TSTP Solution File: ITP066^1 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP066^1 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n021.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 : Mon May 20 22:22:35 EDT 2024

% Result   : Theorem 21.14s 6.36s
% Output   : Refutation 21.14s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  302
% Syntax   : Number of formulae    :  521 ( 269 unt;  85 typ;   0 def)
%            Number of atoms       : 1021 ( 752 equ;   4 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 4414 ( 168   ~;  17   |; 112   &;3715   @)
%                                         (   0 <=>; 402  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Number of types       :   33 (  32 usr)
%            Number of type conns  :  303 ( 303   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   57 (  53 usr;  17 con; 0-6 aty)
%            Number of variables   : 1497 (  88   ^1375   !;  34   ?;1497   :)

% Comments : 
%------------------------------------------------------------------------------
thf(set_Pr1066112583iset_a_type,type,
    set_Pr1066112583iset_a: $tType ).

thf(produc741964647iset_a_type,type,
    produc741964647iset_a: $tType ).

thf(set_Pr146455751iset_a_type,type,
    set_Pr146455751iset_a: $tType ).

thf(set_Pr295926343Tree_a_type,type,
    set_Pr295926343Tree_a: $tType ).

thf(produc558600423iset_a_type,type,
    produc558600423iset_a: $tType ).

thf(produc1136638567Tree_a_type,type,
    produc1136638567Tree_a: $tType ).

thf(set_Pr1821581383od_a_a_type,type,
    set_Pr1821581383od_a_a: $tType ).

thf(set_Pr220561863Tree_a_type,type,
    set_Pr220561863Tree_a: $tType ).

thf(produc2061001575od_a_a_type,type,
    produc2061001575od_a_a: $tType ).

thf(set_Pr720484615iset_a_type,type,
    set_Pr720484615iset_a: $tType ).

thf(produc1254371559Tree_a_type,type,
    produc1254371559Tree_a: $tType ).

thf(produc1293660967iset_a_type,type,
    produc1293660967iset_a: $tType ).

thf(set_Pr1948701895od_a_a_type,type,
    set_Pr1948701895od_a_a: $tType ).

thf(multis782275565iset_a_type,type,
    multis782275565iset_a: $tType ).

thf(produc1572603623od_a_a_type,type,
    produc1572603623od_a_a: $tType ).

thf(set_Pr158363655iset_a_type,type,
    set_Pr158363655iset_a: $tType ).

thf(produc1127127335iset_a_type,type,
    produc1127127335iset_a: $tType ).

thf(multis2082063201Tree_a_type,type,
    multis2082063201Tree_a: $tType ).

thf(set_Pr1070816379Tree_a_type,type,
    set_Pr1070816379Tree_a: $tType ).

thf(set_Pr921883667ee_a_a_type,type,
    set_Pr921883667ee_a_a: $tType ).

thf(multis599418605od_a_a_type,type,
    multis599418605od_a_a: $tType ).

thf(produc143150363Tree_a_type,type,
    produc143150363Tree_a: $tType ).

thf(produc981471411ee_a_a_type,type,
    produc981471411ee_a_a: $tType ).

thf(set_Product_prod_a_a_type,type,
    set_Product_prod_a_a: $tType ).

thf(multiset_multiset_a_type,type,
    multiset_multiset_a: $tType ).

thf(set_multiset_a_type,type,
    set_multiset_a: $tType ).

thf(product_prod_a_a_type,type,
    product_prod_a_a: $tType ).

thf(multiset_a_type,type,
    multiset_a: $tType ).

thf(list_a_type,type,
    list_a: $tType ).

thf(tree_a_type,type,
    tree_a: $tType ).

thf(set_a_type,type,
    set_a: $tType ).

thf(a_type,type,
    a: $tType ).

thf(plus_plus_multiset_a_type,type,
    plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).

thf(zero_zero_multiset_a_type,type,
    zero_zero_multiset_a: multiset_a ).

thf(nO_MAT1617603563iset_a_type,type,
    nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).

thf(heapIm970322378pify_a_type,type,
    heapIm970322378pify_a: tree_a > tree_a ).

thf(heapIm229596386mpty_a_type,type,
    heapIm229596386mpty_a: tree_a > $o ).

thf(heapIm1057938560list_a_type,type,
    heapIm1057938560list_a: list_a > tree_a ).

thf(heapIm915766629_max_a_type,type,
    heapIm915766629_max_a: tree_a > produc143150363Tree_a ).

thf(heapIm1140443833left_a_type,type,
    heapIm1140443833left_a: tree_a > tree_a ).

thf(heapIm1637418125tree_a_type,type,
    heapIm1637418125tree_a: list_a > tree_a ).

thf(heapIm837449470Leaf_a_type,type,
    heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).

thf(heapIm1257206334ight_a_type,type,
    heapIm1257206334ight_a: tree_a > tree_a ).

thf(heapIm1091024090Down_a_type,type,
    heapIm1091024090Down_a: tree_a > tree_a ).

thf(heap_Tree_a_a_type,type,
    heap_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).

thf(heap_axioms_Tree_a_a_type,type,
    heap_axioms_Tree_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).

thf(heap_axioms_a_a_type,type,
    heap_axioms_a_a: ( a > $o ) > ( list_a > a ) > ( a > multiset_a ) > ( a > tree_a ) > ( a > product_prod_a_a ) > $o ).

thf(e_a_type,type,
    e_a: tree_a ).

thf(t_a_type,type,
    t_a: a > tree_a > tree_a > tree_a ).

thf(in_tree_a_type,type,
    in_tree_a: a > tree_a > $o ).

thf(is_heap_a_type,type,
    is_heap_a: tree_a > $o ).

thf(multiset_a2_type,type,
    multiset_a2: tree_a > multiset_a ).

thf(val_a_type,type,
    val_a: tree_a > a ).

thf(if_Pro1144176865Tree_a_type,type,
    if_Pro1144176865Tree_a: $o > produc143150363Tree_a > produc143150363Tree_a > produc143150363Tree_a ).

thf(lattic146396397_Max_a_type,type,
    lattic146396397_Max_a: set_a > a ).

thf(add_mset_a_type,type,
    add_mset_a: a > multiset_a > multiset_a ).

thf(is_empty_a_type,type,
    is_empty_a: multiset_a > $o ).

thf(set_mset_a_type,type,
    set_mset_a: multiset_a > set_a ).

thf(ord_le1748873858_a_a_o_type,type,
    ord_le1748873858_a_a_o: ( tree_a > a > $o ) > ( tree_a > a > $o ) > $o ).

thf(ord_le864074675ee_a_a_type,type,
    ord_le864074675ee_a_a: set_Pr921883667ee_a_a > set_Pr921883667ee_a_a > $o ).

thf(ord_less_eq_a_type,type,
    ord_less_eq_a: a > a > $o ).

thf(produc649172771ee_a_a_type,type,
    produc649172771ee_a_a: tree_a > a > produc981471411ee_a_a ).

thf(produc686083979Tree_a_type,type,
    produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).

thf(product_Pair_a_a_type,type,
    product_Pair_a_a: a > a > product_prod_a_a ).

thf(product_fst_Tree_a_a_type,type,
    product_fst_Tree_a_a: produc981471411ee_a_a > tree_a ).

thf(product_fst_a_Tree_a_type,type,
    product_fst_a_Tree_a: produc143150363Tree_a > a ).

thf(product_snd_Tree_a_a_type,type,
    product_snd_Tree_a_a: produc981471411ee_a_a > a ).

thf(product_snd_a_Tree_a_type,type,
    product_snd_a_Tree_a: produc143150363Tree_a > tree_a ).

thf(produc1478296771ee_a_a_type,type,
    produc1478296771ee_a_a: produc981471411ee_a_a > produc143150363Tree_a ).

thf(produc1515207979Tree_a_type,type,
    produc1515207979Tree_a: produc143150363Tree_a > produc981471411ee_a_a ).

thf(collection_Tree_a_a_type,type,
    collection_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > $o ).

thf(member1092636252ee_a_a_type,type,
    member1092636252ee_a_a: produc981471411ee_a_a > set_Pr921883667ee_a_a > $o ).

thf(member_a_type,type,
    member_a: a > set_a > $o ).

thf(l1_type,type,
    l1: tree_a ).

thf(l2_type,type,
    l2: tree_a ).

thf(r1_type,type,
    r1: tree_a ).

thf(r2_type,type,
    r2: tree_a ).

thf(t_type,type,
    t: tree_a ).

thf(t2_type,type,
    t2: tree_a ).

thf(t_a2_type,type,
    t_a2: tree_a ).

thf(v1_type,type,
    v1: a ).

thf(v2_type,type,
    v2: a ).

thf(v_type,type,
    v: a ).

thf(v_a_type,type,
    v_a: a ).

thf(v3_type,type,
    v3: a ).

thf(97,axiom,
    ! [A: tree_a,B: a] :
      ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_snd__conv) ).

thf(606,plain,
    ! [A: tree_a,B: a] :
      ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(205,axiom,
    ! [A: produc143150363Tree_a,B: a,C: tree_a] :
      ( ( A
        = ( produc686083979Tree_a @ B @ C ) )
     => ( ( product_fst_a_Tree_a @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_fstI) ).

thf(1024,plain,
    ! [A: produc143150363Tree_a,B: a,C: tree_a] :
      ( ( A
        = ( produc686083979Tree_a @ B @ C ) )
     => ( ( product_fst_a_Tree_a @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(81,axiom,
    ( produc1478296771ee_a_a
    = ( ^ [A: produc981471411ee_a_a] : ( produc686083979Tree_a @ ( product_snd_Tree_a_a @ A ) @ ( product_fst_Tree_a_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_prod_Oswap__def) ).

thf(555,plain,
    ( produc1478296771ee_a_a
    = ( ^ [A: produc981471411ee_a_a] : ( produc686083979Tree_a @ ( product_snd_Tree_a_a @ A ) @ ( product_fst_Tree_a_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(113,axiom,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( A @ ( add_mset_a @ C @ D ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_multiset__induct) ).

thf(666,plain,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( A @ ( add_mset_a @ C @ D ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(39,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_union__assoc) ).

thf(333,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(207,axiom,
    ! [A: a,B: tree_a] :
      ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_fst__conv) ).

thf(1031,plain,
    ! [A: a,B: tree_a] :
      ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(30,axiom,
    ! [A: tree_a,B: a,C: tree_a] :
      ( ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
        = C )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_fst__eqD) ).

thf(308,plain,
    ! [A: tree_a,B: a,C: tree_a] :
      ( ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
        = C )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(10,axiom,
    ! [A: produc981471411ee_a_a,B: tree_a,C: a] :
      ( ( A
        = ( produc649172771ee_a_a @ B @ C ) )
     => ( ( product_fst_Tree_a_a @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_fstI) ).

thf(247,plain,
    ! [A: produc981471411ee_a_a,B: tree_a,C: a] :
      ( ( A
        = ( produc649172771ee_a_a @ B @ C ) )
     => ( ( product_fst_Tree_a_a @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(76,axiom,
    ! [A: a,B: a,C: multiset_a] :
      ( ( add_mset_a @ A @ ( add_mset_a @ B @ C ) )
      = ( add_mset_a @ B @ ( add_mset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_add__mset__commute) ).

thf(511,plain,
    ! [A: a,B: a,C: multiset_a] :
      ( ( add_mset_a @ A @ ( add_mset_a @ B @ C ) )
      = ( add_mset_a @ B @ ( add_mset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(17,axiom,
    ! [A: product_prod_a_a] :
    ? [B: a,C: a] :
      ( A
      = ( product_Pair_a_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_surj__pair) ).

thf(269,plain,
    ! [A: product_prod_a_a] :
    ? [B: a,C: a] :
      ( A
      = ( product_Pair_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(94,axiom,
    ! [A: produc143150363Tree_a,B: a,C: tree_a] :
      ( ( A
        = ( produc686083979Tree_a @ B @ C ) )
     => ( ( product_snd_a_Tree_a @ A )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_sndI) ).

thf(597,plain,
    ! [A: produc143150363Tree_a,B: a,C: tree_a] :
      ( ( A
        = ( produc686083979Tree_a @ B @ C ) )
     => ( ( product_snd_a_Tree_a @ A )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(131,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) )
      = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_add_Oleft__commute) ).

