%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP069^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n006.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 : Wed Feb 25 08:55:18 AM UTC 2026
% Result : Theorem 3.04s 1.86s
% Output : Refutation 3.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 7
% Syntax : Number of formulae : 34 ( 20 unt; 0 typ; 2 def)
% Number of atoms : 117 ( 37 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 292 ( 22 ~; 16 |; 0 &; 243 @)
% ( 2 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 9 con; 0-3 aty)
% Number of variables : 9 ( 0 ^; 9 !; 0 ?; 9 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
multiset_a: $tType ).
thf(type_def_7,type,
tree_a: $tType ).
thf(type_def_8,type,
a: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
multiset_multiset_a: $tType ).
thf(type_def_11,type,
set_Product_prod_a_a: $tType ).
thf(type_def_12,type,
set_Pr158363655iset_a: $tType ).
thf(type_def_13,type,
set_multiset_a: $tType ).
thf(type_def_14,type,
produc1127127335iset_a: $tType ).
thf(type_def_15,type,
product_prod_a_a: $tType ).
thf(func_def_0,type,
set_Pr158363655iset_a: $tType ).
thf(func_def_1,type,
produc1127127335iset_a: $tType ).
thf(func_def_2,type,
set_Product_prod_a_a: $tType ).
thf(func_def_3,type,
multiset_multiset_a: $tType ).
thf(func_def_4,type,
set_multiset_a: $tType ).
thf(func_def_5,type,
product_prod_a_a: $tType ).
thf(func_def_6,type,
multiset_a: $tType ).
thf(func_def_7,type,
tree_a: $tType ).
thf(func_def_8,type,
set_a: $tType ).
thf(func_def_9,type,
a: $tType ).
thf(func_def_10,type,
minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_11,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_12,type,
zero_zero_multiset_a: multiset_a ).
thf(func_def_13,type,
nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
thf(func_def_14,type,
heapIm970322378pify_a: tree_a > tree_a ).
thf(func_def_15,type,
heapIm1140443833left_a: tree_a > tree_a ).
thf(func_def_16,type,
heapIm1257206334ight_a: tree_a > tree_a ).
thf(func_def_17,type,
heapIm1091024090Down_a: tree_a > tree_a ).
thf(func_def_18,type,
e_a: tree_a ).
thf(func_def_19,type,
t_a: a > tree_a > tree_a > tree_a ).
thf(func_def_20,type,
in_tree_a: a > tree_a > $o ).
thf(func_def_21,type,
is_heap_a: tree_a > $o ).
thf(func_def_22,type,
multiset_a2: tree_a > multiset_a ).
thf(func_def_23,type,
val_a: tree_a > a ).
thf(func_def_24,type,
lattic146396397_Max_a: set_a > a ).
thf(func_def_25,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(func_def_26,type,
comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(func_def_27,type,
fold_m382157835iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(func_def_28,type,
fold_m364285649iset_a: ( a > multiset_a > multiset_a ) > multiset_a > multiset_a > multiset_a ).
thf(func_def_29,type,
is_empty_a: multiset_a > $o ).
thf(func_def_30,type,
mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_31,type,
mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_32,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(func_def_33,type,
set_mset_a: multiset_a > set_a ).
thf(func_def_34,type,
ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
thf(func_def_35,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_36,type,
order_Greatest_a: ( a > $o ) > a ).
thf(func_def_37,type,
produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
thf(func_def_38,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_39,type,
irrefl_a: set_Product_prod_a_a > $o ).
thf(func_def_40,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_41,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_42,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(func_def_43,type,
member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
thf(func_def_44,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_45,type,
member_a: a > set_a > $o ).
thf(func_def_46,type,
l: tree_a ).
thf(func_def_47,type,
r: tree_a ).
thf(func_def_48,type,
v: a ).
thf(f394,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f389,f390,f393]) ).
thf(f393,plain,
~ spl0_2,
inference(avatar_contradiction_clause,[status(thm)],[f392]) ).
thf(f392,plain,
( $false
| ~ spl0_2 ),
inference(subsumption_resolution,[status(thm)],[f391,f373]) ).
thf(f373,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f358]) ).
thf(f358,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) )
!= $true ),
inference(flattening,[status(thm)],[f351]) ).
thf(f351,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f350]) ).
thf(f350,plain,
~ ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) ),
inference(rectify,[status(thm)],[f144]) ).
thf(f144,negated_conjecture,
~ ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) ),
inference(negated_conjecture,[status(cth)],[f143]) ).
thf(f143,conjecture,
is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ),
file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',conj_0) ).
thf(f391,plain,
( ( ( is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) )
= $true )
| ~ spl0_2 ),
inference(forward_demodulation,[status(thm)],[f388,f371]) ).
thf(f371,plain,
! [X2: tree_a,X0: a,X1: tree_a] :
( ( heapIm1091024090Down_a @ ( t_a @ X0 @ ( heapIm970322378pify_a @ X2 ) @ ( heapIm970322378pify_a @ X1 ) ) )
= ( heapIm970322378pify_a @ ( t_a @ X0 @ X2 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f359]) ).
thf(f359,plain,
! [X0: a,X1: tree_a,X2: tree_a] :
( ( heapIm1091024090Down_a @ ( t_a @ X0 @ ( heapIm970322378pify_a @ X2 ) @ ( heapIm970322378pify_a @ X1 ) ) )
= ( heapIm970322378pify_a @ ( t_a @ X0 @ X2 @ X1 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X0: a,X2: tree_a,X1: tree_a] :
( ( heapIm970322378pify_a @ ( t_a @ X0 @ X1 @ X2 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X0 @ ( heapIm970322378pify_a @ X1 ) @ ( heapIm970322378pify_a @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_3_heapify_Osimps_I2_J) ).
thf(f388,plain,
( ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true )
| ~ spl0_2 ),
inference(avatar_component_clause,[status(thm)],[f386]) ).
thf(f386,definition,
( spl0_2
<=> ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition]) ).
thf(f390,plain,
spl0_1,
inference(avatar_split_clause,[status(thm)],[f375,f382]) ).
thf(f382,definition,
( spl0_1
<=> ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition]) ).
thf(f375,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f195]) ).
thf(f195,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
= $true ),
inference(fool_elimination,[status(thm)],[f194]) ).
thf(f194,plain,
is_heap_a @ ( heapIm970322378pify_a @ r ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
is_heap_a @ ( heapIm970322378pify_a @ r ),
file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_0_T_Ohyps_I2_J) ).
thf(f389,plain,
( ~ spl0_1
| spl0_2 ),
inference(avatar_split_clause,[status(thm)],[f380,f386,f382]) ).
thf(f380,plain,
( ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
!= $true )
| ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true ) ),
inference(subsumption_resolution,[status(thm)],[f379,f370]) ).
thf(f370,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f239]) ).
thf(f239,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
= $true ),
inference(fool_elimination,[status(thm)],[f238]) ).
thf(f238,plain,
is_heap_a @ ( heapIm970322378pify_a @ l ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
is_heap_a @ ( heapIm970322378pify_a @ l ),
file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_1_T_Ohyps_I1_J) ).
thf(f379,plain,
( ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
!= $true )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f372]) ).
thf(f372,plain,
( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
!= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
!= $true )
| ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f361]) ).
thf(f361,plain,
( ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true )
| ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
!= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
!= $true )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
!= $true ) ),
inference(flattening,[status(thm)],[f360]) ).
thf(f360,plain,
( ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true )
| ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
!= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
!= $true )
| ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f341]) ).
thf(f341,plain,
( ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
= $true )
=> ( ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
= $true )
=> ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
=> ( ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f340]) ).
thf(f340,plain,
( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
=> ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
=> ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
=> ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
=> ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
=> ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
=> ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP069^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.34 % Computer : n006.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Feb 24 08:25:30 EST 2026
% 0.14/0.34 % CPUTime :
% 0.14/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP069^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer HeapImperative problem prob_637__5346010_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : HeapImperative/prob_637__5346010_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 192 ( 72 unt; 49 typ; 0 def)
% 0.19/0.46 % Number of atoms : 402 ( 200 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.19/0.46 % Number of connectives : 1738 ( 55 ~; 10 |; 49 &;1455 @)
% 0.19/0.46 % ( 0 <=>; 169 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 20 ( 7 avg)
% 0.19/0.46 % Number of types : 11 ( 10 usr)
% 0.19/0.46 % Number of type conns : 79 ( 79 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 40 ( 39 usr; 5 con; 0-3 aty)
% 0.19/0.46 % Number of variables : 469 ( 20 ^; 434 !; 15 ?; 469 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:05.115
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (10)
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.19/0.46 set_Pr158363655iset_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 0.19/0.46 produc1127127335iset_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.19/0.46 set_Product_prod_a_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.19/0.46 multiset_multiset_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.19/0.46 set_multiset_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.19/0.46 product_prod_a_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 multiset_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Heap__OTree_Itf__a_J,type,
% 0.19/0.46 tree_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.19/0.46 set_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_tf__a,type,
% 0.19/0.46 a: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (39)
% 0.19/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 zero_zero_multiset_a: multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_HOL_ONO__MATCH_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
% 0.19/0.46 heapIm970322378pify_a: tree_a > tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
% 0.19/0.46 heapIm1140443833left_a: tree_a > tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
% 0.19/0.46 heapIm1257206334ight_a: tree_a > tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
% 0.19/0.46 heapIm1091024090Down_a: tree_a > tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_OTree_OE_001tf__a,type,
% 0.19/0.46 e_a: tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_OTree_OT_001tf__a,type,
% 0.19/0.46 t_a: a > tree_a > tree_a > tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_Oin__tree_001tf__a,type,
% 0.19/0.46 in_tree_a: a > tree_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_Ois__heap_001tf__a,type,
% 0.19/0.46 is_heap_a: tree_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_Omultiset_001tf__a,type,
% 0.19/0.46 multiset_a2: tree_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Heap_Oval_001tf__a,type,
% 0.19/0.46 val_a: tree_a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
% 0.19/0.46 lattic146396397_Max_a: set_a > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
% 0.19/0.46 add_mset_a: a > multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 fold_m382157835iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Ofold__mset_001tf__a_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 fold_m364285649iset_a: ( a > multiset_a > multiset_a ) > multiset_a > multiset_a > multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Ois__empty_001tf__a,type,
% 0.19/0.46 is_empty_a: multiset_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Omult1_001tf__a,type,
% 0.19/0.46 mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Omult_001tf__a,type,
% 0.19/0.46 mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 0.19/0.46 set_mset_a: multiset_a > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.19/0.46 ord_less_eq_a: a > a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
% 0.19/0.46 order_Greatest_a: ( a > $o ) > a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.19/0.46 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Relation_Oirrefl_001tf__a,type,
% 0.19/0.46 irrefl_a: set_Product_prod_a_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Relation_Otrans_001tf__a,type,
% 0.19/0.46 trans_a: set_Product_prod_a_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.19/0.46 collect_a: ( a > $o ) > set_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.19/0.46 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 0.19/0.46 member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.19/0.46 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001tf__a,type,
% 0.19/0.46 member_a: a > set_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_l____,type,
% 0.19/0.46 l: tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_r____,type,
% 0.19/0.46 r: tree_a ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_v____,type,
% 0.19/0.46 v: a ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (142)
% 0.19/0.46 thf(fact_0_T_Ohyps_I2_J,axiom,
% 0.19/0.46 is_heap_a @ ( heapIm970322378pify_a @ r ) ).
% 0.19/0.46
% 0.19/0.46 % T.hyps(2)
% 0.19/0.46 thf(fact_1_T_Ohyps_I1_J,axiom,
% 0.19/0.46 is_heap_a @ ( heapIm970322378pify_a @ l ) ).
% 0.19/0.46
% 0.19/0.46 % T.hyps(1)
% 0.19/0.46 thf(fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062,axiom,
% 0.19/0.46 ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
% 0.19/0.46 => ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
% 0.19/0.46 => ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
% 0.19/0.46 = ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
% 0.19/0.46 => ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<lbrakk>is_heap (heapify l); is_heap (heapify r); T v (heapify l) (heapify r) = T v (heapify l) (heapify r)\<rbrakk> \<Longrightarrow> is_heap (siftDown (T v (heapify l) (heapify r)))\<close>
% 0.19/0.46 thf(fact_3_heapify_Osimps_I2_J,axiom,
% 0.19/0.46 ! [V: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 0.19/0.46 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % heapify.simps(2)
% 0.19/0.46 thf(fact_4_siftDown__heap__is__heap,axiom,
% 0.19/0.46 ! [L: tree_a,R: tree_a,T: tree_a,V: a] :
% 0.19/0.46 ( ( is_heap_a @ L )
% 0.19/0.46 => ( ( is_heap_a @ R )
% 0.19/0.46 => ( ( T
% 0.19/0.46 = ( t_a @ V @ L @ R ) )
% 0.19/0.46 => ( is_heap_a @ ( heapIm1091024090Down_a @ T ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown_heap_is_heap
% 0.19/0.46 thf(fact_5_Tree_Oinject,axiom,
% 0.19/0.46 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 0.19/0.46 ( ( ( t_a @ X21 @ X22 @ X23 )
% 0.19/0.46 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 0.19/0.46 = ( ( X21 = Y21 )
% 0.19/0.46 & ( X22 = Y22 )
% 0.19/0.46 & ( X23 = Y23 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Tree.inject
% 0.19/0.46 thf(fact_6_left_Osimps,axiom,
% 0.19/0.46 ! [V: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 0.19/0.46 = L ) ).
% 0.19/0.46
% 0.19/0.46 % left.simps
% 0.19/0.46 thf(fact_7_right_Osimps,axiom,
% 0.19/0.46 ! [V: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 0.19/0.46 = R ) ).
% 0.19/0.46
% 0.19/0.46 % right.simps
% 0.19/0.46 thf(fact_8_is__heap_Osimps_I2_J,axiom,
% 0.19/0.46 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(2)
% 0.19/0.46 thf(fact_9_heapify_Osimps_I1_J,axiom,
% 0.19/0.46 ( ( heapIm970322378pify_a @ e_a )
% 0.19/0.46 = e_a ) ).
% 0.19/0.46
% 0.19/0.46 % heapify.simps(1)
% 0.19/0.46 thf(fact_10_siftDown_Ocases,axiom,
% 0.19/0.46 ! [X: tree_a] :
% 0.19/0.46 ( ( X != e_a )
% 0.19/0.46 => ( ! [V2: a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ e_a @ e_a ) )
% 0.19/0.46 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.19/0.46 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.19/0.46 => ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.cases
% 0.19/0.46 thf(fact_11_in__tree_Osimps_I2_J,axiom,
% 0.19/0.46 ! [V: a,V3: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( in_tree_a @ V @ ( t_a @ V3 @ L @ R ) )
% 0.19/0.46 = ( ( V = V3 )
% 0.19/0.46 | ( in_tree_a @ V @ L )
% 0.19/0.46 | ( in_tree_a @ V @ R ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_tree.simps(2)
% 0.19/0.46 thf(fact_12_siftDown_Osimps_I2_J,axiom,
% 0.19/0.46 ! [V: a] :
% 0.19/0.46 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 0.19/0.46 = ( t_a @ V @ e_a @ e_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(2)
% 0.19/0.46 thf(fact_13_in__tree_Osimps_I1_J,axiom,
% 0.19/0.46 ! [V: a] :
% 0.19/0.46 ~ ( in_tree_a @ V @ e_a ) ).
