TSTP Solution File: SWW542_5 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SWW542_5 : TPTP v8.2.0. Released v6.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue May 21 06:51:13 EDT 2024

% Result   : Theorem 11.39s 3.81s
% Output   : Refutation 11.39s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  163
% Syntax   : Number of formulae    :  290 (  99 unt;  38 typ;   0 def)
%            Number of atoms       :  680 ( 216 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 3451 (  93   ~;  16   |;  74   &;2997   @)
%                                         (  27 <=>; 244  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   7 avg)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  :   35 (  35   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   38 (  36 usr;   9 con; 0-5 aty)
%            Number of variables   :  624 (   0   ^ 576   !;  18   ?; 624   :)
%                                         (  30  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(a1_type,type,
    a1: $tType ).

thf(bool_type,type,
    bool: $tType ).

thf(int_type,type,
    int: $tType ).

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

thf(zero_type,type,
    zero: 
      !>[TA: $tType] : $o ).

thf(ord_type,type,
    ord: 
      !>[TA: $tType] : $o ).

thf(semiring_1_type,type,
    semiring_1: 
      !>[TA: $tType] : $o ).

thf(linorder_type,type,
    linorder: 
      !>[TA: $tType] : $o ).

thf(preorder_type,type,
    preorder: 
      !>[TA: $tType] : $o ).

thf(semiring_char_0_type,type,
    semiring_char_0: 
      !>[TA: $tType] : $o ).

thf(linordered_semidom_type,type,
    linordered_semidom: 
      !>[TA: $tType] : $o ).

thf(zero_zero_type,type,
    zero_zero: 
      !>[TA: $tType] : TA ).

thf(huffma675207370phabet_type,type,
    huffma675207370phabet: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > ( fun @ TA @ bool ) ) ).

thf(huffma1518433673istent_type,type,
    huffma1518433673istent: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > $o ) ).

thf(huffma1134658180e_cost_type,type,
    huffma1134658180e_cost: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > nat ) ).

thf(huffma410068972_depth_type,type,
    huffma410068972_depth: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > TA > nat ) ).

thf(huffma1352802255e_freq_type,type,
    huffma1352802255e_freq: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > ( fun @ TA @ nat ) ) ).

thf(huffma945805758height_type,type,
    huffma945805758height: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > nat ) ).

thf(huffma1393970616ptimum_type,type,
    huffma1393970616ptimum: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > $o ) ).

thf(huffma1401021291ibling_type,type,
    huffma1401021291ibling: 
      !>[TA: $tType] : ( ( huffma1450048681e_tree @ TA ) > TA > TA ) ).

thf(huffma1146269203erNode_type,type,
    huffma1146269203erNode: 
      !>[TA: $tType] : ( nat > ( huffma1450048681e_tree @ TA ) > ( huffma1450048681e_tree @ TA ) > ( huffma1450048681e_tree @ TA ) ) ).

thf(huffma2021818691e_Leaf_type,type,
    huffma2021818691e_Leaf: 
      !>[TA: $tType] : ( nat > TA > ( huffma1450048681e_tree @ TA ) ) ).

thf(huffma107959123e_case_type,type,
    huffma107959123e_case: 
      !>[TA: $tType,TB: $tType] : ( ( fun @ nat @ ( fun @ TB @ TA ) ) > ( fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TB ) @ ( fun @ ( huffma1450048681e_tree @ TB ) @ TA ) ) ) > ( huffma1450048681e_tree @ TB ) > TA ) ).

thf(huffma1280178957ee_rec_type,type,
    huffma1280178957ee_rec: 
      !>[TA: $tType,TB: $tType] : ( ( fun @ nat @ ( fun @ TB @ TA ) ) > ( fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TB ) @ ( fun @ ( huffma1450048681e_tree @ TB ) @ ( fun @ TA @ ( fun @ TA @ TA ) ) ) ) ) > ( huffma1450048681e_tree @ TB ) > TA ) ).

thf(if_type,type,
    if: 
      !>[TA: $tType] : ( bool > TA > TA > TA ) ).

thf(semiring_1_of_nat_type,type,
    semiring_1_of_nat: 
      !>[TA: $tType] : ( nat > TA ) ).

thf(size_size_type,type,
    size_size: 
      !>[TA: $tType] : ( TA > nat ) ).

thf(ord_less_type,type,
    ord_less: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(ord_less_eq_type,type,
    ord_less_eq: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(aa_type,type,
    aa: 
      !>[TA: $tType,TB: $tType] : ( ( fun @ TB @ TA ) > TB > TA ) ).

thf(fFalse_type,type,
    fFalse: bool ).

thf(fTrue_type,type,
    fTrue: bool ).

thf(member_type,type,
    member: 
      !>[TA: $tType] : ( TA > ( fun @ TA @ bool ) > $o ) ).

thf(pp_type,type,
    pp: bool > $o ).

thf(a_type,type,
    a: a1 ).

thf(t_1_type,type,
    t_1: huffma1450048681e_tree @ a1 ).

thf(t_2_type,type,
    t_2: huffma1450048681e_tree @ a1 ).

thf(w_type,type,
    w: nat ).

thf(90,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_not__less0) ).

thf(436,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
        <=> ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_of__nat__less__iff) ).

thf(184,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
           => ( ord_less @ nat @ B @ A ) )
          & ( ( ord_less @ nat @ B @ A )
           => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(31,axiom,
    linordered_semidom @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semidom) ).

thf(228,plain,
    linordered_semidom @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(79,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
    <=> ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_le__0__eq) ).

thf(386,plain,
    ! [A: nat] :
      ( ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
       => ( A
          = ( zero_zero @ nat ) ) )
      & ( ( A
          = ( zero_zero @ nat ) )
       => ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(13,axiom,
    ord @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Oord) ).

thf(158,plain,
    ord @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(103,axiom,
    ! [TA: $tType,A: TA,B: nat] : ( huffma1518433673istent @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_consistent_Osimps_I1_J) ).

thf(479,plain,
    ! [TA: $tType,A: TA,B: nat] : ( huffma1518433673istent @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(115,axiom,
    ! [TA: $tType,TB: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ),E: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma107959123e_case @ TA @ TB @ E @ D @ ( huffma1146269203erNode @ TA @ C @ B @ A ) )
      = ( aa @ ( huffma1450048681e_tree @ TA ) @ TB @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) @ ( aa @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ) @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_tree_Osimps_I6_J) ).