thf(730,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) )
      = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(19,axiom,
    collection_Tree_a_a @ e_a @ heapIm229596386mpty_a @ heapIm1057938560list_a @ multiset_a2,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_HS_OCollection__axioms) ).

thf(276,plain,
    collection_Tree_a_a @ e_a @ heapIm229596386mpty_a @ heapIm1057938560list_a @ multiset_a2,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(54,axiom,
    ! [A: produc981471411ee_a_a] :
      ~ ! [B: tree_a,C: a] :
          ( A
         != ( produc649172771ee_a_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_old_Oprod_Oexhaust) ).

thf(385,plain,
    ! [A: produc981471411ee_a_a] :
      ~ ! [B: tree_a,C: a] :
          ( A
         != ( produc649172771ee_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(126,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) )
      = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_union__lcomm) ).

thf(715,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) )
      = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(59,axiom,
    ! [A: a > multiset_a,B: a > tree_a,C: list_a > a,D: a > $o,E: a > product_prod_a_a] :
      ( ! [F: a] :
          ( ( A @ F )
          = ( multiset_a2 @ ( B @ F ) ) )
     => ( ! [F: list_a] : ( is_heap_a @ ( B @ ( C @ F ) ) )
       => ( ! [F: a] :
              ( ( ( B @ F )
                = e_a )
              = ( D @ F ) )
         => ( ! [F: a,G: a,H: a] :
                ( ~ ( D @ F )
               => ( ( ( product_Pair_a_a @ G @ H )
                    = ( E @ F ) )
                 => ( ( add_mset_a @ G @ ( A @ H ) )
                    = ( A @ F ) ) ) )
           => ( ! [F: a,G: a,H: a] :
                  ( ~ ( D @ F )
                 => ( ( is_heap_a @ ( B @ F ) )
                   => ( ( ( product_Pair_a_a @ G @ H )
                        = ( E @ F ) )
                     => ( is_heap_a @ ( B @ H ) ) ) ) )
             => ( ! [F: a,G: a,H: a] :
                    ( ~ ( D @ F )
                   => ( ( ( product_Pair_a_a @ G @ H )
                        = ( E @ F ) )
                     => ( G
                        = ( val_a @ ( B @ F ) ) ) ) )
               => ( heap_axioms_a_a @ D @ C @ A @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_Heap__axioms_Ointro) ).

thf(404,plain,
    ! [A: a > multiset_a,B: a > tree_a,C: list_a > a,D: a > $o,E: a > product_prod_a_a] :
      ( ! [F: a] :
          ( ( A @ F )
          = ( multiset_a2 @ ( B @ F ) ) )
     => ( ! [F: list_a] : ( is_heap_a @ ( B @ ( C @ F ) ) )
       => ( ! [F: a] :
              ( ( ( B @ F )
                = e_a )
              = ( D @ F ) )
         => ( ! [F: a,G: a,H: a] :
                ( ~ ( D @ F )
               => ( ( ( product_Pair_a_a @ G @ H )
                    = ( E @ F ) )
                 => ( ( add_mset_a @ G @ ( A @ H ) )
                    = ( A @ F ) ) ) )
           => ( ! [F: a,G: a,H: a] :
                  ( ~ ( D @ F )
                 => ( ( is_heap_a @ ( B @ F ) )
                   => ( ( ( product_Pair_a_a @ G @ H )
                        = ( E @ F ) )
                     => ( is_heap_a @ ( B @ H ) ) ) ) )
             => ( ! [F: a,G: a,H: a] :
                    ( ~ ( D @ F )
                   => ( ( ( product_Pair_a_a @ G @ H )
                        = ( E @ F ) )
                     => ( G
                        = ( val_a @ ( B @ F ) ) ) ) )
               => ( heap_axioms_a_a @ D @ C @ A @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(124,axiom,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ B ) @ C )
      = ( add_mset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_union__mset__add__mset__left) ).

thf(709,plain,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ B ) @ C )
      = ( add_mset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(34,axiom,
    ! [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
      ( ( ( ( product_fst_Tree_a_a @ A )
          = ( product_fst_Tree_a_a @ B ) )
        & ( ( product_snd_Tree_a_a @ A )
          = ( product_snd_Tree_a_a @ B ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_prod_Oexpand) ).

thf(320,plain,
    ! [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
      ( ( ( ( product_fst_Tree_a_a @ A )
          = ( product_fst_Tree_a_a @ B ) )
        & ( ( product_snd_Tree_a_a @ A )
          = ( product_snd_Tree_a_a @ B ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(60,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_union__left__cancel) ).

thf(436,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(108,axiom,
    ! [A: produc981471411ee_a_a] :
      ( ( product_snd_a_Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = ( product_fst_Tree_a_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_snd__swap) ).

thf(651,plain,
    ! [A: produc981471411ee_a_a] :
      ( ( product_snd_a_Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = ( product_fst_Tree_a_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(13,axiom,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ! [C: tree_a,D: a] :
          ( ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A )
         => ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) )
     => ( ord_le864074675ee_a_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_257_subrelI) ).

thf(257,plain,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ! [C: tree_a,D: a] :
          ( ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A )
         => ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) )
     => ( ord_le864074675ee_a_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(55,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
        = C )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_snd__eqD) ).

thf(389,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
        = C )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(129,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_comm__monoid__add__class_Oadd__0) ).

thf(724,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(50,axiom,
    ! [A: produc981471411ee_a_a > $o,B: produc981471411ee_a_a] :
      ( ! [C: tree_a,D: a] : ( A @ ( produc649172771ee_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_old_Oprod_Oinducts) ).

thf(372,plain,
    ! [A: produc981471411ee_a_a > $o,B: produc981471411ee_a_a] :
      ( ! [C: tree_a,D: a] : ( A @ ( produc649172771ee_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(138,axiom,
    ! [A: produc981471411ee_a_a] :
    ? [B: tree_a,C: a] :
      ( A
      = ( produc649172771ee_a_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_surj__pair) ).

thf(751,plain,
    ! [A: produc981471411ee_a_a] :
    ? [B: tree_a,C: a] :
      ( A
      = ( produc649172771ee_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(154,axiom,
    ! [A: multiset_a,B: multiset_a,C: a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( ( A
            = ( add_mset_a @ C @ zero_zero_multiset_a ) )
          & ( B = zero_zero_multiset_a ) )
        | ( ( A = zero_zero_multiset_a )
          & ( B
            = ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_union__is__single) ).

thf(802,plain,
    ! [A: multiset_a,B: multiset_a,C: a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( ( A
            = ( add_mset_a @ C @ zero_zero_multiset_a ) )
          & ( B = zero_zero_multiset_a ) )
        | ( ( A = zero_zero_multiset_a )
          & ( B
            = ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(43,axiom,
    ! [A: produc981471411ee_a_a,B: tree_a,C: a] :
      ( ( A
        = ( produc649172771ee_a_a @ B @ C ) )
     => ( ( product_snd_Tree_a_a @ A )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_sndI) ).

thf(344,plain,
    ! [A: produc981471411ee_a_a,B: tree_a,C: a] :
      ( ( A
        = ( produc649172771ee_a_a @ B @ C ) )
     => ( ( product_snd_Tree_a_a @ A )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(149,axiom,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_multiset_Osimps_I1_J) ).

thf(789,plain,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(210,axiom,
    ! [A: a,B: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
     => ( ( ( t_a @ v1 @ l1 @ r1 )
         != e_a )
       => ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ ( multiset_a2 @ B ) )
          = ( multiset_a2 @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9__C4__2_Ohyps_C_I1_J) ).

thf(1040,plain,
    ! [A: a,B: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
     => ( ( ( t_a @ v1 @ l1 @ r1 )
         != e_a )
       => ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ ( multiset_a2 @ B ) )
          = ( multiset_a2 @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(106,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_add_Oassoc) ).

thf(646,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(77,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( B = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_add__cancel__right__right) ).

thf(514,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( B = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(192,axiom,
    ( ( produc686083979Tree_a @ v @ t2 )
    = ( heapIm837449470Leaf_a @ t ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_assms_I1_J) ).

thf(966,plain,
    ( ( produc686083979Tree_a @ v @ t2 )
    = ( heapIm837449470Leaf_a @ t ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(179,axiom,
    ( ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) )
   != e_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2__C4__2_Oprems_C_I2_J) ).

thf(905,plain,
    ( ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) )
   != e_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(44,axiom,
    ! [A: tree_a] :
      ( ( multiset_a2 @ ( heapIm1091024090Down_a @ A ) )
      = ( multiset_a2 @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_siftDown__multiset) ).

thf(348,plain,
    ! [A: tree_a] :
      ( ( multiset_a2 @ ( heapIm1091024090Down_a @ A ) )
      = ( multiset_a2 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(88,axiom,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_Pair__inject) ).

thf(574,plain,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(21,axiom,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_old_Oprod_Oinducts) ).

thf(280,plain,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(176,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( val_a @ ( t_a @ A @ B @ C ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_val_Osimps) ).

thf(877,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( val_a @ ( t_a @ A @ B @ C ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(3,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F )
       => ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_Heap_Ointro) ).

thf(221,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F )
       => ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(172,axiom,
    ! [A: tree_a] :
      ( ( is_heap_a @ A )
     => ( ( A != e_a )
       => ( is_heap_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_removeLeaf__heap__is__heap) ).

thf(866,plain,
    ! [A: tree_a] :
      ( ( is_heap_a @ A )
     => ( ( A != e_a )
       => ( is_heap_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(111,axiom,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $false @ A @ B )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T) ).

thf(660,plain,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $false @ A @ B )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(31,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: list_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( is_heap_a @ ( E @ ( C @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_Heap_Ois__heap__of__list) ).

thf(312,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: list_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( is_heap_a @ ( E @ ( C @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(196,axiom,
    ! [A: a] : ( is_heap_a @ ( t_a @ A @ e_a @ e_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_is__heap_Osimps_I2_J) ).

thf(981,plain,
    ! [A: a] : ( is_heap_a @ ( t_a @ A @ e_a @ e_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(142,axiom,
    ! [A: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_heapify__heap__is__heap) ).

thf(766,plain,
    ! [A: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(7,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( collection_Tree_a_a @ A @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_Heap_Oaxioms_I1_J) ).

thf(233,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( collection_Tree_a_a @ A @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(182,axiom,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( ( ( product_fst_a_Tree_a @ A )
          = ( product_fst_a_Tree_a @ B ) )
        & ( ( product_snd_a_Tree_a @ A )
          = ( product_snd_a_Tree_a @ B ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_prod_Oexpand) ).

thf(925,plain,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( ( ( product_fst_a_Tree_a @ A )
          = ( product_fst_a_Tree_a @ B ) )
        & ( ( product_snd_a_Tree_a @ A )
          = ( product_snd_a_Tree_a @ B ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(157,axiom,
    ! [A: a,B: a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ C ) )
      = ( ( B = A )
        & ( C = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_single__eq__add__mset) ).

thf(810,plain,
    ! [A: a,B: a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ C ) )
      = ( ( B = A )
        & ( C = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(53,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_multiset_a @ A @ C )
        = ( plus_plus_multiset_a @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_add__mono__thms__linordered__semiring_I4_J) ).

thf(381,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_multiset_a @ A @ C )
        = ( plus_plus_multiset_a @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(48,axiom,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_prod_Oinject) ).

thf(366,plain,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(187,axiom,
    ( produc1515207979Tree_a
    = ( ^ [A: produc143150363Tree_a] : ( produc649172771ee_a_a @ ( product_snd_a_Tree_a @ A ) @ ( product_fst_a_Tree_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_prod_Oswap__def) ).

thf(942,plain,
    ( produc1515207979Tree_a
    = ( ^ [A: produc143150363Tree_a] : ( produc649172771ee_a_a @ ( product_snd_a_Tree_a @ A ) @ ( product_fst_a_Tree_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(61,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_subset__mset_Ozero__eq__add__iff__both__eq__0) ).

thf(439,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(212,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_left_Osimps) ).