% 0.19/0.46
% 0.19/0.46 % in_tree.simps(1)
% 0.19/0.46 thf(fact_14_siftDown_Osimps_I1_J,axiom,
% 0.19/0.46 ( ( heapIm1091024090Down_a @ e_a )
% 0.19/0.46 = e_a ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(1)
% 0.19/0.46 thf(fact_15_siftDown__in__tree__set,axiom,
% 0.19/0.46 ( in_tree_a
% 0.19/0.46 = ( ^ [V4: a,T2: tree_a] : ( in_tree_a @ V4 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown_in_tree_set
% 0.19/0.46 thf(fact_16_is__heap_Ocases,axiom,
% 0.19/0.46 ! [X: tree_a] :
% 0.19/0.46 ( ( X != e_a )
% 0.19/0.46 => ( ! [V2: a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ e_a @ e_a ) )
% 0.19/0.46 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.19/0.46 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.19/0.46 => ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 0.19/0.46 ( X
% 0.19/0.46 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.cases
% 0.19/0.46 thf(fact_17_Tree_Oexhaust,axiom,
% 0.19/0.46 ! [Y: tree_a] :
% 0.19/0.46 ( ( Y != e_a )
% 0.19/0.46 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 0.19/0.46 ( Y
% 0.19/0.46 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Tree.exhaust
% 0.19/0.46 thf(fact_18_Tree_Oinduct,axiom,
% 0.19/0.46 ! [P: tree_a > $o,Tree: tree_a] :
% 0.19/0.46 ( ( P @ e_a )
% 0.19/0.46 => ( ! [X1: a,X2: tree_a,X3: tree_a] :
% 0.19/0.46 ( ( P @ X2 )
% 0.19/0.46 => ( ( P @ X3 )
% 0.19/0.46 => ( P @ ( t_a @ X1 @ X2 @ X3 ) ) ) )
% 0.19/0.46 => ( P @ Tree ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Tree.induct
% 0.19/0.46 thf(fact_19_Tree_Odistinct_I1_J,axiom,
% 0.19/0.46 ! [X21: a,X22: tree_a,X23: tree_a] :
% 0.19/0.46 ( e_a
% 0.19/0.46 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Tree.distinct(1)
% 0.19/0.46 thf(fact_20_is__heap_Osimps_I1_J,axiom,
% 0.19/0.46 is_heap_a @ e_a ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(1)
% 0.19/0.46 thf(fact_21_siftDown__in__tree,axiom,
% 0.19/0.46 ! [T: tree_a] :
% 0.19/0.46 ( ( T != e_a )
% 0.19/0.46 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown_in_tree
% 0.19/0.46 thf(fact_22_siftDown_Osimps_I4_J,axiom,
% 0.19/0.46 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 0.19/0.46 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(4)
% 0.19/0.46 thf(fact_23_siftDown_Osimps_I3_J,axiom,
% 0.19/0.46 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 0.19/0.46 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.19/0.46 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ e_a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(3)
% 0.19/0.46 thf(fact_24_siftDown_Osimps_I6_J,axiom,
% 0.19/0.46 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a,V: a] :
% 0.19/0.46 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(6)
% 0.19/0.46 thf(fact_25_siftDown_Osimps_I5_J,axiom,
% 0.19/0.46 ! [Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 0.19/0.46 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
% 0.19/0.46 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 = ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown.simps(5)
% 0.19/0.46 thf(fact_26_siftDown__Node,axiom,
% 0.19/0.46 ! [T: tree_a,V: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( T
% 0.19/0.46 = ( t_a @ V @ L @ R ) )
% 0.19/0.46 => ? [L2: tree_a,V5: a,R2: tree_a] :
% 0.19/0.46 ( ( ( heapIm1091024090Down_a @ T )
% 0.19/0.46 = ( t_a @ V5 @ L2 @ R2 ) )
% 0.19/0.46 & ( ord_less_eq_a @ V @ V5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % siftDown_Node
% 0.19/0.46 thf(fact_27_is__heap_Osimps_I6_J,axiom,
% 0.19/0.46 ! [V: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.19/0.46 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.19/0.46 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(6)
% 0.19/0.46 thf(fact_28_is__heap_Osimps_I5_J,axiom,
% 0.19/0.46 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.19/0.46 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.19/0.46 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 0.19/0.46 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(5)
% 0.19/0.46 thf(fact_29_is__heap__max,axiom,
% 0.19/0.46 ! [V: a,T: tree_a] :
% 0.19/0.46 ( ( in_tree_a @ V @ T )
% 0.19/0.46 => ( ( is_heap_a @ T )
% 0.19/0.46 => ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap_max
% 0.19/0.46 thf(fact_30_val_Osimps,axiom,
% 0.19/0.46 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 0.19/0.46 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 0.19/0.46 = V ) ).
% 0.19/0.46
% 0.19/0.46 % val.simps
% 0.19/0.46 thf(fact_31_is__heap_Osimps_I4_J,axiom,
% 0.19/0.46 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.19/0.46 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.19/0.46 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(4)
% 0.19/0.46 thf(fact_32_is__heap_Osimps_I3_J,axiom,
% 0.19/0.46 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.19/0.46 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.19/0.46 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 0.19/0.46 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_heap.simps(3)
% 0.19/0.46 thf(fact_33_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: a,P: a > $o] :
% 0.19/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_34_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_a] :
% 0.19/0.46 ( ( collect_a
% 0.19/0.46 @ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_35_order__refl,axiom,
% 0.19/0.46 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_36_heap__top__geq,axiom,
% 0.19/0.46 ! [A: a,T: tree_a] :
% 0.19/0.46 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
% 0.19/0.46 => ( ( is_heap_a @ T )
% 0.19/0.46 => ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % heap_top_geq
% 0.19/0.46 thf(fact_37_order__subst1,axiom,
% 0.19/0.46 ! [A: a,F: a > a,B: a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.46 => ( ! [X5: a,Y2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_38_order__subst2,axiom,
% 0.19/0.46 ! [A: a,B: a,F: a > a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X5: a,Y2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_39_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: a,F: a > a,B: a,C: a] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.46 => ( ! [X5: a,Y2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_40_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: a,B: a,F: a > a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: a,Y2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_41_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.19/0.46 = ( ^ [X4: a,Y4: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X4 @ Y4 )
% 0.19/0.46 & ( ord_less_eq_a @ Y4 @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_42_antisym,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 => ( ( ord_less_eq_a @ Y @ X )
% 0.19/0.46 => ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_43_linear,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 | ( ord_less_eq_a @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_44_eq__refl,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ( X = Y )
% 0.19/0.46 => ( ord_less_eq_a @ X @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_45_le__cases,axiom,
% 0.19/0.46 ! [X: a,Y: a] :
% 0.19/0.46 ( ~ ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_a @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_46_order_Otrans,axiom,
% 0.19/0.46 ! [A: a,B: a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.46 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_47_le__cases3,axiom,
% 0.19/0.46 ! [X: a,Y: a,Z2: a] :
% 0.19/0.46 ( ( ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_a @ Y @ Z2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ Y @ X )
% 0.19/0.46 => ~ ( ord_less_eq_a @ X @ Z2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ X @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq_a @ Z2 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ Z2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_a @ Y @ X ) )
% 0.19/0.46 => ( ( ( ord_less_eq_a @ Y @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq_a @ Z2 @ X ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_a @ Z2 @ X )
% 0.19/0.46 => ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_48_antisym__conv,axiom,
% 0.19/0.46 ! [Y: a,X: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ Y @ X )
% 0.19/0.46 => ( ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 = ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_49_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.19/0.46 = ( ^ [A3: a,B2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A3 @ B2 )
% 0.19/0.46 & ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_50_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: a,B: a,C: a] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_a @ B @ C )
% 0.19/0.46 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_51_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: a,B: a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_52_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_a @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_53_order__trans,axiom,
% 0.19/0.46 ! [X: a,Y: a,Z2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ X @ Y )
% 0.19/0.46 => ( ( ord_less_eq_a @ Y @ Z2 )
% 0.19/0.46 => ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_54_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_55_linorder__wlog,axiom,
% 0.19/0.46 ! [P: a > a > $o,A: a,B: a] :
% 0.19/0.46 ( ! [A4: a,B3: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: a,B3: a] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_56_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: a,A: a,C: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_a @ C @ B )
% 0.19/0.46 => ( ord_less_eq_a @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_57_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.19/0.46 = ( ^ [A3: a,B2: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ B2 @ A3 )
% 0.19/0.46 & ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_58_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [B: a,A: a] :
% 0.19/0.46 ( ( ord_less_eq_a @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_a @ A @ B )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_59_add__mset__add__mset__same__iff,axiom,
% 0.19/0.46 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ A2 )
% 0.19/0.46 = ( add_mset_a @ A @ B4 ) )
% 0.19/0.46 = ( A2 = B4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_add_mset_same_iff
% 0.19/0.46 thf(fact_60_multi__self__add__other__not__self,axiom,
% 0.19/0.46 ! [M: multiset_a,X: a] :
% 0.19/0.46 ( M
% 0.19/0.46 != ( add_mset_a @ X @ M ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_self_add_other_not_self
% 0.19/0.46 thf(fact_61_mset__add,axiom,
% 0.19/0.46 ! [A: a,A2: multiset_a] :
% 0.19/0.46 ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 0.19/0.46 => ~ ! [B5: multiset_a] :
% 0.19/0.46 ( A2
% 0.19/0.46 != ( add_mset_a @ A @ B5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mset_add
% 0.19/0.46 thf(fact_62_multi__member__split,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a] :
% 0.19/0.46 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.19/0.46 => ? [A5: multiset_a] :
% 0.19/0.46 ( M
% 0.19/0.46 = ( add_mset_a @ X @ A5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_member_split
% 0.19/0.46 thf(fact_63_insert__noteq__member,axiom,
% 0.19/0.46 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ B @ B4 )
% 0.19/0.46 = ( add_mset_a @ C @ C2 ) )
% 0.19/0.46 => ( ( B != C )
% 0.19/0.46 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_noteq_member
% 0.19/0.46 thf(fact_64_union__single__eq__member,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ X @ M )
% 0.19/0.46 = N )
% 0.19/0.46 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_single_eq_member
% 0.19/0.46 thf(fact_65_heap__top__max,axiom,
% 0.19/0.46 ! [T: tree_a] :
% 0.19/0.46 ( ( T != e_a )
% 0.19/0.46 => ( ( is_heap_a @ T )
% 0.19/0.46 => ( ( val_a @ T )
% 0.19/0.46 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % heap_top_max
% 0.19/0.46 thf(fact_66_add__mset__commute,axiom,
% 0.19/0.46 ! [X: a,Y: a,M: multiset_a] :
% 0.19/0.46 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M ) )
% 0.19/0.46 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_commute
% 0.19/0.46 thf(fact_67_add__eq__conv__ex,axiom,
% 0.19/0.46 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ M )
% 0.19/0.46 = ( add_mset_a @ B @ N ) )
% 0.19/0.46 = ( ( ( M = N )
% 0.19/0.46 & ( A = B ) )
% 0.19/0.46 | ? [K: multiset_a] :
% 0.19/0.46 ( ( M
% 0.19/0.46 = ( add_mset_a @ B @ K ) )
% 0.19/0.46 & ( N
% 0.19/0.46 = ( add_mset_a @ A @ K ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_eq_conv_ex
% 0.19/0.46 thf(fact_68_Greatest__equality,axiom,
% 0.19/0.46 ! [P: a > $o,X: a] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( P @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ Y2 @ X ) )
% 0.19/0.46 => ( ( order_Greatest_a @ P )
% 0.19/0.46 = X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Greatest_equality
% 0.19/0.46 thf(fact_69_GreatestI2__order,axiom,
% 0.19/0.46 ! [P: a > $o,X: a,Q: a > $o] :
% 0.19/0.46 ( ( P @ X )
% 0.19/0.46 => ( ! [Y2: a] :
% 0.19/0.46 ( ( P @ Y2 )
% 0.19/0.46 => ( ord_less_eq_a @ Y2 @ X ) )
% 0.19/0.46 => ( ! [X5: a] :
% 0.19/0.46 ( ( P @ X5 )
% 0.19/0.46 => ( ! [Y5: a] :
% 0.19/0.46 ( ( P @ Y5 )
% 0.19/0.46 => ( ord_less_eq_a @ Y5 @ X5 ) )
% 0.19/0.46 => ( Q @ X5 ) ) )
% 0.19/0.46 => ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % GreatestI2_order
% 0.19/0.46 thf(fact_70_multiset__induct__min,axiom,
% 0.19/0.46 ! [P: multiset_a > $o,M: multiset_a] :
% 0.19/0.46 ( ( P @ zero_zero_multiset_a )
% 0.19/0.46 => ( ! [X5: a,M2: multiset_a] :
% 0.19/0.46 ( ( P @ M2 )
% 0.19/0.46 => ( ! [Xa: a] :
% 0.19/0.46 ( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
% 0.19/0.46 => ( ord_less_eq_a @ X5 @ Xa ) )
% 0.19/0.46 => ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_induct_min
% 0.19/0.46 thf(fact_71_multiset__induct__max,axiom,
% 0.19/0.46 ! [P: multiset_a > $o,M: multiset_a] :
% 0.19/0.46 ( ( P @ zero_zero_multiset_a )
% 0.19/0.46 => ( ! [X5: a,M2: multiset_a] :
% 0.19/0.46 ( ( P @ M2 )
% 0.19/0.46 => ( ! [Xa: a] :
% 0.19/0.46 ( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
% 0.19/0.46 => ( ord_less_eq_a @ Xa @ X5 ) )
% 0.19/0.46 => ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_induct_max
% 0.19/0.46 thf(fact_72_add__mset__eq__singleton__iff,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a,Y: a] :
% 0.19/0.46 ( ( ( add_mset_a @ X @ M )
% 0.19/0.46 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 0.19/0.46 = ( ( M = zero_zero_multiset_a )
% 0.19/0.46 & ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_eq_singleton_iff
% 0.19/0.46 thf(fact_73_single__eq__add__mset,axiom,
% 0.19/0.46 ! [A: a,B: a,M: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.19/0.46 = ( add_mset_a @ B @ M ) )
% 0.19/0.46 = ( ( B = A )
% 0.19/0.46 & ( M = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % single_eq_add_mset
% 0.19/0.46 thf(fact_74_add__mset__eq__single,axiom,
% 0.19/0.46 ! [B: a,M: multiset_a,A: a] :
% 0.19/0.46 ( ( ( add_mset_a @ B @ M )
% 0.19/0.46 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.19/0.46 = ( ( B = A )
% 0.19/0.46 & ( M = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_eq_single
% 0.19/0.46 thf(fact_75_single__eq__single,axiom,
% 0.19/0.46 ! [A: a,B: a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.19/0.46 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % single_eq_single
% 0.19/0.46 thf(fact_76_multiset__cases,axiom,
% 0.19/0.46 ! [M: multiset_a] :
% 0.19/0.46 ( ( M != zero_zero_multiset_a )
% 0.19/0.46 => ~ ! [X5: a,N2: multiset_a] :
% 0.19/0.46 ( M
% 0.19/0.46 != ( add_mset_a @ X5 @ N2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_cases
% 0.19/0.46 thf(fact_77_multiset__induct,axiom,
% 0.19/0.46 ! [P: multiset_a > $o,M: multiset_a] :
% 0.19/0.46 ( ( P @ zero_zero_multiset_a )
% 0.19/0.46 => ( ! [X5: a,M2: multiset_a] :
% 0.19/0.46 ( ( P @ M2 )
% 0.19/0.46 => ( P @ ( add_mset_a @ X5 @ M2 ) ) )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_induct
% 0.19/0.46 thf(fact_78_multiset__induct2,axiom,
% 0.19/0.46 ! [P: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 0.19/0.46 => ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
% 0.19/0.46 ( ( P @ M2 @ N2 )
% 0.19/0.46 => ( P @ ( add_mset_a @ A4 @ M2 ) @ N2 ) )
% 0.19/0.46 => ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
% 0.19/0.46 ( ( P @ M2 @ N2 )
% 0.19/0.46 => ( P @ M2 @ ( add_mset_a @ A4 @ N2 ) ) )
% 0.19/0.46 => ( P @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_induct2
% 0.19/0.46 thf(fact_79_empty__not__add__mset,axiom,
% 0.19/0.46 ! [A: a,A2: multiset_a] :
% 0.19/0.46 ( zero_zero_multiset_a
% 0.19/0.46 != ( add_mset_a @ A @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_not_add_mset
% 0.19/0.46 thf(fact_80_multi__nonempty__split,axiom,
% 0.19/0.46 ! [M: multiset_a] :
% 0.19/0.46 ( ( M != zero_zero_multiset_a )
% 0.19/0.46 => ? [A5: multiset_a,A4: a] :
% 0.19/0.46 ( M
% 0.19/0.46 = ( add_mset_a @ A4 @ A5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_nonempty_split
% 0.19/0.46 thf(fact_81_multiset__nonemptyE,axiom,
% 0.19/0.46 ! [A2: multiset_a] :
% 0.19/0.46 ( ( A2 != zero_zero_multiset_a )
% 0.19/0.46 => ~ ! [X5: a] :
% 0.19/0.46 ~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset_nonemptyE
% 0.19/0.46 thf(fact_82_multi__member__last,axiom,
% 0.19/0.46 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_member_last
% 0.19/0.46 thf(fact_83_multiset_Osimps_I1_J,axiom,
% 0.19/0.46 ( ( multiset_a2 @ e_a )
% 0.19/0.46 = zero_zero_multiset_a ) ).