thf(518,plain,
    ! [TA: $tType,TB: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ),E: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma107959123e_case @ TA @ TB @ E @ D @ ( huffma1146269203erNode @ TA @ C @ B @ A ) )
      = ( aa @ ( huffma1450048681e_tree @ TA ) @ TB @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) @ ( aa @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ) @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(74,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_less__imp__le__nat) ).

thf(365,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(4,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ fTrue @ B @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_1_1_T) ).

thf(138,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ fTrue @ B @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(104,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ( member @ TA @ C @ ( huffma675207370phabet @ TA @ D ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ D @ A ) @ C )
          = ( huffma1401021291ibling @ TA @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_height__gt__0__in__alphabet__imp__sibling__left) ).

thf(481,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ( member @ TA @ C @ ( huffma675207370phabet @ TA @ D ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ D @ A ) @ C )
          = ( huffma1401021291ibling @ TA @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(72,axiom,
    ! [A: fun @ nat @ ( fun @ nat @ bool ),B: nat,C: nat] :
      ( ( ( ord_less @ nat @ C @ B )
       => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
       => ( ( ( ord_less @ nat @ B @ C )
           => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
         => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_nat__less__cases) ).

thf(352,plain,
    ! [A: fun @ nat @ ( fun @ nat @ bool ),B: nat,C: nat] :
      ( ( ( ord_less @ nat @ C @ B )
       => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
       => ( ( ( ord_less @ nat @ B @ C )
           => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) )
         => ( pp @ ( aa @ nat @ bool @ ( aa @ nat @ ( fun @ nat @ bool ) @ A @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(92,axiom,
    semiring_char_0 @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Nat_Osemiring__char__0) ).

thf(449,plain,
    semiring_char_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(39,axiom,
    ! [A: fun @ int @ bool] :
      ( ! [B: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
         => ( pp @ ( aa @ int @ bool @ A @ B ) ) )
    <=> ! [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_transfer__int__nat__quantifiers_I1_J) ).

thf(247,plain,
    ! [A: fun @ int @ bool] :
      ( ( ! [B: int] :
            ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
           => ( pp @ ( aa @ int @ bool @ A @ B ) ) )
       => ! [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) ) )
      & ( ! [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) )
       => ! [B: int] :
            ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
           => ( pp @ ( aa @ int @ bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(44,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: nat,C: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ),D: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma1280178957ee_rec @ TA @ TB @ D @ C @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( aa @ TA @ TB @ ( aa @ nat @ ( fun @ TA @ TB ) @ D @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_tree_Orecs_I1_J) ).

thf(269,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: nat,C: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ),D: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma1280178957ee_rec @ TA @ TB @ D @ C @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( aa @ TA @ TB @ ( aa @ nat @ ( fun @ TA @ TB ) @ D @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(25,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
    <=> ( ord_less @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_zless__int) ).

thf(202,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
       => ( ord_less @ nat @ B @ A ) )
      & ( ( ord_less @ nat @ B @ A )
       => ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(28,axiom,
    ! [A: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_zero__zle__int) ).

thf(217,plain,
    ! [A: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(47,axiom,
    linorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Olinorder) ).

thf(276,plain,
    linorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(78,axiom,
    ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(385,plain,
    ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(100,axiom,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( huffma945805758height @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_height_Osimps_I1_J) ).

thf(472,plain,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( huffma945805758height @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(101,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_less__eq__nat_Osimps_I1_J) ).

thf(475,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(20,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
    <=> ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_zle__int) ).

thf(179,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
       => ( ord_less_eq @ nat @ B @ A ) )
      & ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(63,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_order__refl) ).

thf(321,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(77,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
    <=> ( ( ord_less_eq @ nat @ B @ A )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_nat__less__le) ).

thf(376,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ B @ A )
       => ( ( ord_less_eq @ nat @ B @ A )
          & ( B != A ) ) )
      & ( ( ( ord_less_eq @ nat @ B @ A )
          & ( B != A ) )
       => ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(40,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
    <=> ( ( ord_less @ nat @ B @ A )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_le__eq__less__or__eq) ).

thf(253,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq @ nat @ B @ A )
       => ( ( ord_less @ nat @ B @ A )
          | ( B = A ) ) )
      & ( ( ( ord_less @ nat @ B @ A )
          | ( B = A ) )
       => ( ord_less_eq @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
         => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_of__nat__less__imp__less) ).

thf(238,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
         => ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(8,axiom,
    zero @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ozero) ).

thf(147,plain,
    zero @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(30,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ B )
        = ( semiring_1_of_nat @ int @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_int__int__eq) ).

thf(221,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(42,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ C @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ nat @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_le__trans) ).

thf(265,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ C @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ nat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(60,axiom,
    ! [A: nat,B: nat] :
      ( ( B != A )
     => ( ~ ( ord_less @ nat @ B @ A )
       => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_linorder__neqE__nat) ).

thf(312,plain,
    ! [A: nat,B: nat] :
      ( ( B != A )
     => ( ~ ( ord_less @ nat @ B @ A )
       => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(34,axiom,
    ( ( zero_zero @ int )
    = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_transfer__int__nat__numerals_I1_J) ).

thf(233,plain,
    ( ( zero_zero @ int )
    = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(118,axiom,
    ! [TA: $tType,A: nat,B: huffma1450048681e_tree @ TA,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ~ ( member @ TA @ C @ ( huffma675207370phabet @ TA @ B ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ A @ B @ D ) @ C )
          = ( huffma1401021291ibling @ TA @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_height__gt__0__notin__alphabet__imp__sibling__right) ).

thf(534,plain,
    ! [TA: $tType,A: nat,B: huffma1450048681e_tree @ TA,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ~ ( member @ TA @ C @ ( huffma675207370phabet @ TA @ B ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ A @ B @ D ) @ C )
          = ( huffma1401021291ibling @ TA @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(113,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: TA,E: nat] :
      ( ( huffma2021818691e_Leaf @ TA @ E @ D )
     != ( huffma1146269203erNode @ TA @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_tree_Osimps_I3_J) ).

thf(512,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: TA,E: nat] :
      ( ( huffma2021818691e_Leaf @ TA @ E @ D )
     != ( huffma1146269203erNode @ TA @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(91,axiom,
    ! [A: nat,B: nat] :
      ( ( B != A )
    <=> ( ( ord_less @ nat @ B @ A )
        | ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_nat__neq__iff) ).