thf(1047,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(152,axiom,
    ! [A: a,B: multiset_a,C: a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( B = zero_zero_multiset_a )
        & ( A = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_add__mset__eq__singleton__iff) ).

thf(798,plain,
    ! [A: a,B: multiset_a,C: a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( B = zero_zero_multiset_a )
        & ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(215,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ E @ F @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_removeLeaf_Osimps_I4_J) ).

thf(1057,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ E @ F @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(136,axiom,
    ( add_mset_a
    = ( ^ [A: a,B: multiset_a] : ( plus_plus_multiset_a @ B @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_add__mset__add__single) ).

thf(746,plain,
    ( add_mset_a
    = ( ^ [A: a,B: multiset_a] : ( plus_plus_multiset_a @ B @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(146,axiom,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_add__mset__add__mset__same__iff) ).

thf(778,plain,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(161,axiom,
    ! [A: tree_a,B: a,C: a] :
      ( ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
        = C )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_snd__eqD) ).

thf(820,plain,
    ! [A: tree_a,B: a,C: a] :
      ( ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
        = C )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(14,axiom,
    ! [A: a,B: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ? [C: multiset_a] :
          ( B
          = ( add_mset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_285_multi__member__split) ).

thf(260,plain,
    ! [A: a,B: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ? [C: multiset_a] :
          ( B
          = ( add_mset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(171,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( e_a
     != ( t_a @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_Tree_Odistinct_I1_J) ).

thf(862,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( e_a
     != ( t_a @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(66,axiom,
    ! [A: a,B: tree_a] :
      ( ( in_tree_a @ A @ B )
     => ( ( is_heap_a @ B )
       => ( ord_less_eq_a @ A @ ( val_a @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_is__heap__max) ).

thf(467,plain,
    ! [A: a,B: tree_a] :
      ( ( in_tree_a @ A @ B )
     => ( ( is_heap_a @ B )
       => ( ord_less_eq_a @ A @ ( val_a @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(11,axiom,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ B @ C ) ) )
      = ( ( member_a @ A @ ( set_mset_a @ B ) )
        | ( member_a @ A @ ( set_mset_a @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_295_union__iff) ).

thf(252,plain,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ B @ C ) ) )
      = ( ( member_a @ A @ ( set_mset_a @ B ) )
        | ( member_a @ A @ ( set_mset_a @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(170,axiom,
    ! [A: a > $o,B: a,C: tree_a > $o,D: tree_a] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ B @ D ) ) )
          & ( C @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_conjI__realizer) ).

thf(852,plain,
    ! [A: a > $o,B: a,C: tree_a > $o,D: tree_a] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ B @ D ) ) )
          & ( C @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(64,axiom,
    ! [A: multiset_a,B: a] :
      ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ A )
     => ( ( add_mset_a @ B @ A )
        = ( plus_plus_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_add__mset__replicate__mset__safe) ).

thf(461,plain,
    ! [A: multiset_a,B: a] :
      ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ A )
     => ( ( add_mset_a @ B @ A )
        = ( plus_plus_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(70,axiom,
    ! [A: a,B: tree_a] :
      ( ( produc1515207979Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = ( produc649172771ee_a_a @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_swap__simp) ).

thf(493,plain,
    ! [A: a,B: tree_a] :
      ( ( produc1515207979Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = ( produc649172771ee_a_a @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(140,axiom,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_Pair__inject) ).

thf(757,plain,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(116,axiom,
    ! [A: a,B: multiset_a,C: a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ C @ zero_zero_multiset_a ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_315_multi__member__skip) ).

thf(682,plain,
    ! [A: a,B: multiset_a,C: a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ C @ zero_zero_multiset_a ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(4,axiom,
    ( heap_Tree_a_a
    = ( ^ [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
          ( ( collection_Tree_a_a @ A @ B @ C @ D )
          & ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_Heap__def) ).

thf(224,plain,
    ( heap_Tree_a_a
    = ( ^ [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
          ( ( collection_Tree_a_a @ A @ B @ C @ D )
          & ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(168,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ E @ F @ G ) ) @ A )
        & ( is_heap_a @ ( t_a @ E @ F @ G ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_is__heap_Osimps_I5_J) ).

thf(847,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ E @ F @ G ) ) @ A )
        & ( is_heap_a @ ( t_a @ E @ F @ G ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(85,axiom,
    ( ( heapIm1091024090Down_a @ e_a )
    = e_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_siftDown_Osimps_I1_J) ).

thf(566,plain,
    ( ( heapIm1091024090Down_a @ e_a )
    = e_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(36,axiom,
    ( ( (=) @ produc981471411ee_a_a )
    = ( ^ [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
          ( ( ( product_fst_Tree_a_a @ A )
            = ( product_fst_Tree_a_a @ B ) )
          & ( ( product_snd_Tree_a_a @ A )
            = ( product_snd_Tree_a_a @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_prod__eq__iff) ).

thf(324,plain,
    ( ( (=) @ produc981471411ee_a_a )
    = ( ^ [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
          ( ( ( product_fst_Tree_a_a @ A )
            = ( product_fst_Tree_a_a @ B ) )
          & ( ( product_snd_Tree_a_a @ A )
            = ( product_snd_Tree_a_a @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(195,axiom,
    ! [A: a,B: tree_a,C: tree_a,D: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) )
          = ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) )
          = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ D @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_siftDown_Osimps_I4_J) ).

thf(974,plain,
    ! [A: a,B: tree_a,C: tree_a,D: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) )
          = ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ e_a @ ( t_a @ A @ B @ C ) ) )
          = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ D @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(135,axiom,
    ! [A: a,B: multiset_a] :
      ( zero_zero_multiset_a
     != ( add_mset_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_empty__not__add__mset) ).

thf(742,plain,
    ! [A: a,B: multiset_a] :
      ( zero_zero_multiset_a
     != ( add_mset_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(175,axiom,
    ! [A: produc143150363Tree_a] :
      ( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_prod_Ocollapse) ).

thf(874,plain,
    ! [A: produc143150363Tree_a] :
      ( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(33,axiom,
    ! [A: tree_a,B: produc981471411ee_a_a] :
      ( ( A
        = ( product_fst_Tree_a_a @ B ) )
      = ( ? [C: a] :
            ( B
            = ( produc649172771ee_a_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_eq__fst__iff) ).

thf(317,plain,
    ! [A: tree_a,B: produc981471411ee_a_a] :
      ( ( A
        = ( product_fst_Tree_a_a @ B ) )
      = ( ? [C: a] :
            ( B
            = ( produc649172771ee_a_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(100,axiom,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_prod_Oinject) ).

thf(615,plain,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(47,axiom,
    ! [A: tree_a > $o,B: tree_a,C: a > $o,D: a] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ B @ D ) ) )
          & ( C @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_conjI__realizer) ).

thf(356,plain,
    ! [A: tree_a > $o,B: tree_a,C: a > $o,D: a] :
      ( ( A @ B )
     => ( ( C @ D )
       => ( ( A @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ B @ D ) ) )
          & ( C @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(173,axiom,
    ( ( produc686083979Tree_a @ v_a @ t_a2 )
    = ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0__C4__2_Oprems_C_I1_J) ).

thf(869,plain,
    ( ( produc686083979Tree_a @ v_a @ t_a2 )
    = ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(72,axiom,
    ! [A: tree_a] :
      ( ( heapIm229596386mpty_a @ A )
     => ( A = e_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_HS_Ois__empty__inj) ).

thf(499,plain,
    ! [A: tree_a] :
      ( ( heapIm229596386mpty_a @ A )
     => ( A = e_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(199,axiom,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ~ ! [B: a,C: tree_a,D: tree_a] :
            ( A
           != ( t_a @ B @ C @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_Tree_Oexhaust) ).

thf(989,plain,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ~ ! [B: a,C: tree_a,D: tree_a] :
            ( A
           != ( t_a @ B @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(103,axiom,
    ( heap_axioms_a_a
    = ( ^ [A: a > $o,B: list_a > a,C: a > multiset_a,D: a > tree_a,E: a > product_prod_a_a] :
          ( ! [F: a] :
              ( ( C @ F )
              = ( multiset_a2 @ ( D @ F ) ) )
          & ! [F: list_a] : ( is_heap_a @ ( D @ ( B @ F ) ) )
          & ! [F: a] :
              ( ( ( D @ F )
                = e_a )
              = ( A @ F ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( ( product_Pair_a_a @ G @ H )
                  = ( E @ F ) )
               => ( ( add_mset_a @ G @ ( C @ H ) )
                  = ( C @ F ) ) ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( is_heap_a @ ( D @ F ) )
               => ( ( ( product_Pair_a_a @ G @ H )
                    = ( E @ F ) )
                 => ( is_heap_a @ ( D @ H ) ) ) ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( ( product_Pair_a_a @ G @ H )
                  = ( E @ F ) )
               => ( G
                  = ( val_a @ ( D @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_Heap__axioms__def) ).

thf(637,plain,
    ( heap_axioms_a_a
    = ( ^ [A: a > $o,B: list_a > a,C: a > multiset_a,D: a > tree_a,E: a > product_prod_a_a] :
          ( ! [F: a] :
              ( ( C @ F )
              = ( multiset_a2 @ ( D @ F ) ) )
          & ! [F: list_a] : ( is_heap_a @ ( D @ ( B @ F ) ) )
          & ! [F: a] :
              ( ( ( D @ F )
                = e_a )
              = ( A @ F ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( ( product_Pair_a_a @ G @ H )
                  = ( E @ F ) )
               => ( ( add_mset_a @ G @ ( C @ H ) )
                  = ( C @ F ) ) ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( is_heap_a @ ( D @ F ) )
               => ( ( ( product_Pair_a_a @ G @ H )
                    = ( E @ F ) )
                 => ( is_heap_a @ ( D @ H ) ) ) ) )
          & ! [F: a,G: a,H: a] :
              ( ~ ( A @ F )
             => ( ( ( product_Pair_a_a @ G @ H )
                  = ( E @ F ) )
               => ( G
                  = ( val_a @ ( D @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(69,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_add__left__imp__eq) ).

thf(489,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(115,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_add__right__cancel) ).

thf(679,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(123,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_empty__eq__union) ).

thf(706,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ A @ B ) )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(56,axiom,
    ! [A: tree_a > a > $o,B: tree_a,C: a,D: produc981471411ee_a_a] :
      ( ( A @ B @ C )
     => ( ( D
          = ( produc649172771ee_a_a @ B @ C ) )
       => ( A @ ( product_fst_Tree_a_a @ D ) @ ( product_snd_Tree_a_a @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_BNF__Greatest__Fixpoint_Osubst__Pair) ).

thf(393,plain,
    ! [A: tree_a > a > $o,B: tree_a,C: a,D: produc981471411ee_a_a] :
      ( ( A @ B @ C )
     => ( ( D
          = ( produc649172771ee_a_a @ B @ C ) )
       => ( A @ ( product_fst_Tree_a_a @ D ) @ ( product_snd_Tree_a_a @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(109,axiom,
    ! [A: a,B: multiset_a,C: a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( A = C )
        & ( B = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_add__mset__eq__single) ).

thf(654,plain,
    ! [A: a,B: multiset_a,C: a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ zero_zero_multiset_a ) )
      = ( ( A = C )
        & ( B = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(128,axiom,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ( is_heap_a @ A )
       => ( ( val_a @ A )
          = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_341_heap__top__max) ).

thf(721,plain,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ( is_heap_a @ A )
       => ( ( val_a @ A )
          = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(216,axiom,
    ( t_a2
    = ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062) ).

thf(1060,plain,
    ( t_a2
    = ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(213,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm970322378pify_a @ ( t_a @ A @ B @ C ) )
      = ( heapIm1091024090Down_a @ ( t_a @ A @ ( heapIm970322378pify_a @ B ) @ ( heapIm970322378pify_a @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_heapify_Osimps_I2_J) ).

thf(1050,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm970322378pify_a @ ( t_a @ A @ B @ C ) )
      = ( heapIm1091024090Down_a @ ( t_a @ A @ ( heapIm970322378pify_a @ B ) @ ( heapIm970322378pify_a @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(16,axiom,
    ! [A: tree_a,B: a] :
      ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_fst__conv) ).