% 0.19/0.46
% 0.19/0.46 % multiset.simps(1)
% 0.19/0.46 thf(fact_84_Multiset_Ois__empty__def,axiom,
% 0.19/0.46 ( is_empty_a
% 0.19/0.46 = ( ^ [A6: multiset_a] : ( A6 = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Multiset.is_empty_def
% 0.19/0.46 thf(fact_85_multiset_Osimps_I2_J,axiom,
% 0.19/0.46 ! [V: a,L: tree_a,R: tree_a] :
% 0.19/0.46 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 0.19/0.46 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multiset.simps(2)
% 0.19/0.46 thf(fact_86_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 0.19/0.46 ! [X: multiset_a,Y: multiset_a] :
% 0.19/0.46 ( ( zero_zero_multiset_a
% 0.19/0.46 = ( plus_plus_multiset_a @ X @ Y ) )
% 0.19/0.46 = ( ( X = zero_zero_multiset_a )
% 0.19/0.46 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_mset.zero_eq_add_iff_both_eq_0
% 0.19/0.46 thf(fact_87_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 0.19/0.46 ! [X: multiset_a,Y: multiset_a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ X @ Y )
% 0.19/0.46 = zero_zero_multiset_a )
% 0.19/0.46 = ( ( X = zero_zero_multiset_a )
% 0.19/0.46 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_mset.add_eq_0_iff_both_eq_0
% 0.19/0.46 thf(fact_88_union__eq__empty,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ M @ N )
% 0.19/0.46 = zero_zero_multiset_a )
% 0.19/0.46 = ( ( M = zero_zero_multiset_a )
% 0.19/0.46 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_eq_empty
% 0.19/0.46 thf(fact_89_empty__eq__union,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( zero_zero_multiset_a
% 0.19/0.46 = ( plus_plus_multiset_a @ M @ N ) )
% 0.19/0.46 = ( ( M = zero_zero_multiset_a )
% 0.19/0.46 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_eq_union
% 0.19/0.46 thf(fact_90_union__mset__add__mset__right,axiom,
% 0.19/0.46 ! [A2: multiset_a,A: a,B4: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ A2 @ ( add_mset_a @ A @ B4 ) )
% 0.19/0.46 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_mset_add_mset_right
% 0.19/0.46 thf(fact_91_union__mset__add__mset__left,axiom,
% 0.19/0.46 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
% 0.19/0.46 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_mset_add_mset_left
% 0.19/0.46 thf(fact_92_add__right__mono,axiom,
% 0.19/0.46 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.19/0.46 ( ( ord_le1199012836iset_a @ A @ B )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_mono
% 0.19/0.46 thf(fact_93_add__left__mono,axiom,
% 0.19/0.46 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.19/0.46 ( ( ord_le1199012836iset_a @ A @ B )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_mono
% 0.19/0.46 thf(fact_94_add__mono,axiom,
% 0.19/0.46 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 0.19/0.46 ( ( ord_le1199012836iset_a @ A @ B )
% 0.19/0.46 => ( ( ord_le1199012836iset_a @ C @ D )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono
% 0.19/0.46 thf(fact_95_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.46 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 0.19/0.46 ( ( ( ord_le1199012836iset_a @ I @ J )
% 0.19/0.46 & ( ord_le1199012836iset_a @ K2 @ L ) )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(1)
% 0.19/0.46 thf(fact_96_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.46 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_le1199012836iset_a @ K2 @ L ) )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(2)
% 0.19/0.46 thf(fact_97_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.46 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 0.19/0.46 ( ( ( ord_le1199012836iset_a @ I @ J )
% 0.19/0.46 & ( K2 = L ) )
% 0.19/0.46 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(3)
% 0.19/0.46 thf(fact_98_empty__neutral_I2_J,axiom,
% 0.19/0.46 ! [X: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 0.19/0.46 = X ) ).
% 0.19/0.46
% 0.19/0.46 % empty_neutral(2)
% 0.19/0.46 thf(fact_99_empty__neutral_I1_J,axiom,
% 0.19/0.46 ! [X: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 0.19/0.46 = X ) ).
% 0.19/0.46
% 0.19/0.46 % empty_neutral(1)
% 0.19/0.46 thf(fact_100_union__iff,axiom,
% 0.19/0.46 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.19/0.46 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B4 ) ) )
% 0.19/0.46 = ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 0.19/0.46 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_iff
% 0.19/0.46 thf(fact_101_union__assoc,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a,K3: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K3 )
% 0.19/0.46 = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_assoc
% 0.19/0.46 thf(fact_102_union__lcomm,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a,K3: multiset_a] :
% 0.19/0.46 ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) )
% 0.19/0.46 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_lcomm
% 0.19/0.46 thf(fact_103_union__commute,axiom,
% 0.19/0.46 ( plus_plus_multiset_a
% 0.19/0.46 = ( ^ [M3: multiset_a,N3: multiset_a] : ( plus_plus_multiset_a @ N3 @ M3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_commute
% 0.19/0.46 thf(fact_104_union__left__cancel,axiom,
% 0.19/0.46 ! [K3: multiset_a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ K3 @ M )
% 0.19/0.46 = ( plus_plus_multiset_a @ K3 @ N ) )
% 0.19/0.46 = ( M = N ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_left_cancel
% 0.19/0.46 thf(fact_105_union__right__cancel,axiom,
% 0.19/0.46 ! [M: multiset_a,K3: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ M @ K3 )
% 0.19/0.46 = ( plus_plus_multiset_a @ N @ K3 ) )
% 0.19/0.46 = ( M = N ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_right_cancel
% 0.19/0.46 thf(fact_106_multi__union__self__other__eq,axiom,
% 0.19/0.46 ! [A2: multiset_a,X6: multiset_a,Y6: multiset_a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ A2 @ X6 )
% 0.19/0.46 = ( plus_plus_multiset_a @ A2 @ Y6 ) )
% 0.19/0.46 => ( X6 = Y6 ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_union_self_other_eq
% 0.19/0.46 thf(fact_107_add__mset__add__single,axiom,
% 0.19/0.46 ( add_mset_a
% 0.19/0.46 = ( ^ [A3: a,A6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( add_mset_a @ A3 @ zero_zero_multiset_a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_add_single
% 0.19/0.46 thf(fact_108_union__is__single,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a,A: a] :
% 0.19/0.46 ( ( ( plus_plus_multiset_a @ M @ N )
% 0.19/0.46 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.19/0.46 = ( ( ( M
% 0.19/0.46 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.19/0.46 & ( N = zero_zero_multiset_a ) )
% 0.19/0.46 | ( ( M = zero_zero_multiset_a )
% 0.19/0.46 & ( N
% 0.19/0.46 = ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_is_single
% 0.19/0.46 thf(fact_109_single__is__union,axiom,
% 0.19/0.46 ! [A: a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.19/0.46 = ( plus_plus_multiset_a @ M @ N ) )
% 0.19/0.46 = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.19/0.46 = M )
% 0.19/0.46 & ( N = zero_zero_multiset_a ) )
% 0.19/0.46 | ( ( M = zero_zero_multiset_a )
% 0.19/0.46 & ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.19/0.46 = N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % single_is_union
% 0.19/0.46 thf(fact_110_multi__member__skip,axiom,
% 0.19/0.46 ! [X: a,XS: multiset_a,Y: a] :
% 0.19/0.46 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 0.19/0.46 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_member_skip
% 0.19/0.46 thf(fact_111_multi__member__this,axiom,
% 0.19/0.46 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % multi_member_this
% 0.19/0.46 thf(fact_112_mult1E,axiom,
% 0.19/0.46 ! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
% 0.19/0.46 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
% 0.19/0.46 => ~ ! [A4: a,M0: multiset_a] :
% 0.19/0.46 ( ( M
% 0.19/0.46 = ( add_mset_a @ A4 @ M0 ) )
% 0.19/0.46 => ! [K4: multiset_a] :
% 0.19/0.46 ( ( N
% 0.19/0.46 = ( plus_plus_multiset_a @ M0 @ K4 ) )
% 0.19/0.46 => ~ ! [B6: a] :
% 0.19/0.46 ( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
% 0.19/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A4 ) @ R ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult1E
% 0.19/0.46 thf(fact_113_mult1I,axiom,
% 0.19/0.46 ! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K3: multiset_a,R: set_Product_prod_a_a] :
% 0.19/0.46 ( ( M
% 0.19/0.46 = ( add_mset_a @ A @ M02 ) )
% 0.19/0.46 => ( ( N
% 0.19/0.46 = ( plus_plus_multiset_a @ M02 @ K3 ) )
% 0.19/0.46 => ( ! [B3: a] :
% 0.19/0.46 ( ( member_a @ B3 @ ( set_mset_a @ K3 ) )
% 0.19/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
% 0.19/0.46 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult1I
% 0.19/0.46 thf(fact_114_less__add,axiom,
% 0.19/0.46 ! [N: multiset_a,A: a,M02: multiset_a,R: set_Product_prod_a_a] :
% 0.19/0.46 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R ) )
% 0.19/0.46 => ( ? [M2: multiset_a] :
% 0.19/0.46 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R ) )
% 0.19/0.46 & ( N
% 0.19/0.46 = ( add_mset_a @ A @ M2 ) ) )
% 0.19/0.46 | ? [K4: multiset_a] :
% 0.19/0.46 ( ! [B6: a] :
% 0.19/0.46 ( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
% 0.19/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R ) )
% 0.19/0.46 & ( N
% 0.19/0.46 = ( plus_plus_multiset_a @ M02 @ K4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add
% 0.19/0.46 thf(fact_115_one__step__implies__mult,axiom,
% 0.19/0.46 ! [J2: multiset_a,K3: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
% 0.19/0.46 ( ( J2 != zero_zero_multiset_a )
% 0.19/0.46 => ( ! [X5: a] :
% 0.19/0.46 ( ( member_a @ X5 @ ( set_mset_a @ K3 ) )
% 0.19/0.46 => ? [Xa: a] :
% 0.19/0.46 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 0.19/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ R ) ) )
% 0.19/0.46 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K3 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % one_step_implies_mult
% 0.19/0.46 thf(fact_116_not__less__empty,axiom,
% 0.19/0.46 ! [M: multiset_a,R: set_Product_prod_a_a] :
% 0.19/0.46 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_empty
% 0.19/0.46 thf(fact_117_mult1__union,axiom,
% 0.19/0.46 ! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
% 0.19/0.46 ( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
% 0.19/0.46 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult1_union
% 0.19/0.46 thf(fact_118_mult__implies__one__step,axiom,
% 0.19/0.46 ! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( trans_a @ R )
% 0.19/0.46 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
% 0.19/0.46 => ? [I3: multiset_a,J3: multiset_a] :
% 0.19/0.46 ( ( N
% 0.19/0.46 = ( plus_plus_multiset_a @ I3 @ J3 ) )
% 0.19/0.46 & ? [K4: multiset_a] :
% 0.19/0.46 ( ( M
% 0.19/0.46 = ( plus_plus_multiset_a @ I3 @ K4 ) )
% 0.19/0.46 & ( J3 != zero_zero_multiset_a )
% 0.19/0.46 & ! [X7: a] :
% 0.19/0.46 ( ( member_a @ X7 @ ( set_mset_a @ K4 ) )
% 0.19/0.46 => ? [Xa2: a] :
% 0.19/0.46 ( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
% 0.19/0.46 & ( member449909584od_a_a @ ( product_Pair_a_a @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult_implies_one_step
% 0.19/0.46 thf(fact_119_subset__mset_Osum__mset__0__iff,axiom,
% 0.19/0.46 ! [M: multiset_multiset_a] :
% 0.19/0.46 ( ( ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
% 0.19/0.46 = zero_zero_multiset_a )
% 0.19/0.46 = ( ! [X4: multiset_a] :
% 0.19/0.46 ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
% 0.19/0.46 => ( X4 = zero_zero_multiset_a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_mset.sum_mset_0_iff
% 0.19/0.46 thf(fact_120_mult__cancel__add__mset,axiom,
% 0.19/0.46 ! [S: set_Product_prod_a_a,Uu: a,X6: multiset_a,Y6: multiset_a] :
% 0.19/0.46 ( ( trans_a @ S )
% 0.19/0.46 => ( ( irrefl_a @ S )
% 0.19/0.46 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X6 ) @ ( add_mset_a @ Uu @ Y6 ) ) @ ( mult_a @ S ) )
% 0.19/0.46 = ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult_cancel_add_mset
% 0.19/0.46 thf(fact_121_mult__cancel,axiom,
% 0.19/0.46 ! [S: set_Product_prod_a_a,X6: multiset_a,Z3: multiset_a,Y6: multiset_a] :
% 0.19/0.46 ( ( trans_a @ S )
% 0.19/0.46 => ( ( irrefl_a @ S )
% 0.19/0.46 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X6 @ Z3 ) @ ( plus_plus_multiset_a @ Y6 @ Z3 ) ) @ ( mult_a @ S ) )
% 0.19/0.46 = ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mult_cancel
% 0.19/0.46 thf(fact_122_in__mset__fold__plus__iff,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a,NN: multiset_multiset_a] :
% 0.19/0.46 ( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
% 0.19/0.46 = ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.19/0.46 | ? [N3: multiset_a] :
% 0.19/0.46 ( ( member_multiset_a @ N3 @ ( set_mset_multiset_a @ NN ) )
% 0.19/0.46 & ( member_a @ X @ ( set_mset_a @ N3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_mset_fold_plus_iff
% 0.19/0.46 thf(fact_123_union__fold__mset__add__mset,axiom,
% 0.19/0.46 ( plus_plus_multiset_a
% 0.19/0.46 = ( fold_m364285649iset_a @ add_mset_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_fold_mset_add_mset
% 0.19/0.46 thf(fact_124_add__mset__replicate__mset__safe,axiom,
% 0.19/0.46 ! [M: multiset_a,A: a] :
% 0.19/0.46 ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
% 0.19/0.46 => ( ( add_mset_a @ A @ M )
% 0.19/0.46 = ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_replicate_mset_safe
% 0.19/0.46 thf(fact_125_insert__DiffM2,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a] :
% 0.19/0.46 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.19/0.46 => ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_DiffM2
% 0.19/0.46 thf(fact_126_diff__diff__add__mset,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a,P: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ P )
% 0.19/0.46 = ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_add_mset
% 0.19/0.46 thf(fact_127_add__mset__remove__trivial,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_remove_trivial
% 0.19/0.46 thf(fact_128_diff__add__mset__swap,axiom,
% 0.19/0.46 ! [B: a,A2: multiset_a,M: multiset_a] :
% 0.19/0.46 ( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
% 0.19/0.46 => ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A2 )
% 0.19/0.46 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_mset_swap
% 0.19/0.46 thf(fact_129_diff__union__swap2,axiom,
% 0.19/0.46 ! [Y: a,M: multiset_a,X: a] :
% 0.19/0.46 ( ( member_a @ Y @ ( set_mset_a @ M ) )
% 0.19/0.46 => ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 0.19/0.46 = ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_union_swap2
% 0.19/0.46 thf(fact_130_insert__DiffM,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a] :
% 0.19/0.46 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.19/0.46 => ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_DiffM
% 0.19/0.46 thf(fact_131_in__diffD,axiom,
% 0.19/0.46 ! [A: a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
% 0.19/0.46 => ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_diffD
% 0.19/0.46 thf(fact_132_add__mset__diff__bothsides,axiom,
% 0.19/0.46 ! [A: a,M: multiset_a,A2: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ A @ A2 ) )
% 0.19/0.46 = ( minus_1649712171iset_a @ M @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mset_diff_bothsides
% 0.19/0.46 thf(fact_133_Multiset_Odiff__cancel,axiom,
% 0.19/0.46 ! [A2: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ A2 @ A2 )
% 0.19/0.46 = zero_zero_multiset_a ) ).