thf(439,plain,
    ! [A: nat,B: nat] :
      ( ( ( B != A )
       => ( ( ord_less @ nat @ B @ A )
          | ( ord_less @ nat @ A @ B ) ) )
      & ( ( ( ord_less @ nat @ B @ A )
          | ( ord_less @ nat @ A @ B ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(68,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
    <=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_neq0__conv) ).

thf(334,plain,
    ! [A: nat] :
      ( ( ( A
         != ( zero_zero @ nat ) )
       => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( A
         != ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(57,axiom,
    ! [TA: $tType,TB: $tType,A: fun @ TA @ TB,B: fun @ TA @ TB] :
      ( ! [C: TA] :
          ( ( aa @ TA @ TB @ B @ C )
          = ( aa @ TA @ TB @ A @ C ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_ext) ).

thf(302,plain,
    ! [TA: $tType,TB: $tType,A: fun @ TA @ TB,B: fun @ TA @ TB] :
      ( ! [C: TA] :
          ( ( aa @ TA @ TB @ B @ C )
          = ( aa @ TA @ TB @ A @ C ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(71,axiom,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( huffma1134658180e_cost @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_cost_Osimps_I1_J) ).

thf(349,plain,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( huffma1134658180e_cost @ TA @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(123,axiom,
    ! [TA: $tType,TB: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ),E: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ ( huffma1146269203erNode @ TA @ C @ B @ A ) )
      = ( aa @ TB @ TB @ ( aa @ TB @ ( fun @ TB @ TB ) @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) @ ( aa @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ) @ D @ C ) @ B ) @ A ) @ ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ B ) ) @ ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_tree_Orecs_I2_J) ).

thf(561,plain,
    ! [TA: $tType,TB: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ),E: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ ( huffma1146269203erNode @ TA @ C @ B @ A ) )
      = ( aa @ TB @ TB @ ( aa @ TB @ ( fun @ TB @ TB ) @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) @ ( aa @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) @ ( aa @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ TB @ ( fun @ TB @ TB ) ) ) ) @ D @ C ) @ B ) @ A ) @ ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ B ) ) @ ( huffma1280178957ee_rec @ TA @ TB @ E @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(95,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_le__refl) ).

thf(461,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(108,axiom,
    ! [TA: $tType,A: TA,B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA,E: nat] :
      ( ( huffma1146269203erNode @ TA @ E @ D @ C )
     != ( huffma2021818691e_Leaf @ TA @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_tree_Osimps_I4_J) ).

thf(493,plain,
    ! [TA: $tType,A: TA,B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA,E: nat] :
      ( ( huffma1146269203erNode @ TA @ E @ D @ C )
     != ( huffma2021818691e_Leaf @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_of__nat__less__0__iff) ).

thf(229,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(125,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA,E: nat,F: TA] :
      ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
          = ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ F ) ) )
      & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) ) )
       => ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ A ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
              = ( huffma1401021291ibling @ TA @ A @ F ) ) )
          & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ A ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
              = F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_sibling_Osimps_I3_J) ).

thf(566,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA,E: nat,F: TA] :
      ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
          = ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ F ) ) )
      & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ E @ D @ C ) ) )
       => ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ A ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
              = ( huffma1401021291ibling @ TA @ A @ F ) ) )
          & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ A ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ ( huffma1146269203erNode @ TA @ E @ D @ C ) @ A ) @ F )
              = F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(105,axiom,
    ! [TA: $tType,A: nat,B: huffma1450048681e_tree @ TA,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ( member @ TA @ C @ ( huffma675207370phabet @ TA @ B ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ A @ B @ D ) @ C )
          = ( huffma1401021291ibling @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_height__gt__0__in__alphabet__imp__sibling__right) ).

thf(484,plain,
    ! [TA: $tType,A: nat,B: huffma1450048681e_tree @ TA,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ( member @ TA @ C @ ( huffma675207370phabet @ TA @ B ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ A @ B @ D ) @ C )
          = ( huffma1401021291ibling @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(17,axiom,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( A
           != ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_nonneg__eq__int) ).

thf(169,plain,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( A
           != ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(5,axiom,
    ! [A: bool] :
      ( ( A = fTrue )
      | ( A = fFalse ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_3_1_T) ).

thf(141,plain,
    ! [A: bool] :
      ( ( A = fTrue )
      | ( A = fFalse ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(85,axiom,
    ! [TA: $tType,A: nat,B: TA,C: TA] :
      ( ( ( C = B )
       => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ ( huffma2021818691e_Leaf @ TA @ A @ B ) ) @ C )
          = A ) )
      & ( ( C != B )
       => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ ( huffma2021818691e_Leaf @ TA @ A @ B ) ) @ C )
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_freq_Osimps_I1_J) ).

thf(420,plain,
    ! [TA: $tType,A: nat,B: TA,C: TA] :
      ( ( ( C = B )
       => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ ( huffma2021818691e_Leaf @ TA @ A @ B ) ) @ C )
          = A ) )
      & ( ( C != B )
       => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ ( huffma2021818691e_Leaf @ TA @ A @ B ) ) @ C )
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(110,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: nat,E: huffma1450048681e_tree @ TA,F: TA] :
      ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ E ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
          = ( huffma1401021291ibling @ TA @ E @ F ) ) )
      & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ E ) )
       => ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
              = ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) @ F ) ) )
          & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
              = F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_sibling_Osimps_I4_J) ).

thf(499,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: nat,E: huffma1450048681e_tree @ TA,F: TA] :
      ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ E ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
          = ( huffma1401021291ibling @ TA @ E @ F ) ) )
      & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ E ) )
       => ( ( ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
              = ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) @ F ) ) )
          & ( ~ ( member @ TA @ F @ ( huffma675207370phabet @ TA @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ E @ ( huffma1146269203erNode @ TA @ C @ B @ A ) ) @ F )
              = F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(66,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_le0) ).

thf(328,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(23,axiom,
    ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
    = ( zero_zero @ int ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_int__0) ).

thf(193,plain,
    ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
    = ( zero_zero @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(119,axiom,
    ! [TA: $tType,A: TA,B: huffma1450048681e_tree @ TA] :
      ( ( ( huffma945805758height @ TA @ B )
        = ( zero_zero @ nat ) )
     => ( ( huffma1401021291ibling @ TA @ B @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_height__0__imp__sibling__id) ).