thf(266,plain,
    ! [A: tree_a,B: a] :
      ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(155,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = zero_zero_multiset_a )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_subset__mset_Oadd__eq__0__iff__both__eq__0) ).

thf(805,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = zero_zero_multiset_a )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(75,axiom,
    ! [A: produc143150363Tree_a] :
    ? [B: a,C: tree_a] :
      ( A
      = ( produc686083979Tree_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_surj__pair) ).

thf(508,plain,
    ! [A: produc143150363Tree_a] :
    ? [B: a,C: tree_a] :
      ( A
      = ( produc686083979Tree_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(110,axiom,
    ! [A: produc143150363Tree_a > $o,B: produc143150363Tree_a] :
      ( ! [C: a,D: tree_a] : ( A @ ( produc686083979Tree_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_old_Oprod_Oinducts) ).

thf(657,plain,
    ! [A: produc143150363Tree_a > $o,B: produc143150363Tree_a] :
      ( ! [C: a,D: tree_a] : ( A @ ( produc686083979Tree_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(57,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_add__left__cancel) ).

thf(398,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(151,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_add__right__imp__eq) ).

thf(794,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(127,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_verit__sum__simplify) ).

thf(718,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(101,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_empty__neutral_I2_J) ).

thf(618,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(162,axiom,
    ( heapIm229596386mpty_a
    = ( ^ [A: tree_a] : ( A = e_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_hs__is__empty__def) ).

thf(824,plain,
    ( heapIm229596386mpty_a
    = ( ^ [A: tree_a] : ( A = e_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(26,axiom,
    ! [A: product_prod_a_a] :
      ~ ! [B: a,C: a] :
          ( A
         != ( product_Pair_a_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_old_Oprod_Oexhaust) ).

thf(296,plain,
    ! [A: product_prod_a_a] :
      ~ ! [B: a,C: a] :
          ( A
         != ( product_Pair_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(95,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_add_Oleft__neutral) ).

thf(601,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(51,axiom,
    ! [A: tree_a] :
      ( ( multiset_a2 @ ( heapIm970322378pify_a @ A ) )
      = ( multiset_a2 @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_multiset__heapify) ).

thf(375,plain,
    ! [A: tree_a] :
      ( ( multiset_a2 @ ( heapIm970322378pify_a @ A ) )
      = ( multiset_a2 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(132,axiom,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $true @ A @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T) ).

thf(733,plain,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( if_Pro1144176865Tree_a @ $true @ A @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(160,axiom,
    heapIm229596386mpty_a @ e_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_HS_Ois__empty__empty) ).

thf(819,plain,
    heapIm229596386mpty_a @ e_a,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(203,axiom,
    ! [A: tree_a] :
      ( ! [B: a] :
          ( A
         != ( t_a @ B @ e_a @ e_a ) )
     => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
            ( A
           != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                ( A
               != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) )
           => ( ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) )
             => ( A = e_a ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_removeLeaf_Ocases) ).

thf(1007,plain,
    ! [A: tree_a] :
      ( ! [B: a] :
          ( A
         != ( t_a @ B @ e_a @ e_a ) )
     => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
            ( A
           != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                ( A
               != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) )
           => ( ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) )
             => ( A = e_a ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(23,axiom,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = C )
     => ( member_a @ A @ ( set_mset_a @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_275_union__single__eq__member) ).

thf(286,plain,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = C )
     => ( member_a @ A @ ( set_mset_a @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(163,axiom,
    ! [A: a,B: tree_a,C: tree_a,D: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) )
          = ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) )
          = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1091024090Down_a @ ( t_a @ D @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) @ e_a ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_siftDown_Osimps_I3_J) ).

thf(826,plain,
    ! [A: a,B: tree_a,C: tree_a,D: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) )
          = ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ D )
       => ( ( heapIm1091024090Down_a @ ( t_a @ D @ ( t_a @ A @ B @ C ) @ e_a ) )
          = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1091024090Down_a @ ( t_a @ D @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) @ e_a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(87,axiom,
    ( plus_plus_multiset_a
    = ( ^ [A: multiset_a,B: multiset_a] : ( plus_plus_multiset_a @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_union__commute) ).

thf(572,plain,
    ( plus_plus_multiset_a
    = ( ^ [A: multiset_a,B: multiset_a] : ( plus_plus_multiset_a @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(8,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ( D @ G )
        = ( multiset_a2 @ ( E @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_Heap_Omultiset) ).

thf(236,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ( D @ G )
        = ( multiset_a2 @ ( E @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(189,axiom,
    ! [A: produc143150363Tree_a] :
      ( ( product_snd_Tree_a_a @ ( produc1515207979Tree_a @ A ) )
      = ( product_fst_a_Tree_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_snd__swap) ).

thf(947,plain,
    ! [A: produc143150363Tree_a] :
      ( ( product_snd_Tree_a_a @ ( produc1515207979Tree_a @ A ) )
      = ( product_fst_a_Tree_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(63,axiom,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( ! [E: a] :
                  ( ( member_a @ E @ ( set_mset_a @ D ) )
                 => ( ord_less_eq_a @ C @ E ) )
             => ( A @ ( add_mset_a @ C @ D ) ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_309_multiset__induct__min) ).

thf(445,plain,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( ! [E: a] :
                  ( ( member_a @ E @ ( set_mset_a @ D ) )
                 => ( ord_less_eq_a @ C @ E ) )
             => ( A @ ( add_mset_a @ C @ D ) ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(185,axiom,
    ! [A: a,B: produc143150363Tree_a] :
      ( ( A
        = ( product_fst_a_Tree_a @ B ) )
      = ( ? [C: tree_a] :
            ( B
            = ( produc686083979Tree_a @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_eq__fst__iff) ).

thf(936,plain,
    ! [A: a,B: produc143150363Tree_a] :
      ( ( A
        = ( product_fst_a_Tree_a @ B ) )
      = ( ? [C: tree_a] :
            ( B
            = ( produc686083979Tree_a @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(1,conjecture,
    ( v_a
    = ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ( v_a
   != ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(219,plain,
    ( v_a
   != ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(204,axiom,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a,G: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1257206334ight_a @ ( t_a @ D @ E @ F ) ) ) ) @ ( t_a @ A @ B @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( t_a @ D @ E @ F ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_siftDown_Osimps_I6_J) ).

thf(1011,plain,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a,G: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1257206334ight_a @ ( t_a @ D @ E @ F ) ) ) ) @ ( t_a @ A @ B @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( t_a @ D @ E @ F ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(133,axiom,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_old_Oprod_Oinject) ).

thf(736,plain,
    ! [A: tree_a,B: a,C: tree_a,D: a] :
      ( ( ( produc649172771ee_a_a @ A @ B )
        = ( produc649172771ee_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(201,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ e_a ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_is__heap_Osimps_I4_J) ).

thf(996,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ e_a ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(6,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( D @ A )
        = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_Collection_Omultiset__empty) ).

thf(230,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( D @ A )
        = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(28,axiom,
    nO_MAT1617603563iset_a = nO_MAT1617603563iset_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_NO__MATCH__cong) ).

thf(303,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(141,axiom,
    ( is_empty_a
    = ( ^ [A: multiset_a] : ( A = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_Multiset_Ois__empty__def) ).

thf(764,plain,
    ( is_empty_a
    = ( ^ [A: multiset_a] : ( A = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(96,axiom,
    ( ( heapIm970322378pify_a @ e_a )
    = e_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_heapify_Osimps_I1_J) ).

thf(604,plain,
    ( ( heapIm970322378pify_a @ e_a )
    = e_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(177,axiom,
    ! [A: produc143150363Tree_a] :
      ( A
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_prod_Oexhaust__sel) ).

thf(880,plain,
    ! [A: produc143150363Tree_a] :
      ( A
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(107,axiom,
    ( heap_axioms_Tree_a_a
    = ( ^ [A: tree_a > $o,B: list_a > tree_a,C: tree_a > multiset_a,D: tree_a > tree_a,E: tree_a > produc143150363Tree_a] :
          ( ! [F: tree_a] :
              ( ( C @ F )
              = ( multiset_a2 @ ( D @ F ) ) )
          & ! [F: list_a] : ( is_heap_a @ ( D @ ( B @ F ) ) )
          & ! [F: tree_a] :
              ( ( ( D @ F )
                = e_a )
              = ( A @ F ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( ( produc686083979Tree_a @ G @ H )
                  = ( E @ F ) )
               => ( ( add_mset_a @ G @ ( C @ H ) )
                  = ( C @ F ) ) ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( is_heap_a @ ( D @ F ) )
               => ( ( ( produc686083979Tree_a @ G @ H )
                    = ( E @ F ) )
                 => ( is_heap_a @ ( D @ H ) ) ) ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( ( produc686083979Tree_a @ G @ H )
                  = ( E @ F ) )
               => ( G
                  = ( val_a @ ( D @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_Heap__axioms__def) ).

thf(649,plain,
    ( heap_axioms_Tree_a_a
    = ( ^ [A: tree_a > $o,B: list_a > tree_a,C: tree_a > multiset_a,D: tree_a > tree_a,E: tree_a > produc143150363Tree_a] :
          ( ! [F: tree_a] :
              ( ( C @ F )
              = ( multiset_a2 @ ( D @ F ) ) )
          & ! [F: list_a] : ( is_heap_a @ ( D @ ( B @ F ) ) )
          & ! [F: tree_a] :
              ( ( ( D @ F )
                = e_a )
              = ( A @ F ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( ( produc686083979Tree_a @ G @ H )
                  = ( E @ F ) )
               => ( ( add_mset_a @ G @ ( C @ H ) )
                  = ( C @ F ) ) ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( is_heap_a @ ( D @ F ) )
               => ( ( ( produc686083979Tree_a @ G @ H )
                    = ( E @ F ) )
                 => ( is_heap_a @ ( D @ H ) ) ) ) )
          & ! [F: tree_a,G: a,H: tree_a] :
              ( ~ ( A @ F )
             => ( ( ( produc686083979Tree_a @ G @ H )
                  = ( E @ F ) )
               => ( G
                  = ( val_a @ ( D @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(211,axiom,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ! [B: a] :
            ( A
           != ( t_a @ B @ e_a @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
                ( A
               != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
           => ~ ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_siftDown_Ocases) ).

thf(1043,plain,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ! [B: a] :
            ( A
           != ( t_a @ B @ e_a @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
                ( A
               != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
           => ~ ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(80,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = zero_zero_multiset_a )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_union__eq__empty) ).

thf(552,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = zero_zero_multiset_a )
      = ( ( A = zero_zero_multiset_a )
        & ( B = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(159,axiom,
    ! [A: multiset_a,B: a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( add_mset_a @ B @ C ) )
      = ( add_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_union__mset__add__mset__right) ).

thf(816,plain,
    ! [A: multiset_a,B: a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ ( add_mset_a @ B @ C ) )
      = ( add_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(98,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( is_heap_a @ ( E @ G ) )
         => ( ( ( produc686083979Tree_a @ H @ I )
              = ( F @ G ) )
           => ( is_heap_a @ ( E @ I ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_Heap_Oremove__max__is__heap) ).

thf(609,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( is_heap_a @ ( E @ G ) )
         => ( ( ( produc686083979Tree_a @ H @ I )
              = ( F @ G ) )
           => ( is_heap_a @ ( E @ I ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(35,axiom,
    ! [A: $o] :
      ( ( A = $true )
      | ( A = $false ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T) ).

thf(323,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(93,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ A ) )
      = ( B = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_add__cancel__right__left) ).

thf(594,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ A ) )
      = ( B = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(102,axiom,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( ! [E: a] :
                  ( ( member_a @ E @ ( set_mset_a @ D ) )
                 => ( ord_less_eq_a @ E @ C ) )
             => ( A @ ( add_mset_a @ C @ D ) ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_310_multiset__induct__max) ).

thf(621,plain,
    ! [A: multiset_a > $o,B: multiset_a] :
      ( ( A @ zero_zero_multiset_a )
     => ( ! [C: a,D: multiset_a] :
            ( ( A @ D )
           => ( ! [E: a] :
                  ( ( member_a @ E @ ( set_mset_a @ D ) )
                 => ( ord_less_eq_a @ E @ C ) )
             => ( A @ ( add_mset_a @ C @ D ) ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(9,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_Pair__inject) ).