% 0.19/0.46
% 0.19/0.46 % Multiset.diff_cancel
% 0.19/0.46 thf(fact_134_diff__empty,axiom,
% 0.19/0.46 ! [M: multiset_a] :
% 0.19/0.46 ( ( ( minus_1649712171iset_a @ M @ zero_zero_multiset_a )
% 0.19/0.46 = M )
% 0.19/0.46 & ( ( minus_1649712171iset_a @ zero_zero_multiset_a @ M )
% 0.19/0.46 = zero_zero_multiset_a ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_empty
% 0.19/0.46 thf(fact_135_union__single__eq__diff,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ X @ M )
% 0.19/0.46 = N )
% 0.19/0.46 => ( M
% 0.19/0.46 = ( minus_1649712171iset_a @ N @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % union_single_eq_diff
% 0.19/0.46 thf(fact_136_add__eq__conv__diff,axiom,
% 0.19/0.46 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 0.19/0.46 ( ( ( add_mset_a @ A @ M )
% 0.19/0.46 = ( add_mset_a @ B @ N ) )
% 0.19/0.46 = ( ( ( M = N )
% 0.19/0.46 & ( A = B ) )
% 0.19/0.46 | ( ( M
% 0.19/0.46 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 0.19/0.46 & ( N
% 0.19/0.46 = ( add_mset_a @ A @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_eq_conv_diff
% 0.19/0.46 thf(fact_137_diff__union__swap,axiom,
% 0.19/0.46 ! [A: a,B: a,M: multiset_a] :
% 0.19/0.46 ( ( A != B )
% 0.19/0.46 => ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 0.19/0.46 = ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_union_swap
% 0.19/0.46 thf(fact_138_diff__union__cancelR,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ M @ N ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_union_cancelR
% 0.19/0.46 thf(fact_139_diff__union__cancelL,axiom,
% 0.19/0.46 ! [N: multiset_a,M: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ N @ M ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_union_cancelL
% 0.19/0.46 thf(fact_140_Multiset_Odiff__add,axiom,
% 0.19/0.46 ! [M: multiset_a,N: multiset_a,Q: multiset_a] :
% 0.19/0.46 ( ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ Q ) )
% 0.19/0.46 = ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ Q ) ) ).
% 0.19/0.46
% 0.19/0.46 % Multiset.diff_add
% 0.19/0.46 thf(fact_141_diff__single__trivial,axiom,
% 0.19/0.46 ! [X: a,M: multiset_a] :
% 0.19/0.46 ( ~ ( member_a @ X @ ( set_mset_a @ M ) )
% 0.19/0.46 => ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_single_trivial
% 0.19/0.46
% 0.19/0.46 % Conjectures (1)
% 0.19/0.46 thf(conj_0,conjecture,
% 0.19/0.46 is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 2.78/1.78 ---- Embedded in THF ---
% 2.78/1.78 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.78/1.78 %%% Generated on Tue Feb 24 08:25:31 EST 2026
% 2.78/1.78 %%% using '$$dhol' embedding, version 1.3.0.
% 2.78/1.78 %%% Logic specification used:
% 2.78/1.78 %%% thf(spec, logic, $$dhol).
% 2.78/1.78
% 2.78/1.78 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2DTF2057.p
% See solution above
% 2.78/1.78 ------------------------
% 2.78/1.79 ---- Cleaned THF ---
% 2.78/1.79 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.78/1.79 set_Pr158363655iset_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.78/1.79 produc1127127335iset_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.78/1.79 set_Product_prod_a_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.78/1.79 multiset_multiset_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.78/1.79 set_multiset_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.78/1.79 product_prod_a_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 multiset_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Heap__OTree_Itf__a_J,type,
% 2.78/1.79 tree_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.78/1.79 set_a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(ty_n_tf__a,type,
% 2.78/1.79 a: $tType ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 zero_zero_multiset_a: multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_HOL_ONO__MATCH_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
% 2.78/1.79 heapIm970322378pify_a: tree_a > tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
% 2.78/1.79 heapIm1140443833left_a: tree_a > tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
% 2.78/1.79 heapIm1257206334ight_a: tree_a > tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
% 2.78/1.79 heapIm1091024090Down_a: tree_a > tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_OTree_OE_001tf__a,type,
% 2.78/1.79 e_a: tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_OTree_OT_001tf__a,type,
% 2.78/1.79 t_a: a > tree_a > tree_a > tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_Oin__tree_001tf__a,type,
% 2.78/1.79 in_tree_a: a > tree_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_Ois__heap_001tf__a,type,
% 2.78/1.79 is_heap_a: tree_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_Omultiset_001tf__a,type,
% 2.78/1.79 multiset_a2: tree_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Heap_Oval_001tf__a,type,
% 2.78/1.79 val_a: tree_a > a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
% 2.78/1.79 lattic146396397_Max_a: set_a > a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
% 2.78/1.79 add_mset_a: a > multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 fold_m382157835iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Ofold__mset_001tf__a_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 fold_m364285649iset_a: ( a > multiset_a > multiset_a ) > multiset_a > multiset_a > multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Ois__empty_001tf__a,type,
% 2.78/1.79 is_empty_a: multiset_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Omult1_001tf__a,type,
% 2.78/1.79 mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Omult_001tf__a,type,
% 2.78/1.79 mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 2.78/1.79 set_mset_a: multiset_a > set_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 2.78/1.79 ord_less_eq_a: a > a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
% 2.78/1.79 order_Greatest_a: ( a > $o ) > a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 2.78/1.79 product_Pair_a_a: a > a > product_prod_a_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Relation_Oirrefl_001tf__a,type,
% 2.78/1.79 irrefl_a: set_Product_prod_a_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Relation_Otrans_001tf__a,type,
% 2.78/1.79 trans_a: set_Product_prod_a_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.78/1.79 collect_a: ( a > $o ) > set_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.78/1.79 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.78/1.79 member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.78/1.79 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_c_member_001tf__a,type,
% 2.78/1.79 member_a: a > set_a > $o ).
% 2.78/1.79
% 2.78/1.79 thf(sy_v_l____,type,
% 2.78/1.79 l: tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_v_r____,type,
% 2.78/1.79 r: tree_a ).
% 2.78/1.79
% 2.78/1.79 thf(sy_v_v____,type,
% 2.78/1.79 v: a ).
% 2.78/1.79
% 2.78/1.79 thf(fact_0_T_Ohyps_I2_J,axiom,
% 2.78/1.79 is_heap_a @ ( heapIm970322378pify_a @ r ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_1_T_Ohyps_I1_J,axiom,
% 2.78/1.79 is_heap_a @ ( heapIm970322378pify_a @ l ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062,axiom,
% 2.78/1.79 ( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
% 2.78/1.79 => ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
% 2.78/1.79 => ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
% 2.78/1.79 = ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
% 2.78/1.79 => ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_3_heapify_Osimps_I2_J,axiom,
% 2.78/1.79 ! [V: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 2.78/1.79 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_4_siftDown__heap__is__heap,axiom,
% 2.78/1.79 ! [L: tree_a,R: tree_a,T: tree_a,V: a] :
% 2.78/1.79 ( ( is_heap_a @ L )
% 2.78/1.79 => ( ( is_heap_a @ R )
% 2.78/1.79 => ( ( T
% 2.78/1.79 = ( t_a @ V @ L @ R ) )
% 2.78/1.79 => ( is_heap_a @ ( heapIm1091024090Down_a @ T ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_5_Tree_Oinject,axiom,
% 2.78/1.79 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 2.78/1.79 ( ( ( t_a @ X21 @ X22 @ X23 )
% 2.78/1.79 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 2.78/1.79 = ( ( X21 = Y21 )
% 2.78/1.79 & ( X22 = Y22 )
% 2.78/1.79 & ( X23 = Y23 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_6_left_Osimps,axiom,
% 2.78/1.79 ! [V: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 2.78/1.79 = L ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_7_right_Osimps,axiom,
% 2.78/1.79 ! [V: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 2.78/1.79 = R ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_8_is__heap_Osimps_I2_J,axiom,
% 2.78/1.79 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_9_heapify_Osimps_I1_J,axiom,
% 2.78/1.79 ( ( heapIm970322378pify_a @ e_a )
% 2.78/1.79 = e_a ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_10_siftDown_Ocases,axiom,
% 2.78/1.79 ! [X: tree_a] :
% 2.78/1.79 ( ( X != e_a )
% 2.78/1.79 => ( ! [V2: a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ e_a @ e_a ) )
% 2.78/1.79 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.78/1.79 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.78/1.79 => ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_11_in__tree_Osimps_I2_J,axiom,
% 2.78/1.79 ! [V: a,V3: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( in_tree_a @ V @ ( t_a @ V3 @ L @ R ) )
% 2.78/1.79 = ( ( V = V3 )
% 2.78/1.79 | ( in_tree_a @ V @ L )
% 2.78/1.79 | ( in_tree_a @ V @ R ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_12_siftDown_Osimps_I2_J,axiom,
% 2.78/1.79 ! [V: a] :
% 2.78/1.79 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 2.78/1.79 = ( t_a @ V @ e_a @ e_a ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_13_in__tree_Osimps_I1_J,axiom,
% 2.78/1.79 ! [V: a] :
% 2.78/1.79 ~ ( in_tree_a @ V @ e_a ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_14_siftDown_Osimps_I1_J,axiom,
% 2.78/1.79 ( ( heapIm1091024090Down_a @ e_a )
% 2.78/1.79 = e_a ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_15_siftDown__in__tree__set,axiom,
% 2.78/1.79 ( in_tree_a
% 2.78/1.79 = ( ^ [V4: a,T2: tree_a] : ( in_tree_a @ V4 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_16_is__heap_Ocases,axiom,
% 2.78/1.79 ! [X: tree_a] :
% 2.78/1.79 ( ( X != e_a )
% 2.78/1.79 => ( ! [V2: a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ e_a @ e_a ) )
% 2.78/1.79 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.78/1.79 => ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.78/1.79 => ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 2.78/1.79 ( X
% 2.78/1.79 != ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_17_Tree_Oexhaust,axiom,
% 2.78/1.79 ! [Y: tree_a] :
% 2.78/1.79 ( ( Y != e_a )
% 2.78/1.79 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 2.78/1.79 ( Y
% 2.78/1.79 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_18_Tree_Oinduct,axiom,
% 2.78/1.79 ! [P: tree_a > $o,Tree: tree_a] :
% 2.78/1.79 ( ( P @ e_a )
% 2.78/1.79 => ( ! [X1: a,X2: tree_a,X3: tree_a] :
% 2.78/1.79 ( ( P @ X2 )
% 2.78/1.79 => ( ( P @ X3 )
% 2.78/1.79 => ( P @ ( t_a @ X1 @ X2 @ X3 ) ) ) )
% 2.78/1.79 => ( P @ Tree ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_19_Tree_Odistinct_I1_J,axiom,
% 2.78/1.79 ! [X21: a,X22: tree_a,X23: tree_a] :
% 2.78/1.79 ( e_a
% 2.78/1.79 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_20_is__heap_Osimps_I1_J,axiom,
% 2.78/1.79 is_heap_a @ e_a ).
