D_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(A_3,C_6)),insert_nat(A_3,D_1))) ) ).
tff(fact_333_insert__mono,axiom,
! [A_3: x_a,C_6: fun_a_bool,D_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_6),D_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(A_3,C_6)),insert_a(A_3,D_1))) ) ).
tff(fact_334_image__insert,axiom,
! [F: fun_a_pname,A_3: x_a,B_6: fun_a_bool] : image_a_pname(F,insert_a(A_3,B_6)) = insert_pname(hAPP_a_pname(F,A_3),image_a_pname(F,B_6)) ).
tff(fact_335_image__insert,axiom,
! [F: fun_a_nat,A_3: x_a,B_6: fun_a_bool] : image_a_nat(F,insert_a(A_3,B_6)) = insert_nat(hAPP_a_nat(F,A_3),image_a_nat(F,B_6)) ).
tff(fact_336_image__insert,axiom,
! [F: fun_nat_a,A_3: nat,B_6: fun_nat_bool] : image_nat_a(F,insert_nat(A_3,B_6)) = insert_a(hAPP_nat_a(F,A_3),image_nat_a(F,B_6)) ).
tff(fact_337_image__insert,axiom,
! [F: fun_pname_a,A_3: pname,B_6: fun_pname_bool] : image_pname_a(F,insert_pname(A_3,B_6)) = insert_a(hAPP_pname_a(F,A_3),image_pname_a(F,B_6)) ).
tff(fact_338_insert__image,axiom,
! [F: fun_pname_pname,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( insert_pname(hAPP_pname_pname(F,X_3),image_pname_pname(F,A_1)) = image_pname_pname(F,A_1) ) ) ).
tff(fact_339_insert__image,axiom,
! [F: fun_pname_nat,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( insert_nat(hAPP_pname_nat(F,X_3),image_pname_nat(F,A_1)) = image_pname_nat(F,A_1) ) ) ).
tff(fact_340_insert__image,axiom,
! [F: fun_a_pname,X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( insert_pname(hAPP_a_pname(F,X_3),image_a_pname(F,A_1)) = image_a_pname(F,A_1) ) ) ).
tff(fact_341_insert__image,axiom,
! [F: fun_a_nat,X_3: x_a,A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_3),A_1))
=> ( insert_nat(hAPP_a_nat(F,X_3),image_a_nat(F,A_1)) = image_a_nat(F,A_1) ) ) ).
tff(fact_342_insert__image,axiom,
! [F: fun_nat_a,X_3: nat,A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_3),A_1))
=> ( insert_a(hAPP_nat_a(F,X_3),image_nat_a(F,A_1)) = image_nat_a(F,A_1) ) ) ).
tff(fact_343_insert__image,axiom,
! [F: fun_pname_a,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( insert_a(hAPP_pname_a(F,X_3),image_pname_a(F,A_1)) = image_pname_a(F,A_1) ) ) ).
tff(fact_344_subset__image__iff,axiom,
! [B_6: fun_a_bool,F: fun_nat_a,A_1: fun_nat_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_nat_a(F,A_1)))
<=> ? [AA: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,AA),A_1))
& ( B_6 = image_nat_a(F,AA) ) ) ) ).
tff(fact_345_subset__image__iff,axiom,
! [B_6: fun_pname_bool,F: fun_a_pname,A_1: fun_a_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_6),image_a_pname(F,A_1)))
<=> ? [AA: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,AA),A_1))
& ( B_6 = image_a_pname(F,AA) ) ) ) ).
tff(fact_346_subset__image__iff,axiom,
! [B_6: fun_nat_bool,F: fun_a_nat,A_1: fun_a_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_6),image_a_nat(F,A_1)))
<=> ? [AA: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,AA),A_1))
& ( B_6 = image_a_nat(F,AA) ) ) ) ).
tff(fact_347_subset__image__iff,axiom,
! [B_6: fun_a_bool,F: fun_pname_a,A_1: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_6),image_pname_a(F,A_1)))
<=> ? [AA: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,AA),A_1))
& ( B_6 = image_pname_a(F,AA) ) ) ) ).
tff(fact_348_image__mono,axiom,
! [F: fun_a_pname,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,image_a_pname(F,A_1)),image_a_pname(F,B_6))) ) ).
tff(fact_349_image__mono,axiom,
! [F: fun_a_nat,A_1: fun_a_bool,B_6: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_a_nat(F,A_1)),image_a_nat(F,B_6))) ) ).
tff(fact_350_image__mono,axiom,
! [F: fun_nat_a,A_1: fun_nat_bool,B_6: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_nat_a(F,A_1)),image_nat_a(F,B_6))) ) ).
tff(fact_351_image__mono,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool,B_6: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A_1)),image_pname_a(F,B_6))) ) ).
tff(fact_352_imageE,axiom,
! [B_1: pname,F: fun_nat_pname,A_1: fun_nat_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),image_nat_pname(F,A_1)))
=> ~ ! [X_1: nat] :
( ( B_1 = hAPP_nat_pname(F,X_1) )
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1)) ) ) ).
tff(fact_353_imageE,axiom,
! [B_1: x_a,F: fun_nat_a,A_1: fun_nat_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_nat_a(F,A_1)))
=> ~ ! [X_1: nat] :
( ( B_1 = hAPP_nat_a(F,X_1) )
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1)) ) ) ).
tff(fact_354_imageE,axiom,
! [B_1: nat,F: fun_pname_nat,A_1: fun_pname_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_pname_nat(F,A_1)))
=> ~ ! [X_1: pname] :
( ( B_1 = hAPP_pname_nat(F,X_1) )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1)) ) ) ).
tff(fact_355_imageE,axiom,
! [B_1: nat,F: fun_a_nat,A_1: fun_a_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),image_a_nat(F,A_1)))
=> ~ ! [X_1: x_a] :
( ( B_1 = hAPP_a_nat(F,X_1) )
=> ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1)) ) ) ).
tff(fact_356_imageE,axiom,
! [B_1: x_a,F: fun_pname_a,A_1: fun_pname_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A_1)))
=> ~ ! [X_1: pname] :
( ( B_1 = hAPP_pname_a(F,X_1) )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1)) ) ) ).
tff(fact_357_subsetI,axiom,
! [B_6: fun_nat_bool,A_1: fun_nat_bool] :
( ! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B_6)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_6)) ) ).
tff(fact_358_subsetI,axiom,
! [B_6: fun_pname_bool,A_1: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B_6)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),B_6)) ) ).
tff(fact_359_subsetI,axiom,
! [B_6: fun_a_bool,A_1: fun_a_bool] :
( ! [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B_6)) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),B_6)) ) ).
tff(fact_360_image__subsetI,axiom,
! [F: fun_pname_nat,B_6: fun_nat_bool,A_1: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_pname_nat(F,X_1)),B_6)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_pname_nat(F,A_1)),B_6)) ) ).
tff(fact_361_image__subsetI,axiom,
! [F: fun_pname_pname,B_6: fun_pname_bool,A_1: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_pname_pname(F,X_1)),B_6)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,image_pname_pname(F,A_1)),B_6)) ) ).
tff(fact_362_image__subsetI,axiom,
! [F: fun_a_nat,B_6: fun_nat_bool,A_1: fun_a_bool] :
( ! [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_a_nat(F,X_1)),B_6)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_a_nat(F,A_1)),B_6)) ) ).
tff(fact_363_image__subsetI,axiom,
! [F: fun_a_pname,B_6: fun_pname_bool,A_1: fun_a_bool] :
( ! [X_1: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_a_pname(F,X_1)),B_6)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,image_a_pname(F,A_1)),B_6)) ) ).
tff(fact_364_image__subsetI,axiom,
! [F: fun_nat_pname,B_6: fun_pname_bool,A_1: fun_nat_bool] :
( ! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,hAPP_nat_pname(F,X_1)),B_6)) )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,image_nat_pname(F,A_1)),B_6)) ) ).
tff(fact_365_image__subsetI,axiom,
! [F: fun_nat_a,B_6: fun_a_bool,A_1: fun_nat_bool] :
( ! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_nat_a(F,X_1)),B_6)) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_nat_a(F,A_1)),B_6)) ) ).
tff(fact_366_image__subsetI,axiom,
! [F: fun_pname_a,B_6: fun_a_bool,A_1: fun_pname_bool] :
( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_1)),B_6)) )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A_1)),B_6)) ) ).
tff(fact_367_order__refl,axiom,
! [X_3: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),X_3)) ).
tff(fact_368_order__refl,axiom,
! [X_3: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),X_3)) ).
tff(fact_369_order__refl,axiom,
! [X_3: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),X_3)) ).
tff(fact_370_order__refl,axiom,
! [X_3: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),X_3)) ).
tff(fact_371_order__refl,axiom,
! [X_10: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_10),X_10)) ).
tff(fact_372_finite__nat__set__iff__bounded__le,axiom,
! [N_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_1))
<=> ? [M_2: nat] :
! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),M_2)) ) ) ).
tff(fact_373_assms_I3_J,axiom,
! [G: fun_a_bool,C_1: com] :
( hBOOL(wt(C_1))
=> ( ! [X_1: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),u))
=> hBOOL(hAPP_fun_a_bool_bool(p(G),insert_a(hAPP_pname_a(mgt_call,X_1),bot_bot_fun_a_bool))) )
=> hBOOL(hAPP_fun_a_bool_bool(p(G),insert_a(mgt(C_1),bot_bot_fun_a_bool))) ) ) ).
tff(fact_374_le__fun__def,axiom,
! [F: fun_pname_bool,G_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F),G_1))
<=> ! [X_1: pname] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_pname_bool(F,X_1)),hAPP_pname_bool(G_1,X_1))) ) ).
tff(fact_375_le__fun__def,axiom,
! [F: fun_nat_bool,G_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
<=> ! [X_1: nat] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G_1,X_1))) ) ).
tff(fact_376_le__fun__def,axiom,
! [F: fun_a_bool,G_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
<=> ! [X_1: x_a] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_1)),hAPP_a_bool(G_1,X_1))) ) ).
tff(fact_377_le__funD,axiom,
! [X_3: pname,F: fun_pname_bool,G_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_pname_bool(F,X_3)),hAPP_pname_bool(G_1,X_3))) ) ).
tff(fact_378_le__funD,axiom,
! [X_3: nat,F: fun_nat_bool,G_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_3)),hAPP_nat_bool(G_1,X_3))) ) ).
tff(fact_379_le__funD,axiom,
! [X_3: x_a,F: fun_a_bool,G_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_3)),hAPP_a_bool(G_1,X_3))) ) ).
