%------------------------------------------------------------------------------
% File     : DAT429^1 : TPTP v9.2.1. Released v9.2.0.
% Domain   : Data Structures
% Problem  : Reversal of a red-black-tree is an involution, bad tree
% Version  : Especial.
% English  :

% 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    : RedBlackTree/bad_tree.p [Rot25]

% Status   : Theorem
% Rating   : ? v9.2.0
% Syntax   : Number of formulae    :   11 (   1 unt;  10 typ;   0 def)
%            Number of atoms       :    1 (   1 equ;   0 cnn)
%            Maximal formula atoms :    1 (   1 avg)
%            Number of connectives :   36 (   0   ~;   0   |;   0   &;  36   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   9 avg)
%            Number of types       :    2 (   2 usr)
%            Number of type decls  :   10 (   3 !>P;   2 !>D)
%            Number of type conns  :   10 (  10   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8 usr;   3 con; 0-7 aty)
%            Number of variables   :   15 (   0   ^;   5   !;   3   ?;  15   :)
%                                         (   7  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : DH1_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
thf(color_type,type,
    color: $tType ).

thf(black_type,type,
    black: color ).

thf(red_type,type,
    red: color ).

thf(nat_type,type,
    nat: $tType ).

thf(zero_type,type,
    zero: nat ).

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

thf(tree_type,type,
    tree: $tType > color > nat > $tType ).

thf(leaf_type,type,
    leaf: 
      !>[A: $tType] : ( tree @ A @ black @ zero ) ).

thf(redTree_type,type,
    rt: 
      !>[A: $tType,N: nat] : ( ( tree @ A @ black @ N ) > A > ( tree @ A @ black @ N ) > ( tree @ A @ red @ N ) ) ).

thf(blackTree_type,type,
    bt: 
      !>[A: $tType,N: nat,C1: color,C2: color] : ( ( tree @ A @ C1 @ N ) > A > ( tree @ A @ C2 @ N ) > ( tree @ A @ black @ ( suc @ N ) ) ) ).

thf(bad_tree,conjecture,
    ! [A: $tType,N: nat,C1: color,C2: color,T: tree @ A @ red @ N] :
    ? [X: A,T1: tree @ A @ C1 @ N,T2: tree @ A @ C1 @ N] :
      ( T
      = ( rt @ A @ N @ T1 @ X @ T2 ) ) ).

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