% 2.78/1.79
% 2.78/1.79 thf(fact_21_siftDown__in__tree,axiom,
% 2.78/1.79 ! [T: tree_a] :
% 2.78/1.79 ( ( T != e_a )
% 2.78/1.79 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_22_siftDown_Osimps_I4_J,axiom,
% 2.78/1.79 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 2.78/1.79 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_23_siftDown_Osimps_I3_J,axiom,
% 2.78/1.79 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 2.78/1.79 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.78/1.79 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ e_a ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_24_siftDown_Osimps_I6_J,axiom,
% 2.78/1.79 ! [Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a,V: a] :
% 2.78/1.79 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_25_siftDown_Osimps_I5_J,axiom,
% 2.78/1.79 ! [Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
% 2.78/1.79 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 = ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
% 2.78/1.79 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 = ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_26_siftDown__Node,axiom,
% 2.78/1.79 ! [T: tree_a,V: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( T
% 2.78/1.79 = ( t_a @ V @ L @ R ) )
% 2.78/1.79 => ? [L2: tree_a,V5: a,R2: tree_a] :
% 2.78/1.79 ( ( ( heapIm1091024090Down_a @ T )
% 2.78/1.79 = ( t_a @ V5 @ L2 @ R2 ) )
% 2.78/1.79 & ( ord_less_eq_a @ V @ V5 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_27_is__heap_Osimps_I6_J,axiom,
% 2.78/1.79 ! [V: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.78/1.79 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.78/1.79 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_28_is__heap_Osimps_I5_J,axiom,
% 2.78/1.79 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.78/1.79 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.78/1.79 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 2.78/1.79 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_29_is__heap__max,axiom,
% 2.78/1.79 ! [V: a,T: tree_a] :
% 2.78/1.79 ( ( in_tree_a @ V @ T )
% 2.78/1.79 => ( ( is_heap_a @ T )
% 2.78/1.79 => ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_30_val_Osimps,axiom,
% 2.78/1.79 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 2.78/1.79 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 2.78/1.79 = V ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_31_is__heap_Osimps_I4_J,axiom,
% 2.78/1.79 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.78/1.79 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.78/1.79 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_32_is__heap_Osimps_I3_J,axiom,
% 2.78/1.79 ! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.78/1.79 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.78/1.79 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
% 2.78/1.79 & ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_33_mem__Collect__eq,axiom,
% 2.78/1.79 ! [A: a,P: a > $o] :
% 2.78/1.79 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.78/1.79 = ( P @ A ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_34_Collect__mem__eq,axiom,
% 2.78/1.79 ! [A2: set_a] :
% 2.78/1.79 ( ( collect_a
% 2.78/1.79 @ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
% 2.78/1.79 = A2 ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_35_order__refl,axiom,
% 2.78/1.79 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_36_heap__top__geq,axiom,
% 2.78/1.79 ! [A: a,T: tree_a] :
% 2.78/1.79 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
% 2.78/1.79 => ( ( is_heap_a @ T )
% 2.78/1.79 => ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_37_order__subst1,axiom,
% 2.78/1.79 ! [A: a,F: a > a,B: a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 2.78/1.79 => ( ( ord_less_eq_a @ B @ C )
% 2.78/1.79 => ( ! [X5: a,Y2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.78/1.79 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_38_order__subst2,axiom,
% 2.78/1.79 ! [A: a,B: a,F: a > a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 2.78/1.79 => ( ! [X5: a,Y2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_39_ord__eq__le__subst,axiom,
% 2.78/1.79 ! [A: a,F: a > a,B: a,C: a] :
% 2.78/1.79 ( ( A
% 2.78/1.79 = ( F @ B ) )
% 2.78/1.79 => ( ( ord_less_eq_a @ B @ C )
% 2.78/1.79 => ( ! [X5: a,Y2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.78/1.79 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_40_ord__le__eq__subst,axiom,
% 2.78/1.79 ! [A: a,B: a,F: a > a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( ( ( F @ B )
% 2.78/1.79 = C )
% 2.78/1.79 => ( ! [X5: a,Y2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.78/1.79 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_41_eq__iff,axiom,
% 2.78/1.79 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.78/1.79 = ( ^ [X4: a,Y4: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X4 @ Y4 )
% 2.78/1.79 & ( ord_less_eq_a @ Y4 @ X4 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_42_antisym,axiom,
% 2.78/1.79 ! [X: a,Y: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 => ( ( ord_less_eq_a @ Y @ X )
% 2.78/1.79 => ( X = Y ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_43_linear,axiom,
% 2.78/1.79 ! [X: a,Y: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 | ( ord_less_eq_a @ Y @ X ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_44_eq__refl,axiom,
% 2.78/1.79 ! [X: a,Y: a] :
% 2.78/1.79 ( ( X = Y )
% 2.78/1.79 => ( ord_less_eq_a @ X @ Y ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_45_le__cases,axiom,
% 2.78/1.79 ! [X: a,Y: a] :
% 2.78/1.79 ( ~ ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 => ( ord_less_eq_a @ Y @ X ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_46_order_Otrans,axiom,
% 2.78/1.79 ! [A: a,B: a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( ( ord_less_eq_a @ B @ C )
% 2.78/1.79 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_47_le__cases3,axiom,
% 2.78/1.79 ! [X: a,Y: a,Z2: a] :
% 2.78/1.79 ( ( ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 => ~ ( ord_less_eq_a @ Y @ Z2 ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ Y @ X )
% 2.78/1.79 => ~ ( ord_less_eq_a @ X @ Z2 ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ X @ Z2 )
% 2.78/1.79 => ~ ( ord_less_eq_a @ Z2 @ Y ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ Z2 @ Y )
% 2.78/1.79 => ~ ( ord_less_eq_a @ Y @ X ) )
% 2.78/1.79 => ( ( ( ord_less_eq_a @ Y @ Z2 )
% 2.78/1.79 => ~ ( ord_less_eq_a @ Z2 @ X ) )
% 2.78/1.79 => ~ ( ( ord_less_eq_a @ Z2 @ X )
% 2.78/1.79 => ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_48_antisym__conv,axiom,
% 2.78/1.79 ! [Y: a,X: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ Y @ X )
% 2.78/1.79 => ( ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 = ( X = Y ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_49_order__class_Oorder_Oeq__iff,axiom,
% 2.78/1.79 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.78/1.79 = ( ^ [A3: a,B2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A3 @ B2 )
% 2.78/1.79 & ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_50_ord__eq__le__trans,axiom,
% 2.78/1.79 ! [A: a,B: a,C: a] :
% 2.78/1.79 ( ( A = B )
% 2.78/1.79 => ( ( ord_less_eq_a @ B @ C )
% 2.78/1.79 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_51_ord__le__eq__trans,axiom,
% 2.78/1.79 ! [A: a,B: a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( ( B = C )
% 2.78/1.79 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_52_order__class_Oorder_Oantisym,axiom,
% 2.78/1.79 ! [A: a,B: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( ( ord_less_eq_a @ B @ A )
% 2.78/1.79 => ( A = B ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_53_order__trans,axiom,
% 2.78/1.79 ! [X: a,Y: a,Z2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ X @ Y )
% 2.78/1.79 => ( ( ord_less_eq_a @ Y @ Z2 )
% 2.78/1.79 => ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_54_dual__order_Orefl,axiom,
% 2.78/1.79 ! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_55_linorder__wlog,axiom,
% 2.78/1.79 ! [P: a > a > $o,A: a,B: a] :
% 2.78/1.79 ( ! [A4: a,B3: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ A4 @ B3 )
% 2.78/1.79 => ( P @ A4 @ B3 ) )
% 2.78/1.79 => ( ! [A4: a,B3: a] :
% 2.78/1.79 ( ( P @ B3 @ A4 )
% 2.78/1.79 => ( P @ A4 @ B3 ) )
% 2.78/1.79 => ( P @ A @ B ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_56_dual__order_Otrans,axiom,
% 2.78/1.79 ! [B: a,A: a,C: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ B @ A )
% 2.78/1.79 => ( ( ord_less_eq_a @ C @ B )
% 2.78/1.79 => ( ord_less_eq_a @ C @ A ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_57_dual__order_Oeq__iff,axiom,
% 2.78/1.79 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.78/1.79 = ( ^ [A3: a,B2: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ B2 @ A3 )
% 2.78/1.79 & ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_58_dual__order_Oantisym,axiom,
% 2.78/1.79 ! [B: a,A: a] :
% 2.78/1.79 ( ( ord_less_eq_a @ B @ A )
% 2.78/1.79 => ( ( ord_less_eq_a @ A @ B )
% 2.78/1.79 => ( A = B ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_59_add__mset__add__mset__same__iff,axiom,
% 2.78/1.79 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ A2 )
% 2.78/1.79 = ( add_mset_a @ A @ B4 ) )
% 2.78/1.79 = ( A2 = B4 ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_60_multi__self__add__other__not__self,axiom,
% 2.78/1.79 ! [M: multiset_a,X: a] :
% 2.78/1.79 ( M
% 2.78/1.79 != ( add_mset_a @ X @ M ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_61_mset__add,axiom,
% 2.78/1.79 ! [A: a,A2: multiset_a] :
% 2.78/1.79 ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 2.78/1.79 => ~ ! [B5: multiset_a] :
% 2.78/1.79 ( A2
% 2.78/1.79 != ( add_mset_a @ A @ B5 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_62_multi__member__split,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a] :
% 2.78/1.79 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.78/1.79 => ? [A5: multiset_a] :
% 2.78/1.79 ( M
% 2.78/1.79 = ( add_mset_a @ X @ A5 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_63_insert__noteq__member,axiom,
% 2.78/1.79 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ B @ B4 )
% 2.78/1.79 = ( add_mset_a @ C @ C2 ) )
% 2.78/1.79 => ( ( B != C )
% 2.78/1.79 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_64_union__single__eq__member,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ X @ M )
% 2.78/1.79 = N )
% 2.78/1.79 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_65_heap__top__max,axiom,
% 2.78/1.79 ! [T: tree_a] :
% 2.78/1.79 ( ( T != e_a )
% 2.78/1.79 => ( ( is_heap_a @ T )
% 2.78/1.79 => ( ( val_a @ T )
% 2.78/1.79 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_66_add__mset__commute,axiom,
% 2.78/1.79 ! [X: a,Y: a,M: multiset_a] :
% 2.78/1.79 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M ) )
% 2.78/1.79 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_67_add__eq__conv__ex,axiom,
% 2.78/1.79 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ M )
% 2.78/1.79 = ( add_mset_a @ B @ N ) )
% 2.78/1.79 = ( ( ( M = N )
% 2.78/1.79 & ( A = B ) )
% 2.78/1.79 | ? [K: multiset_a] :
% 2.78/1.79 ( ( M
% 2.78/1.79 = ( add_mset_a @ B @ K ) )
% 2.78/1.79 & ( N
% 2.78/1.79 = ( add_mset_a @ A @ K ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_68_Greatest__equality,axiom,
% 2.78/1.79 ! [P: a > $o,X: a] :
% 2.78/1.79 ( ( P @ X )
% 2.78/1.79 => ( ! [Y2: a] :
% 2.78/1.79 ( ( P @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ Y2 @ X ) )
% 2.78/1.79 => ( ( order_Greatest_a @ P )
% 2.78/1.79 = X ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_69_GreatestI2__order,axiom,
% 2.78/1.79 ! [P: a > $o,X: a,Q: a > $o] :
% 2.78/1.79 ( ( P @ X )
% 2.78/1.79 => ( ! [Y2: a] :
% 2.78/1.79 ( ( P @ Y2 )
% 2.78/1.79 => ( ord_less_eq_a @ Y2 @ X ) )
% 2.78/1.79 => ( ! [X5: a] :
% 2.78/1.79 ( ( P @ X5 )
% 2.78/1.79 => ( ! [Y5: a] :
% 2.78/1.79 ( ( P @ Y5 )
% 2.78/1.79 => ( ord_less_eq_a @ Y5 @ X5 ) )
% 2.78/1.79 => ( Q @ X5 ) ) )
% 2.78/1.79 => ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_70_multiset__induct__min,axiom,
% 2.78/1.79 ! [P: multiset_a > $o,M: multiset_a] :
% 2.78/1.79 ( ( P @ zero_zero_multiset_a )
% 2.78/1.79 => ( ! [X5: a,M2: multiset_a] :
% 2.78/1.79 ( ( P @ M2 )
% 2.78/1.79 => ( ! [Xa: a] :
% 2.78/1.79 ( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
% 2.78/1.79 => ( ord_less_eq_a @ X5 @ Xa ) )
% 2.78/1.79 => ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
% 2.78/1.79 => ( P @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_71_multiset__induct__max,axiom,
% 2.78/1.79 ! [P: multiset_a > $o,M: multiset_a] :
% 2.78/1.79 ( ( P @ zero_zero_multiset_a )
% 2.78/1.79 => ( ! [X5: a,M2: multiset_a] :
% 2.78/1.79 ( ( P @ M2 )
% 2.78/1.79 => ( ! [Xa: a] :
% 2.78/1.79 ( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
% 2.78/1.79 => ( ord_less_eq_a @ Xa @ X5 ) )
% 2.78/1.79 => ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
% 2.78/1.79 => ( P @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_72_add__mset__eq__singleton__iff,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a,Y: a] :
% 2.78/1.79 ( ( ( add_mset_a @ X @ M )
% 2.78/1.79 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 2.78/1.79 = ( ( M = zero_zero_multiset_a )
% 2.78/1.79 & ( X = Y ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_73_single__eq__add__mset,axiom,
% 2.78/1.79 ! [A: a,B: a,M: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.78/1.79 = ( add_mset_a @ B @ M ) )
% 2.78/1.79 = ( ( B = A )
% 2.78/1.79 & ( M = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_74_add__mset__eq__single,axiom,
% 2.78/1.79 ! [B: a,M: multiset_a,A: a] :
% 2.78/1.79 ( ( ( add_mset_a @ B @ M )
% 2.78/1.79 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.78/1.79 = ( ( B = A )
% 2.78/1.79 & ( M = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_75_single__eq__single,axiom,
% 2.78/1.79 ! [A: a,B: a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.78/1.79 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 2.78/1.79 = ( A = B ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_76_multiset__cases,axiom,
% 2.78/1.79 ! [M: multiset_a] :
% 2.78/1.79 ( ( M != zero_zero_multiset_a )
% 2.78/1.79 => ~ ! [X5: a,N2: multiset_a] :
% 2.78/1.79 ( M
% 2.78/1.79 != ( add_mset_a @ X5 @ N2 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_77_multiset__induct,axiom,
% 2.78/1.79 ! [P: multiset_a > $o,M: multiset_a] :
% 2.78/1.79 ( ( P @ zero_zero_multiset_a )
% 2.78/1.79 => ( ! [X5: a,M2: multiset_a] :
% 2.78/1.79 ( ( P @ M2 )
% 2.78/1.79 => ( P @ ( add_mset_a @ X5 @ M2 ) ) )
% 2.78/1.79 => ( P @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_78_multiset__induct2,axiom,
% 2.78/1.79 ! [P: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 2.78/1.79 => ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
% 2.78/1.79 ( ( P @ M2 @ N2 )
% 2.78/1.79 => ( P @ ( add_mset_a @ A4 @ M2 ) @ N2 ) )
% 2.78/1.79 => ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
% 2.78/1.79 ( ( P @ M2 @ N2 )
% 2.78/1.79 => ( P @ M2 @ ( add_mset_a @ A4 @ N2 ) ) )
% 2.78/1.79 => ( P @ M @ N ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_79_empty__not__add__mset,axiom,
% 2.78/1.79 ! [A: a,A2: multiset_a] :
% 2.78/1.79 ( zero_zero_multiset_a
% 2.78/1.79 != ( add_mset_a @ A @ A2 ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_80_multi__nonempty__split,axiom,
% 2.78/1.79 ! [M: multiset_a] :
% 2.78/1.79 ( ( M != zero_zero_multiset_a )
% 2.78/1.79 => ? [A5: multiset_a,A4: a] :
% 2.78/1.79 ( M
% 2.78/1.79 = ( add_mset_a @ A4 @ A5 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_81_multiset__nonemptyE,axiom,
% 2.78/1.79 ! [A2: multiset_a] :
% 2.78/1.79 ( ( A2 != zero_zero_multiset_a )
% 2.78/1.79 => ~ ! [X5: a] :
% 2.78/1.79 ~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_82_multi__member__last,axiom,
% 2.78/1.79 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_83_multiset_Osimps_I1_J,axiom,
% 2.78/1.79 ( ( multiset_a2 @ e_a )
% 2.78/1.79 = zero_zero_multiset_a ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_84_Multiset_Ois__empty__def,axiom,
% 2.78/1.79 ( is_empty_a
% 2.78/1.79 = ( ^ [A6: multiset_a] : ( A6 = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_85_multiset_Osimps_I2_J,axiom,