thf(537,plain,
    ! [TA: $tType,A: TA,B: huffma1450048681e_tree @ TA] :
      ( ( ( huffma945805758height @ TA @ B )
        = ( zero_zero @ nat ) )
     => ( ( huffma1401021291ibling @ TA @ B @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(80,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ B @ A )
        | ( B = A ) )
     => ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_less__or__eq__imp__le) ).

thf(393,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ B @ A )
        | ( B = A ) )
     => ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(114,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ B ) )
     => ( ( huffma1518433673istent @ TA @ B )
       => ( ( ( huffma675207370phabet @ TA @ B )
            = ( huffma675207370phabet @ TA @ A ) )
         => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_height__gt__0__alphabet__eq__imp__height__gt__0) ).

thf(515,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ B ) )
     => ( ( huffma1518433673istent @ TA @ B )
       => ( ( ( huffma675207370phabet @ TA @ B )
            = ( huffma675207370phabet @ TA @ A ) )
         => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(18,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
    <=> ( ord_less @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_transfer__int__nat__relations_I2_J) ).

thf(173,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
       => ( ord_less @ nat @ B @ A ) )
      & ( ( ord_less @ nat @ B @ A )
       => ( ord_less @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(38,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_less__not__refl) ).

thf(244,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(81,axiom,
    ! [A: nat,B: nat] :
      ( ( B = A )
     => ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_eq__imp__le) ).

thf(398,plain,
    ! [A: nat,B: nat] :
      ( ( B = A )
     => ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(122,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ! [B: nat,C: TA] :
          ( A
         != ( huffma2021818691e_Leaf @ TA @ B @ C ) )
     => ~ ! [B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA] :
            ( A
           != ( huffma1146269203erNode @ TA @ B @ C @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_tree_Oexhaust) ).

thf(558,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ! [B: nat,C: TA] :
          ( A
         != ( huffma2021818691e_Leaf @ TA @ B @ C ) )
     => ~ ! [B: nat,C: huffma1450048681e_tree @ TA,D: huffma1450048681e_tree @ TA] :
            ( A
           != ( huffma1146269203erNode @ TA @ B @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(84,axiom,
    ! [TA: $tType,A: TA,B: nat,C: TA,D: nat] :
      ( ( ( huffma2021818691e_Leaf @ TA @ D @ C )
        = ( huffma2021818691e_Leaf @ TA @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_tree_Osimps_I1_J) ).

thf(412,plain,
    ! [TA: $tType,A: TA,B: nat,C: TA,D: nat] :
      ( ( ( ( huffma2021818691e_Leaf @ TA @ D @ C )
          = ( huffma2021818691e_Leaf @ TA @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( huffma2021818691e_Leaf @ TA @ D @ C )
          = ( huffma2021818691e_Leaf @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(15,axiom,
    ! [A: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) ).

thf(166,plain,
    ! [A: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(9,axiom,
    linorder @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Olinorder) ).

thf(148,plain,
    linorder @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(50,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( fun @ TB @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Oord) ).

thf(279,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( fun @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(126,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ~ ( member @ TA @ C @ ( huffma675207370phabet @ TA @ D ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ D @ A ) @ C )
          = ( huffma1401021291ibling @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_height__gt__0__notin__alphabet__imp__sibling__left) ).

thf(573,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: nat,C: TA,D: huffma1450048681e_tree @ TA] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ TA @ D ) )
     => ( ~ ( member @ TA @ C @ ( huffma675207370phabet @ TA @ D ) )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ B @ D @ A ) @ C )
          = ( huffma1401021291ibling @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(55,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_less__zeroE) ).

thf(296,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(10,axiom,
    preorder @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Opreorder) ).

thf(149,plain,
    preorder @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(82,axiom,
    ! [TA: $tType,A: TA,B: TA,C: nat] :
      ( ( huffma410068972_depth @ TA @ ( huffma2021818691e_Leaf @ TA @ C @ B ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_depth_Osimps_I1_J) ).

thf(402,plain,
    ! [TA: $tType,A: TA,B: TA,C: nat] :
      ( ( huffma410068972_depth @ TA @ ( huffma2021818691e_Leaf @ TA @ C @ B ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(53,axiom,
    ! [A: int,B: int] :
      ( ( ord_less @ int @ B @ A )
    <=> ( ( ord_less_eq @ int @ B @ A )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_less__int__def) ).

thf(286,plain,
    ! [A: int,B: int] :
      ( ( ( ord_less @ int @ B @ A )
       => ( ( ord_less_eq @ int @ B @ A )
          & ( B != A ) ) )
      & ( ( ( ord_less_eq @ int @ B @ A )
          & ( B != A ) )
       => ( ord_less @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(51,axiom,
    semiring_1 @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__1) ).

thf(281,plain,
    semiring_1 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_of__nat__0__le__iff) ).

thf(144,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(97,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: nat,C: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ),D: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma107959123e_case @ TA @ TB @ D @ C @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( aa @ TA @ TB @ ( aa @ nat @ ( fun @ TA @ TB ) @ D @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_tree_Osimps_I5_J) ).

thf(466,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: nat,C: fun @ nat @ ( fun @ ( huffma1450048681e_tree @ TA ) @ ( fun @ ( huffma1450048681e_tree @ TA ) @ TB ) ),D: fun @ nat @ ( fun @ TA @ TB )] :
      ( ( huffma107959123e_case @ TA @ TB @ D @ C @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( aa @ TA @ TB @ ( aa @ nat @ ( fun @ TA @ TB ) @ D @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(120,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( huffma1393970616ptimum @ TA @ A )
    <=> ! [B: huffma1450048681e_tree @ TA] :
          ( ( huffma1518433673istent @ TA @ B )
         => ( ( ( huffma675207370phabet @ TA @ A )
              = ( huffma675207370phabet @ TA @ B ) )
           => ( ( ( huffma1352802255e_freq @ TA @ A )
                = ( huffma1352802255e_freq @ TA @ B ) )
             => ( ord_less_eq @ nat @ ( huffma1134658180e_cost @ TA @ A ) @ ( huffma1134658180e_cost @ TA @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_optimum__def) ).

