%------------------------------------------------------------------------------
% File     : NUN023^2 : TPTP v8.2.0. Released v6.4.0.
% Domain   : Number Theory
% Problem  : Function h s.t. h(0) = 1, h(1) = 0, with witness
% Version  : Especial.
% English  : Using an axiomatiztion of if-then-else, find the if-then-else
%            term that expresses the function H.

% Refs     : [Rie16] Riener (2016), Email to Geoff Sutcliffe
% Source   : [TPTP]
% Names    : ntape6-0-with-witness.tptp [Rie16]

% Status   : Theorem
% Rating   : 0.30 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0
% Syntax   : Number of formulae    :    5 (   0 unt;   4 typ;   0 def)
%            Number of atoms       :    6 (   6 equ;   0 cnn)
%            Maximal formula atoms :    5 (   6 avg)
%            Number of connectives :   22 (   1   ~;   0   |;   3   &;  15   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   9 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    5 (   4 usr;   1 con; 0-3 aty)
%            Number of variables   :    8 (   0   ^;   7   !;   1   ?;   8   :)
% SPC      : TH0_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
thf(n6,type,
    zero: $i ).

thf(n7,type,
    s: $i > $i ).

thf(n8,type,
    ite: $o > $i > $i > $i ).

thf(n9,type,
    h: $i > $i ).

thf(n10,conjecture,
    ( ( ! [X100: $o,U: $i,V: $i] :
          ( X100
         => ( ( ite @ X100 @ U @ V )
            = U ) )
      & ! [X100: $o,U: $i,V: $i] :
          ( ~ X100
         => ( ( ite @ X100 @ U @ V )
            = V ) )
      & ! [X: $i] :
          ( ( h @ X )
          = ( ite @ ( X = zero ) @ ( s @ zero ) @ zero ) ) )
   => ? [H: $i > $i] :
        ( ( ( H @ zero )
          = ( s @ zero ) )
        & ( ( H @ ( s @ zero ) )
          = zero ) ) ) ).

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