% 2.78/1.79 ! [V: a,L: tree_a,R: tree_a] :
% 2.78/1.79 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 2.78/1.79 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_86_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 2.78/1.79 ! [X: multiset_a,Y: multiset_a] :
% 2.78/1.79 ( ( zero_zero_multiset_a
% 2.78/1.79 = ( plus_plus_multiset_a @ X @ Y ) )
% 2.78/1.79 = ( ( X = zero_zero_multiset_a )
% 2.78/1.79 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_87_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 2.78/1.79 ! [X: multiset_a,Y: multiset_a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ X @ Y )
% 2.78/1.79 = zero_zero_multiset_a )
% 2.78/1.79 = ( ( X = zero_zero_multiset_a )
% 2.78/1.79 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_88_union__eq__empty,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ M @ N )
% 2.78/1.79 = zero_zero_multiset_a )
% 2.78/1.79 = ( ( M = zero_zero_multiset_a )
% 2.78/1.79 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_89_empty__eq__union,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( zero_zero_multiset_a
% 2.78/1.79 = ( plus_plus_multiset_a @ M @ N ) )
% 2.78/1.79 = ( ( M = zero_zero_multiset_a )
% 2.78/1.79 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_90_union__mset__add__mset__right,axiom,
% 2.78/1.79 ! [A2: multiset_a,A: a,B4: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ A2 @ ( add_mset_a @ A @ B4 ) )
% 2.78/1.79 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_91_union__mset__add__mset__left,axiom,
% 2.78/1.79 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
% 2.78/1.79 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_92_add__right__mono,axiom,
% 2.78/1.79 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.78/1.79 ( ( ord_le1199012836iset_a @ A @ B )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_93_add__left__mono,axiom,
% 2.78/1.79 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.78/1.79 ( ( ord_le1199012836iset_a @ A @ B )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_94_add__mono,axiom,
% 2.78/1.79 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 2.78/1.79 ( ( ord_le1199012836iset_a @ A @ B )
% 2.78/1.79 => ( ( ord_le1199012836iset_a @ C @ D )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_95_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.78/1.79 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 2.78/1.79 ( ( ( ord_le1199012836iset_a @ I @ J )
% 2.78/1.79 & ( ord_le1199012836iset_a @ K2 @ L ) )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_96_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.78/1.79 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 2.78/1.79 ( ( ( I = J )
% 2.78/1.79 & ( ord_le1199012836iset_a @ K2 @ L ) )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_97_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.78/1.79 ! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
% 2.78/1.79 ( ( ( ord_le1199012836iset_a @ I @ J )
% 2.78/1.79 & ( K2 = L ) )
% 2.78/1.79 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_98_empty__neutral_I2_J,axiom,
% 2.78/1.79 ! [X: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 2.78/1.79 = X ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_99_empty__neutral_I1_J,axiom,
% 2.78/1.79 ! [X: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 2.78/1.79 = X ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_100_union__iff,axiom,
% 2.78/1.79 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.78/1.79 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B4 ) ) )
% 2.78/1.79 = ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 2.78/1.79 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_101_union__assoc,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a,K3: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K3 )
% 2.78/1.79 = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_102_union__lcomm,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a,K3: multiset_a] :
% 2.78/1.79 ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) )
% 2.78/1.79 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K3 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_103_union__commute,axiom,
% 2.78/1.79 ( plus_plus_multiset_a
% 2.78/1.79 = ( ^ [M3: multiset_a,N3: multiset_a] : ( plus_plus_multiset_a @ N3 @ M3 ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_104_union__left__cancel,axiom,
% 2.78/1.79 ! [K3: multiset_a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ K3 @ M )
% 2.78/1.79 = ( plus_plus_multiset_a @ K3 @ N ) )
% 2.78/1.79 = ( M = N ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_105_union__right__cancel,axiom,
% 2.78/1.79 ! [M: multiset_a,K3: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ M @ K3 )
% 2.78/1.79 = ( plus_plus_multiset_a @ N @ K3 ) )
% 2.78/1.79 = ( M = N ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_106_multi__union__self__other__eq,axiom,
% 2.78/1.79 ! [A2: multiset_a,X6: multiset_a,Y6: multiset_a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ A2 @ X6 )
% 2.78/1.79 = ( plus_plus_multiset_a @ A2 @ Y6 ) )
% 2.78/1.79 => ( X6 = Y6 ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_107_add__mset__add__single,axiom,
% 2.78/1.79 ( add_mset_a
% 2.78/1.79 = ( ^ [A3: a,A6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( add_mset_a @ A3 @ zero_zero_multiset_a ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_108_union__is__single,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a,A: a] :
% 2.78/1.79 ( ( ( plus_plus_multiset_a @ M @ N )
% 2.78/1.79 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.78/1.79 = ( ( ( M
% 2.78/1.79 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.78/1.79 & ( N = zero_zero_multiset_a ) )
% 2.78/1.79 | ( ( M = zero_zero_multiset_a )
% 2.78/1.79 & ( N
% 2.78/1.79 = ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_109_single__is__union,axiom,
% 2.78/1.79 ! [A: a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.78/1.79 = ( plus_plus_multiset_a @ M @ N ) )
% 2.78/1.79 = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.78/1.79 = M )
% 2.78/1.79 & ( N = zero_zero_multiset_a ) )
% 2.78/1.79 | ( ( M = zero_zero_multiset_a )
% 2.78/1.79 & ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.78/1.79 = N ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_110_multi__member__skip,axiom,
% 2.78/1.79 ! [X: a,XS: multiset_a,Y: a] :
% 2.78/1.79 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 2.78/1.79 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_111_multi__member__this,axiom,
% 2.78/1.79 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_112_mult1E,axiom,
% 2.78/1.79 ! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
% 2.78/1.79 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
% 2.78/1.79 => ~ ! [A4: a,M0: multiset_a] :
% 2.78/1.79 ( ( M
% 2.78/1.79 = ( add_mset_a @ A4 @ M0 ) )
% 2.78/1.79 => ! [K4: multiset_a] :
% 2.78/1.79 ( ( N
% 2.78/1.79 = ( plus_plus_multiset_a @ M0 @ K4 ) )
% 2.78/1.79 => ~ ! [B6: a] :
% 2.78/1.79 ( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
% 2.78/1.79 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A4 ) @ R ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_113_mult1I,axiom,
% 2.78/1.79 ! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K3: multiset_a,R: set_Product_prod_a_a] :
% 2.78/1.79 ( ( M
% 2.78/1.79 = ( add_mset_a @ A @ M02 ) )
% 2.78/1.79 => ( ( N
% 2.78/1.79 = ( plus_plus_multiset_a @ M02 @ K3 ) )
% 2.78/1.79 => ( ! [B3: a] :
% 2.78/1.79 ( ( member_a @ B3 @ ( set_mset_a @ K3 ) )
% 2.78/1.79 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
% 2.78/1.79 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_114_less__add,axiom,
% 2.78/1.79 ! [N: multiset_a,A: a,M02: multiset_a,R: set_Product_prod_a_a] :
% 2.78/1.79 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R ) )
% 2.78/1.79 => ( ? [M2: multiset_a] :
% 2.78/1.79 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R ) )
% 2.78/1.79 & ( N
% 2.78/1.79 = ( add_mset_a @ A @ M2 ) ) )
% 2.78/1.79 | ? [K4: multiset_a] :
% 2.78/1.79 ( ! [B6: a] :
% 2.78/1.79 ( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
% 2.78/1.79 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R ) )
% 2.78/1.79 & ( N
% 2.78/1.79 = ( plus_plus_multiset_a @ M02 @ K4 ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_115_one__step__implies__mult,axiom,
% 2.78/1.79 ! [J2: multiset_a,K3: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
% 2.78/1.79 ( ( J2 != zero_zero_multiset_a )
% 2.78/1.79 => ( ! [X5: a] :
% 2.78/1.79 ( ( member_a @ X5 @ ( set_mset_a @ K3 ) )
% 2.78/1.79 => ? [Xa: a] :
% 2.78/1.79 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 2.78/1.79 & ( member449909584od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ R ) ) )
% 2.78/1.79 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K3 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_116_not__less__empty,axiom,
% 2.78/1.79 ! [M: multiset_a,R: set_Product_prod_a_a] :
% 2.78/1.79 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_117_mult1__union,axiom,
% 2.78/1.79 ! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
% 2.78/1.79 ( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
% 2.78/1.79 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_118_mult__implies__one__step,axiom,
% 2.78/1.79 ! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( trans_a @ R )
% 2.78/1.79 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
% 2.78/1.79 => ? [I3: multiset_a,J3: multiset_a] :
% 2.78/1.79 ( ( N
% 2.78/1.79 = ( plus_plus_multiset_a @ I3 @ J3 ) )
% 2.78/1.79 & ? [K4: multiset_a] :
% 2.78/1.79 ( ( M
% 2.78/1.79 = ( plus_plus_multiset_a @ I3 @ K4 ) )
% 2.78/1.79 & ( J3 != zero_zero_multiset_a )
% 2.78/1.79 & ! [X7: a] :
% 2.78/1.79 ( ( member_a @ X7 @ ( set_mset_a @ K4 ) )
% 2.78/1.79 => ? [Xa2: a] :
% 2.78/1.79 ( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
% 2.78/1.79 & ( member449909584od_a_a @ ( product_Pair_a_a @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_119_subset__mset_Osum__mset__0__iff,axiom,
% 2.78/1.79 ! [M: multiset_multiset_a] :
% 2.78/1.79 ( ( ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
% 2.78/1.79 = zero_zero_multiset_a )
% 2.78/1.79 = ( ! [X4: multiset_a] :
% 2.78/1.79 ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
% 2.78/1.79 => ( X4 = zero_zero_multiset_a ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_120_mult__cancel__add__mset,axiom,
% 2.78/1.79 ! [S: set_Product_prod_a_a,Uu: a,X6: multiset_a,Y6: multiset_a] :
% 2.78/1.79 ( ( trans_a @ S )
% 2.78/1.79 => ( ( irrefl_a @ S )
% 2.78/1.79 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X6 ) @ ( add_mset_a @ Uu @ Y6 ) ) @ ( mult_a @ S ) )
% 2.78/1.79 = ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_121_mult__cancel,axiom,
% 2.78/1.79 ! [S: set_Product_prod_a_a,X6: multiset_a,Z3: multiset_a,Y6: multiset_a] :
% 2.78/1.79 ( ( trans_a @ S )
% 2.78/1.79 => ( ( irrefl_a @ S )
% 2.78/1.79 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X6 @ Z3 ) @ ( plus_plus_multiset_a @ Y6 @ Z3 ) ) @ ( mult_a @ S ) )
% 2.78/1.79 = ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_122_in__mset__fold__plus__iff,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a,NN: multiset_multiset_a] :
% 2.78/1.79 ( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
% 2.78/1.79 = ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.78/1.79 | ? [N3: multiset_a] :
% 2.78/1.79 ( ( member_multiset_a @ N3 @ ( set_mset_multiset_a @ NN ) )
% 2.78/1.79 & ( member_a @ X @ ( set_mset_a @ N3 ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_123_union__fold__mset__add__mset,axiom,
% 2.78/1.79 ( plus_plus_multiset_a
% 2.78/1.79 = ( fold_m364285649iset_a @ add_mset_a ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_124_add__mset__replicate__mset__safe,axiom,
% 2.78/1.79 ! [M: multiset_a,A: a] :
% 2.78/1.79 ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
% 2.78/1.79 => ( ( add_mset_a @ A @ M )
% 2.78/1.79 = ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_125_insert__DiffM2,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a] :
% 2.78/1.79 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.78/1.79 => ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.78/1.79 = M ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_126_diff__diff__add__mset,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a,P: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ P )
% 2.78/1.79 = ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_127_add__mset__remove__trivial,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.78/1.79 = M ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_128_diff__add__mset__swap,axiom,
% 2.78/1.79 ! [B: a,A2: multiset_a,M: multiset_a] :
% 2.78/1.79 ( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
% 2.78/1.79 => ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A2 )
% 2.78/1.79 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A2 ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_129_diff__union__swap2,axiom,
% 2.78/1.79 ! [Y: a,M: multiset_a,X: a] :
% 2.78/1.79 ( ( member_a @ Y @ ( set_mset_a @ M ) )
% 2.78/1.79 => ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 2.78/1.79 = ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_130_insert__DiffM,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a] :
% 2.78/1.79 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.78/1.79 => ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
% 2.78/1.79 = M ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_131_in__diffD,axiom,
% 2.78/1.79 ! [A: a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
% 2.78/1.79 => ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_132_add__mset__diff__bothsides,axiom,
% 2.78/1.79 ! [A: a,M: multiset_a,A2: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ A @ A2 ) )
% 2.78/1.79 = ( minus_1649712171iset_a @ M @ A2 ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_133_Multiset_Odiff__cancel,axiom,
% 2.78/1.79 ! [A2: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ A2 @ A2 )
% 2.78/1.79 = zero_zero_multiset_a ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_134_diff__empty,axiom,
% 2.78/1.79 ! [M: multiset_a] :
% 2.78/1.79 ( ( ( minus_1649712171iset_a @ M @ zero_zero_multiset_a )
% 2.78/1.79 = M )
% 2.78/1.79 & ( ( minus_1649712171iset_a @ zero_zero_multiset_a @ M )
% 2.78/1.79 = zero_zero_multiset_a ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_135_union__single__eq__diff,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ X @ M )
% 2.78/1.79 = N )
% 2.78/1.79 => ( M
% 2.78/1.79 = ( minus_1649712171iset_a @ N @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_136_add__eq__conv__diff,axiom,
% 2.78/1.79 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 2.78/1.79 ( ( ( add_mset_a @ A @ M )
% 2.78/1.79 = ( add_mset_a @ B @ N ) )
% 2.78/1.79 = ( ( ( M = N )
% 2.78/1.79 & ( A = B ) )
% 2.78/1.79 | ( ( M
% 2.78/1.79 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 2.78/1.79 & ( N
% 2.78/1.79 = ( add_mset_a @ A @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_137_diff__union__swap,axiom,
% 2.78/1.79 ! [A: a,B: a,M: multiset_a] :
% 2.78/1.79 ( ( A != B )
% 2.78/1.79 => ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 2.78/1.79 = ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_138_diff__union__cancelR,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ M @ N ) @ N )
% 2.78/1.79 = M ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_139_diff__union__cancelL,axiom,
% 2.78/1.79 ! [N: multiset_a,M: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ N @ M ) @ N )
% 2.78/1.79 = M ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_140_Multiset_Odiff__add,axiom,
% 2.78/1.79 ! [M: multiset_a,N: multiset_a,Q: multiset_a] :
% 2.78/1.79 ( ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ Q ) )
% 2.78/1.79 = ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ Q ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(fact_141_diff__single__trivial,axiom,
% 2.78/1.79 ! [X: a,M: multiset_a] :
% 2.78/1.79 ( ~ ( member_a @ X @ ( set_mset_a @ M ) )
% 2.78/1.79 => ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.78/1.79 = M ) ) ).