thf(540,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( ( huffma1393970616ptimum @ TA @ A )
       => ! [B: huffma1450048681e_tree @ TA] :
            ( ( huffma1518433673istent @ TA @ B )
           => ( ( ( huffma675207370phabet @ TA @ A )
                = ( huffma675207370phabet @ TA @ B ) )
             => ( ( ( huffma1352802255e_freq @ TA @ A )
                  = ( huffma1352802255e_freq @ TA @ B ) )
               => ( ord_less_eq @ nat @ ( huffma1134658180e_cost @ TA @ A ) @ ( huffma1134658180e_cost @ TA @ B ) ) ) ) ) )
      & ( ! [B: huffma1450048681e_tree @ TA] :
            ( ( huffma1518433673istent @ TA @ B )
           => ( ( ( huffma675207370phabet @ TA @ A )
                = ( huffma675207370phabet @ TA @ B ) )
             => ( ( ( huffma1352802255e_freq @ TA @ A )
                  = ( huffma1352802255e_freq @ TA @ B ) )
               => ( ord_less_eq @ nat @ ( huffma1134658180e_cost @ TA @ A ) @ ( huffma1134658180e_cost @ TA @ B ) ) ) ) )
       => ( huffma1393970616ptimum @ TA @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(14,axiom,
    ! [A: nat,B: nat,C: bool] :
      ( ( ( pp @ C )
       => ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ ( if @ nat @ C @ B @ A ) ) ) )
      & ( ~ ( pp @ C )
       => ( ( semiring_1_of_nat @ int @ A )
          = ( semiring_1_of_nat @ int @ ( if @ nat @ C @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_int__if__cong) ).

thf(159,plain,
    ! [A: nat,B: nat,C: bool] :
      ( ( ( pp @ C )
       => ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ ( if @ nat @ C @ B @ A ) ) ) )
      & ( ~ ( pp @ C )
       => ( ( semiring_1_of_nat @ int @ A )
          = ( semiring_1_of_nat @ int @ ( if @ nat @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(22,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( semiring_1_of_nat @ TA @ B )
            = ( semiring_1_of_nat @ TA @ A ) )
        <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_of__nat__eq__iff) ).

thf(187,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( ( semiring_1_of_nat @ TA @ B )
              = ( semiring_1_of_nat @ TA @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( semiring_1_of_nat @ TA @ B )
              = ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(29,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_zero__le__imp__of__nat) ).

thf(219,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(73,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_less__irrefl__nat) ).

thf(362,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(62,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: fun @ TA @ TB,B: fun @ TA @ TB] :
          ( ( ord_less_eq @ ( fun @ TA @ TB ) @ B @ A )
        <=> ! [C: TA] : ( ord_less_eq @ TB @ ( aa @ TA @ TB @ B @ C ) @ ( aa @ TA @ TB @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_le__fun__def) ).

thf(318,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: fun @ TA @ TB,B: fun @ TA @ TB] :
          ( ( ( ord_less_eq @ ( fun @ TA @ TB ) @ B @ A )
           => ! [C: TA] : ( ord_less_eq @ TB @ ( aa @ TA @ TB @ B @ C ) @ ( aa @ TA @ TB @ A @ C ) ) )
          & ( ! [C: TA] : ( ord_less_eq @ TB @ ( aa @ TA @ TB @ B @ C ) @ ( aa @ TA @ TB @ A @ C ) )
           => ( ord_less_eq @ ( fun @ TA @ TB ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(102,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ B @ A )
          | ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_linorder__linear) ).

thf(477,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ B @ A )
          | ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(75,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A ) @ ( zero_zero @ int ) )
    <=> ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_int__le__0__conv) ).

thf(367,plain,
    ! [A: nat] :
      ( ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A ) @ ( zero_zero @ int ) )
       => ( A
          = ( zero_zero @ nat ) ) )
      & ( ( A
          = ( zero_zero @ nat ) )
       => ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A ) @ ( zero_zero @ int ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(64,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
     => ( ( ord_less_eq @ nat @ A @ B )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_le__antisym) ).

thf(323,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
     => ( ( ord_less_eq @ nat @ A @ B )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(37,axiom,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_zero__less__imp__eq__int) ).

thf(240,plain,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(12,axiom,
    ! [A: nat] :
      ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A ) @ ( zero_zero @ int ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_int__less__0__conv) ).

thf(155,plain,
    ! [A: nat] :
      ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A ) @ ( zero_zero @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(67,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_less__not__refl2) ).

thf(330,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(116,axiom,
    ! [TA: $tType,A: fun @ TA @ bool,B: TA] :
      ( ( member @ TA @ B @ A )
    <=> ( pp @ ( aa @ TA @ bool @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_mem__def) ).

thf(521,plain,
    ! [TA: $tType,A: fun @ TA @ bool,B: TA] :
      ( ( ( member @ TA @ B @ A )
       => ( pp @ ( aa @ TA @ bool @ A @ B ) ) )
      & ( ( pp @ ( aa @ TA @ bool @ A @ B ) )
       => ( member @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(49,axiom,
    ord @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oord) ).

thf(278,plain,
    ord @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(3,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ fFalse @ B @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_2_1_T) ).

thf(135,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ fFalse @ B @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(56,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_of__nat__0) ).

thf(299,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(43,axiom,
    ! [TA: $tType,A: TA,B: huffma1450048681e_tree @ TA] : ( ord_less_eq @ nat @ ( huffma410068972_depth @ TA @ B @ A ) @ ( huffma945805758height @ TA @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_depth__le__height) ).

thf(267,plain,
    ! [TA: $tType,A: TA,B: huffma1450048681e_tree @ TA] : ( ord_less_eq @ nat @ ( huffma410068972_depth @ TA @ B @ A ) @ ( huffma945805758height @ TA @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(87,axiom,
    linorder @ bool,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Olinorder) ).

thf(431,plain,
    linorder @ bool,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(107,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: TA] :
      ( ~ ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
     => ( ( huffma1401021291ibling @ TA @ A @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_notin__alphabet__imp__sibling__id) ).

thf(490,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: TA] :
      ( ~ ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
     => ( ( huffma1401021291ibling @ TA @ A @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(86,axiom,
    ! [A: nat] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) )
    <=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_zero__less__int__conv) ).

thf(426,plain,
    ! [A: nat] :
      ( ( ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) )
       => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( ord_less @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(76,axiom,
    ~ ( pp @ fFalse ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_pp_1_1_U) ).

thf(374,plain,
    ~ ( pp @ fFalse ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(106,axiom,
    ! [TA: $tType,A: TA,B: TA,C: nat] :
      ( ( huffma1401021291ibling @ TA @ ( huffma2021818691e_Leaf @ TA @ C @ B ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_sibling_Osimps_I1_J) ).