tff(fact_380_le__funE,axiom,
! [X_3: pname,F: fun_pname_bool,G_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_pname_bool(F,X_3)),hAPP_pname_bool(G_1,X_3))) ) ).
tff(fact_381_le__funE,axiom,
! [X_3: nat,F: fun_nat_bool,G_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_3)),hAPP_nat_bool(G_1,X_3))) ) ).
tff(fact_382_le__funE,axiom,
! [X_3: x_a,F: fun_a_bool,G_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F),G_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_a_bool(F,X_3)),hAPP_a_bool(G_1,X_3))) ) ).
tff(fact_383_emptyE,axiom,
! [A_3: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),bot_bot_fun_nat_bool)) ).
tff(fact_384_emptyE,axiom,
! [A_3: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),bot_bo844097828e_bool)) ).
tff(fact_385_emptyE,axiom,
! [A_3: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),bot_bot_fun_a_bool)) ).
tff(fact_386_finite_OemptyI,axiom,
hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,bot_bo1701429464l_bool)) ).
tff(fact_387_finite_OemptyI,axiom,
hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,bot_bo1649642514l_bool)) ).
tff(fact_388_finite_OemptyI,axiom,
hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,bot_bo1389914601l_bool)) ).
tff(fact_389_finite_OemptyI,axiom,
hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,bot_bot_fun_a_bool)) ).
tff(fact_390_finite_OemptyI,axiom,
hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,bot_bo844097828e_bool)) ).
tff(fact_391_finite_OemptyI,axiom,
hBOOL(hAPP_f54304608l_bool(finite_finite_nat,bot_bot_fun_nat_bool)) ).
tff(fact_392_empty__subsetI,axiom,
! [A_1: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,bot_bo844097828e_bool),A_1)) ).
tff(fact_393_empty__subsetI,axiom,
! [A_1: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_1)) ).
tff(fact_394_empty__subsetI,axiom,
! [A_1: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A_1)) ).
tff(fact_395_equals0D,axiom,
! [A_3: nat,A_1: fun_nat_bool] :
( ( A_1 = bot_bot_fun_nat_bool )
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_3),A_1)) ) ).
tff(fact_396_equals0D,axiom,
! [A_3: pname,A_1: fun_pname_bool] :
( ( A_1 = bot_bo844097828e_bool )
=> ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_3),A_1)) ) ).
tff(fact_397_equals0D,axiom,
! [A_3: x_a,A_1: fun_a_bool] :
( ( A_1 = bot_bot_fun_a_bool )
=> ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_3),A_1)) ) ).
tff(fact_398_Collect__empty__eq,axiom,
! [Pa: fun_pname_bool] :
( ( collect_pname(Pa) = bot_bo844097828e_bool )
<=> ! [X_1: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X_1)) ) ).
tff(fact_399_Collect__empty__eq,axiom,
! [Pa: fun_fu425979586l_bool] :
( ( collect_fun_nat_bool(Pa) = bot_bo1701429464l_bool )
<=> ! [X_1: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(Pa,X_1)) ) ).
tff(fact_400_Collect__empty__eq,axiom,
! [Pa: fun_fu1430349052l_bool] :
( ( collec1974731493e_bool(Pa) = bot_bo1649642514l_bool )
<=> ! [X_1: fun_pname_bool] : ~ hBOOL(hAPP_f1664156314l_bool(Pa,X_1)) ) ).
tff(fact_401_Collect__empty__eq,axiom,
! [Pa: fun_fun_a_bool_bool] :
( ( collect_fun_a_bool(Pa) = bot_bo1389914601l_bool )
<=> ! [X_1: fun_a_bool] : ~ hBOOL(hAPP_fun_a_bool_bool(Pa,X_1)) ) ).
tff(fact_402_Collect__empty__eq,axiom,
! [Pa: fun_a_bool] :
( ( collect_a(Pa) = bot_bot_fun_a_bool )
<=> ! [X_1: x_a] : ~ hBOOL(hAPP_a_bool(Pa,X_1)) ) ).
tff(fact_403_Collect__empty__eq,axiom,
! [Pa: fun_nat_bool] :
( ( collect_nat(Pa) = bot_bot_fun_nat_bool )
<=> ! [X_1: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_1)) ) ).
tff(fact_404_empty__iff,axiom,
! [C_1: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),bot_bot_fun_nat_bool)) ).
tff(fact_405_empty__iff,axiom,
! [C_1: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),bot_bo844097828e_bool)) ).
tff(fact_406_empty__iff,axiom,
! [C_1: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),bot_bot_fun_a_bool)) ).
tff(fact_407_empty__Collect__eq,axiom,
! [Pa: fun_pname_bool] :
( ( bot_bo844097828e_bool = collect_pname(Pa) )
<=> ! [X_1: pname] : ~ hBOOL(hAPP_pname_bool(Pa,X_1)) ) ).
tff(fact_408_empty__Collect__eq,axiom,
! [Pa: fun_fu425979586l_bool] :
( ( bot_bo1701429464l_bool = collect_fun_nat_bool(Pa) )
<=> ! [X_1: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(Pa,X_1)) ) ).
tff(fact_409_empty__Collect__eq,axiom,
! [Pa: fun_fu1430349052l_bool] :
( ( bot_bo1649642514l_bool = collec1974731493e_bool(Pa) )
<=> ! [X_1: fun_pname_bool] : ~ hBOOL(hAPP_f1664156314l_bool(Pa,X_1)) ) ).
tff(fact_410_empty__Collect__eq,axiom,
! [Pa: fun_fun_a_bool_bool] :
( ( bot_bo1389914601l_bool = collect_fun_a_bool(Pa) )
<=> ! [X_1: fun_a_bool] : ~ hBOOL(hAPP_fun_a_bool_bool(Pa,X_1)) ) ).
tff(fact_411_empty__Collect__eq,axiom,
! [Pa: fun_a_bool] :
( ( bot_bot_fun_a_bool = collect_a(Pa) )
<=> ! [X_1: x_a] : ~ hBOOL(hAPP_a_bool(Pa,X_1)) ) ).
tff(fact_412_empty__Collect__eq,axiom,
! [Pa: fun_nat_bool] :
( ( bot_bot_fun_nat_bool = collect_nat(Pa) )
<=> ! [X_1: nat] : ~ hBOOL(hAPP_nat_bool(Pa,X_1)) ) ).
tff(fact_413_ex__in__conv,axiom,
! [A_1: fun_nat_bool] :
( ? [X_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
<=> ( A_1 != bot_bot_fun_nat_bool ) ) ).
tff(fact_414_ex__in__conv,axiom,
! [A_1: fun_pname_bool] :
( ? [X_1: pname] : hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
<=> ( A_1 != bot_bo844097828e_bool ) ) ).
tff(fact_415_ex__in__conv,axiom,
! [A_1: fun_a_bool] :
( ? [X_1: x_a] : hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
<=> ( A_1 != bot_bot_fun_a_bool ) ) ).
tff(fact_416_all__not__in__conv,axiom,
! [A_1: fun_nat_bool] :
( ! [X_1: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1))
<=> ( A_1 = bot_bot_fun_nat_bool ) ) ).
tff(fact_417_all__not__in__conv,axiom,
! [A_1: fun_pname_bool] :
( ! [X_1: pname] : ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A_1))
<=> ( A_1 = bot_bo844097828e_bool ) ) ).
tff(fact_418_all__not__in__conv,axiom,
! [A_1: fun_a_bool] :
( ! [X_1: x_a] : ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A_1))
<=> ( A_1 = bot_bot_fun_a_bool ) ) ).
tff(fact_419_empty__def,axiom,
bot_bo844097828e_bool = collect_pname(cOMBK_bool_pname(fFalse)) ).
tff(fact_420_empty__def,axiom,
bot_bo1701429464l_bool = collect_fun_nat_bool(cOMBK_1994329625t_bool(fFalse)) ).
tff(fact_421_empty__def,axiom,
bot_bo1649642514l_bool = collec1974731493e_bool(cOMBK_1857069011e_bool(fFalse)) ).
tff(fact_422_empty__def,axiom,
bot_bo1389914601l_bool = collect_fun_a_bool(cOMBK_324466864a_bool(fFalse)) ).
tff(fact_423_empty__def,axiom,
bot_bot_fun_a_bool = collect_a(cOMBK_bool_a(fFalse)) ).
tff(fact_424_empty__def,axiom,
bot_bot_fun_nat_bool = collect_nat(cOMBK_bool_nat(fFalse)) ).
tff(fact_425_bot__fun__def,axiom,
! [X_1: pname] :
( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_426_bot__fun__def,axiom,
! [X_1: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_427_bot__fun__def,axiom,
! [X_1: x_a] :
( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_428_bot__apply,axiom,
! [X_3: pname] :
( hBOOL(hAPP_pname_bool(bot_bo844097828e_bool,X_3))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_429_bot__apply,axiom,
! [X_3: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_3))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_430_bot__apply,axiom,
! [X_3: x_a] :
( hBOOL(hAPP_a_bool(bot_bot_fun_a_bool,X_3))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_431_le__bot,axiom,
! [A_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_3),bot_bo844097828e_bool))
=> ( A_3 = bot_bo844097828e_bool ) ) ).
tff(fact_432_le__bot,axiom,
! [A_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_3),bot_bot_bool))
=> ( hBOOL(A_3)
<=> hBOOL(bot_bot_bool) ) ) ).
tff(fact_433_le__bot,axiom,
! [A_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_3),bot_bot_fun_nat_bool))
=> ( A_3 = bot_bot_fun_nat_bool ) ) ).
tff(fact_434_le__bot,axiom,
! [A_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_3),bot_bot_fun_a_bool))
=> ( A_3 = bot_bot_fun_a_bool ) ) ).
tff(fact_435_le__bot,axiom,
! [A_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),bot_bot_nat))
=> ( A_9 = bot_bot_nat ) ) ).
tff(fact_436_bot__unique,axiom,
! [A_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_3),bot_bo844097828e_bool))
<=> ( A_3 = bot_bo844097828e_bool ) ) ).
tff(fact_437_bot__unique,axiom,
! [A_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_3),bot_bot_bool))
<=> ( hBOOL(A_3)
<=> hBOOL(bot_bot_bool) ) ) ).
tff(fact_438_bot__unique,axiom,
! [A_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_3),bot_bot_fun_nat_bool))
<=> ( A_3 = bot_bot_fun_nat_bool ) ) ).
tff(fact_439_bot__unique,axiom,
! [A_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_3),bot_bot_fun_a_bool))
<=> ( A_3 = bot_bot_fun_a_bool ) ) ).