% 2.78/1.79
% 2.78/1.79 thf(conj_0,conjecture,
% 2.78/1.79 is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) ).
% 2.78/1.79 ------------------------
% 3.04/1.81 % (2183)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/4Mi)
% 3.04/1.81 % (2186)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/2Mi)
% 3.04/1.81 % (2188)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/18Mi)
% 3.04/1.81 % (2185)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/2Mi)
% 3.04/1.81 % (2184)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/27Mi)
% 3.04/1.81 % (2181)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/183Mi)
% 3.04/1.81 % (2187)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/275Mi)
% 3.04/1.81 % (2186)Instruction limit reached!
% 3.04/1.81 % (2186)------------------------------
% 3.04/1.81 % (2186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.81 % (2186)Termination reason: Unknown
% 3.04/1.81 % (2186)Termination phase: shuffling
% 3.04/1.81
% 3.04/1.81 % (2186)Memory used [KB]: 1151
% 3.04/1.81 % (2186)Time elapsed: 0.002 s
% 3.04/1.81 % (2186)Instructions burned: 2 (million)
% 3.04/1.81 % (2186)------------------------------
% 3.04/1.81 % (2186)------------------------------
% 3.04/1.81 % (2183)Instruction limit reached!
% 3.04/1.81 % (2183)------------------------------
% 3.04/1.81 % (2183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.81 % (2183)Termination reason: Unknown
% 3.04/1.81 % (2183)Termination phase: shuffling
% 3.04/1.81
% 3.04/1.81 % (2183)Memory used [KB]: 1279
% 3.04/1.81 % (2183)Time elapsed: 0.003 s
% 3.04/1.81 % (2183)Instructions burned: 5 (million)
% 3.04/1.81 % (2183)------------------------------
% 3.04/1.81 % (2183)------------------------------
% 3.04/1.81 % (2185)Instruction limit reached!
% 3.04/1.81 % (2185)------------------------------
% 3.04/1.81 % (2185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.81 % (2185)Termination reason: Unknown
% 3.04/1.81 % (2185)Termination phase: shuffling
% 3.04/1.81
% 3.04/1.81 % (2185)Memory used [KB]: 1279
% 3.04/1.81 % (2185)Time elapsed: 0.003 s
% 3.04/1.81 % (2185)Instructions burned: 4 (million)
% 3.04/1.81 % (2185)------------------------------
% 3.04/1.81 % (2185)------------------------------
% 3.04/1.82 % (2188)Instruction limit reached!
% 3.04/1.82 % (2188)------------------------------
% 3.04/1.82 % (2188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.82 % (2188)Termination reason: Unknown
% 3.04/1.82 % (2188)Termination phase: Property scanning
% 3.04/1.82
% 3.04/1.82 % (2188)Memory used [KB]: 1407
% 3.04/1.82 % (2188)Time elapsed: 0.007 s
% 3.04/1.82 % (2188)Instructions burned: 18 (million)
% 3.04/1.82 % (2188)------------------------------
% 3.04/1.82 % (2188)------------------------------
% 3.04/1.82 % (2184)Instruction limit reached!
% 3.04/1.82 % (2184)------------------------------
% 3.04/1.82 % (2184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.82 % (2184)Termination reason: Unknown
% 3.04/1.82 % (2184)Termination phase: Preprocessing 3
% 3.04/1.82
% 3.04/1.82 % (2184)Memory used [KB]: 1535
% 3.04/1.82 % (2184)Time elapsed: 0.010 s
% 3.04/1.82 % (2184)Instructions burned: 28 (million)
% 3.04/1.82 % (2184)------------------------------
% 3.04/1.82 % (2184)------------------------------
% 3.04/1.82 % (2193)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/3Mi)
% 3.04/1.82 % (2187)Refutation not found, incomplete strategy
% 3.04/1.82 % (2187)------------------------------
% 3.04/1.82 % (2187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.82 % (2187)Termination reason: Refutation not found, incomplete strategy
% 3.04/1.82
% 3.04/1.82
% 3.04/1.82 % (2187)Memory used [KB]: 6012
% 3.04/1.82 % (2187)Time elapsed: 0.011 s
% 3.04/1.82 % (2187)Instructions burned: 32 (million)
% 3.04/1.82 % (2187)------------------------------
% 3.04/1.82 % (2187)------------------------------
% 3.04/1.82 % (2194)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/37Mi)
% 3.04/1.82 % (2195)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/15Mi)
% 3.04/1.82 % (2193)Instruction limit reached!
% 3.04/1.82 % (2193)------------------------------
% 3.04/1.82 % (2193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.82 % (2193)Termination reason: Unknown
% 3.04/1.82 % (2193)Termination phase: shuffling
% 3.04/1.82
% 3.04/1.82 % (2193)Memory used [KB]: 1151
% 3.04/1.82 % (2193)Time elapsed: 0.002 s
% 3.04/1.82 % (2193)Instructions burned: 3 (million)
% 3.04/1.82 % (2193)------------------------------
% 3.04/1.82 % (2193)------------------------------
% 3.04/1.83 % (2197)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/3Mi)
% 3.04/1.83 % (2195)Instruction limit reached!
% 3.04/1.83 % (2195)------------------------------
% 3.04/1.83 % (2195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.83 % (2195)Termination reason: Unknown
% 3.04/1.83 % (2195)Termination phase: Property scanning
% 3.04/1.83
% 3.04/1.83 % (2195)Memory used [KB]: 1407
% 3.04/1.83 % (2195)Time elapsed: 0.006 s
% 3.04/1.83 % (2195)Instructions burned: 16 (million)
% 3.04/1.83 % (2195)------------------------------
% 3.04/1.83 % (2195)------------------------------
% 3.04/1.83 % (2197)Instruction limit reached!
% 3.04/1.83 % (2197)------------------------------
% 3.04/1.83 % (2197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.83 % (2197)Termination reason: Unknown
% 3.04/1.83 % (2197)Termination phase: shuffling
% 3.04/1.83
% 3.04/1.83 % (2197)Memory used [KB]: 1151
% 3.04/1.83 % (2197)Time elapsed: 0.002 s
% 3.04/1.83 % (2197)Instructions burned: 3 (million)
% 3.04/1.83 % (2197)------------------------------
% 3.04/1.83 % (2197)------------------------------
% 3.04/1.83 % (2199)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/1041Mi)
% 3.04/1.83 % (2201)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/7Mi)
% 3.04/1.83 % (2202)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/16Mi)
% 3.04/1.83 % (2194)Instruction limit reached!
% 3.04/1.83 % (2194)------------------------------
% 3.04/1.83 % (2194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.83 % (2194)Termination reason: Unknown
% 3.04/1.83 % (2194)Termination phase: Property scanning
% 3.04/1.83
% 3.04/1.83 % (2194)Memory used [KB]: 1791
% 3.04/1.83 % (2194)Time elapsed: 0.011 s
% 3.04/1.83 % (2194)Instructions burned: 37 (million)
% 3.04/1.83 % (2194)------------------------------
% 3.04/1.83 % (2194)------------------------------
% 3.04/1.83 % (2201)Instruction limit reached!
% 3.04/1.83 % (2201)------------------------------
% 3.04/1.83 % (2201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.83 % (2201)Termination reason: Unknown
% 3.04/1.83 % (2201)Termination phase: shuffling
% 3.04/1.83
% 3.04/1.83 % (2201)Memory used [KB]: 1279
% 3.04/1.83 % (2201)Time elapsed: 0.004 s
% 3.04/1.83 % (2201)Instructions burned: 9 (million)
% 3.04/1.83 % (2201)------------------------------
% 3.04/1.83 % (2201)------------------------------
% 3.04/1.84 % (2205)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/3Mi)
% 3.04/1.84 % (2206)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/3Mi)
% 3.04/1.84 % (2205)Instruction limit reached!
% 3.04/1.84 % (2205)------------------------------
% 3.04/1.84 % (2205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.84 % (2205)Termination reason: Unknown
% 3.04/1.84 % (2205)Termination phase: shuffling
% 3.04/1.84
% 3.04/1.84 % (2205)Memory used [KB]: 1151
% 3.04/1.84 % (2205)Time elapsed: 0.002 s
% 3.04/1.84 % (2205)Instructions burned: 3 (million)
% 3.04/1.84 % (2205)------------------------------
% 3.04/1.84 % (2205)------------------------------
% 3.04/1.84 % (2202)Instruction limit reached!
% 3.04/1.84 % (2202)------------------------------
% 3.04/1.84 % (2202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.84 % (2202)Termination reason: Unknown
% 3.04/1.84 % (2202)Termination phase: Property scanning
% 3.04/1.84
% 3.04/1.84 % (2202)Memory used [KB]: 1407
% 3.04/1.84 % (2202)Time elapsed: 0.007 s
% 3.04/1.84 % (2202)Instructions burned: 18 (million)
% 3.04/1.84 % (2202)------------------------------
% 3.04/1.84 % (2202)------------------------------
% 3.04/1.84 % (2206)Instruction limit reached!
% 3.04/1.84 % (2206)------------------------------
% 3.04/1.84 % (2206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.84 % (2206)Termination reason: Unknown
% 3.04/1.84 % (2206)Termination phase: shuffling
% 3.04/1.84
% 3.04/1.84 % (2206)Memory used [KB]: 1151
% 3.04/1.84 % (2206)Time elapsed: 0.002 s
% 3.04/1.84 % (2206)Instructions burned: 3 (million)
% 3.04/1.84 % (2206)------------------------------
% 3.04/1.84 % (2206)------------------------------
% 3.04/1.84 % (2209)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/7Mi)
% 3.04/1.84 % (2210)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/3Mi)
% 3.04/1.84 % (2209)Instruction limit reached!
% 3.04/1.84 % (2209)------------------------------
% 3.04/1.84 % (2209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.84 % (2209)Termination reason: Unknown
% 3.04/1.84 % (2209)Termination phase: shuffling
% 3.04/1.84
% 3.04/1.84 % (2209)Memory used [KB]: 1279
% 3.04/1.84 % (2209)Time elapsed: 0.004 s
% 3.04/1.84 % (2209)Instructions burned: 9 (million)
% 3.04/1.84 % (2209)------------------------------
% 3.04/1.84 % (2209)------------------------------
% 3.04/1.84 % (2210)Instruction limit reached!
% 3.04/1.84 % (2210)------------------------------
% 3.04/1.84 % (2210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.84 % (2210)Termination reason: Unknown
% 3.04/1.84 % (2210)Termination phase: shuffling
% 3.04/1.84
% 3.04/1.84 % (2210)Memory used [KB]: 1279
% 3.04/1.84 % (2210)Time elapsed: 0.002 s
% 3.04/1.84 % (2210)Instructions burned: 3 (million)
% 3.04/1.84 % (2210)------------------------------
% 3.04/1.84 % (2210)------------------------------
% 3.04/1.85 % (2214)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/4Mi)
% 3.04/1.85 % (2215)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/18Mi)
% 3.04/1.85 % (2216)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/710Mi)
% 3.04/1.85 % (2214)Instruction limit reached!
% 3.04/1.85 % (2214)------------------------------
% 3.04/1.85 % (2214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.85 % (2214)Termination reason: Unknown
% 3.04/1.85 % (2214)Termination phase: shuffling
% 3.04/1.85
% 3.04/1.85 % (2214)Memory used [KB]: 1279
% 3.04/1.85 % (2214)Time elapsed: 0.003 s
% 3.04/1.85 % (2214)Instructions burned: 6 (million)
% 3.04/1.85 % (2214)------------------------------
% 3.04/1.85 % (2214)------------------------------
% 3.04/1.85 % (2215)Instruction limit reached!
% 3.04/1.85 % (2215)------------------------------
% 3.04/1.85 % (2215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.85 % (2215)Termination reason: Unknown
% 3.04/1.85 % (2215)Termination phase: Property scanning
% 3.04/1.85
% 3.04/1.85 % (2215)Memory used [KB]: 1407
% 3.04/1.85 % (2215)Time elapsed: 0.007 s
% 3.04/1.85 % (2215)Instructions burned: 20 (million)
% 3.04/1.85 % (2215)------------------------------
% 3.04/1.85 % (2215)------------------------------
% 3.04/1.85 % (2221)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/6Mi)
% 3.04/1.85 % (2222)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_2057 for (2999ds/902Mi)
% 3.04/1.86 % (2221)Instruction limit reached!
