%------------------------------------------------------------------------------
% File     : DAT347^1 : TPTP v9.2.1. Released v9.2.0.
% Domain   : Data Structures
% Problem  : Head of a list with first element 0 is 0
% Version  : Especial.
% English  : Introduces the unfailing head function on fixed length lists of 
%            length >= 1 and states that the head of a list with first element 
%            0 is 0.

% 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    : ChoiceList/dchoice_list_hd.p [Rot25]

% Status   : Theorem
% Rating   : ? v9.2.0
% Syntax   : Number of formulae    :    9 (   2 unt;   7 typ;   0 def)
%            Number of atoms       :    3 (   3 equ;   0 cnn)
%            Maximal formula atoms :    1 (   1 avg)
%            Number of connectives :   18 (   0   ~;   0   |;   0   &;  18   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    4 (   3 avg)
%            Number of types       :    1 (   1 usr)
%            Number of type decls  :    7 (   0 !>P;   2 !>D)
%            Number of type conns  :    5 (   5   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   6 usr;   2 con; 0-3 aty)
%            Number of variables   :    6 (   0   ^;   3   !;   0   ?;   6   :)
%                                         (   2  !>;   0  ?*;   0  @-;   1  @+)
% SPC      : DH0_THM_EQU_NAR

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

thf(zero_type,type,
    zero: nat ).

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

thf(list_type,type,
    list: nat > $tType ).

thf(nil_type,type,
    nil: list @ zero ).

thf(cons_type,type,
    cons: 
      !>[N: nat] : ( nat > ( list @ N ) > ( list @ ( suc @ N ) ) ) ).

thf(hd_type,type,
    hd: 
      !>[LENMINUSONE: nat] : ( ( list @ ( suc @ LENMINUSONE ) ) > nat ) ).

thf(hd,axiom,
    ! [LEN: nat,H: nat,L: list @ LEN] :
      ( ( hd @ LEN @ ( cons @ LEN @ H @ L ) )
      = H ) ).

thf(c,conjecture,
    ( ( hd @ zero
      @ @+[X: list @ ( suc @ zero )] :
          ( ( hd @ zero @ X )
          = zero ) )
    = zero ) ).

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