tff(fact_440_bot__unique,axiom,
! [A_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),bot_bot_nat))
<=> ( A_3 = bot_bot_nat ) ) ).
tff(fact_441_bot__least,axiom,
! [A_3: fun_pname_bool] : hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,bot_bo844097828e_bool),A_3)) ).
tff(fact_442_bot__least,axiom,
! [A_3: bool] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,bot_bot_bool),A_3)) ).
tff(fact_443_bot__least,axiom,
! [A_3: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,bot_bot_fun_nat_bool),A_3)) ).
tff(fact_444_bot__least,axiom,
! [A_3: fun_a_bool] : hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,bot_bot_fun_a_bool),A_3)) ).
tff(fact_445_bot__least,axiom,
! [A_8: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,bot_bot_nat),A_8)) ).
tff(fact_446_singleton__inject,axiom,
! [A_3: pname,B_1: pname] :
( ( insert_pname(A_3,bot_bo844097828e_bool) = insert_pname(B_1,bot_bo844097828e_bool) )
=> ( A_3 = B_1 ) ) ).
tff(fact_447_singleton__inject,axiom,
! [A_3: nat,B_1: nat] :
( ( insert_nat(A_3,bot_bot_fun_nat_bool) = insert_nat(B_1,bot_bot_fun_nat_bool) )
=> ( A_3 = B_1 ) ) ).
tff(fact_448_singleton__inject,axiom,
! [A_3: x_a,B_1: x_a] :
( ( insert_a(A_3,bot_bot_fun_a_bool) = insert_a(B_1,bot_bot_fun_a_bool) )
=> ( A_3 = B_1 ) ) ).
tff(fact_449_singletonE,axiom,
! [B_1: nat,A_3: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),insert_nat(A_3,bot_bot_fun_nat_bool)))
=> ( B_1 = A_3 ) ) ).
tff(fact_450_singletonE,axiom,
! [B_1: pname,A_3: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),insert_pname(A_3,bot_bo844097828e_bool)))
=> ( B_1 = A_3 ) ) ).
tff(fact_451_singletonE,axiom,
! [B_1: x_a,A_3: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),insert_a(A_3,bot_bot_fun_a_bool)))
=> ( B_1 = A_3 ) ) ).
tff(fact_452_doubleton__eq__iff,axiom,
! [A_3: pname,B_1: pname,C_1: pname,D: pname] :
( ( insert_pname(A_3,insert_pname(B_1,bot_bo844097828e_bool)) = insert_pname(C_1,insert_pname(D,bot_bo844097828e_bool)) )
<=> ( ( ( A_3 = C_1 )
& ( B_1 = D ) )
| ( ( A_3 = D )
& ( B_1 = C_1 ) ) ) ) ).
tff(fact_453_doubleton__eq__iff,axiom,
! [A_3: nat,B_1: nat,C_1: nat,D: nat] :
( ( insert_nat(A_3,insert_nat(B_1,bot_bot_fun_nat_bool)) = insert_nat(C_1,insert_nat(D,bot_bot_fun_nat_bool)) )
<=> ( ( ( A_3 = C_1 )
& ( B_1 = D ) )
| ( ( A_3 = D )
& ( B_1 = C_1 ) ) ) ) ).
tff(fact_454_doubleton__eq__iff,axiom,
! [A_3: x_a,B_1: x_a,C_1: x_a,D: x_a] :
( ( insert_a(A_3,insert_a(B_1,bot_bot_fun_a_bool)) = insert_a(C_1,insert_a(D,bot_bot_fun_a_bool)) )
<=> ( ( ( A_3 = C_1 )
& ( B_1 = D ) )
| ( ( A_3 = D )
& ( B_1 = C_1 ) ) ) ) ).
tff(fact_455_singleton__iff,axiom,
! [B_1: nat,A_3: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),insert_nat(A_3,bot_bot_fun_nat_bool)))
<=> ( B_1 = A_3 ) ) ).
tff(fact_456_singleton__iff,axiom,
! [B_1: pname,A_3: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,B_1),insert_pname(A_3,bot_bo844097828e_bool)))
<=> ( B_1 = A_3 ) ) ).
tff(fact_457_singleton__iff,axiom,
! [B_1: x_a,A_3: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),insert_a(A_3,bot_bot_fun_a_bool)))
<=> ( B_1 = A_3 ) ) ).
tff(fact_458_insert__not__empty,axiom,
! [A_3: pname,A_1: fun_pname_bool] : insert_pname(A_3,A_1) != bot_bo844097828e_bool ).
tff(fact_459_insert__not__empty,axiom,
! [A_3: nat,A_1: fun_nat_bool] : insert_nat(A_3,A_1) != bot_bot_fun_nat_bool ).
tff(fact_460_insert__not__empty,axiom,
! [A_3: x_a,A_1: fun_a_bool] : insert_a(A_3,A_1) != bot_bot_fun_a_bool ).
tff(fact_461_empty__not__insert,axiom,
! [A_3: pname,A_1: fun_pname_bool] : bot_bo844097828e_bool != insert_pname(A_3,A_1) ).
tff(fact_462_empty__not__insert,axiom,
! [A_3: nat,A_1: fun_nat_bool] : bot_bot_fun_nat_bool != insert_nat(A_3,A_1) ).
tff(fact_463_empty__not__insert,axiom,
! [A_3: x_a,A_1: fun_a_bool] : bot_bot_fun_a_bool != insert_a(A_3,A_1) ).
tff(fact_464_subset__empty,axiom,
! [A_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),bot_bot_fun_nat_bool))
<=> ( A_1 = bot_bot_fun_nat_bool ) ) ).
tff(fact_465_subset__empty,axiom,
! [A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),bot_bo844097828e_bool))
<=> ( A_1 = bot_bo844097828e_bool ) ) ).
tff(fact_466_subset__empty,axiom,
! [A_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),bot_bot_fun_a_bool))
<=> ( A_1 = bot_bot_fun_a_bool ) ) ).
tff(fact_467_image__is__empty,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool] :
( ( image_pname_a(F,A_1) = bot_bot_fun_a_bool )
<=> ( A_1 = bot_bo844097828e_bool ) ) ).
tff(fact_468_image__empty,axiom,
! [F: fun_pname_a] : image_pname_a(F,bot_bo844097828e_bool) = bot_bot_fun_a_bool ).
tff(fact_469_empty__is__image,axiom,
! [F: fun_pname_a,A_1: fun_pname_bool] :
( ( bot_bot_fun_a_bool = image_pname_a(F,A_1) )
<=> ( A_1 = bot_bo844097828e_bool ) ) ).
tff(fact_470_Collect__conv__if,axiom,
! [Pa: fun_nat_bool,A_3: nat] :
( ( hBOOL(hAPP_nat_bool(Pa,A_3))
=> ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_3)),Pa)) = insert_nat(A_3,bot_bot_fun_nat_bool) ) )
& ( ~ hBOOL(hAPP_nat_bool(Pa,A_3))
=> ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_3)),Pa)) = bot_bot_fun_nat_bool ) ) ) ).
tff(fact_471_Collect__conv__if,axiom,
! [Pa: fun_pname_bool,A_3: pname] :
( ( hBOOL(hAPP_pname_bool(Pa,A_3))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_3)),Pa)) = insert_pname(A_3,bot_bo844097828e_bool) ) )
& ( ~ hBOOL(hAPP_pname_bool(Pa,A_3))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_3)),Pa)) = bot_bo844097828e_bool ) ) ) ).
tff(fact_472_Collect__conv__if,axiom,
! [Pa: fun_a_bool,A_3: x_a] :
( ( hBOOL(hAPP_a_bool(Pa,A_3))
=> ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_3)),Pa)) = insert_a(A_3,bot_bot_fun_a_bool) ) )
& ( ~ hBOOL(hAPP_a_bool(Pa,A_3))
=> ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_3)),Pa)) = bot_bot_fun_a_bool ) ) ) ).
tff(fact_473_Collect__conv__if,axiom,
! [Pa: fun_fu425979586l_bool,A_3: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(Pa,A_3))
=> ( collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_3)),Pa)) = insert_fun_nat_bool(A_3,bot_bo1701429464l_bool) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(Pa,A_3))
=> ( collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_3)),Pa)) = bot_bo1701429464l_bool ) ) ) ).
tff(fact_474_Collect__conv__if,axiom,
! [Pa: fun_fu1430349052l_bool,A_3: fun_pname_bool] :
( ( hBOOL(hAPP_f1664156314l_bool(Pa,A_3))
=> ( collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_3)),Pa)) = insert1325755072e_bool(A_3,bot_bo1649642514l_bool) ) )
& ( ~ hBOOL(hAPP_f1664156314l_bool(Pa,A_3))
=> ( collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_3)),Pa)) = bot_bo1649642514l_bool ) ) ) ).
tff(fact_475_Collect__conv__if,axiom,
! [Pa: fun_fun_a_bool_bool,A_3: fun_a_bool] :
( ( hBOOL(hAPP_fun_a_bool_bool(Pa,A_3))
=> ( collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_3)),Pa)) = insert_fun_a_bool(A_3,bot_bo1389914601l_bool) ) )
& ( ~ hBOOL(hAPP_fun_a_bool_bool(Pa,A_3))
=> ( collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_3)),Pa)) = bot_bo1389914601l_bool ) ) ) ).
tff(fact_476_Collect__conv__if2,axiom,
! [Pa: fun_nat_bool,A_3: nat] :
( ( hBOOL(hAPP_nat_bool(Pa,A_3))
=> ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_3)),Pa)) = insert_nat(A_3,bot_bot_fun_nat_bool) ) )
& ( ~ hBOOL(hAPP_nat_bool(Pa,A_3))
=> ( collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_n1699378549t_bool(fequal_nat,A_3)),Pa)) = bot_bot_fun_nat_bool ) ) ) ).
tff(fact_477_Collect__conv__if2,axiom,
! [Pa: fun_pname_bool,A_3: pname] :
( ( hBOOL(hAPP_pname_bool(Pa,A_3))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(fequal_pname,A_3)),Pa)) = insert_pname(A_3,bot_bo844097828e_bool) ) )
& ( ~ hBOOL(hAPP_pname_bool(Pa,A_3))
=> ( collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_p61793385e_bool(fequal_pname,A_3)),Pa)) = bot_bo844097828e_bool ) ) ) ).
