%------------------------------------------------------------------------------
% File     : ITP001^2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, bushy and chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% Source   : [BG+19]
% Names    : HL4001^2.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   16 (   2 unt;   9 typ;   0 def)
%            Number of atoms       :   19 (   5 equ;   0 cnn)
%            Maximal formula atoms :    5 (   1 avg)
%            Number of connectives :   61 (   0   ~;   0   |;   0   &;  51   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   7 avg;  51 nst)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   13 (  13   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8 usr;   2 con; 0-2 aty)
%            Number of variables   :   19 (   0   ^  19   !;   0   ?;  19   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments : 
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(del_tp,type,
    del: $tType ).

thf(bool,type,
    bool: del ).

thf(ind,type,
    ind: del ).

thf(arr,type,
    arr: del > del > del ).

thf(mem,type,
    mem: $i > del > $o ).

thf(ap,type,
    ap: $i > $i > $i ).

thf(lam,type,
    lam: del > ( $i > $i ) > $i ).

thf(p,type,
    p: $i > $o ).

thf(stp_inj_o,type,
    inj__o: $o > $i ).

thf(stp_inj_surj_o,axiom,
    ! [X: $o] :
      ( ( p @ ( inj__o @ X ) )
      = X ) ).

thf(stp_inj_mem_o,axiom,
    ! [X: $o] : ( mem @ ( inj__o @ X ) @ bool ) ).

thf(stp_iso_mem_o,axiom,
    ! [X: $i] :
      ( ( mem @ X @ bool )
     => ( X
        = ( inj__o @ ( p @ X ) ) ) ) ).

thf(ap_tp,axiom,
    ! [A: del,B: del,F: $i] :
      ( ( mem @ F @ ( arr @ A @ B ) )
     => ! [X: $i] :
          ( ( mem @ X @ A )
         => ( mem @ ( ap @ F @ X ) @ B ) ) ) ).

thf(lam_tp,axiom,
    ! [A: del,B: del,F: $i > $i] :
      ( ! [X: $i] :
          ( ( mem @ X @ A )
         => ( mem @ ( F @ X ) @ B ) )
     => ( mem @ ( lam @ A @ F ) @ ( arr @ A @ B ) ) ) ).

thf(funcext,axiom,
    ! [A: del,B: del,F: $i] :
      ( ( mem @ F @ ( arr @ A @ B ) )
     => ! [G: $i] :
          ( ( mem @ G @ ( arr @ A @ B ) )
         => ( ! [X: $i] :
                ( ( mem @ X @ A )
               => ( ( ap @ F @ X )
                  = ( ap @ G @ X ) ) )
           => ( F = G ) ) ) ) ).

thf(beta,axiom,
    ! [A: del,F: $i > $i,X: $i] :
      ( ( mem @ X @ A )
     => ( ( ap @ ( lam @ A @ F ) @ X )
        = ( F @ X ) ) ) ).

%------------------------------------------------------------------------------