ITP001 Axioms: ITP104^5.ax
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% File : ITP104^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : ucord^2.ax [Gau20]
% : HL4104^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 9 ( 1 unt; 1 typ; 0 def)
% Number of atoms : 213 ( 1 equ; 0 cnn)
% Maximal formula atoms : 62 ( 23 avg)
% Number of connectives : 250 ( 3 ~; 0 |; 0 &; 245 @)
% ( 1 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 11 avg; 245 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 20 ( 19 usr; 18 con; 0-2 aty)
% Number of variables : 11 ( 2 ^ 9 !; 0 ?; 11 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
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thf(tp_c_2Eucord_2Eomega1,type,
c_2Eucord_2Eomega1: del > $i ).
thf(mem_c_2Eucord_2Eomega1,axiom,
! [A_27a: del] : ( mem @ ( c_2Eucord_2Eomega1 @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ).
thf(conj_thm_2Eucord_2Eucinf__uncountable,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).
thf(conj_thm_2Eucord_2EUnum__cardlt__ucinf,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ).
thf(conj_thm_2Eucord_2EUnum__cardle__ucinf,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).
thf(conj_thm_2Eucord_2Eucord__sup__exists__lemma,axiom,
! [A_27a: del] :
( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0a ) ) ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).
thf(ax_thm_2Eucord_2Eomega1__def,axiom,
! [A_27a: del] :
( ( c_2Eucord_2Eomega1 @ A_27a )
= ( ap @ ( c_2Eordinal_2Esup @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eucord_2Ex__lt__omega1__countable,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0x ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Eucord_2Eomega1__not__countable,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ( ty_2Esum_2Esum @ A_27a @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) ) ) ).
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