tff(fact_478_Collect__conv__if2,axiom,
! [Pa: fun_a_bool,A_3: x_a] :
( ( hBOOL(hAPP_a_bool(Pa,A_3))
=> ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_3)),Pa)) = insert_a(A_3,bot_bot_fun_a_bool) ) )
& ( ~ hBOOL(hAPP_a_bool(Pa,A_3))
=> ( collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,hAPP_a_fun_a_bool(fequal_a,A_3)),Pa)) = bot_bot_fun_a_bool ) ) ) ).
tff(fact_479_Collect__conv__if2,axiom,
! [Pa: fun_fu425979586l_bool,A_3: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(Pa,A_3))
=> ( collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f103356543l_bool(fequal_fun_nat_bool,A_3)),Pa)) = insert_fun_nat_bool(A_3,bot_bo1701429464l_bool) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(Pa,A_3))
=> ( collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,hAPP_f103356543l_bool(fequal_fun_nat_bool,A_3)),Pa)) = bot_bo1701429464l_bool ) ) ) ).
tff(fact_480_Collect__conv__if2,axiom,
! [Pa: fun_fu1430349052l_bool,A_3: fun_pname_bool] :
( ( hBOOL(hAPP_f1664156314l_bool(Pa,A_3))
=> ( collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f434788991l_bool(fequal533582459e_bool,A_3)),Pa)) = insert1325755072e_bool(A_3,bot_bo1649642514l_bool) ) )
& ( ~ hBOOL(hAPP_f1664156314l_bool(Pa,A_3))
=> ( collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,hAPP_f434788991l_bool(fequal533582459e_bool,A_3)),Pa)) = bot_bo1649642514l_bool ) ) ) ).
tff(fact_481_Collect__conv__if2,axiom,
! [Pa: fun_fun_a_bool_bool,A_3: fun_a_bool] :
( ( hBOOL(hAPP_fun_a_bool_bool(Pa,A_3))
=> ( collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f1631501043l_bool(fequal_fun_a_bool,A_3)),Pa)) = insert_fun_a_bool(A_3,bot_bo1389914601l_bool) ) )
& ( ~ hBOOL(hAPP_fun_a_bool_bool(Pa,A_3))
=> ( collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,hAPP_f1631501043l_bool(fequal_fun_a_bool,A_3)),Pa)) = bot_bo1389914601l_bool ) ) ) ).
tff(fact_482_singleton__conv,axiom,
! [A_3: nat] : collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_3)) = insert_nat(A_3,bot_bot_fun_nat_bool) ).
tff(fact_483_singleton__conv,axiom,
! [A_3: pname] : collect_pname(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_3)) = insert_pname(A_3,bot_bo844097828e_bool) ).
tff(fact_484_singleton__conv,axiom,
! [A_3: x_a] : collect_a(hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_3)) = insert_a(A_3,bot_bot_fun_a_bool) ).
tff(fact_485_singleton__conv,axiom,
! [A_3: fun_nat_bool] : collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_3)) = insert_fun_nat_bool(A_3,bot_bo1701429464l_bool) ).
tff(fact_486_singleton__conv,axiom,
! [A_3: fun_pname_bool] : collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_3)) = insert1325755072e_bool(A_3,bot_bo1649642514l_bool) ).
tff(fact_487_singleton__conv,axiom,
! [A_3: fun_a_bool] : collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_3)) = insert_fun_a_bool(A_3,bot_bo1389914601l_bool) ).
tff(fact_488_singleton__conv2,axiom,
! [A_3: nat] : collect_nat(hAPP_n1699378549t_bool(fequal_nat,A_3)) = insert_nat(A_3,bot_bot_fun_nat_bool) ).
tff(fact_489_singleton__conv2,axiom,
! [A_3: pname] : collect_pname(hAPP_p61793385e_bool(fequal_pname,A_3)) = insert_pname(A_3,bot_bo844097828e_bool) ).
tff(fact_490_singleton__conv2,axiom,
! [A_3: x_a] : collect_a(hAPP_a_fun_a_bool(fequal_a,A_3)) = insert_a(A_3,bot_bot_fun_a_bool) ).
tff(fact_491_singleton__conv2,axiom,
! [A_3: fun_nat_bool] : collect_fun_nat_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,A_3)) = insert_fun_nat_bool(A_3,bot_bo1701429464l_bool) ).
tff(fact_492_singleton__conv2,axiom,
! [A_3: fun_pname_bool] : collec1974731493e_bool(hAPP_f434788991l_bool(fequal533582459e_bool,A_3)) = insert1325755072e_bool(A_3,bot_bo1649642514l_bool) ).
tff(fact_493_singleton__conv2,axiom,
! [A_3: fun_a_bool] : collect_fun_a_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,A_3)) = insert_fun_a_bool(A_3,bot_bo1389914601l_bool) ).
tff(fact_494_subset__singletonD,axiom,
! [A_1: fun_nat_bool,X_3: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),insert_nat(X_3,bot_bot_fun_nat_bool)))
=> ( ( A_1 = bot_bot_fun_nat_bool )
| ( A_1 = insert_nat(X_3,bot_bot_fun_nat_bool) ) ) ) ).
tff(fact_495_subset__singletonD,axiom,
! [A_1: fun_pname_bool,X_3: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_1),insert_pname(X_3,bot_bo844097828e_bool)))
=> ( ( A_1 = bot_bo844097828e_bool )
| ( A_1 = insert_pname(X_3,bot_bo844097828e_bool) ) ) ) ).
tff(fact_496_subset__singletonD,axiom,
! [A_1: fun_a_bool,X_3: x_a] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_1),insert_a(X_3,bot_bot_fun_a_bool)))
=> ( ( A_1 = bot_bot_fun_a_bool )
| ( A_1 = insert_a(X_3,bot_bot_fun_a_bool) ) ) ) ).
tff(fact_497_image__constant,axiom,
! [C_1: x_a,X_3: pname,A_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_3),A_1))
=> ( image_pname_a(cOMBK_a_pname(C_1),A_1) = insert_a(C_1,bot_bot_fun_a_bool) ) ) ).
tff(fact_498_image__constant__conv,axiom,
! [C_1: x_a,A_1: fun_pname_bool] :
( ( ( A_1 = bot_bo844097828e_bool )
=> ( image_pname_a(cOMBK_a_pname(C_1),A_1) = bot_bot_fun_a_bool ) )
& ( ( A_1 != bot_bo844097828e_bool )
=> ( image_pname_a(cOMBK_a_pname(C_1),A_1) = insert_a(C_1,bot_bot_fun_a_bool) ) ) ) ).
tff(fact_499_linorder__le__cases,axiom,
! [X_9: nat,Y_8: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_9),Y_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_8),X_9)) ) ).
tff(fact_500_xt1_I6_J,axiom,
! [Z_1: bool,Y_2: bool,X_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_3))
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Z_1),Y_2))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Z_1),X_3)) ) ) ).
tff(fact_501_xt1_I6_J,axiom,
! [Z_1: fun_nat_bool,Y_2: fun_nat_bool,X_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),Y_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_1),X_3)) ) ) ).
tff(fact_502_xt1_I6_J,axiom,
! [Z_1: fun_pname_bool,Y_2: fun_pname_bool,X_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Z_1),Y_2))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Z_1),X_3)) ) ) ).
tff(fact_503_xt1_I6_J,axiom,
! [Z_3: nat,Y_7: nat,X_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_7),X_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_3),Y_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_3),X_8)) ) ) ).
tff(fact_504_xt1_I6_J,axiom,
! [Z_1: fun_a_bool,Y_2: fun_a_bool,X_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),X_3))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_1),Y_2))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Z_1),X_3)) ) ) ).
tff(fact_505_xt1_I5_J,axiom,
! [Y_2: bool,X_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_3))
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2))
=> ( hBOOL(X_3)
<=> hBOOL(Y_2) ) ) ) ).
tff(fact_506_xt1_I5_J,axiom,
! [Y_2: fun_nat_bool,X_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_507_xt1_I5_J,axiom,
! [Y_2: fun_pname_bool,X_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_508_xt1_I5_J,axiom,
! [Y_6: nat,X_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_6),X_7))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_7),Y_6))
=> ( X_7 = Y_6 ) ) ) ).
tff(fact_509_xt1_I5_J,axiom,
! [Y_2: fun_a_bool,X_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),X_3))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_510_order__trans,axiom,
! [Z_1: bool,X_3: bool,Y_2: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2))
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),Z_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Z_1)) ) ) ).
tff(fact_511_order__trans,axiom,
! [Z_1: fun_nat_bool,X_3: fun_nat_bool,Y_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),Z_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Z_1)) ) ) ).
tff(fact_512_order__trans,axiom,
! [Z_1: fun_pname_bool,X_3: fun_pname_bool,Y_2: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),Z_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Z_1)) ) ) ).
tff(fact_513_order__trans,axiom,
! [Z_2: nat,X_6: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Y_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_5),Z_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_6),Z_2)) ) ) ).
tff(fact_514_order__trans,axiom,
! [Z_1: fun_a_bool,X_3: fun_a_bool,Y_2: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),Z_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Z_1)) ) ) ).
tff(fact_515_order__antisym,axiom,
! [X_3: bool,Y_2: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2))
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_3))
=> ( hBOOL(X_3)
<=> hBOOL(Y_2) ) ) ) ).
tff(fact_516_order__antisym,axiom,
! [X_3: fun_nat_bool,Y_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_3))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_517_order__antisym,axiom,
! [X_3: fun_pname_bool,Y_2: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),X_3))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_518_order__antisym,axiom,
! [X_5: nat,Y_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_5),Y_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_4),X_5))
=> ( X_5 = Y_4 ) ) ) ).
tff(fact_519_order__antisym,axiom,
! [X_3: fun_a_bool,Y_2: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),X_3))
=> ( X_3 = Y_2 ) ) ) ).
tff(fact_520_xt1_I4_J,axiom,
! [C_1: bool,B_1: bool,A_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B_1),A_3))
=> ( ( hBOOL(B_1)
<=> hBOOL(C_1) )
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,C_1),A_3)) ) ) ).
tff(fact_521_xt1_I4_J,axiom,
! [C_1: fun_nat_bool,B_1: fun_nat_bool,A_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_3))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_1),A_3)) ) ) ).
tff(fact_522_xt1_I4_J,axiom,
! [C_1: fun_pname_bool,B_1: fun_pname_bool,A_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_1),A_3))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_1),A_3)) ) ) ).
tff(fact_523_xt1_I4_J,axiom,
! [C_5: nat,B_5: nat,A_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_5),A_7))
=> ( ( B_5 = C_5 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_5),A_7)) ) ) ).
tff(fact_524_xt1_I4_J,axiom,
! [C_1: fun_a_bool,B_1: fun_a_bool,A_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_1),A_3))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_1),A_3)) ) ) ).