% 3.04/1.86 % (2221)------------------------------
% 3.04/1.86 % (2221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.86 % (2221)Termination reason: Unknown
% 3.04/1.86 % (2221)Termination phase: shuffling
% 3.04/1.86
% 3.04/1.86 % (2221)Memory used [KB]: 1279
% 3.04/1.86 % (2221)Time elapsed: 0.003 s
% 3.04/1.86 % (2221)Instructions burned: 7 (million)
% 3.04/1.86 % (2221)------------------------------
% 3.04/1.86 % (2221)------------------------------
% 3.04/1.86 % (2216)First to succeed.
% 3.04/1.86 % (2216)Refutation found. Thanks to Tanya!
% 3.04/1.86 % SZS status Theorem for DTF2THF_2057
% 3.04/1.86 % SZS output start Proof for DTF2THF_2057
% 3.04/1.86 thf(type_def_5, type, multiset_a: $tType).
% 3.04/1.86 thf(type_def_7, type, tree_a: $tType).
% 3.04/1.86 thf(type_def_8, type, a: $tType).
% 3.04/1.86 thf(type_def_9, type, set_a: $tType).
% 3.04/1.86 thf(type_def_10, type, multiset_multiset_a: $tType).
% 3.04/1.86 thf(type_def_11, type, set_Product_prod_a_a: $tType).
% 3.04/1.86 thf(type_def_12, type, set_Pr158363655iset_a: $tType).
% 3.04/1.86 thf(type_def_13, type, set_multiset_a: $tType).
% 3.04/1.86 thf(type_def_14, type, produc1127127335iset_a: $tType).
% 3.04/1.86 thf(type_def_15, type, product_prod_a_a: $tType).
% 3.04/1.86 thf(func_def_0, type, set_Pr158363655iset_a: $tType).
% 3.04/1.86 thf(func_def_1, type, produc1127127335iset_a: $tType).
% 3.04/1.86 thf(func_def_2, type, set_Product_prod_a_a: $tType).
% 3.04/1.86 thf(func_def_3, type, multiset_multiset_a: $tType).
% 3.04/1.86 thf(func_def_4, type, set_multiset_a: $tType).
% 3.04/1.86 thf(func_def_5, type, product_prod_a_a: $tType).
% 3.04/1.86 thf(func_def_6, type, multiset_a: $tType).
% 3.04/1.86 thf(func_def_7, type, tree_a: $tType).
% 3.04/1.86 thf(func_def_8, type, set_a: $tType).
% 3.04/1.86 thf(func_def_9, type, a: $tType).
% 3.04/1.86 thf(func_def_10, type, minus_1649712171iset_a: multiset_a > multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_11, type, plus_plus_multiset_a: multiset_a > multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_12, type, zero_zero_multiset_a: multiset_a).
% 3.04/1.86 thf(func_def_13, type, nO_MAT1617603563iset_a: multiset_a > multiset_a > $o).
% 3.04/1.86 thf(func_def_14, type, heapIm970322378pify_a: tree_a > tree_a).
% 3.04/1.86 thf(func_def_15, type, heapIm1140443833left_a: tree_a > tree_a).
% 3.04/1.86 thf(func_def_16, type, heapIm1257206334ight_a: tree_a > tree_a).
% 3.04/1.86 thf(func_def_17, type, heapIm1091024090Down_a: tree_a > tree_a).
% 3.04/1.86 thf(func_def_18, type, e_a: tree_a).
% 3.04/1.86 thf(func_def_19, type, t_a: a > tree_a > tree_a > tree_a).
% 3.04/1.86 thf(func_def_20, type, in_tree_a: a > tree_a > $o).
% 3.04/1.86 thf(func_def_21, type, is_heap_a: tree_a > $o).
% 3.04/1.86 thf(func_def_22, type, multiset_a2: tree_a > multiset_a).
% 3.04/1.86 thf(func_def_23, type, val_a: tree_a > a).
% 3.04/1.86 thf(func_def_24, type, lattic146396397_Max_a: set_a > a).
% 3.04/1.86 thf(func_def_25, type, add_mset_a: a > multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_26, type, comm_m315775925iset_a: (multiset_a > multiset_a > multiset_a) > multiset_a > multiset_multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_27, type, fold_m382157835iset_a: (multiset_a > multiset_a > multiset_a) > multiset_a > multiset_multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_28, type, fold_m364285649iset_a: (a > multiset_a > multiset_a) > multiset_a > multiset_a > multiset_a).
% 3.04/1.86 thf(func_def_29, type, is_empty_a: multiset_a > $o).
% 3.04/1.86 thf(func_def_30, type, mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 3.04/1.86 thf(func_def_31, type, mult_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 3.04/1.86 thf(func_def_32, type, set_mset_multiset_a: multiset_multiset_a > set_multiset_a).
% 3.04/1.86 thf(func_def_33, type, set_mset_a: multiset_a > set_a).
% 3.04/1.86 thf(func_def_34, type, ord_le1199012836iset_a: multiset_a > multiset_a > $o).
% 3.04/1.86 thf(func_def_35, type, ord_less_eq_a: a > a > $o).
% 3.04/1.86 thf(func_def_36, type, order_Greatest_a: (a > $o) > a).
% 3.04/1.86 thf(func_def_37, type, produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a).
% 3.04/1.86 thf(func_def_38, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.04/1.86 thf(func_def_39, type, irrefl_a: set_Product_prod_a_a > $o).
% 3.04/1.86 thf(func_def_40, type, trans_a: set_Product_prod_a_a > $o).
% 3.04/1.86 thf(func_def_41, type, collect_a: (a > $o) > set_a).
% 3.04/1.86 thf(func_def_42, type, member_multiset_a: multiset_a > set_multiset_a > $o).
% 3.04/1.86 thf(func_def_43, type, member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o).
% 3.04/1.86 thf(func_def_44, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.04/1.86 thf(func_def_45, type, member_a: a > set_a > $o).
% 3.04/1.86 thf(func_def_46, type, l: tree_a).
% 3.04/1.86 thf(func_def_47, type, r: tree_a).
% 3.04/1.86 thf(func_def_48, type, v: a).
% 3.04/1.86 thf(f394,plain,(
% 3.04/1.86 $false),
% 3.04/1.86 inference(avatar_sat_refutation,[status(thm)],[f389,f390,f393])).
% 3.04/1.86 thf(f393,plain,(
% 3.04/1.86 ~spl0_2),
% 3.04/1.86 inference(avatar_contradiction_clause,[status(thm)],[f392])).
% 3.04/1.86 thf(f392,plain,(
% 3.04/1.86 $false | ~spl0_2),
% 3.04/1.86 inference(subsumption_resolution,[status(thm)],[f391,f373])).
% 3.04/1.86 thf(f373,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r))) != $true)),
% 3.04/1.86 inference(cnf_transformation,[status(thm)],[f358])).
% 3.04/1.86 thf(f358,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r))) != $true)),
% 3.04/1.86 inference(flattening,[status(thm)],[f351])).
% 3.04/1.86 thf(f351,plain,(
% 3.04/1.86 ~((is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r))) = $true)),
% 3.04/1.86 inference(fool_elimination,[status(thm)],[f350])).
% 3.04/1.86 thf(f350,plain,(
% 3.04/1.86 ~(is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r)))),
% 3.04/1.86 inference(rectify,[status(thm)],[f144])).
% 3.04/1.86 thf(f144,negated_conjecture,(
% 3.04/1.86 ~(is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r)))),
% 3.04/1.86 inference(negated_conjecture,[status(cth)],[f143])).
% 3.04/1.86 thf(f143,conjecture,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r)))),
% 3.04/1.86 file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',conj_0)).
% 3.04/1.86 thf(f391,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ (t_a @ v @ l @ r))) = $true) | ~spl0_2),
% 3.04/1.86 inference(forward_demodulation,[status(thm)],[f388,f371])).
% 3.04/1.86 thf(f371,plain,(
% 3.04/1.86 ( ! [X2 : tree_a,X0 : a,X1 : tree_a] : (((heapIm1091024090Down_a @ (t_a @ X0 @ (heapIm970322378pify_a @ X2) @ (heapIm970322378pify_a @ X1))) = (heapIm970322378pify_a @ (t_a @ X0 @ X2 @ X1)))) )),
% 3.04/1.86 inference(cnf_transformation,[status(thm)],[f359])).
% 3.04/1.86 thf(f359,plain,(
% 3.04/1.86 ! [X0 : a,X1 : tree_a,X2 : tree_a] : ((heapIm1091024090Down_a @ (t_a @ X0 @ (heapIm970322378pify_a @ X2) @ (heapIm970322378pify_a @ X1))) = (heapIm970322378pify_a @ (t_a @ X0 @ X2 @ X1)))),
% 3.04/1.86 inference(rectify,[status(thm)],[f4])).
% 3.04/1.86 thf(f4,axiom,(
% 3.04/1.86 ! [X0 : a,X2 : tree_a,X1 : tree_a] : ((heapIm970322378pify_a @ (t_a @ X0 @ X1 @ X2)) = (heapIm1091024090Down_a @ (t_a @ X0 @ (heapIm970322378pify_a @ X1) @ (heapIm970322378pify_a @ X2))))),
% 3.04/1.86 file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_3_heapify_Osimps_I2_J)).
% 3.04/1.86 thf(f388,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true) | ~spl0_2),
% 3.04/1.86 inference(avatar_component_clause,[status(thm)],[f386])).
% 3.04/1.86 thf(f386,definition,(
% 3.04/1.86 spl0_2 <=> ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true)),
% 3.04/1.86 introduced(definition,[new_symbols(naming,[spl0_2])],[avatar_definition])).
% 3.04/1.86 thf(f390,plain,(
% 3.04/1.86 spl0_1),
% 3.04/1.86 inference(avatar_split_clause,[status(thm)],[f375,f382])).
% 3.04/1.86 thf(f382,definition,(
% 3.04/1.86 spl0_1 <=> ((is_heap_a @ (heapIm970322378pify_a @ r)) = $true)),
% 3.04/1.86 introduced(definition,[new_symbols(naming,[spl0_1])],[avatar_definition])).
% 3.04/1.86 thf(f375,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ r)) = $true)),
% 3.04/1.86 inference(cnf_transformation,[status(thm)],[f195])).
% 3.04/1.86 thf(f195,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ r)) = $true)),
% 3.04/1.86 inference(fool_elimination,[status(thm)],[f194])).
% 3.04/1.86 thf(f194,plain,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ r))),
% 3.04/1.86 inference(rectify,[status(thm)],[f1])).
% 3.04/1.86 thf(f1,axiom,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ r))),
% 3.04/1.86 file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_0_T_Ohyps_I2_J)).
% 3.04/1.86 thf(f389,plain,(
% 3.04/1.86 ~spl0_1 | spl0_2),
% 3.04/1.86 inference(avatar_split_clause,[status(thm)],[f380,f386,f382])).
% 3.04/1.86 thf(f380,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ r)) != $true) | ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true)),
% 3.04/1.86 inference(subsumption_resolution,[status(thm)],[f379,f370])).
% 3.04/1.86 thf(f370,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ l)) = $true)),
% 3.04/1.86 inference(cnf_transformation,[status(thm)],[f239])).
% 3.04/1.86 thf(f239,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ l)) = $true)),
% 3.04/1.86 inference(fool_elimination,[status(thm)],[f238])).
% 3.04/1.86 thf(f238,plain,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ l))),
% 3.04/1.86 inference(rectify,[status(thm)],[f2])).
% 3.04/1.86 thf(f2,axiom,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ l))),
% 3.04/1.86 file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_1_T_Ohyps_I1_J)).
% 3.04/1.86 thf(f379,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true) | ((is_heap_a @ (heapIm970322378pify_a @ r)) != $true) | ((is_heap_a @ (heapIm970322378pify_a @ l)) != $true)),
% 3.04/1.86 inference(trivial_inequality_removal,[status(thm)],[f372])).
% 3.04/1.86 thf(f372,plain,(
% 3.04/1.86 ((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) != (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))) | ((is_heap_a @ (heapIm970322378pify_a @ r)) != $true) | ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true) | ((is_heap_a @ (heapIm970322378pify_a @ l)) != $true)),
% 3.04/1.86 inference(cnf_transformation,[status(thm)],[f361])).
% 3.04/1.86 thf(f361,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true) | ((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) != (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))) | ((is_heap_a @ (heapIm970322378pify_a @ l)) != $true) | ((is_heap_a @ (heapIm970322378pify_a @ r)) != $true)),
% 3.04/1.86 inference(flattening,[status(thm)],[f360])).
% 3.04/1.86 thf(f360,plain,(
% 3.04/1.86 ((((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true) | ((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) != (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) | ((is_heap_a @ (heapIm970322378pify_a @ r)) != $true)) | ((is_heap_a @ (heapIm970322378pify_a @ l)) != $true)),
% 3.04/1.86 inference(ennf_transformation,[status(thm)],[f341])).
% 3.04/1.86 thf(f341,plain,(
% 3.04/1.86 ((is_heap_a @ (heapIm970322378pify_a @ l)) = $true) => (((is_heap_a @ (heapIm970322378pify_a @ r)) = $true) => (((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) = (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))) => ((is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)))) = $true)))),
% 3.04/1.86 inference(fool_elimination,[status(thm)],[f340])).
% 3.04/1.86 thf(f340,plain,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ l)) => ((is_heap_a @ (heapIm970322378pify_a @ r)) => (((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) = (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))) => (is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))))))),
% 3.04/1.86 inference(rectify,[status(thm)],[f3])).
% 3.04/1.86 thf(f3,axiom,(
% 3.04/1.86 (is_heap_a @ (heapIm970322378pify_a @ l)) => ((is_heap_a @ (heapIm970322378pify_a @ r)) => (((t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r)) = (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))) => (is_heap_a @ (heapIm1091024090Down_a @ (t_a @ v @ (heapIm970322378pify_a @ l) @ (heapIm970322378pify_a @ r))))))),
% 3.04/1.86 file('/export/starexec/sandbox/tmp/tmp.HAOGau68XK/DTF2THF_2057.p',fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062)).
% 3.04/1.86 % SZS output end Proof for DTF2THF_2057
% 3.04/1.86 % (2216)------------------------------
% 3.04/1.86 % (2216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.86 % (2216)Termination reason: Refutation
% 3.04/1.86
% 3.04/1.86 % (2216)Memory used [KB]: 6012
% 3.04/1.86 % (2216)Time elapsed: 0.010 s
% 3.04/1.86 % (2216)Instructions burned: 29 (million)
% 3.04/1.86 % (2216)------------------------------
% 3.04/1.86 % (2216)------------------------------
% 3.04/1.86 % (2177)Success in time 0.054 s
%------------------------------------------------------------------------------