thf(240,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(206,axiom,
    ! [A: tree_a > a > $o,B: tree_a,C: a] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ C @ B ) ) @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_exI__realizer) ).

thf(1029,plain,
    ! [A: tree_a > a > $o,B: tree_a,C: a] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ C @ B ) ) @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(38,axiom,
    ! [A: a > tree_a > $o,B: produc981471411ee_a_a] :
      ( ( A @ ( product_snd_Tree_a_a @ B ) @ ( product_fst_Tree_a_a @ B ) )
     => ~ ! [C: tree_a,D: a] :
            ~ ( A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_exE__realizer_H) ).

thf(330,plain,
    ! [A: a > tree_a > $o,B: produc981471411ee_a_a] :
      ( ( A @ ( product_snd_Tree_a_a @ B ) @ ( product_fst_Tree_a_a @ B ) )
     => ~ ! [C: tree_a,D: a] :
            ~ ( A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(117,axiom,
    ! [A: a,B: a] :
      ( ( A = B )
      | ~ ( ord_less_eq_a @ A @ B )
      | ~ ( ord_less_eq_a @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_verit__la__disequality) ).

thf(685,plain,
    ! [A: a,B: a] :
      ( ( A = B )
      | ~ ( ord_less_eq_a @ A @ B )
      | ~ ( ord_less_eq_a @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(197,axiom,
    ! [A: produc143150363Tree_a] :
      ( A
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_surjective__pairing) ).

thf(983,plain,
    ! [A: produc143150363Tree_a] :
      ( A
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(143,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = B )
      = ( A = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_add__cancel__left__left) ).

thf(768,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = B )
      = ( A = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(178,axiom,
    ! [A: tree_a > $o,B: tree_a] :
      ( ! [C: a] : ( A @ ( t_a @ C @ e_a @ e_a ) )
     => ( ! [C: a,D: a,E: tree_a,F: tree_a] :
            ( ( A @ ( t_a @ D @ E @ F ) )
           => ( ( A @ ( t_a @ D @ E @ F ) )
             => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ e_a ) ) ) )
       => ( ! [C: a,D: a,E: tree_a,F: tree_a] :
              ( ( A @ ( t_a @ D @ E @ F ) )
             => ( ( A @ ( t_a @ D @ E @ F ) )
               => ( A @ ( t_a @ C @ e_a @ ( t_a @ D @ E @ F ) ) ) ) )
         => ( ! [C: a,D: a,E: tree_a,F: tree_a,G: a,H: tree_a,I: tree_a] :
                ( ( A @ ( t_a @ D @ E @ F ) )
               => ( ( A @ ( t_a @ D @ E @ F ) )
                 => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ ( t_a @ G @ H @ I ) ) ) ) )
           => ( ! [C: a,D: a,E: tree_a,F: tree_a,G: a,H: tree_a,I: tree_a] :
                  ( ( A @ ( t_a @ D @ E @ F ) )
                 => ( ( A @ ( t_a @ D @ E @ F ) )
                   => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ ( t_a @ G @ H @ I ) ) ) ) )
             => ( ( A @ e_a )
               => ( A @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_removeLeaf_Oinduct) ).

thf(883,plain,
    ! [A: tree_a > $o,B: tree_a] :
      ( ! [C: a] : ( A @ ( t_a @ C @ e_a @ e_a ) )
     => ( ! [C: a,D: a,E: tree_a,F: tree_a] :
            ( ( A @ ( t_a @ D @ E @ F ) )
           => ( ( A @ ( t_a @ D @ E @ F ) )
             => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ e_a ) ) ) )
       => ( ! [C: a,D: a,E: tree_a,F: tree_a] :
              ( ( A @ ( t_a @ D @ E @ F ) )
             => ( ( A @ ( t_a @ D @ E @ F ) )
               => ( A @ ( t_a @ C @ e_a @ ( t_a @ D @ E @ F ) ) ) ) )
         => ( ! [C: a,D: a,E: tree_a,F: tree_a,G: a,H: tree_a,I: tree_a] :
                ( ( A @ ( t_a @ D @ E @ F ) )
               => ( ( A @ ( t_a @ D @ E @ F ) )
                 => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ ( t_a @ G @ H @ I ) ) ) ) )
           => ( ! [C: a,D: a,E: tree_a,F: tree_a,G: a,H: tree_a,I: tree_a] :
                  ( ( A @ ( t_a @ D @ E @ F ) )
                 => ( ( A @ ( t_a @ D @ E @ F ) )
                   => ( A @ ( t_a @ C @ ( t_a @ D @ E @ F ) @ ( t_a @ G @ H @ I ) ) ) ) )
             => ( ( A @ e_a )
               => ( A @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(198,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ E @ F @ G ) ) @ A )
        & ( is_heap_a @ ( t_a @ E @ F @ G ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_is__heap_Osimps_I6_J) ).

thf(986,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ E @ F @ G ) ) @ A )
        & ( is_heap_a @ ( t_a @ E @ F @ G ) )
        & ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(84,axiom,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ( ord_le1748873858_a_a_o
        @ ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A )
        @ ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) )
      = ( ord_le864074675ee_a_a @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_pred__subset__eq2) ).

thf(563,plain,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ( ord_le1748873858_a_a_o
        @ ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A )
        @ ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) )
      = ( ord_le864074675ee_a_a @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(5,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( B @ E )
       => ( ( D @ E )
          = zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_Collection_Ois__empty__as__list) ).

thf(226,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a] :
      ( ( collection_Tree_a_a @ A @ B @ C @ D )
     => ( ( B @ E )
       => ( ( D @ E )
          = zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(74,axiom,
    ! [A: produc981471411ee_a_a] :
      ( A
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_surjective__pairing) ).

thf(505,plain,
    ! [A: produc981471411ee_a_a] :
      ( A
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(181,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_right_Osimps) ).

thf(922,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(208,axiom,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( ( product_fst_a_Tree_a @ A )
        = ( product_fst_a_Tree_a @ B ) )
     => ( ( ( product_snd_a_Tree_a @ A )
          = ( product_snd_a_Tree_a @ B ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_prod__eqI) ).

thf(1034,plain,
    ! [A: produc143150363Tree_a,B: produc143150363Tree_a] :
      ( ( ( product_fst_a_Tree_a @ A )
        = ( product_fst_a_Tree_a @ B ) )
     => ( ( ( product_snd_a_Tree_a @ A )
          = ( product_snd_a_Tree_a @ B ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(188,axiom,
    ! [A: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) )
       != e_a )
     => ( ( A != e_a )
       => ( ( val_a @ A )
          = ( val_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_removeLeaf__val__val) ).

thf(944,plain,
    ! [A: tree_a] :
      ( ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) )
       != e_a )
     => ( ( A != e_a )
       => ( ( val_a @ A )
          = ( val_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(112,axiom,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_siftDown__in__tree) ).

thf(663,plain,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(12,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_Heap_Oaxioms_I2_J) ).

thf(255,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( heap_axioms_Tree_a_a @ B @ C @ D @ E @ F ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(68,axiom,
    ! [A: a,B: multiset_a] : ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_320_multi__member__this) ).

thf(487,plain,
    ! [A: a,B: multiset_a] : ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(186,axiom,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a] :
      ( ( ( t_a @ A @ B @ C )
        = ( t_a @ D @ E @ F ) )
      = ( ( A = D )
        & ( B = E )
        & ( C = F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_Tree_Oinject) ).

thf(939,plain,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a] :
      ( ( ( t_a @ A @ B @ C )
        = ( t_a @ D @ E @ F ) )
      = ( ( A = D )
        & ( B = E )
        & ( C = F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(139,axiom,
    ! [A: produc981471411ee_a_a] :
      ( A
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_prod_Oexhaust__sel) ).

thf(754,plain,
    ! [A: produc981471411ee_a_a] :
      ( A
      = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(193,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ e_a @ ( t_a @ B @ C @ D ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ e_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_removeLeaf_Osimps_I3_J) ).

thf(968,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ e_a @ ( t_a @ B @ C @ D ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ e_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(165,axiom,
    ! [A: a,B: tree_a,C: a] :
      ( ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
        = C )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_fst__eqD) ).

thf(836,plain,
    ! [A: a,B: tree_a,C: a] :
      ( ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
        = C )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(194,axiom,
    ! [A: a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ e_a @ e_a ) )
      = ( produc686083979Tree_a @ A @ e_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_removeLeaf_Osimps_I1_J) ).

thf(971,plain,
    ! [A: a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ e_a @ e_a ) )
      = ( produc686083979Tree_a @ A @ e_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(134,axiom,
    ! [A: produc143150363Tree_a > $o,B: produc143150363Tree_a] :
      ( ! [C: a,D: tree_a] : ( A @ ( produc686083979Tree_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_prod__cases) ).

thf(739,plain,
    ! [A: produc143150363Tree_a > $o,B: produc143150363Tree_a] :
      ( ! [C: a,D: tree_a] : ( A @ ( produc686083979Tree_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(15,axiom,
    ! [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
      ( ( ( product_fst_Tree_a_a @ A )
        = ( product_fst_Tree_a_a @ B ) )
     => ( ( ( product_snd_Tree_a_a @ A )
          = ( product_snd_Tree_a_a @ B ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_prod__eqI) ).

thf(263,plain,
    ! [A: produc981471411ee_a_a,B: produc981471411ee_a_a] :
      ( ( ( product_fst_Tree_a_a @ A )
        = ( product_fst_Tree_a_a @ B ) )
     => ( ( ( product_snd_Tree_a_a @ A )
          = ( product_snd_Tree_a_a @ B ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(200,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ e_a @ ( t_a @ B @ C @ D ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_is__heap_Osimps_I3_J) ).

thf(993,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( is_heap_a @ ( t_a @ A @ e_a @ ( t_a @ B @ C @ D ) ) )
      = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ B @ C @ D ) ) @ A )
        & ( is_heap_a @ ( t_a @ B @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(125,axiom,
    ! [A: a,B: tree_a] :
      ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_snd__conv) ).

thf(712,plain,
    ! [A: a,B: tree_a] :
      ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ A @ B ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(82,axiom,
    ! [A: produc981471411ee_a_a] :
      ( ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_prod_Ocollapse) ).

thf(557,plain,
    ! [A: produc981471411ee_a_a] :
      ( ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(214,axiom,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ! [B: a] :
            ( A
           != ( t_a @ B @ e_a @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
                ( A
               != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
           => ~ ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_is__heap_Ocases) ).

thf(1053,plain,
    ! [A: tree_a] :
      ( ( A != e_a )
     => ( ! [B: a] :
            ( A
           != ( t_a @ B @ e_a @ e_a ) )
       => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
              ( A
             != ( t_a @ B @ e_a @ ( t_a @ C @ D @ E ) ) )
         => ( ! [B: a,C: a,D: tree_a,E: tree_a] :
                ( A
               != ( t_a @ B @ ( t_a @ C @ D @ E ) @ e_a ) )
           => ~ ! [B: a,C: a,D: tree_a,E: tree_a,F: a,G: tree_a,H: tree_a] :
                  ( A
                 != ( t_a @ B @ ( t_a @ C @ D @ E ) @ ( t_a @ F @ G @ H ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(150,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_empty__neutral_I1_J) ).

thf(791,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(90,axiom,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = A )
      = ( B = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_add__cancel__left__right) ).

thf(585,plain,
    ! [A: multiset_a,B: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = A )
      = ( B = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(73,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_add_Oright__neutral) ).

thf(502,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(166,axiom,
    ! [A: a > tree_a > $o,B: a,C: tree_a,D: produc143150363Tree_a] :
      ( ( A @ B @ C )
     => ( ( D
          = ( produc686083979Tree_a @ B @ C ) )
       => ( A @ ( product_fst_a_Tree_a @ D ) @ ( product_snd_a_Tree_a @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_BNF__Greatest__Fixpoint_Osubst__Pair) ).