tff(fact_525_ord__le__eq__trans,axiom,
! [C_1: bool,A_3: bool,B_1: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_3),B_1))
=> ( ( hBOOL(B_1)
<=> hBOOL(C_1) )
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_3),C_1)) ) ) ).
tff(fact_526_ord__le__eq__trans,axiom,
! [C_1: fun_nat_bool,A_3: fun_nat_bool,B_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_3),B_1))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_3),C_1)) ) ) ).
tff(fact_527_ord__le__eq__trans,axiom,
! [C_1: fun_pname_bool,A_3: fun_pname_bool,B_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_3),B_1))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_3),C_1)) ) ) ).
tff(fact_528_ord__le__eq__trans,axiom,
! [C_4: nat,A_6: nat,B_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_6),B_4))
=> ( ( B_4 = C_4 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_6),C_4)) ) ) ).
tff(fact_529_ord__le__eq__trans,axiom,
! [C_1: fun_a_bool,A_3: fun_a_bool,B_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_3),B_1))
=> ( ( B_1 = C_1 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_3),C_1)) ) ) ).
tff(fact_530_xt1_I3_J,axiom,
! [C_1: bool,B_1: bool,A_3: bool] :
( ( hBOOL(A_3)
<=> hBOOL(B_1) )
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,C_1),B_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,C_1),A_3)) ) ) ).
tff(fact_531_xt1_I3_J,axiom,
! [C_1: fun_nat_bool,A_3: fun_nat_bool,B_1: fun_nat_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_1),B_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C_1),A_3)) ) ) ).
tff(fact_532_xt1_I3_J,axiom,
! [C_1: fun_pname_bool,A_3: fun_pname_bool,B_1: fun_pname_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_1),B_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_1),A_3)) ) ) ).
tff(fact_533_xt1_I3_J,axiom,
! [C_3: nat,A_5: nat,B_3: nat] :
( ( A_5 = B_3 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),B_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_3),A_5)) ) ) ).
tff(fact_534_xt1_I3_J,axiom,
! [C_1: fun_a_bool,A_3: fun_a_bool,B_1: fun_a_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_1),B_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C_1),A_3)) ) ) ).
tff(fact_535_ord__eq__le__trans,axiom,
! [C_1: bool,B_1: bool,A_3: bool] :
( ( hBOOL(A_3)
<=> hBOOL(B_1) )
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,B_1),C_1))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,A_3),C_1)) ) ) ).
tff(fact_536_ord__eq__le__trans,axiom,
! [C_1: fun_nat_bool,A_3: fun_nat_bool,B_1: fun_nat_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),C_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_3),C_1)) ) ) ).
tff(fact_537_ord__eq__le__trans,axiom,
! [C_1: fun_pname_bool,A_3: fun_pname_bool,B_1: fun_pname_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B_1),C_1))
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A_3),C_1)) ) ) ).
tff(fact_538_ord__eq__le__trans,axiom,
! [C_2: nat,A_4: nat,B_2: nat] :
( ( A_4 = B_2 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_2),C_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_4),C_2)) ) ) ).
tff(fact_539_ord__eq__le__trans,axiom,
! [C_1: fun_a_bool,A_3: fun_a_bool,B_1: fun_a_bool] :
( ( A_3 = B_1 )
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B_1),C_1))
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A_3),C_1)) ) ) ).
tff(fact_540_order__antisym__conv,axiom,
! [Y_2: bool,X_3: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_3))
=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2))
<=> ( hBOOL(X_3)
<=> hBOOL(Y_2) ) ) ) ).
tff(fact_541_order__antisym__conv,axiom,
! [Y_2: fun_nat_bool,X_3: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2))
<=> ( X_3 = Y_2 ) ) ) ).
tff(fact_542_order__antisym__conv,axiom,
! [Y_2: fun_pname_bool,X_3: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),X_3))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2))
<=> ( X_3 = Y_2 ) ) ) ).
tff(fact_543_order__antisym__conv,axiom,
! [Y_2: nat,X_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Y_2))
<=> ( X_3 = Y_2 ) ) ) ).
tff(fact_544_order__antisym__conv,axiom,
! [Y_2: fun_a_bool,X_3: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),X_3))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2))
<=> ( X_3 = Y_2 ) ) ) ).
tff(fact_545_order__eq__refl,axiom,
! [Y_2: bool,X_3: bool] :
( ( hBOOL(X_3)
<=> hBOOL(Y_2) )
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2)) ) ).
tff(fact_546_order__eq__refl,axiom,
! [X_3: fun_nat_bool,Y_2: fun_nat_bool] :
( ( X_3 = Y_2 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2)) ) ).
tff(fact_547_order__eq__refl,axiom,
! [X_3: fun_pname_bool,Y_2: fun_pname_bool] :
( ( X_3 = Y_2 )
=> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2)) ) ).
tff(fact_548_order__eq__refl,axiom,
! [X_4: nat,Y_3: nat] :
( ( X_4 = Y_3 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Y_3)) ) ).
tff(fact_549_order__eq__refl,axiom,
! [X_3: fun_a_bool,Y_2: fun_a_bool] :
( ( X_3 = Y_2 )
=> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2)) ) ).
tff(fact_550_order__eq__iff,axiom,
! [Y_2: bool,X_3: bool] :
( ( hBOOL(X_3)
<=> hBOOL(Y_2) )
<=> ( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,X_3),Y_2))
& hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,Y_2),X_3)) ) ) ).
tff(fact_551_order__eq__iff,axiom,
! [X_3: fun_nat_bool,Y_2: fun_nat_bool] :
( ( X_3 = Y_2 )
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_3),Y_2))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Y_2),X_3)) ) ) ).
tff(fact_552_order__eq__iff,axiom,
! [X_3: fun_pname_bool,Y_2: fun_pname_bool] :
( ( X_3 = Y_2 )
<=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_3),Y_2))
& hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,Y_2),X_3)) ) ) ).
tff(fact_553_order__eq__iff,axiom,
! [X_3: nat,Y_2: nat] :
( ( X_3 = Y_2 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_3)) ) ) ).
tff(fact_554_order__eq__iff,axiom,
! [X_3: fun_a_bool,Y_2: fun_a_bool] :
( ( X_3 = Y_2 )
<=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_3),Y_2))
& hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Y_2),X_3)) ) ) ).
tff(fact_555_linorder__linear,axiom,
! [X_2: nat,Y_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_1),X_2)) ) ).
tff(fact_556_finite__subset__induct,axiom,
! [Pa: fun_fu425979586l_bool,A_1: fun_nat_bool,F_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F_1),A_1))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,bot_bot_fun_nat_bool))
=> ( ! [A_2: nat,F_2: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_2))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),A_1))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_2),F_2))
=> ( hBOOL(hAPP_f54304608l_bool(Pa,F_2))
=> hBOOL(hAPP_f54304608l_bool(Pa,insert_nat(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_f54304608l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_557_finite__subset__induct,axiom,
! [Pa: fun_fu1430349052l_bool,A_1: fun_pname_bool,F_1: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,F_1),A_1))
=> ( hBOOL(hAPP_f1664156314l_bool(Pa,bot_bo844097828e_bool))
=> ( ! [A_2: pname,F_2: fun_pname_bool] :
( hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_2))
=> ( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),A_1))
=> ( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_2),F_2))
=> ( hBOOL(hAPP_f1664156314l_bool(Pa,F_2))
=> hBOOL(hAPP_f1664156314l_bool(Pa,insert_pname(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_f1664156314l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_558_finite__subset__induct,axiom,
! [Pa: fun_fun_a_bool_bool,A_1: fun_a_bool,F_1: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,F_1),A_1))
=> ( hBOOL(hAPP_fun_a_bool_bool(Pa,bot_bot_fun_a_bool))
=> ( ! [A_2: x_a,F_2: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_2))
=> ( hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),A_1))
=> ( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_2),F_2))
=> ( hBOOL(hAPP_fun_a_bool_bool(Pa,F_2))
=> hBOOL(hAPP_fun_a_bool_bool(Pa,insert_a(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_fun_a_bool_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_559_finite__subset__induct,axiom,
! [Pa: fun_fu814369080l_bool,A_1: fun_fu425979586l_bool,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,F_1),A_1))
=> ( hBOOL(hAPP_f1637334154l_bool(Pa,bot_bo1701429464l_bool))
=> ( ! [A_2: fun_nat_bool,F_2: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_2))
=> ( hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,A_2),A_1))
=> ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,A_2),F_2))
=> ( hBOOL(hAPP_f1637334154l_bool(Pa,F_2))
=> hBOOL(hAPP_f1637334154l_bool(Pa,insert_fun_nat_bool(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_f1637334154l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_560_finite__subset__induct,axiom,
! [Pa: fun_fu399576434l_bool,A_1: fun_fu1430349052l_bool,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,F_1),A_1))
=> ( hBOOL(hAPP_f1935102916l_bool(Pa,bot_bo1649642514l_bool))
=> ( ! [A_2: fun_pname_bool,F_2: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_2))
=> ( hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,A_2),A_1))
=> ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,A_2),F_2))
=> ( hBOOL(hAPP_f1935102916l_bool(Pa,F_2))
=> hBOOL(hAPP_f1935102916l_bool(Pa,insert1325755072e_bool(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_f1935102916l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_561_finite__subset__induct,axiom,
! [Pa: fun_fu885608257l_bool,A_1: fun_fun_a_bool_bool,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,F_1),A_1))
=> ( hBOOL(hAPP_f621171935l_bool(Pa,bot_bo1389914601l_bool))
=> ( ! [A_2: fun_a_bool,F_2: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_2))
=> ( hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,A_2),A_1))
=> ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,A_2),F_2))
=> ( hBOOL(hAPP_f621171935l_bool(Pa,F_2))
=> hBOOL(hAPP_f621171935l_bool(Pa,insert_fun_a_bool(A_2,F_2))) ) ) ) )
=> hBOOL(hAPP_f621171935l_bool(Pa,F_1)) ) ) ) ) ).
tff(fact_562_assms_I2_J,axiom,
! [Pn: pname,G: fun_a_bool] :
( hBOOL(hAPP_fun_a_bool_bool(p(insert_a(hAPP_pname_a(mgt_call,Pn),G)),insert_a(mgt(the_com(body(Pn))),bot_bot_fun_a_bool)))
=> hBOOL(hAPP_fun_a_bool_bool(p(G),insert_a(hAPP_pname_a(mgt_call,Pn),bot_bot_fun_a_bool))) ) ).
