%------------------------------------------------------------------------------
% File     : PHI005^4 : TPTP v9.0.0. Released v6.4.0.
% Domain   : Philosophy
% Problem  : Necessarily, God exists
% Version  : [Ben16] axioms : Biased.
% English  :

% Refs     : [Ben16] Benzmueller (2016), Email to Geoff Sutcliffe
% Source   : [Ben16]
% Names    : Sledgehammer_T3_S5U_direct_satallax.p [Ben16]

% Status   : Theorem
% Rating   : 0.62 v9.0.0, 0.70 v8.2.0, 0.69 v8.1.0, 0.64 v7.5.0, 0.71 v7.4.0, 0.78 v7.2.0, 0.75 v7.0.0, 0.71 v6.4.0
% Syntax   : Number of formulae    :   16 (   4 unt;   6 typ;   0 def)
%            Number of atoms       :   21 (   3 equ;   0 cnn)
%            Maximal formula atoms :    2 (   2 avg)
%            Number of connectives :   62 (   4   ~;   0   |;   2   &;  47   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   6 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   31 (  31   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    5 (   4 usr;   0 con; 2-3 aty)
%            Number of variables   :   35 (  11   ^;  22   !;   2   ?;  35   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This problem file has been generated by Sledgehammer (satallax
%            translation) in default setting.
%------------------------------------------------------------------------------
%----Could-be-implicit typings (2)
thf(ty_n_t__QML____S5U__O__092__060mu__062,type,
    qML_mu: $tType ).

thf(ty_n_t__QML____S5U__Oi,type,
    qML_i: $tType ).

%----Explicit typings (4)
thf(sy_c_Scott__S5U_OG,type,
    scott_G: qML_mu > qML_i > $o ).

thf(sy_c_Scott__S5U_ONE_001t__QML____S5U__O__092__060mu__062,type,
    scott_NE_QML_mu: qML_mu > qML_i > $o ).

thf(sy_c_Scott__S5U_OP,type,
    scott_P: ( qML_mu > qML_i > $o ) > qML_i > $o ).

thf(sy_c_Scott__S5U_Oess_001t__QML____S5U__O__092__060mu__062,type,
    scott_ess_QML_mu: ( qML_mu > qML_i > $o ) > qML_mu > qML_i > $o ).

%----Relevant facts (9)
thf(fact_0_G__def,axiom,
    ( scott_G
    = ( ^ [X: qML_mu,W: qML_i] :
        ! [Y: qML_mu > qML_i > $o] :
          ( ( scott_P @ Y @ W )
         => ( Y @ X @ W ) ) ) ) ).

% G_def
thf(fact_1_A3,axiom,
    ! [X_1: qML_i] : ( scott_P @ scott_G @ X_1 ) ).

% A3
thf(fact_2_A4,axiom,
    ! [W2: qML_i,X2: qML_mu > qML_i > $o] :
      ( ( scott_P @ X2 @ W2 )
     => ! [X_1: qML_i] : ( scott_P @ X2 @ X_1 ) ) ).

% A4
thf(fact_3_A1b,axiom,
    ! [W2: qML_i,X2: qML_mu > qML_i > $o] :
      ( ~ ( scott_P @ X2 @ W2 )
     => ( scott_P
        @ ^ [Y: qML_mu,Z: qML_i] :
            ~ ( X2 @ Y @ Z )
        @ W2 ) ) ).

% A1b
thf(fact_4_A2,axiom,
    ! [W2: qML_i,X2: qML_mu > qML_i > $o,Xa: qML_mu > qML_i > $o] :
      ( ( ( scott_P @ X2 @ W2 )
        & ! [V: qML_i,Xb: qML_mu] :
            ( ( X2 @ Xb @ V )
           => ( Xa @ Xb @ V ) ) )
     => ( scott_P @ Xa @ W2 ) ) ).

% A2
thf(fact_5_A5,axiom,
    ! [X_1: qML_i] : ( scott_P @ scott_NE_QML_mu @ X_1 ) ).

% A5
thf(fact_6_ess__def,axiom,
    ( scott_ess_QML_mu
    = ( ^ [Phi: qML_mu > qML_i > $o,X: qML_mu,W: qML_i] :
          ( ( Phi @ X @ W )
          & ! [Y: qML_mu > qML_i > $o] :
              ( ( Y @ X @ W )
             => ! [V2: qML_i,Z: qML_mu] :
                  ( ( Phi @ Z @ V2 )
                 => ( Y @ Z @ V2 ) ) ) ) ) ) ).

% ess_def
thf(fact_7_A1a,axiom,
    ! [W2: qML_i,X2: qML_mu > qML_i > $o] :
      ( ( scott_P
        @ ^ [Y: qML_mu,Z: qML_i] :
            ~ ( X2 @ Y @ Z )
        @ W2 )
     => ~ ( scott_P @ X2 @ W2 ) ) ).

% A1a
thf(fact_8_NE__def,axiom,
    ( scott_NE_QML_mu
    = ( ^ [X: qML_mu,W: qML_i] :
        ! [Y: qML_mu > qML_i > $o] :
          ( ( scott_ess_QML_mu @ Y @ X @ W )
         => ! [V2: qML_i] :
            ? [Z: qML_mu] : ( Y @ Z @ V2 ) ) ) ) ).

% NE_def

%----Conjectures (1)
thf(conj_0,conjecture,
    ! [W3: qML_i,V: qML_i] :
    ? [X2: qML_mu] : ( scott_G @ X2 @ V ) ).

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