thf(840,plain,
    ! [A: a > tree_a > $o,B: a,C: tree_a,D: produc143150363Tree_a] :
      ( ( A @ B @ C )
     => ( ( D
          = ( produc686083979Tree_a @ B @ C ) )
       => ( A @ ( product_fst_a_Tree_a @ D ) @ ( product_snd_a_Tree_a @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(41,axiom,
    ! [A: produc981471411ee_a_a] :
      ( ( produc1515207979Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_swap__swap) ).

thf(338,plain,
    ! [A: produc981471411ee_a_a] :
      ( ( produc1515207979Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(156,axiom,
    ! [A: a > tree_a > $o,B: a,C: tree_a] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ C @ B ) ) @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_exI__realizer) ).

thf(808,plain,
    ! [A: a > tree_a > $o,B: a,C: tree_a] :
      ( ( A @ B @ C )
     => ( A @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ C @ B ) ) @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(105,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(643,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
      = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(40,axiom,
    ( plus_plus_multiset_a
    = ( ^ [A: multiset_a,B: multiset_a] : ( plus_plus_multiset_a @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_add_Ocommute) ).

thf(336,plain,
    ( plus_plus_multiset_a
    = ( ^ [A: multiset_a,B: multiset_a] : ( plus_plus_multiset_a @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(122,axiom,
    ! [A: produc143150363Tree_a] :
      ( ( produc1478296771ee_a_a @ ( produc1515207979Tree_a @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_swap__swap) ).

thf(703,plain,
    ! [A: produc143150363Tree_a] :
      ( ( produc1478296771ee_a_a @ ( produc1515207979Tree_a @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(99,axiom,
    ! [A: a,B: tree_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ B ) ) )
     => ( ( is_heap_a @ B )
       => ( ord_less_eq_a @ A @ ( val_a @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_265_heap__top__geq) ).

thf(613,plain,
    ! [A: a,B: tree_a] :
      ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ B ) ) )
     => ( ( is_heap_a @ B )
       => ( ord_less_eq_a @ A @ ( val_a @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(130,axiom,
    ! [A: tree_a] :
      ( ( heapIm229596386mpty_a @ A )
     => ( ( multiset_a2 @ A )
        = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_HS_Ois__empty__as__list) ).

thf(727,plain,
    ! [A: tree_a] :
      ( ( heapIm229596386mpty_a @ A )
     => ( ( multiset_a2 @ A )
        = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(167,axiom,
    ( heapIm915766629_max_a
    = ( ^ [A: tree_a] :
          ( if_Pro1144176865Tree_a
          @ ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) )
            = e_a )
          @ ( produc686083979Tree_a @ ( val_a @ A ) @ e_a )
          @ ( produc686083979Tree_a @ ( val_a @ A ) @ ( heapIm1091024090Down_a @ ( t_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) @ ( heapIm1140443833left_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) @ ( heapIm1257206334ight_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_hs__remove__max__def) ).

thf(845,plain,
    ( heapIm915766629_max_a
    = ( ^ [A: tree_a] :
          ( if_Pro1144176865Tree_a
          @ ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) )
            = e_a )
          @ ( produc686083979Tree_a @ ( val_a @ A ) @ e_a )
          @ ( produc686083979Tree_a @ ( val_a @ A ) @ ( heapIm1091024090Down_a @ ( t_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) @ ( heapIm1140443833left_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) @ ( heapIm1257206334ight_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(65,axiom,
    ( heapIm1057938560list_a
    = ( ^ [A: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_hs__of__list__def) ).

thf(465,plain,
    ( heapIm1057938560list_a
    = ( ^ [A: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(184,axiom,
    ! [A: tree_a,B: tree_a,C: tree_a,D: a] :
      ( ( is_heap_a @ A )
     => ( ( is_heap_a @ B )
       => ( ( C
            = ( t_a @ D @ A @ B ) )
         => ( is_heap_a @ ( heapIm1091024090Down_a @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_siftDown__heap__is__heap) ).

thf(931,plain,
    ! [A: tree_a,B: tree_a,C: tree_a,D: a] :
      ( ( is_heap_a @ A )
     => ( ( is_heap_a @ B )
       => ( ( C
            = ( t_a @ D @ A @ B ) )
         => ( is_heap_a @ ( heapIm1091024090Down_a @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(118,axiom,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ? [B: multiset_a,C: a] :
          ( A
          = ( add_mset_a @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_multi__nonempty__split) ).

thf(688,plain,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ? [B: multiset_a,C: a] :
          ( A
          = ( add_mset_a @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(89,axiom,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ~ ! [B: a,C: multiset_a] :
            ( A
           != ( add_mset_a @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_multiset__cases) ).

thf(581,plain,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ~ ! [B: a,C: multiset_a] :
            ( A
           != ( add_mset_a @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(144,axiom,
    ! [A: multiset_a,B: a] :
      ( A
     != ( add_mset_a @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_multi__self__add__other__not__self) ).

thf(771,plain,
    ! [A: multiset_a,B: a] :
      ( A
     != ( add_mset_a @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(158,axiom,
    ! [A: produc981471411ee_a_a > $o,B: produc981471411ee_a_a] :
      ( ! [C: tree_a,D: a] : ( A @ ( produc649172771ee_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_prod__cases) ).

thf(813,plain,
    ! [A: produc981471411ee_a_a > $o,B: produc981471411ee_a_a] :
      ( ! [C: tree_a,D: a] : ( A @ ( produc649172771ee_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(137,axiom,
    ! [A: a,B: produc981471411ee_a_a] :
      ( ( A
        = ( product_snd_Tree_a_a @ B ) )
      = ( ? [C: tree_a] :
            ( B
            = ( produc649172771ee_a_a @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_eq__snd__iff) ).

thf(748,plain,
    ! [A: a,B: produc981471411ee_a_a] :
      ( ( A
        = ( product_snd_Tree_a_a @ B ) )
      = ( ? [C: tree_a] :
            ( B
            = ( produc649172771ee_a_a @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(29,axiom,
    ! [A: a,B: multiset_a,C: a,D: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ D ) )
     => ( ( A != C )
       => ( member_a @ C @ ( set_mset_a @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_280_insert__noteq__member) ).

thf(304,plain,
    ! [A: a,B: multiset_a,C: a,D: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ D ) )
     => ( ( A != C )
       => ( member_a @ C @ ( set_mset_a @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(217,axiom,
    ! [A: produc981471411ee_a_a] :
      ( ( product_fst_a_Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = ( product_snd_Tree_a_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_fst__swap) ).

thf(1062,plain,
    ! [A: produc981471411ee_a_a] :
      ( ( product_fst_a_Tree_a @ ( produc1478296771ee_a_a @ A ) )
      = ( product_snd_Tree_a_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(45,axiom,
    ! [A: tree_a,B: produc143150363Tree_a] :
      ( ( A
        = ( product_snd_a_Tree_a @ B ) )
      = ( ? [C: a] :
            ( B
            = ( produc686083979Tree_a @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_eq__snd__iff) ).

thf(351,plain,
    ! [A: tree_a,B: produc143150363Tree_a] :
      ( ( A
        = ( product_snd_a_Tree_a @ B ) )
      = ( ? [C: a] :
            ( B
            = ( produc686083979Tree_a @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(58,axiom,
    ! [A: tree_a,B: a] :
      ( ( produc1478296771ee_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = ( produc686083979Tree_a @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_swap__simp) ).

thf(401,plain,
    ! [A: tree_a,B: a] :
      ( ( produc1478296771ee_a_a @ ( produc649172771ee_a_a @ A @ B ) )
      = ( produc686083979Tree_a @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(148,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_union__right__cancel) ).

thf(786,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(147,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ C ) )
     => ( ( plus_plus_multiset_a @ A @ D )
        = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_group__cancel_Oadd1) ).

thf(781,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ C ) )
     => ( ( plus_plus_multiset_a @ A @ D )
        = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(91,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_multi__union__self__other__eq) ).

thf(588,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A @ B )
        = ( plus_plus_multiset_a @ A @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(121,axiom,
    ! [A: produc143150363Tree_a] :
      ~ ! [B: a,C: tree_a] :
          ( A
         != ( produc686083979Tree_a @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_old_Oprod_Oexhaust) ).

thf(699,plain,
    ! [A: produc143150363Tree_a] :
      ~ ! [B: a,C: tree_a] :
          ( A
         != ( produc686083979Tree_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(191,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( in_tree_a @ A @ ( t_a @ B @ C @ D ) )
      = ( ( A = B )
        | ( in_tree_a @ A @ C )
        | ( in_tree_a @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_in__tree_Osimps_I2_J) ).

thf(963,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( in_tree_a @ A @ ( t_a @ B @ C @ D ) )
      = ( ( A = B )
        | ( in_tree_a @ A @ C )
        | ( in_tree_a @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(78,axiom,
    ! [A: tree_a > multiset_a,B: tree_a > tree_a,C: list_a > tree_a,D: tree_a > $o,E: tree_a > produc143150363Tree_a] :
      ( ! [F: tree_a] :
          ( ( A @ F )
          = ( multiset_a2 @ ( B @ F ) ) )
     => ( ! [F: list_a] : ( is_heap_a @ ( B @ ( C @ F ) ) )
       => ( ! [F: tree_a] :
              ( ( ( B @ F )
                = e_a )
              = ( D @ F ) )
         => ( ! [F: tree_a,G: a,H: tree_a] :
                ( ~ ( D @ F )
               => ( ( ( produc686083979Tree_a @ G @ H )
                    = ( E @ F ) )
                 => ( ( add_mset_a @ G @ ( A @ H ) )
                    = ( A @ F ) ) ) )
           => ( ! [F: tree_a,G: a,H: tree_a] :
                  ( ~ ( D @ F )
                 => ( ( is_heap_a @ ( B @ F ) )
                   => ( ( ( produc686083979Tree_a @ G @ H )
                        = ( E @ F ) )
                     => ( is_heap_a @ ( B @ H ) ) ) ) )
             => ( ! [F: tree_a,G: a,H: tree_a] :
                    ( ~ ( D @ F )
                   => ( ( ( produc686083979Tree_a @ G @ H )
                        = ( E @ F ) )
                     => ( G
                        = ( val_a @ ( B @ F ) ) ) ) )
               => ( heap_axioms_Tree_a_a @ D @ C @ A @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_Heap__axioms_Ointro) ).

thf(517,plain,
    ! [A: tree_a > multiset_a,B: tree_a > tree_a,C: list_a > tree_a,D: tree_a > $o,E: tree_a > produc143150363Tree_a] :
      ( ! [F: tree_a] :
          ( ( A @ F )
          = ( multiset_a2 @ ( B @ F ) ) )
     => ( ! [F: list_a] : ( is_heap_a @ ( B @ ( C @ F ) ) )
       => ( ! [F: tree_a] :
              ( ( ( B @ F )
                = e_a )
              = ( D @ F ) )
         => ( ! [F: tree_a,G: a,H: tree_a] :
                ( ~ ( D @ F )
               => ( ( ( produc686083979Tree_a @ G @ H )
                    = ( E @ F ) )
                 => ( ( add_mset_a @ G @ ( A @ H ) )
                    = ( A @ F ) ) ) )
           => ( ! [F: tree_a,G: a,H: tree_a] :
                  ( ~ ( D @ F )
                 => ( ( is_heap_a @ ( B @ F ) )
                   => ( ( ( produc686083979Tree_a @ G @ H )
                        = ( E @ F ) )
                     => ( is_heap_a @ ( B @ H ) ) ) ) )
             => ( ! [F: tree_a,G: a,H: tree_a] :
                    ( ~ ( D @ F )
                   => ( ( ( produc686083979Tree_a @ G @ H )
                        = ( E @ F ) )
                     => ( G
                        = ( val_a @ ( B @ F ) ) ) ) )
               => ( heap_axioms_Tree_a_a @ D @ C @ A @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(86,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( ( produc686083979Tree_a @ H @ I )
            = ( F @ G ) )
         => ( H
            = ( val_a @ ( E @ G ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_Heap_Oremove__max__val) ).

thf(568,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( ( produc686083979Tree_a @ H @ I )
            = ( F @ G ) )
         => ( H
            = ( val_a @ ( E @ G ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(18,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ( ( E @ G )
          = e_a )
        = ( B @ G ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_Heap_Oas__tree__empty) ).

