ITP001 Axioms: ITP105^5.ax
%------------------------------------------------------------------------------
% File : ITP105^5 : 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 : veblen^2.ax [Gau20]
% : HL4105^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 23 ( 0 unt; 5 typ; 0 def)
% Number of atoms : 421 ( 1 equ; 0 cnn)
% Maximal formula atoms : 88 ( 18 avg)
% Number of connectives : 661 ( 5 ~; 0 |; 12 &; 599 @)
% ( 5 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 15 avg; 599 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 35 usr; 30 con; 0-2 aty)
% Number of variables : 65 ( 5 ^ 59 !; 1 ?; 65 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eveblen_2Eclosed,type,
c_2Eveblen_2Eclosed: del > $i ).
thf(mem_c_2Eveblen_2Eclosed,axiom,
! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eclosed @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eveblen_2Eclub,type,
c_2Eveblen_2Eclub: del > $i ).
thf(mem_c_2Eveblen_2Eclub,axiom,
! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eveblen_2Econtinuous,type,
c_2Eveblen_2Econtinuous: del > del > $i ).
thf(mem_c_2Eveblen_2Econtinuous,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ bool ) ) ).
thf(tp_c_2Eveblen_2Estrict__mono,type,
c_2Eveblen_2Estrict__mono: del > del > $i ).
thf(mem_c_2Eveblen_2Estrict__mono,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ bool ) ) ).
thf(tp_c_2Eveblen_2Eunbounded,type,
c_2Eveblen_2Eunbounded: del > $i ).
thf(mem_c_2Eveblen_2Eunbounded,axiom,
! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).
thf(conj_thm_2Eveblen_2Ebetter__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
& ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ V0P @ V4a ) ) ) ) ) ).
thf(ax_thm_2Eveblen_2Eclosed__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eveblen_2Eclosed @ A_27a ) @ V0A ) )
<=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0A ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Eordinal_2Esup @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ V1g @ V3n ) ) @ c_2Ebool_2ET ) ) ) ) )
@ V0A ) ) ) ) ) ) ).
thf(ax_thm_2Eveblen_2Eunbounded__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ V0A ) )
<=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ? [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0A ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ).
thf(ax_thm_2Eveblen_2Eclub__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ V0A ) )
<=> ( ( p @ ( ap @ ( c_2Eveblen_2Eclosed @ A_27a ) @ V0A ) )
& ( p @ ( ap @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ V0A ) ) ) ) ) ).
thf(ax_thm_2Eveblen_2Econtinuous__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b ) @ V0f ) )
<=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1A ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1A ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V0f ) @ V1A ) ) ) ) ) ) ) ).
thf(ax_thm_2Eveblen_2Estrict__mono__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b ) @ V0f ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ V0f @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Enrange__IN__Uinf,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V0f @ V1n ) ) @ c_2Ebool_2ET ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Eincreasing,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a ) @ V0f ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V1x ) ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Eclubs__exist,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a ) @ V0f ) ) )
=> ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Emono__natI,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ! [V1n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Esup__mem__INTER,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V0A @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Eordinal_2Esup @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ V1f @ V7n ) ) @ c_2Ebool_2ET ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) @ ( ap @ V0A @ V8n ) ) @ c_2Ebool_2ET ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Eoleast__leq,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Eveblen_2Eclub__INTER,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V0A @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Eveblen_2Eclub @ A_27a )
@ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) @ ( ap @ V0A @ V3n ) ) @ c_2Ebool_2ET ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------