tff(fact_563_finite__induct,axiom,
! [Pa: fun_fu814369080l_bool,F_1: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1))
=> ( hBOOL(hAPP_f1637334154l_bool(Pa,bot_bo1701429464l_bool))
=> ( ! [X_1: fun_nat_bool,F_2: fun_fu425979586l_bool] :
( hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_2))
=> ( ~ hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_1),F_2))
=> ( hBOOL(hAPP_f1637334154l_bool(Pa,F_2))
=> hBOOL(hAPP_f1637334154l_bool(Pa,insert_fun_nat_bool(X_1,F_2))) ) ) )
=> hBOOL(hAPP_f1637334154l_bool(Pa,F_1)) ) ) ) ).
tff(fact_564_finite__induct,axiom,
! [Pa: fun_fu399576434l_bool,F_1: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1))
=> ( hBOOL(hAPP_f1935102916l_bool(Pa,bot_bo1649642514l_bool))
=> ( ! [X_1: fun_pname_bool,F_2: fun_fu1430349052l_bool] :
( hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_2))
=> ( ~ hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_1),F_2))
=> ( hBOOL(hAPP_f1935102916l_bool(Pa,F_2))
=> hBOOL(hAPP_f1935102916l_bool(Pa,insert1325755072e_bool(X_1,F_2))) ) ) )
=> hBOOL(hAPP_f1935102916l_bool(Pa,F_1)) ) ) ) ).
tff(fact_565_finite__induct,axiom,
! [Pa: fun_fu885608257l_bool,F_1: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1))
=> ( hBOOL(hAPP_f621171935l_bool(Pa,bot_bo1389914601l_bool))
=> ( ! [X_1: fun_a_bool,F_2: fun_fun_a_bool_bool] :
( hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_2))
=> ( ~ hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_1),F_2))
=> ( hBOOL(hAPP_f621171935l_bool(Pa,F_2))
=> hBOOL(hAPP_f621171935l_bool(Pa,insert_fun_a_bool(X_1,F_2))) ) ) )
=> hBOOL(hAPP_f621171935l_bool(Pa,F_1)) ) ) ) ).
tff(fact_566_finite__less__ub,axiom,
! [U: nat,F: fun_nat_nat] :
( ! [N_2: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(cOMBB_800536526ol_nat(ord_less_eq_nat,F)),U)))) ) ).
tff(fact_567_assms_I4_J,axiom,
! [Pn: pname] :
( hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,Pn),u))
=> hBOOL(wt(the_com(body(Pn)))) ) ).
tff(fact_568_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N) ).
tff(fact_569_diff__Suc__1,axiom,
! [N: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,N)),one_one_nat) = N ).
tff(fact_570_less__eq__nat_Osimps_I2_J,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),Na))
<=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M_1)),Na)) ) ).
tff(fact_571_add__Suc__right,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N)) ).
tff(fact_572_add__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),N)) ).
tff(fact_573_add__Suc__shift,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(suc,N)) ).
tff(fact_574_nat__add__right__cancel,axiom,
! [M_1: nat,K_2: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),K_2) = hAPP_nat_nat(plus_plus_nat(Na),K_2) )
<=> ( M_1 = Na ) ) ).
tff(fact_575_nat__add__left__cancel,axiom,
! [K_2: nat,M_1: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(K_2),M_1) = hAPP_nat_nat(plus_plus_nat(K_2),Na) )
<=> ( M_1 = Na ) ) ).
tff(fact_576_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] : hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M),N)),K) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(plus_plus_nat(N),K)) ).
tff(fact_577_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] : hAPP_nat_nat(plus_plus_nat(X),hAPP_nat_nat(plus_plus_nat(Y),Z)) = hAPP_nat_nat(plus_plus_nat(Y),hAPP_nat_nat(plus_plus_nat(X),Z)) ).
tff(fact_578_nat__add__commute,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(plus_plus_nat(M),N) = hAPP_nat_nat(plus_plus_nat(N),M) ).
tff(fact_579_diff__add__inverse2,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),N)),N) = M ).
tff(fact_580_diff__add__inverse,axiom,
! [N: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N),M)),N) = M ).
tff(fact_581_diff__diff__left,axiom,
! [I: nat,J_2: nat,K: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I),J_2)),K) = hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(plus_plus_nat(J_2),K)) ).
tff(fact_582_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(K),M)),hAPP_nat_nat(plus_plus_nat(K),N)) = hAPP_nat_nat(minus_minus_nat(M),N) ).
tff(fact_583_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(plus_plus_nat(N),K)) = hAPP_nat_nat(minus_minus_nat(M),N) ).
tff(fact_584_le__add2,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(plus_plus_nat(M),N))) ).
tff(fact_585_le__add1,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(plus_plus_nat(N),M))) ).
tff(fact_586_le__iff__add,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
<=> ? [K_1: nat] : Na = hAPP_nat_nat(plus_plus_nat(M_1),K_1) ) ).
tff(fact_587_nat__add__left__cancel__le,axiom,
! [K_2: nat,M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K_2),M_1)),hAPP_nat_nat(plus_plus_nat(K_2),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)) ) ).
tff(fact_588_trans__le__add1,axiom,
! [M: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(J_2),M))) ) ).
tff(fact_589_trans__le__add2,axiom,
! [M: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(plus_plus_nat(M),J_2))) ) ).
tff(fact_590_add__le__mono1,axiom,
! [K: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))) ) ).
tff(fact_591_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J_2),L))) ) ) ).
tff(fact_592_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ).
tff(fact_593_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_594_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K)),N))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ) ).
tff(fact_595_diff__add__assoc2,axiom,
! [I: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_2),I)),K) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_2),K)),I) ) ) ).
tff(fact_596_add__diff__assoc2,axiom,
! [I: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_2),K)),I) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_2),I)),K) ) ) ).
tff(fact_597_diff__add__assoc,axiom,
! [I: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J_2)),K) = hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J_2),K)) ) ) ).
tff(fact_598_le__imp__diff__is__add,axiom,
! [K_2: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( ( hAPP_nat_nat(minus_minus_nat(J_1),I_1) = K_2 )
<=> ( J_1 = hAPP_nat_nat(plus_plus_nat(K_2),I_1) ) ) ) ).
tff(fact_599_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),N)),N) = M ) ) ).
tff(fact_600_le__diff__conv2,axiom,
! [I_1: nat,K_2: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_2)),J_1)) ) ) ).
tff(fact_601_add__diff__assoc,axiom,
! [I: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(plus_plus_nat(I),hAPP_nat_nat(minus_minus_nat(J_2),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),J_2)),K) ) ) ).
tff(fact_602_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(plus_plus_nat(N),hAPP_nat_nat(minus_minus_nat(M),N)) = M ) ) ).
tff(fact_603_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N),M)),K))) ) ).
tff(fact_604_le__diff__conv,axiom,
! [J_1: nat,K_2: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(J_1),K_2)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(plus_plus_nat(I_1),K_2))) ) ).
tff(fact_605_diff__diff__right,axiom,
! [I: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(I),hAPP_nat_nat(minus_minus_nat(J_2),K)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I),K)),J_2) ) ) ).
tff(fact_606_Suc__eq__plus1,axiom,
! [N: nat] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(plus_plus_nat(N),one_one_nat) ).
tff(fact_607_Suc__eq__plus1__left,axiom,
! [N: nat] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(plus_plus_nat(one_one_nat),N) ).
tff(fact_608_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J_2),K))),M) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(plus_plus_nat(K),M)) ) ) ).
tff(fact_609_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2))
=> ( hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(J_2),K))) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K)),hAPP_nat_nat(suc,J_2)) ) ) ).
tff(fact_610_termination__basic__simps_I3_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z))) ) ).
tff(fact_611_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z))) ) ).
tff(fact_612_lessI,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).
tff(fact_613_Suc__mono,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) ) ).
tff(fact_614_finite__Collect__less__nat,axiom,
! [K_2: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),K_2)))) ).
tff(fact_615_termination__basic__simps_I1_J,axiom,
! [Z: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z))) ) ).
tff(fact_616_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(plus_plus_nat(Y),Z))) ) ).
tff(fact_617_add__lessD1,axiom,
! [I: nat,J_2: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J_2)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
tff(fact_618_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> ( ( hAPP_nat_nat(plus_plus_nat(M),L) = hAPP_nat_nat(plus_plus_nat(K),N) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_619_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J_2),L))) ) ) ).
tff(fact_620_add__less__mono1,axiom,
! [K: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),K)),hAPP_nat_nat(plus_plus_nat(J_2),K))) ) ).
tff(fact_621_trans__less__add2,axiom,
! [M: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(M),J_2))) ) ).
tff(fact_622_trans__less__add1,axiom,
! [M: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(plus_plus_nat(J_2),M))) ) ).
tff(fact_623_nat__add__left__cancel__less,axiom,
! [K_2: nat,M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K_2),M_1)),hAPP_nat_nat(plus_plus_nat(K_2),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) ) ).
tff(fact_624_not__add__less2,axiom,
! [J_2: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J_2),I)),I)) ).
tff(fact_625_not__add__less1,axiom,
! [I: nat,J_2: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I),J_2)),I)) ).
tff(fact_626_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_627_Suc__lessD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_628_less__SucE,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( M = N ) ) ) ).
tff(fact_629_less__trans__Suc,axiom,
! [K: nat,I: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K)) ) ) ).
tff(fact_630_Suc__lessI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( ( hAPP_nat_nat(suc,M) != N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) ) ) ).
tff(fact_631_less__SucI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_632_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
=> ( M = N ) ) ) ).
tff(fact_633_not__less__less__Suc__eq,axiom,
! [Na: nat,M_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M_1)))
<=> ( Na = M_1 ) ) ) ).
tff(fact_634_Suc__less__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) ) ).
tff(fact_635_less__Suc__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
| ( M_1 = Na ) ) ) ).
tff(fact_636_not__less__eq,axiom,
! [M_1: nat,Na: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M_1))) ) ).
tff(fact_637_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| ( M = N ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_638_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( ( M != N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_639_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_640_le__eq__less__or__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
| ( M_1 = Na ) ) ) ).
tff(fact_641_nat__less__le,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
& ( M_1 != Na ) ) ) ).
tff(fact_642_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(L),N)),hAPP_nat_nat(minus_minus_nat(L),M))) ) ) ).
tff(fact_643_less__imp__diff__less,axiom,
! [N: nat,J_2: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(J_2),N)),K)) ) ).
tff(fact_644_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
tff(fact_645_less__not__refl,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_646_nat__neq__iff,axiom,
! [M_1: nat,Na: nat] :
( ( M_1 != Na )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M_1)) ) ) ).