thf(272,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ( ( E @ G )
          = e_a )
        = ( B @ G ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(79,axiom,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_old_Oprod_Oinject) ).

thf(549,plain,
    ! [A: a,B: tree_a,C: a,D: tree_a] :
      ( ( ( produc686083979Tree_a @ A @ B )
        = ( produc686083979Tree_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(24,axiom,
    ! [A: a,B: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ~ ! [C: multiset_a] :
            ( B
           != ( add_mset_a @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_290_mset__add) ).

thf(290,plain,
    ! [A: a,B: multiset_a] :
      ( ( member_a @ A @ ( set_mset_a @ B ) )
     => ~ ! [C: multiset_a] :
            ( B
           != ( add_mset_a @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(52,axiom,
    ! [A: a,B: a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ zero_zero_multiset_a ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_single__eq__single) ).

thf(378,plain,
    ! [A: a,B: a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( add_mset_a @ B @ zero_zero_multiset_a ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(27,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_old_Oprod_Oinject) ).

thf(300,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(202,axiom,
    ! [A: tree_a,B: a,C: tree_a,D: tree_a] :
      ( ( A
        = ( t_a @ B @ C @ D ) )
     => ? [E: tree_a,F: a,G: tree_a] :
          ( ( ( heapIm1091024090Down_a @ A )
            = ( t_a @ F @ E @ G ) )
          & ( ord_less_eq_a @ B @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_siftDown__Node) ).

thf(999,plain,
    ! [A: tree_a,B: a,C: tree_a,D: tree_a] :
      ( ( A
        = ( t_a @ B @ C @ D ) )
     => ? [E: tree_a,F: a,G: tree_a] :
          ( ( ( heapIm1091024090Down_a @ A )
            = ( t_a @ F @ E @ G ) )
          & ( ord_less_eq_a @ B @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(32,axiom,
    ( nO_MAT1617603563iset_a
    = ( ^ [A: multiset_a,B: multiset_a] : $true ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_NO__MATCH__def) ).

thf(315,plain,
    ( nO_MAT1617603563iset_a
    = ( ^ [A: multiset_a,B: multiset_a] : $true ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(164,axiom,
    ! [A: a] :
      ( ( heapIm1091024090Down_a @ ( t_a @ A @ e_a @ e_a ) )
      = ( t_a @ A @ e_a @ e_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_siftDown_Osimps_I2_J) ).

thf(833,plain,
    ! [A: a] :
      ( ( heapIm1091024090Down_a @ ( t_a @ A @ e_a @ e_a ) )
      = ( t_a @ A @ e_a @ e_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(190,axiom,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a,G: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1257206334ight_a @ ( t_a @ D @ E @ F ) ) ) ) @ ( t_a @ A @ B @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( t_a @ D @ E @ F ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_siftDown_Osimps_I5_J) ).

thf(950,plain,
    ! [A: a,B: tree_a,C: tree_a,D: a,E: tree_a,F: tree_a,G: a] :
      ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ D @ E @ F ) ) @ ( heapIm1257206334ight_a @ ( t_a @ D @ E @ F ) ) ) ) @ ( t_a @ A @ B @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( val_a @ ( t_a @ D @ E @ F ) ) )
       => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) ) )
          & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ G )
           => ( ( heapIm1091024090Down_a @ ( t_a @ G @ ( t_a @ D @ E @ F ) @ ( t_a @ A @ B @ C ) ) )
              = ( t_a @ ( val_a @ ( t_a @ A @ B @ C ) ) @ ( t_a @ D @ E @ F ) @ ( heapIm1091024090Down_a @ ( t_a @ G @ ( heapIm1140443833left_a @ ( t_a @ A @ B @ C ) ) @ ( heapIm1257206334ight_a @ ( t_a @ A @ B @ C ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(37,axiom,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ~ ! [B: a] :
            ~ ( member_a @ B @ ( set_mset_a @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_300_multiset__nonemptyE) ).

thf(326,plain,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ~ ! [B: a] :
            ~ ( member_a @ B @ ( set_mset_a @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(42,axiom,
    ! [A: a,B: multiset_a,C: a,D: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ D ) )
      = ( ( ( B = D )
          & ( A = C ) )
        | ? [E: multiset_a] :
            ( ( B
              = ( add_mset_a @ C @ E ) )
            & ( D
              = ( add_mset_a @ A @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_add__eq__conv__ex) ).

thf(341,plain,
    ! [A: a,B: multiset_a,C: a,D: multiset_a] :
      ( ( ( add_mset_a @ A @ B )
        = ( add_mset_a @ C @ D ) )
      = ( ( ( B = D )
          & ( A = C ) )
        | ? [E: multiset_a] :
            ( ( B
              = ( add_mset_a @ C @ E ) )
            & ( D
              = ( add_mset_a @ A @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(218,axiom,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( multiset_a2 @ ( t_a @ A @ B @ C ) )
      = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ B ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_multiset_Osimps_I2_J) ).

thf(1065,plain,
    ! [A: a,B: tree_a,C: tree_a] :
      ( ( multiset_a2 @ ( t_a @ A @ B @ C ) )
      = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ B ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(120,axiom,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ( ( ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A ) )
        = ( ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_pred__equals__eq2) ).

thf(696,plain,
    ! [A: set_Pr921883667ee_a_a,B: set_Pr921883667ee_a_a] :
      ( ( ( ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ A ) )
        = ( ^ [C: tree_a,D: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ C @ D ) @ B ) ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(22,axiom,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_prod__cases) ).

thf(283,plain,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(183,axiom,
    ! [A: tree_a > a > $o,B: produc143150363Tree_a] :
      ( ( A @ ( product_snd_a_Tree_a @ B ) @ ( product_fst_a_Tree_a @ B ) )
     => ~ ! [C: a,D: tree_a] :
            ~ ( A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_exE__realizer_H) ).

thf(928,plain,
    ! [A: tree_a > a > $o,B: produc143150363Tree_a] :
      ( ( A @ ( product_snd_a_Tree_a @ B ) @ ( product_fst_a_Tree_a @ B ) )
     => ~ ! [C: a,D: tree_a] :
            ~ ( A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(25,axiom,
    ! [A: a] : ( member_a @ A @ ( set_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_multi__member__last) ).

thf(294,plain,
    ! [A: a] : ( member_a @ A @ ( set_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(153,axiom,
    is_heap_a @ e_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_is__heap_Osimps_I1_J) ).

thf(801,plain,
    is_heap_a @ e_a,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(180,axiom,
    ! [A: tree_a > $o,B: tree_a] :
      ( ( A @ e_a )
     => ( ! [C: a,D: tree_a,E: tree_a] :
            ( ( A @ D )
           => ( ( A @ E )
             => ( A @ ( t_a @ C @ D @ E ) ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_Tree_Oinduct) ).

thf(908,plain,
    ! [A: tree_a > $o,B: tree_a] :
      ( ( A @ e_a )
     => ( ! [C: a,D: tree_a,E: tree_a] :
            ( ( A @ D )
           => ( ( A @ E )
             => ( A @ ( t_a @ C @ D @ E ) ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(169,axiom,
    ( ( (=) @ produc143150363Tree_a )
    = ( ^ [A: produc143150363Tree_a,B: produc143150363Tree_a] :
          ( ( ( product_fst_a_Tree_a @ A )
            = ( product_fst_a_Tree_a @ B ) )
          & ( ( product_snd_a_Tree_a @ A )
            = ( product_snd_a_Tree_a @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_prod__eq__iff) ).

thf(850,plain,
    ( ( (=) @ produc143150363Tree_a )
    = ( ^ [A: produc143150363Tree_a,B: produc143150363Tree_a] :
          ( ( ( product_fst_a_Tree_a @ A )
            = ( product_fst_a_Tree_a @ B ) )
          & ( ( product_snd_a_Tree_a @ A )
            = ( product_snd_a_Tree_a @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(62,axiom,
    t != e_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_assms_I2_J) ).

thf(442,plain,
    t != e_a,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(174,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ E @ F @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_removeLeaf_Osimps_I5_J) ).

thf(871,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a,E: a,F: tree_a,G: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ ( t_a @ E @ F @ G ) ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ E @ F @ G ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(49,axiom,
    ! [A: produc143150363Tree_a] :
      ( ( product_fst_Tree_a_a @ ( produc1515207979Tree_a @ A ) )
      = ( product_snd_a_Tree_a @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_fst__swap) ).

thf(369,plain,
    ! [A: produc143150363Tree_a] :
      ( ( product_fst_Tree_a_a @ ( produc1515207979Tree_a @ A ) )
      = ( product_snd_a_Tree_a @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(20,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_prod_Oinject) ).

thf(277,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(119,axiom,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ C ) )
     => ( ( plus_plus_multiset_a @ D @ A )
        = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_group__cancel_Oadd2) ).

thf(691,plain,
    ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
      ( ( A
        = ( plus_plus_multiset_a @ B @ C ) )
     => ( ( plus_plus_multiset_a @ D @ A )
        = ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(67,axiom,
    ! [A: multiset_a > multiset_a > $o,B: multiset_a,C: multiset_a] :
      ( ( A @ zero_zero_multiset_a @ zero_zero_multiset_a )
     => ( ! [D: a,E: multiset_a,F: multiset_a] :
            ( ( A @ E @ F )
           => ( A @ ( add_mset_a @ D @ E ) @ F ) )
       => ( ! [D: a,E: multiset_a,F: multiset_a] :
              ( ( A @ E @ F )
             => ( A @ E @ ( add_mset_a @ D @ F ) ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_multiset__induct2) ).

thf(469,plain,
    ! [A: multiset_a > multiset_a > $o,B: multiset_a,C: multiset_a] :
      ( ( A @ zero_zero_multiset_a @ zero_zero_multiset_a )
     => ( ! [D: a,E: multiset_a,F: multiset_a] :
            ( ( A @ E @ F )
           => ( A @ ( add_mset_a @ D @ E ) @ F ) )
       => ( ! [D: a,E: multiset_a,F: multiset_a] :
              ( ( A @ E @ F )
             => ( A @ E @ ( add_mset_a @ D @ F ) ) )
         => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(46,axiom,
    ( in_tree_a
    = ( ^ [A: a,B: tree_a] : ( in_tree_a @ A @ ( heapIm1091024090Down_a @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_siftDown__in__tree__set) ).

thf(354,plain,
    ( in_tree_a
    = ( ^ [A: a,B: tree_a] : ( in_tree_a @ A @ ( heapIm1091024090Down_a @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(104,axiom,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( ( produc686083979Tree_a @ H @ I )
            = ( F @ G ) )
         => ( ( add_mset_a @ H @ ( D @ I ) )
            = ( D @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_Heap_Oremove__max__multiset_H) ).

thf(639,plain,
    ! [A: tree_a,B: tree_a > $o,C: list_a > tree_a,D: tree_a > multiset_a,E: tree_a > tree_a,F: tree_a > produc143150363Tree_a,G: tree_a,H: a,I: tree_a] :
      ( ( heap_Tree_a_a @ A @ B @ C @ D @ E @ F )
     => ( ~ ( B @ G )
       => ( ( ( produc686083979Tree_a @ H @ I )
            = ( F @ G ) )
         => ( ( add_mset_a @ H @ ( D @ I ) )
            = ( D @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(114,axiom,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( plus_plus_multiset_a @ B @ C ) )
      = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = B )
          & ( C = zero_zero_multiset_a ) )
        | ( ( B = zero_zero_multiset_a )
          & ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_single__is__union) ).

thf(676,plain,
    ! [A: a,B: multiset_a,C: multiset_a] :
      ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
        = ( plus_plus_multiset_a @ B @ C ) )
      = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = B )
          & ( C = zero_zero_multiset_a ) )
        | ( ( B = zero_zero_multiset_a )
          & ( ( add_mset_a @ A @ zero_zero_multiset_a )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(92,axiom,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_HS_Omultiset__empty) ).

thf(592,plain,
    ( ( multiset_a2 @ e_a )
    = zero_zero_multiset_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(71,axiom,
    ! [A: a] :
      ~ ( in_tree_a @ A @ e_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_in__tree_Osimps_I1_J) ).