thf(487,plain,
    ! [TA: $tType,A: TA,B: TA,C: nat] :
      ( ( huffma1401021291ibling @ TA @ ( huffma2021818691e_Leaf @ TA @ C @ B ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(54,axiom,
    preorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Opreorder) ).

thf(295,plain,
    preorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(52,axiom,
    ! [A: nat,B: fun @ nat @ bool] :
      ( ~ ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
     => ( ( pp @ ( aa @ nat @ bool @ B @ A ) )
       => ? [C: nat] :
            ( ( ord_less_eq @ nat @ C @ A )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ C )
               => ~ ( pp @ ( aa @ nat @ bool @ B @ D ) ) )
            & ( pp @ ( aa @ nat @ bool @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_ex__least__nat__le) ).

thf(282,plain,
    ! [A: nat,B: fun @ nat @ bool] :
      ( ~ ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
     => ( ( pp @ ( aa @ nat @ bool @ B @ A ) )
       => ? [C: nat] :
            ( ( ord_less_eq @ nat @ C @ A )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ C )
               => ~ ( pp @ ( aa @ nat @ bool @ B @ D ) ) )
            & ( pp @ ( aa @ nat @ bool @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(11,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
    <=> ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_transfer__int__nat__relations_I3_J) ).

thf(150,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) )
       => ( ord_less_eq @ nat @ B @ A ) )
      & ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ B ) @ ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(96,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_less__nat__zero__code) ).

thf(463,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(1,conjecture,
    ( ( ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
     => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
        = ( huffma1401021291ibling @ a1 @ t_1 @ a ) ) )
    & ( ~ ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
     => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
        = ( huffma1401021291ibling @ a1 @ t_2 @ a ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).

thf(2,negated_conjecture,
    ~ ( ( ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
       => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
          = ( huffma1401021291ibling @ a1 @ t_1 @ a ) ) )
      & ( ~ ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
       => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
          = ( huffma1401021291ibling @ a1 @ t_2 @ a ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(127,plain,
    ~ ( ( ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
       => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
          = ( huffma1401021291ibling @ a1 @ t_1 @ a ) ) )
      & ( ~ ( member @ a1 @ a @ ( huffma675207370phabet @ a1 @ t_1 ) )
       => ( ( huffma1401021291ibling @ a1 @ ( huffma1146269203erNode @ a1 @ w @ t_1 @ t_2 ) @ a )
          = ( huffma1401021291ibling @ a1 @ t_2 @ a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(99,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
      | ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_nat__le__linear) ).

thf(470,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
      | ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(33,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ B @ A )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_less__imp__of__nat__less) ).

thf(231,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ B @ A )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(93,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_less__not__refl3) ).

thf(450,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(117,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: huffma1450048681e_tree @ TA,E: huffma1450048681e_tree @ TA,F: nat] :
      ( ( ( huffma1146269203erNode @ TA @ F @ E @ D )
        = ( huffma1146269203erNode @ TA @ C @ B @ A ) )
    <=> ( ( F = C )
        & ( E = B )
        & ( D = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_tree_Osimps_I2_J) ).

thf(524,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: huffma1450048681e_tree @ TA,C: nat,D: huffma1450048681e_tree @ TA,E: huffma1450048681e_tree @ TA,F: nat] :
      ( ( ( ( huffma1146269203erNode @ TA @ F @ E @ D )
          = ( huffma1146269203erNode @ TA @ C @ B @ A ) )
       => ( ( F = C )
          & ( E = B )
          & ( D = A ) ) )
      & ( ( ( F = C )
          & ( E = B )
          & ( D = A ) )
       => ( ( huffma1146269203erNode @ TA @ F @ E @ D )
          = ( huffma1146269203erNode @ TA @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(35,axiom,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( A
          = ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_zero__le__imp__eq__int) ).

thf(235,plain,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( A
          = ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(26,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
        <=> ( ord_less_eq @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_of__nat__le__iff) ).

thf(207,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) )
           => ( ord_less_eq @ nat @ B @ A ) )
          & ( ( ord_less_eq @ nat @ B @ A )
           => ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(45,axiom,
    ord @ bool,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oord) ).

thf(272,plain,
    ord @ bool,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(70,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( ( huffma945805758height @ TA @ A )
        = ( zero_zero @ nat ) )
     => ( ( huffma1134658180e_cost @ TA @ A )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_height__0__imp__cost__0) ).

thf(346,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( ( huffma945805758height @ TA @ A )
        = ( zero_zero @ nat ) )
     => ( ( huffma1134658180e_cost @ TA @ A )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(121,axiom,
    ! [TA: $tType,A: TA,B: nat,C: nat,D: nat,E: TA,F: TA] :
      ( ( ( F = E )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( ( F = A )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
              = E ) )
          & ( ( F != A )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
              = F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_sibling_Osimps_I2_J) ).

thf(549,plain,
    ! [TA: $tType,A: TA,B: nat,C: nat,D: nat,E: TA,F: TA] :
      ( ( ( F = E )
       => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( ( F = A )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
              = E ) )
          & ( ( F != A )
           => ( ( huffma1401021291ibling @ TA @ ( huffma1146269203erNode @ TA @ D @ ( huffma2021818691e_Leaf @ TA @ C @ E ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) ) @ F )
              = F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(89,axiom,
    ! [A: nat,B: nat,C: fun @ nat @ nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( aa @ nat @ nat @ C @ D ) @ ( aa @ nat @ nat @ C @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ nat @ ( aa @ nat @ nat @ C @ B ) @ ( aa @ nat @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_less__mono__imp__le__mono) ).

thf(433,plain,
    ! [A: nat,B: nat,C: fun @ nat @ nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( aa @ nat @ nat @ C @ D ) @ ( aa @ nat @ nat @ C @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( ord_less_eq @ nat @ ( aa @ nat @ nat @ C @ B ) @ ( aa @ nat @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(112,axiom,
    ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ a1 @ t_1 ) )
    | ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ a1 @ t_2 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(510,plain,
    ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ a1 @ t_1 ) )
    | ( ord_less @ nat @ ( zero_zero @ nat ) @ ( huffma945805758height @ a1 @ t_2 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(61,axiom,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( size_size @ ( huffma1450048681e_tree @ TA ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_tree_Osize_I3_J) ).

thf(315,plain,
    ! [TA: $tType,A: TA,B: nat] :
      ( ( size_size @ ( huffma1450048681e_tree @ TA ) @ ( huffma2021818691e_Leaf @ TA @ B @ A ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(88,axiom,
    pp @ fTrue,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_pp_2_1_U) ).

