%------------------------------------------------------------------------------
% File     : SYO997^1 : TPTP v9.2.1. Released v9.2.0.
% Domain   : Syntactic
% Problem  : Definition of choice for finite sets of size 2
% Version  : Especial.
% English  : Given there is an element x of type (fin 2) that is in the 
%            predicate p, the choice operator can find an element of fin 2 
%            that makes said predicate true.

% Refs     : [RRB23] Rothgang et al. (2023), Theorem Proving in Dependently
%          : [Rot25] Rothgang (2025), Email to Geoff Sutcliffe
%          : [RK+25] Ranalter et al. (2025), The Dependently Typed Higher-O
% Source   : [Rot25]
% Names    : ChoiceBasic/dchoice_choice_def1.p [Rot25]

% Status   : Theorem
% Rating   : ? v9.2.0
% Syntax   : Number of formulae    :   10 (   1 unt;   7 typ;   0 def)
%            Number of atoms       :    3 (   0 equ;   0 cnn)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :   33 (   0   ~;   0   |;   0   &;  31   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   5 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type decls  :    7 (   0 !>P;   3 !>D)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   6 usr;   1 con; 0-2 aty)
%            Number of variables   :   11 (   0   ^;   6   !;   1   ?;  11   :)
%                                         (   3  !>;   0  ?*;   0  @-;   1  @+)
% SPC      : DH0_THM_NEQ_NAR

% Comments :
%------------------------------------------------------------------------------
thf(nat_type,type,
    nat: $tType ).

thf(zero_type,type,
    zero: nat ).

thf(suc_type,type,
    suc: nat > nat ).

thf(fin_type,type,
    fin: nat > $tType ).

thf(f1_type,type,
    f1: 
      !>[A: nat] : ( fin @ ( suc @ A ) ) ).

thf(fs_type,type,
    fs: 
      !>[A: nat] : ( ( fin @ A ) > ( fin @ ( suc @ A ) ) ) ).

thf(fin_elim,axiom,
    ! [P: !>[N: nat] : ( ( fin @ ( suc @ N ) ) > $o )] :
      ( ! [N: nat] : ( P @ N @ ( f1 @ N ) )
     => ( ! [N: nat,F: fin @ N] : ( P @ N @ ( fs @ N @ F ) )
       => ! [N: nat,F: fin @ ( suc @ N )] : ( P @ N @ F ) ) ) ).

thf(p_type,type,
    p: ( fin @ ( suc @ ( suc @ zero ) ) ) > $o ).

thf(pex,axiom,
    ? [X: fin @ ( suc @ ( suc @ zero ) )] : ( p @ X ) ).

thf(pch,conjecture,
    ( p
    @ @+[X: fin @ ( suc @ ( suc @ zero ) )] : ( p @ X ) ) ).

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