thf(496,plain,
    ! [A: a] :
      ~ ( in_tree_a @ A @ e_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(209,axiom,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ e_a ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ e_a ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_removeLeaf_Osimps_I2_J) ).

thf(1037,plain,
    ! [A: a,B: a,C: tree_a,D: tree_a] :
      ( ( heapIm837449470Leaf_a @ ( t_a @ A @ ( t_a @ B @ C @ D ) @ e_a ) )
      = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ ( t_a @ A @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ B @ C @ D ) ) ) @ e_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(145,axiom,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_add_Ocomm__neutral) ).

thf(775,plain,
    ! [A: multiset_a] :
      ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(83,axiom,
    ! [A: multiset_a] :
      ( ( zero_zero_multiset_a = A )
      = ( A = zero_zero_multiset_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_zero__reorient) ).

thf(560,plain,
    ! [A: multiset_a] :
      ( ( zero_zero_multiset_a = A )
      = ( A = zero_zero_multiset_a ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(1403,plain,
    $false,
    inference(cvc4,[status(thm)],[606,1024,555,666,333,1031,308,247,511,269,597,730,276,385,715,404,709,320,436,651,257,389,724,372,751,802,344,789,1040,646,514,966,905,348,574,280,877,221,866,660,312,981,766,233,925,810,381,366,942,439,1047,798,1057,746,778,820,260,862,467,252,852,461,493,757,682,224,847,566,324,974,742,874,317,615,356,869,499,989,637,489,679,706,393,654,721,1060,1050,266,805,508,657,398,794,718,618,824,296,601,375,733,819,1007,286,826,572,236,947,445,936,219,1011,736,996,230,303,764,604,880,649,1043,552,816,609,323,594,621,240,1029,330,685,983,768,883,986,563,226,505,922,1034,944,663,255,487,939,754,968,836,971,739,263,993,712,557,1053,791,585,502,840,338,808,643,336,703,613,727,845,465,931,688,581,771,813,748,304,1062,351,401,786,781,588,699,963,517,568,272,549,290,378,300,999,315,833,950,326,341,1065,696,283,928,294,801,908,850,442,871,369,277,691,469,354,639,676,592,496,1037,775,560]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : ITP066^1 : TPTP v8.2.0. Released v7.5.0.
% 0.14/0.15  % Command  : run_Leo-III %s %d
% 0.16/0.36  % Computer : n021.cluster.edu
% 0.16/0.36  % Model    : x86_64 x86_64
% 0.16/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36  % Memory   : 8042.1875MB
% 0.16/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36  % CPULimit : 300
% 0.16/0.36  % WCLimit  : 300
% 0.16/0.36  % DateTime : Sat May 18 15:46:39 EDT 2024
% 0.16/0.36  % CPUTime  : 
% 0.94/0.86  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.89/1.10  % [INFO] 	 Parsing done (242ms). 
% 1.89/1.11  % [INFO] 	 Running in sequential loop mode. 
% 2.68/1.31  % [INFO] 	 eprover registered as external prover. 
% 2.68/1.32  % [INFO] 	 cvc4 registered as external prover. 
% 2.68/1.32  % [INFO] 	 Scanning for conjecture ... 
% 2.83/1.42  % [INFO] 	 Found a conjecture (or negated_conjecture) and 356 axioms. Running axiom selection ... 
% 3.19/1.50  % [INFO] 	 Axiom selection finished. Selected 216 axioms (removed 140 axioms). 
% 3.67/1.65  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.67/1.66  % [INFO] 	 Type checking passed. 
% 3.67/1.66  % [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 ... 
% 21.14/6.35  % External prover 'cvc4' found a proof!
% 21.14/6.35  % [INFO] 	 Killing All external provers ... 
% 21.14/6.35  % Time passed: 5831ms (effective reasoning time: 5243ms)
% 21.14/6.35  % 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)>
% 21.14/6.36  % Axioms used in derivation (216): fact_138_add__mset__commute, fact_98_removeLeaf_Ocases, fact_65_old_Oprod_Oexhaust, fact_261_pred__subset__eq2, fact_165_exE__realizer_H, fact_54_prod__cases, fact_178_HS_Ois__empty__empty, fact_195_snd__swap, fact_177_add__mset__replicate__mset__safe, fact_107_add__mset__eq__singleton__iff, fact_96_Tree_Odistinct_I1_J, fact_219_Heap_Omultiset, fact_128_group__cancel_Oadd2, fact_243_heapify_Osimps_I1_J, fact_189_swap__swap, fact_60_Pair__inject, fact_142_empty__neutral_I1_J, fact_120_add__cancel__left__left, fact_131_ab__semigroup__add__class_Oadd__ac_I1_J, fact_8_right_Osimps, fact_228_siftDown_Osimps_I3_J, fact_70_HS_Omultiset__empty, fact_250_Collection_Omultiset__empty, fact_180_HS_Ois__empty__inj, fact_197_fst__swap, fact_79_prod__eq__iff, fact_134_union__left__cancel, fact_229_siftDown_Osimps_I5_J, fact_148_multiset__cases, fact_151_single__is__union, fact_176_HS_Ois__empty__as__list, fact_300_multiset__nonemptyE, fact_14_fstI, fact_23_assms_I2_J, fact_140_add_Ocomm__neutral, fact_104_single__eq__single, fact_155_exI__realizer, fact_4_Tree_Oinject, fact_112_add_Oleft__neutral, fact_5_removeLeaf_Osimps_I5_J, fact_103_union__mset__add__mset__right, fact_77_prod_Oexhaust__sel, fact_295_union__iff, fact_125_add_Oleft__commute, fact_280_insert__noteq__member, fact_239_in__tree_Osimps_I1_J, fact_185_removeLeaf__heap__is__heap, fact_44_snd__eqD, fact_144_multi__nonempty__split, fact_110_subset__mset_Oadd__eq__0__iff__both__eq__0, fact_99_siftDown_Ocases, fact_275_union__single__eq__member, fact_315_multi__member__skip, fact_194_snd__swap, fact_21_fst__conv, fact_204_Heap__axioms_Ointro, fact_242_heapify_Osimps_I2_J, fact_83_prod__eqI, fact_159_conjI__realizer, fact_143_empty__neutral_I2_J, fact_26_prod_Oinject, fact_1_assms_I1_J, fact_34_prod_Ocollapse, fact_173_eq__fst__iff, fact_252_hs__of__list__def, fact_48_snd__conv, fact_198_is__heap_Osimps_I1_J, fact_85_is__heap_Ocases, fact_146_multiset__induct2, fact_249_HS_OCollection__axioms, fact_137_union__assoc, fact_106_single__eq__add__mset, fact_232_siftDown__Node, fact_50_surj__pair, help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T, fact_7_left_Osimps, fact_179_hs__is__empty__def, fact_341_heap__top__max, fact_66_old_Oprod_Oinducts, fact_22_fst__conv, fact_270_multi__member__last, fact_132_multi__union__self__other__eq, fact_56_prod__cases, fact_230_siftDown_Osimps_I1_J, fact_205_val_Osimps, fact_196_fst__swap, fact_40_sndI, fact_113_add__right__cancel, fact_82_prod_Oexpand, fact_253_pred__equals__eq2, fact_10_removeLeaf_Osimps_I1_J, fact_13_fstI, help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T, fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062, fact_265_heap__top__geq, fact_206_is__heap_Osimps_I4_J, fact_135_union__commute, fact_227_siftDown_Osimps_I4_J, fact_240_siftDown__in__tree__set, fact_233_siftDown_Osimps_I2_J, fact_285_multi__member__split, fact_231_siftDown__multiset, fact_41_sndI, fact_117_add__cancel__right__right, fact_310_multiset__induct__max, fact_78_prod_Oexhaust__sel, fact_73_surjective__pairing, fact_28_old_Oprod_Oinject, fact_170_eq__snd__iff, fact_210_verit__la__disequality, fact_58_Pair__inject, fact_17_fst__eqD, fact_220_Heap_Oaxioms_I2_J, fact_183_prod_Oswap__def, fact_115_multi__self__add__other__not__self, fact_74_surjective__pairing, fact_68_old_Oprod_Oinducts, fact_309_multiset__induct__min, fact_53_surj__pair, fact_244_heapify__heap__is__heap, fact_320_multi__member__this, fact_61_Pair__inject, fact_166_exE__realizer_H, fact_248_Heap_Oaxioms_I1_J, fact_184_prod_Oswap__def, fact_192_swap__simp, fact_64_old_Oprod_Oexhaust, fact_101_removeLeaf_Osimps_I3_J, fact_18_fst__eqD, fact_241_hs__remove__max__def, fact_24_prod_Oinject, fact_199_is__heap_Osimps_I2_J, fact_31_old_Oprod_Oinject, fact_52_surj__pair, fact_37_multiset_Osimps_I1_J, fact_126_add_Ocommute, fact_147_multiset__induct, fact_127_add_Oassoc, fact_124_add__left__imp__eq, fact_237_is__heap__max, fact_136_union__lcomm, fact_6_removeLeaf_Osimps_I4_J, fact_116_add__mset__add__mset__same__iff, fact_251_Collection_Ois__empty__as__list, fact_234_siftDown__heap__is__heap, fact_150_union__is__single, fact_49_snd__conv, fact_221_is__heap_Osimps_I6_J, fact_105_add__mset__eq__single, fact_145_empty__not__add__mset, fact_2__C4__2_Oprems_C_I2_J, fact_119_add__cancel__left__right, fact_57_prod__cases, fact_245_multiset__heapify, fact_175_verit__sum__simplify, fact_0__C4__2_Oprems_C_I1_J, fact_290_mset__add, fact_69_old_Oprod_Oinducts, fact_257_subrelI, fact_130_add__mono__thms__linordered__semiring_I4_J, fact_129_group__cancel_Oadd1, fact_100_removeLeaf_Osimps_I2_J, fact_202_Heap__axioms__def, fact_102_union__mset__add__mset__left, fact_45_snd__eqD, fact_149_add__mset__add__single, fact_95_Tree_Oinduct, fact_111_subset__mset_Ozero__eq__add__iff__both__eq__0, fact_246_Heap__def, fact_224_Heap_Oremove__max__multiset_H, fact_123_add__right__imp__eq, fact_84_prod__eqI, fact_222_is__heap_Osimps_I5_J, fact_81_prod_Oexpand, fact_236_siftDown__in__tree, fact_80_prod__eq__iff, fact_200_removeLeaf__val__val, fact_118_add__cancel__right__left, fact_139_add__eq__conv__ex, fact_35_prod_Ocollapse, fact_9__C4__2_Ohyps_C_I1_J, fact_27_prod_Oinject, fact_238_in__tree_Osimps_I2_J, fact_186_NO__MATCH__cong, fact_30_old_Oprod_Oinject, fact_108_empty__eq__union, fact_235_siftDown_Osimps_I6_J, fact_154_exI__realizer, fact_226_Heap_Oremove__max__val, fact_133_union__right__cancel, fact_141_comm__monoid__add__class_Oadd__0, fact_169_eq__snd__iff, fact_218_Heap_Ois__heap__of__list, fact_94_Tree_Oexhaust, fact_109_union__eq__empty, fact_164_Multiset_Ois__empty__def, fact_203_Heap__axioms_Ointro, help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T, fact_122_zero__reorient, fact_188_swap__swap, fact_158_conjI__realizer, fact_163_BNF__Greatest__Fixpoint_Osubst__Pair, fact_201_Heap__axioms__def, fact_162_BNF__Greatest__Fixpoint_Osubst__Pair, fact_114_add__left__cancel, fact_121_add_Oright__neutral, fact_36_multiset_Osimps_I2_J, fact_62_old_Oprod_Oexhaust, fact_207_is__heap_Osimps_I3_J, fact_193_swap__simp, fact_187_NO__MATCH__def, fact_217_Heap_Oas__tree__empty, fact_209_Heap_Oremove__max__is__heap, fact_174_eq__fst__iff, fact_97_removeLeaf_Oinduct, fact_247_Heap_Ointro
% 21.14/6.36  % No. of inferences in proof: 436
% 21.14/6.36  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 5831 ms resp. 5243 ms w/o parsing
% 21.14/6.46  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 21.14/6.46  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------