%------------------------------------------------------------------------------
% File     : SYP000^1 : TPTP v9.2.1. Released v9.2.0.
% Domain   : Syntactic
% Problem  : Choice respects identity for the type (fin 1)
% Version  : Especial.
% English  : Picking an element using the choice operator, such that it is 
%            equal to a certain element of type (fin 1), yields this exact 
%            element.

% 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_eq1.p [Rot25]

% Status   : Theorem
% Rating   : ? v9.2.0
% Syntax   : Number of formulae    :    7 (   1 unt;   6 typ;   0 def)
%            Number of atoms       :    2 (   2 equ;   0 cnn)
%            Maximal formula atoms :    1 (   2 avg)
%            Number of connectives :    9 (   0   ~;   0   |;   0   &;   9   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    1 (   1 avg)
%            Number of types       :    1 (   1 usr)
%            Number of type decls  :    6 (   0 !>P;   2 !>D)
%            Number of type conns  :    3 (   3   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   5 usr;   1 con; 0-2 aty)
%            Number of variables   :    3 (   0   ^;   0   !;   0   ?;   3   :)
%                                         (   2  !>;   0  ?*;   0  @-;   1  @+)
% SPC      : DH0_THM_EQU_NAR

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

thf(zer_type,type,
    zer: nat ).

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

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

thf(zerf_type,type,
    zerf: 
      !>[N: nat] : ( fin @ ( suc @ N ) ) ).

thf(sucf_type,type,
    sucf: 
      !>[N: nat] : ( ( fin @ N ) > ( fin @ ( suc @ N ) ) ) ).

thf(dchoiceex3,conjecture,
    ( ( @+[X: fin @ ( suc @ zer )] :
          ( X
          = ( zerf @ zer ) ) )
    = ( zerf @ zer ) ) ).

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