tff(fact_647_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
tff(fact_648_less__irrefl__nat,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_649_less__not__refl2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
tff(fact_650_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_651_nat__less__cases,axiom,
! [Pa: fun_nat_fun_nat_bool,M_1: nat,Na: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M_1)) )
=> ( ( ( M_1 = Na )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(Pa,Na),M_1)) ) ) ) ).
tff(fact_652_finite__nat__set__iff__bounded,axiom,
! [N_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_1))
<=> ? [M_2: nat] :
! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),M_2)) ) ) ).
tff(fact_653_card__Collect__less__nat,axiom,
! [Na: nat] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),Na))) = Na ).
tff(fact_654_finite__M__bounded__by__nat,axiom,
! [Pa: fun_nat_bool,I_1: nat] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_nat),I_1))))) ).
tff(fact_655_less__add__Suc1,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(I),M)))) ).
tff(fact_656_less__add__Suc2,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),I)))) ).
tff(fact_657_less__iff__Suc__add,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
<=> ? [K_1: nat] : Na = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M_1),K_1)) ) ).
tff(fact_658_less__eq__Suc__le,axiom,
! [Na: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),M_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M_1)) ) ).
tff(fact_659_less__Suc__eq__le,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)) ) ).
tff(fact_660_Suc__le__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) ) ).
tff(fact_661_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_662_Suc__leI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) ) ).
tff(fact_663_le__less__Suc__eq,axiom,
! [M_1: nat,Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,M_1)))
<=> ( Na = M_1 ) ) ) ).
tff(fact_664_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_665_diff__less__Suc,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N)),hAPP_nat_nat(suc,M))) ).
tff(fact_666_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hAPP_nat_nat(plus_plus_nat(N),hAPP_nat_nat(minus_minus_nat(M),N)) = M ) ) ).
tff(fact_667_less__diff__conv,axiom,
! [I_1: nat,J_1: nat,K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_2)),J_1)) ) ).
tff(fact_668_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(A),C)),hAPP_nat_nat(minus_minus_nat(B),C))) ) ) ).
tff(fact_669_less__diff__iff,axiom,
! [Na: nat,K_2: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M_1),K_2)),hAPP_nat_nat(minus_minus_nat(Na),K_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) ) ) ) ).
tff(fact_670_less__eq__Suc__le__raw,axiom,
! [X_1: nat] : hAPP_n1699378549t_bool(ord_less_nat,X_1) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_1)) ).
tff(fact_671_mono__nat__linear__lb,axiom,
! [M_1: nat,K_2: nat,F: fun_nat_nat] :
( ! [M_2: nat,N_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_2)),hAPP_nat_nat(F,N_2))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F,M_1)),K_2)),hAPP_nat_nat(F,hAPP_nat_nat(plus_plus_nat(M_1),K_2)))) ) ).
tff(fact_672_inc__induct,axiom,
! [Pa: fun_nat_bool,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(Pa,J_1))
=> ( ! [I_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1))
=> ( hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,I_2)))
=> hBOOL(hAPP_nat_bool(Pa,I_2)) ) )
=> hBOOL(hAPP_nat_bool(Pa,I_1)) ) ) ) ).
tff(fact_673_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ? [K_1: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(plus_plus_nat(M),K_1)) ) ).
tff(fact_674_bounded__nat__set__is__finite,axiom,
! [Na: nat,N_1: fun_nat_bool] :
( ! [X_1: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Na)) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_1)) ) ).
tff(fact_675_less__mono__imp__le__mono,axiom,
! [I_1: nat,J_1: nat,F: fun_nat_nat] :
( ! [I_2: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))) ) ) ).
tff(fact_676_Suc__lessE,axiom,
! [I: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K))
=> ~ ! [J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( K != hAPP_nat_nat(suc,J) ) ) ) ).
tff(fact_677_lessE,axiom,
! [I: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K))
=> ( ( K != hAPP_nat_nat(suc,I) )
=> ~ ! [J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( K != hAPP_nat_nat(suc,J) ) ) ) ) ).
tff(fact_678_less__zeroE,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_679_le0,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_680_zero__less__Suc,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).
tff(fact_681_add__is__1,axiom,
! [M_1: nat,Na: nat] :
( ( hAPP_nat_nat(plus_plus_nat(M_1),Na) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( Na = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_682_one__is__add,axiom,
! [M_1: nat,Na: nat] :
( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(plus_plus_nat(M_1),Na) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( Na = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_683_diff__add__0,axiom,
! [N: nat,M: nat] : hAPP_nat_nat(minus_minus_nat(N),hAPP_nat_nat(plus_plus_nat(N),M)) = zero_zero_nat ).
tff(fact_684_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hAPP_nat_nat(minus_minus_nat(M),N) = zero_zero_nat ) ) ).
tff(fact_685_diff__is__0__eq,axiom,
! [M_1: nat,Na: nat] :
( ( hAPP_nat_nat(minus_minus_nat(M_1),Na) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)) ) ).
tff(fact_686_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
tff(fact_687_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(minus_minus_nat(M),N) = zero_zero_nat )
=> ( ( hAPP_nat_nat(minus_minus_nat(N),M) = zero_zero_nat )
=> ( M = N ) ) ) ).
tff(fact_688_diff__self__eq__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).
tff(fact_689_minus__nat_Odiff__0,axiom,
! [M: nat] : hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).
tff(fact_690_diff__0__eq__0,axiom,
! [N: nat] : hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N) = zero_zero_nat ).
tff(fact_691_Suc__neq__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_692_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_693_nat_Osimps_I3_J,axiom,
! [Nat_1: nat] : hAPP_nat_nat(suc,Nat_1) != zero_zero_nat ).
tff(fact_694_Suc__not__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_695_nat_Osimps_I2_J,axiom,
! [Nat: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat) ).
tff(fact_696_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_697_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
tff(fact_698_le__0__eq,axiom,
! [Na: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
<=> ( Na = zero_zero_nat ) ) ).
tff(fact_699_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
%----Helper facts (107)
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_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ~ hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000t__a_T,axiom,
! [X: x_a,Y: x_a] :
( ( X != Y )
| hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ~ hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X: pname,Y: pname] :
( ( X != Y )
| hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) ) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000t__a_U,axiom,
! [P: bool,Q: x_a] : hAPP_a_bool(cOMBK_bool_a(P),Q) = P ).
tff(help_COMBK_1_1_COMBK_000t__a_000tc__Com__Opname_U,axiom,
! [P: x_a,Q: pname] : hAPP_pname_a(cOMBK_a_pname(P),Q) = P ).
tff(help_COMBC_1_1_COMBC_000t__a_000t__a_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_a_bool,Q: x_a,R: x_a] : hAPP_a_bool(hAPP_a_fun_a_bool(cOMBC_a_a_bool(P),Q),R) = hAPP_a_bool(hAPP_a_fun_a_bool(P,R),Q) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P: bool,Q: nat] : hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ).
tff(help_fequal_1_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_a_bool,Y: fun_a_bool] :
( ~ hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_a_bool,Y: fun_a_bool] :
( ( X != Y )
| hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),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_COMBC_1_1_COMBC_000t__a_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_nat_bool,Q: nat,R: x_a] : hAPP_a_bool(hAPP_nat_fun_a_bool(cOMBC_a_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_a_fun_nat_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000t__a_000tc__HOL__Obool_U,axiom,
! [P: fun_nat_fun_a_bool,Q: x_a,R: nat] : hAPP_nat_bool(hAPP_a_fun_nat_bool(cOMBC_nat_a_bool(P),Q),R) = hAPP_a_bool(hAPP_nat_fun_a_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a_U,axiom,
! [P: fun_bool_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(cOMBB_bool_bool_a(P,Q),R) = hAPP_bool_bool(P,hAPP_a_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_bool_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(cOMBS_a_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_a_fun_bool_bool(P,R),hAPP_a_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000t__a_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
! [P: fun_a_fun_pname_bool,Q: pname,R: x_a] : hAPP_a_bool(hAPP_p1534023578a_bool(cOMBC_a_pname_bool(P),Q),R) = hAPP_pname_bool(hAPP_a93125764e_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool_U,axiom,
! [P: fun_pname_fun_a_bool,Q: x_a,R: pname] : hAPP_pname_bool(hAPP_a93125764e_bool(cOMBC_pname_a_bool(P),Q),R) = hAPP_a_bool(hAPP_p1534023578a_bool(P,R),Q) ).
tff(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_nat_bool,Y: fun_nat_bool] :
( ( X != Y )
| hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_pname_bool,Y: fun_pname_bool] :
( ~ hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T,axiom,
! [X: fun_pname_bool,Y: fun_pname_bool] :
( ( X != Y )
| hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)) ) ).
tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a_U,axiom,
! [P: fun_a_fun_a_bool,Q: fun_a_a,R: x_a] : hAPP_a_fun_a_bool(cOMBB_a_fun_a_bool_a(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_a_a(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_It__a_Mtc__HOL__Obool_J_U,axiom,
! [P: bool,Q: fun_a_bool] : hAPP_fun_a_bool_bool(cOMBK_324466864a_bool(P),Q) = P ).
tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P: fun_pn406123357t_bool,Q: nat,R: pname] : hAPP_pname_bool(hAPP_n1025906991e_bool(cOMBC_pname_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_p1499970991t_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Com__Opname_000tc__HOL__Obool_U,axiom,
! [P: fun_na936072029e_bool,Q: pname,R: nat] : hAPP_nat_bool(hAPP_p1499970991t_bool(cOMBC_nat_pname_bool(P),Q),R) = hAPP_pname_bool(hAPP_n1025906991e_bool(P,R),Q) ).