thf(432,plain,
    pp @ fTrue,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(7,axiom,
    linordered_semidom @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semidom) ).

thf(146,plain,
    linordered_semidom @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(83,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
        <=> ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_zero__reorient) ).

thf(405,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( ( zero_zero @ TA )
              = A )
           => ( A
              = ( zero_zero @ TA ) ) )
          & ( ( A
              = ( zero_zero @ TA ) )
           => ( ( zero_zero @ TA )
              = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(46,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
        <=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_of__nat__0__less__iff) ).

thf(273,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat] :
          ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
           => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
          & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(16,axiom,
    semiring_1 @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring__1) ).

thf(168,plain,
    semiring_1 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(41,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
     => ( ( B != A )
       => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_le__neq__implies__less) ).

thf(262,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ B @ A )
     => ( ( B != A )
       => ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(124,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
    ? [B: TA] : ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_exists__in__alphabet) ).

thf(564,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
    ? [B: TA] : ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(58,axiom,
    ! [A: nat,B: fun @ nat @ bool] :
      ( ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
           => ( ~ ( pp @ ( aa @ nat @ bool @ B @ C ) )
             => ? [D: nat] :
                  ( ( ord_less @ nat @ D @ C )
                  & ~ ( pp @ ( aa @ nat @ bool @ B @ D ) ) ) ) )
       => ( pp @ ( aa @ nat @ bool @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_infinite__descent0) ).

thf(305,plain,
    ! [A: nat,B: fun @ nat @ bool] :
      ( ( pp @ ( aa @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
           => ( ~ ( pp @ ( aa @ nat @ bool @ B @ C ) )
             => ? [D: nat] :
                  ( ( ord_less @ nat @ D @ C )
                  & ~ ( pp @ ( aa @ nat @ bool @ B @ D ) ) ) ) )
       => ( pp @ ( aa @ nat @ bool @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(27,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ B )
        = ( semiring_1_of_nat @ int @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_transfer__int__nat__relations_I1_J) ).

thf(210,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( semiring_1_of_nat @ int @ B )
          = ( semiring_1_of_nat @ int @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(65,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( fun @ TB @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Opreorder) ).

thf(326,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( fun @ TB @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(69,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( A
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_gr__implies__not0) ).

thf(341,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ B @ A )
     => ( A
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(111,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( huffma1518433673istent @ TA @ A )
     => ? [B: TA] :
          ( ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
          & ( ( huffma410068972_depth @ TA @ A @ B )
            = ( huffma945805758height @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_exists__at__height) ).

thf(506,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA] :
      ( ( huffma1518433673istent @ TA @ A )
     => ? [B: TA] :
          ( ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
          & ( ( huffma410068972_depth @ TA @ A @ B )
            = ( huffma945805758height @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(24,axiom,
    ! [A: nat] :
      ( ( ( semiring_1_of_nat @ int @ A )
        = ( zero_zero @ int ) )
    <=> ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_int__eq__0__conv) ).

thf(195,plain,
    ! [A: nat] :
      ( ( ( ( semiring_1_of_nat @ int @ A )
          = ( zero_zero @ int ) )
       => ( A
          = ( zero_zero @ nat ) ) )
      & ( ( A
          = ( zero_zero @ nat ) )
       => ( ( semiring_1_of_nat @ int @ A )
          = ( zero_zero @ int ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(19,axiom,
    semiring_char_0 @ int,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Nat_Osemiring__char__0) ).

thf(178,plain,
    semiring_char_0 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(48,axiom,
    zero @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ozero) ).

thf(277,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(98,axiom,
    preorder @ bool,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Opreorder) ).

thf(469,plain,
    preorder @ bool,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(59,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_gr0I) ).

thf(309,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(109,axiom,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: TA] :
      ( ~ ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
     => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ A ) @ B )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_notin__alphabet__imp__freq__0) ).

thf(496,plain,
    ! [TA: $tType,A: huffma1450048681e_tree @ TA,B: TA] :
      ( ~ ( member @ TA @ B @ ( huffma675207370phabet @ TA @ A ) )
     => ( ( aa @ TA @ nat @ ( huffma1352802255e_freq @ TA @ A ) @ B )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(94,axiom,
    ! [A: fun @ int @ bool] :
      ( ? [B: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
          & ( pp @ ( aa @ int @ bool @ A @ B ) ) )
    <=> ? [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_transfer__int__nat__quantifiers_I2_J) ).