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(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat,axiom,
! [P: fun_a_fun_a_bool,Q: fun_nat_a,R: nat] : hAPP_nat_fun_a_bool(cOMBB_1885489796ol_nat(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_nat_a(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Oboo,axiom,
! [P: fun_a_1255737515l_bool,Q: fun_a_bool,R: x_a] : hAPP_a_bool(hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opna,axiom,
! [P: fun_a_fun_a_bool,Q: fun_pname_a,R: pname] : hAPP_p1534023578a_bool(cOMBB_1897541054_pname(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_pname_a(Q,R)) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool,axiom,
! [P: bool,Q: fun_nat_bool] : hAPP_f54304608l_bool(cOMBK_1994329625t_bool(P),Q) = P ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obo,axiom,
! [P: bool,Q: fun_pname_bool] : hAPP_f1664156314l_bool(cOMBK_1857069011e_bool(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_a_nat,R: x_a] : hAPP_a_fun_nat_bool(cOMBB_1321347587bool_a(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_a_nat(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__,axiom,
! [P: fun_a_762926252l_bool,Q: fun_nat_bool,R: x_a] : hAPP_a_bool(hAPP_f1549043526a_bool(cOMBC_777206479l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_a1392362872l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__,axiom,
! [P: fun_na1632405922l_bool,Q: fun_a_bool,R: nat] : hAPP_nat_bool(hAPP_f282463732t_bool(cOMBC_1928494297l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_n1414589940l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Nat__Onat_000tc__,axiom,
! [P: fun_fu600484770t_bool,Q: nat,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_n1414589940l_bool(cOMBC_619334683t_bool(P),Q),R) = hAPP_nat_bool(hAPP_f282463732t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_It__a_Mtc__H,axiom,
! [P: fun_bool_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(cOMBB_2140588453a_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_a_bool,R: x_a] : hAPP_a_fun_bool_bool(cOMBB_1972296269bool_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_a_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc_,axiom,
! [P: fun_fu554186387l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(cOMBS_1035972772l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f198738859l_bool(P,R),hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc,axiom,
! [P: fun_a_968466022l_bool,Q: fun_pname_bool,R: x_a] : hAPP_a_bool(hAPP_f1051908748a_bool(cOMBC_1834145417l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_a217006258l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc,axiom,
! [P: fun_pn1038293468l_bool,Q: fun_a_bool,R: pname] : hAPP_pname_bool(hAPP_f1794073134e_bool(cOMBC_445755039l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_p1824510254l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname_000tc,axiom,
! [P: fun_fu2112074076e_bool,Q: pname,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_p1824510254l_bool(cOMBC_1738168533e_bool(P),Q),R) = hAPP_pname_bool(hAPP_f1794073134e_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob,axiom,
! [P: fun_pn800050071e_bool,Q: fun_a_pname,R: x_a] : hAPP_a93125764e_bool(cOMBB_610033911bool_a(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_a_pname(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__034,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_nat_nat,R: nat] : hAPP_n1699378549t_bool(cOMBB_800536526ol_nat(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Nat__Ona,axiom,
! [P: fun_fu821463397t_bool,Q: nat,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_n215258509l_bool(cOMBC_385542954t_bool(P),Q),R) = hAPP_nat_bool(hAPP_f800510211t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Nat__On,axiom,
! [P: fun_bool_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(cOMBB_238756964t_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_035,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__036,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_pname_nat,R: pname] : hAPP_p1499970991t_bool(cOMBB_523834888_pname(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_pname_nat(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo,axiom,
! [P: fun_fu1016514960l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(cOMBS_1187019125l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Oboo,axiom,
! [P: fun_pn1165013435l_bool,Q: fun_nat_bool,R: pname] : hAPP_pname_bool(hAPP_f654413245e_bool(cOMBC_267053842l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_p130839763l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Oboo,axiom,
! [P: fun_na2122364079l_bool,Q: fun_pname_bool,R: nat] : hAPP_nat_bool(hAPP_f1066163005t_bool(cOMBC_386238098l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_n850744903l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Nat__O,axiom,
! [P: fun_fu900310687t_bool,Q: nat,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_n850744903l_bool(cOMBC_1666426608t_bool(P),Q),R) = hAPP_nat_bool(hAPP_f1066163005t_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Com__Opn,axiom,
! [P: fun_fu1492573599e_bool,Q: pname,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_p130839763l_bool(cOMBC_615407716e_bool(P),Q),R) = hAPP_pname_bool(hAPP_f654413245e_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_037,axiom,
! [P: fun_pn800050071e_bool,Q: fun_nat_pname,R: nat] : hAPP_n1025906991e_bool(cOMBB_1212655066ol_nat(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_nat_pname(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Com__Op,axiom,
! [P: fun_bool_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(cOMBB_307249310e_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f1664156314l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_038,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_COMBS_1_1_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O,axiom,
! [P: fun_fu31783638l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(cOMBS_350070575l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1476298914l_bool(P,R),hAPP_f1664156314l_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(hAPP_f759274231e_bool(cOMBC_1058051404l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__O,axiom,
! [P: fun_fu410713561e_bool,Q: pname,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_p338031245l_bool(cOMBC_1004116266e_bool(P),Q),R) = hAPP_pname_bool(hAPP_f759274231e_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_039,axiom,
! [P: fun_pn800050071e_bool,Q: fun_pname_pname,R: pname] : hAPP_p61793385e_bool(cOMBB_542850580_pname(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_pname_pname(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HO,axiom,
! [P: fun_fu1471507361l_bool,Q: fun_a_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(P),Q),R) = hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__040,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_fun_a_bool_nat,R: fun_a_bool] : hAPP_f282463732t_bool(cOMBB_164527437a_bool(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_fun_a_bool_nat(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_041,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_f198738859l_bool(cOMBB_338059395a_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_fun_a_bool_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_042,axiom,
! [P: fun_pn800050071e_bool,Q: fun_fun_a_bool_pname,R: fun_a_bool] : hAPP_f1794073134e_bool(cOMBB_1693087065a_bool(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_f1128469712_pname(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__043,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_fun_nat_bool_nat,R: fun_nat_bool] : hAPP_f800510211t_bool(cOMBB_1250273980t_bool(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_f22106695ol_nat(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__,axiom,
! [P: fun_fu1471507361l_bool,Q: fun_a_fun_a_bool,R: x_a] : hAPP_a85458249l_bool(cOMBB_472261505bool_a(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_a_fun_a_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P: fun_fu1217155507l_bool,Q: fun_nat_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_044,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f1748468828l_bool(cOMBB_444170502t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool__045,axiom,
! [P: fun_nat_fun_nat_bool,Q: fun_fu1668467777ol_nat,R: fun_pname_bool] : hAPP_f1066163005t_bool(cOMBB_1896684278e_bool(P,Q),R) = hAPP_n1699378549t_bool(P,hAPP_f921600141ol_nat(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_046,axiom,
! [P: fun_pn800050071e_bool,Q: fun_fu665170229_pname,R: fun_nat_bool] : hAPP_f654413245e_bool(cOMBB_141086128t_bool(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_f1291551745_pname(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_047,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1476298914l_bool(cOMBB_2095475776e_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f1664156314l_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
! [P: fun_fu802393907l_bool,Q: fun_pname_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(P),Q),R) = hAPP_f1664156314l_bool(hAPP_f434788991l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob_048,axiom,
! [P: fun_pn800050071e_bool,Q: fun_fu1499449723_pname,R: fun_pname_bool] : hAPP_f759274231e_bool(cOMBB_598082538e_bool(P,Q),R) = hAPP_p61793385e_bool(P,hAPP_f1297739591_pname(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It___049,axiom,
! [P: fun_fu1471507361l_bool,Q: fun_nat_fun_a_bool,R: nat] : hAPP_n1414589940l_bool(cOMBB_1823939024ol_nat(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_nat_fun_a_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__,axiom,
! [P: fun_fu386216885l_bool,Q: fun_fun_a_bool_bool,R: fun_a_bool] : hAPP_fun_a_bool_bool(hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f285962445l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It___050,axiom,
! [P: fun_fu1471507361l_bool,Q: fun_pname_fun_a_bool,R: pname] : hAPP_p1824510254l_bool(cOMBB_392435466_pname(P,Q),R) = hAPP_f1631501043l_bool(P,hAPP_p1534023578a_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_,axiom,
! [P: fun_fu1217155507l_bool,Q: fun_a_fun_nat_bool,R: x_a] : hAPP_a1392362872l_bool(cOMBB_743407885bool_a(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_a_fun_nat_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
! [P: fun_fu802393907l_bool,Q: fun_a_fun_pname_bool,R: x_a] : hAPP_a217006258l_bool(cOMBB_1137537805bool_a(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_a93125764e_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__051,axiom,
! [P: fun_fu1217155507l_bool,Q: fun_nat_fun_nat_bool,R: nat] : hAPP_n215258509l_bool(cOMBB_2026977092ol_nat(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_n1699378549t_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__052,axiom,
! [P: fun_fu1436348701l_bool,Q: fun_fu425979586l_bool,R: fun_nat_bool] : hAPP_f54304608l_bool(hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__053,axiom,
! [P: fun_fu1217155507l_bool,Q: fun_pn406123357t_bool,R: pname] : hAPP_p130839763l_bool(cOMBB_928955006_pname(P,Q),R) = hAPP_f103356543l_bool(P,hAPP_p1499970991t_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_054,axiom,
! [P: fun_fu802393907l_bool,Q: fun_na936072029e_bool,R: nat] : hAPP_n850744903l_bool(cOMBB_2117322052ol_nat(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_n1025906991e_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_055,axiom,
! [P: fun_fu931343505l_bool,Q: fun_fu1430349052l_bool,R: fun_pname_bool] : hAPP_f1664156314l_bool(hAPP_f559147733l_bool(cOMBC_1988546018l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f556039215l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_056,axiom,
! [P: fun_fu802393907l_bool,Q: fun_pn800050071e_bool,R: pname] : hAPP_p338031245l_bool(cOMBB_408569982_pname(P,Q),R) = hAPP_f434788991l_bool(P,hAPP_p61793385e_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool,axiom,
! [P: fun_fu418465139l_bool,Q: fun_fun_a_bool_bool,R: fun_fun_a_bool_bool] : hAPP_f621171935l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f1434722111l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H,axiom,
! [P: fun_fu140186515l_bool,Q: fun_fu425979586l_bool,R: fun_fu425979586l_bool] : hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(P),Q),R) = hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_,axiom,
! [P: fun_fu1911931399l_bool,Q: fun_fu1430349052l_bool,R: fun_fu1430349052l_bool] : hAPP_f1935102916l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f510955609l_bool(P,R),Q) ).
%----Conjectures (7)
tff(conj_0,hypothesis,
hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,u)) ).
tff(conj_1,hypothesis,
hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,g),image_pname_a(mgt_call,u))) ).
tff(conj_2,hypothesis,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,na)),hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u)))) ).
tff(conj_3,hypothesis,
hAPP_fun_a_bool_nat(finite_card_a,g) = hAPP_nat_nat(minus_minus_nat(hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(mgt_call,u))),hAPP_nat_nat(suc,na)) ).
tff(conj_4,hypothesis,
hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,pn),u)) ).
tff(conj_5,hypothesis,
~ hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(mgt_call,pn)),g)) ).
tff(conj_6,conjecture,
hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(hAPP_pname_a(mgt_call,pn),g)),image_pname_a(mgt_call,u))) ).
%------------------------------------------------------------------------------