thf(454,plain,
    ! [A: fun @ int @ bool] :
      ( ( ? [B: int] :
            ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
            & ( pp @ ( aa @ int @ bool @ A @ B ) ) )
       => ? [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) ) )
      & ( ? [B: nat] : ( pp @ ( aa @ int @ bool @ A @ ( semiring_1_of_nat @ int @ B ) ) )
       => ? [B: int] :
            ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B )
            & ( pp @ ( aa @ int @ bool @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(621,plain,
    $false,
    inference(cvc4,[status(thm)],[436,184,228,386,158,479,518,365,138,481,352,449,247,269,202,217,276,385,472,475,179,321,376,253,238,147,221,265,312,233,534,512,439,334,302,349,561,461,493,229,566,484,169,141,420,499,328,193,537,393,515,173,244,398,558,412,166,148,279,573,296,149,402,286,281,144,466,540,159,187,219,362,318,477,367,323,240,155,330,521,278,135,299,267,431,490,426,374,487,295,282,150,463,127,470,231,450,524,235,207,272,346,549,433,510,315,432,146,405,273,168,262,564,305,210,326,341,506,195,178,277,469,309,496,454]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13  % Problem  : SWW542_5 : TPTP v8.2.0. Released v6.0.0.
% 0.12/0.16  % Command  : run_Leo-III %s %d
% 0.16/0.37  % Computer : n004.cluster.edu
% 0.16/0.37  % Model    : x86_64 x86_64
% 0.16/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.37  % Memory   : 8042.1875MB
% 0.16/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.37  % CPULimit : 300
% 0.16/0.37  % WCLimit  : 300
% 0.16/0.37  % DateTime : Sat May 18 20:54:39 EDT 2024
% 0.16/0.37  % CPUTime  : 
% 1.02/0.93  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.54/1.13  % [INFO] 	 Parsing done (199ms). 
% 1.54/1.14  % [INFO] 	 Running in sequential loop mode. 
% 2.29/1.33  % [INFO] 	 eprover registered as external prover. 
% 2.29/1.34  % [INFO] 	 cvc4 registered as external prover. 
% 2.29/1.34  % [INFO] 	 Scanning for conjecture ... 
% 2.52/1.43  % [INFO] 	 Found a conjecture (or negated_conjecture) and 124 axioms. Running axiom selection ... 
% 2.77/1.51  % [INFO] 	 Axiom selection finished. Selected 124 axioms (removed 0 axioms). 
% 3.31/1.62  % [INFO] 	 Problem is typed first-order (TPTP TFF). 
% 3.31/1.64  % [INFO] 	 Type checking passed. 
% 3.31/1.65  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 3.80/1.80  % [INFO] 	 [Domain constraints] Detected constraint on bool 
% 3.80/1.80  % [INFO] 	 [Domain constraints] dom(bool) ⊆ {fTrue,fFalse} 
% 11.39/3.81  % External prover 'cvc4' found a proof!
% 11.39/3.81  % [INFO] 	 Killing All external provers ... 
% 11.39/3.81  % Time passed: 3277ms (effective reasoning time: 2669ms)
% 11.39/3.81  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 11.39/3.81  % Axioms used in derivation (124): fact_4_height__gt__0__notin__alphabet__imp__sibling__left, fact_46_of__nat__less__0__iff, fact_63_le__neq__implies__less, arity_HOL_Obool___Orderings_Oord, fact_11_less__zeroE, fact_67_eq__imp__le, arity_HOL_Obool___Orderings_Opreorder, arity_Nat_Onat___Rings_Osemiring__1, fact_85_transfer__int__nat__relations_I3_J, fact_39_tree_Oexhaust, fact_56_le0, fact_61_le__eq__less__or__eq, fact_77_zless__int, arity_Int_Oint___Orderings_Oord, fact_5_height__gt__0__notin__alphabet__imp__sibling__right, fact_3_notin__alphabet__imp__sibling__id, arity_fun___Orderings_Oord, fact_9_neq0__conv, arity_Nat_Onat___Orderings_Opreorder, fact_49_notin__alphabet__imp__freq__0, fact_10_less__nat__zero__code, arity_Int_Oint___Orderings_Opreorder, fact_57_of__nat__le__iff, fact_81_transfer__int__nat__numerals_I1_J, fact_75_mem__def, fact_59_int__int__eq, fact_60_nat__less__le, fact_18_sibling_Osimps_I2_J, fact_32_less__irrefl__nat, fact_29_nat__less__cases, fact_80_transfer__int__nat__relations_I2_J, fact_71_zero__le__imp__of__nat, fact_86_less__int__def, fact_53_tree_Orecs_I2_J, fact_42_infinite__descent0, arity_Int_Oint___Rings_Osemiring__1, arity_HOL_Obool___Orderings_Olinorder, fact_97_le__fun__def, fact_30_less__not__refl3, fact_74_ext, fact_88_zero__zle__int, fact_14_gr__implies__not0, fact_15_gr0I, help_If_1_1_T, fact_25_tree_Osimps_I4_J, fact_45_of__nat__less__iff, fact_51_optimum__def, arity_Nat_Onat___Orderings_Olinorder, fact_66_le__trans, fact_37_depth_Osimps_I1_J, arity_fun___Orderings_Opreorder, fact_12_height__gt__0__alphabet__eq__imp__height__gt__0, fact_68_nat__le__linear, fact_69_le__refl, fact_93_nonneg__eq__int, fact_34_nat__neq__iff, fact_83_transfer__int__nat__quantifiers_I2_J, fact_78_zero__less__imp__eq__int, fact_94_zero__le__imp__eq__int, fact_28_zero__reorient, arity_Int_Oint___Orderings_Olinorder, fact_72_depth__le__height, arity_Nat_Onat___Rings_Olinordered__semidom, fact_84_transfer__int__nat__quantifiers_I1_J, fact_87_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, fact_7_height__gt__0__in__alphabet__imp__sibling__right, fact_98_linorder__linear, fact_38_of__nat__0__less__iff, arity_Nat_Onat___Groups_Ozero, fact_26_tree_Osimps_I3_J, fact_31_less__not__refl2, arity_Nat_Onat___Orderings_Oord, help_If_3_1_T, help_If_2_1_T, fact_82_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, fact_35_less__not__refl, fact_8_height__0__imp__sibling__id, fact_91_transfer__int__nat__relations_I1_J, fact_90_int__le__0__conv, fact_50_zero__less__int__conv, fact_36_tree_Osize_I3_J, arity_Nat_Onat___Nat_Osemiring__char__0, fact_23_cost_Osimps_I1_J, fact_20_tree_Osimps_I1_J, fact_44_of__nat__0, fact_58_int__less__0__conv, fact_16_tree_Osimps_I6_J, fact_47_less__imp__of__nat__less, fact_64_less__or__eq__imp__le, fact_13_not__less0, fact_27_sibling_Osimps_I1_J, fact_54_le__0__eq, fact_22_height_Osimps_I1_J, fact_17_exists__in__alphabet, fact_0_tree_Osimps_I2_J, fact_70_of__nat__0__le__iff, fact_33_linorder__neqE__nat, fact_52_tree_Orecs_I1_J, help_pp_1_1_U, fact_76_int__0, fact_62_less__imp__le__nat, fact_65_le__antisym, fact_43_of__nat__eq__iff, fact_21_consistent_Osimps_I1_J, fact_55_less__eq__nat_Osimps_I1_J, fact_2_sibling_Osimps_I3_J, fact_73_int__eq__0__conv, fact_40_freq_Osimps_I1_J, fact_6_height__gt__0__in__alphabet__imp__sibling__left, conj_0, arity_Int_Oint___Nat_Osemiring__char__0, help_pp_2_1_U, arity_Int_Oint___Rings_Olinordered__semidom, fact_89_zle__int, arity_Int_Oint___Groups_Ozero, fact_95_less__mono__imp__le__mono, fact_92_int__if__cong, fact_79_ex__least__nat__le, fact_48_of__nat__less__imp__less, fact_41_exists__at__height, fact_1_sibling_Osimps_I4_J, fact_24_tree_Osimps_I5_J, fact_19_height__0__imp__cost__0, fact_96_order__refl
% 11.39/3.81  % No. of inferences in proof: 252
% 11.39/3.81  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 3277 ms resp. 2669 ms w/o parsing
% 11.39/3.88  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 11.39/3.88  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------