%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP070^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n022.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 3.14s 1.73s
% Output : Refutation 3.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP070^2 : TPTP v9.2.1. Released v7.5.0.
% 0.10/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.31 % Computer : n022.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.31 % WCLimit : 300
% 0.12/0.31 % DateTime : Tue Feb 24 08:25:36 EST 2026
% 0.12/0.31 % CPUTime :
% 0.12/0.31 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.43 ---- Original DTF file ---
% 0.15/0.43 thf(spec,logic,$$dhol).
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 % File : ITP070^2 : TPTP v9.2.1. Released v7.5.0.
% 0.15/0.43 % Domain : Interactive Theorem Proving
% 0.15/0.43 % Problem : Sledgehammer HeapImperative problem prob_824__5349520_1
% 0.15/0.43 % Version : Especial.
% 0.15/0.43 % English :
% 0.15/0.43
% 0.15/0.43 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.15/0.43 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.15/0.43 % Source : [Des21]
% 0.15/0.43 % Names : HeapImperative/prob_824__5349520_1 [Des21]
% 0.15/0.43
% 0.15/0.43 % Status : Theorem
% 0.15/0.43 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.15/0.43 % Syntax : Number of formulae : 343 ( 90 unt; 66 typ; 0 def)
% 0.15/0.43 % Number of atoms : 861 ( 319 equ; 0 cnn)
% 0.15/0.43 % Maximal formula atoms : 13 ( 3 avg)
% 0.15/0.43 % Number of connectives : 5500 ( 102 ~; 7 |; 61 &;4852 @)
% 0.15/0.43 % ( 0 <=>; 478 =>; 0 <=; 0 <~>)
% 0.15/0.43 % Maximal formula depth : 29 ( 9 avg)
% 0.15/0.43 % Number of types : 2 ( 1 usr)
% 0.15/0.43 % Number of type conns : 322 ( 322 >; 0 *; 0 +; 0 <<)
% 0.15/0.43 % Number of symbols : 68 ( 65 usr; 9 con; 0-8 aty)
% 0.15/0.43 % Number of variables : 1429 ( 66 ^;1282 !; 14 ?;1429 :)
% 0.15/0.43 % ( 67 !>; 0 ?*; 0 @-; 0 @+)
% 0.15/0.43 % SPC : TH1_THM_EQU_NAR
% 0.15/0.43
% 0.15/0.43 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:47.478
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 % Could-be-implicit typings (6)
% 0.15/0.43 thf(ty_t_Product__Type_Oprod,type,
% 0.15/0.43 product_prod: $tType > $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Multiset_Omultiset,type,
% 0.15/0.43 multiset: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_List_Olist,type,
% 0.15/0.43 list: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Heap_OTree,type,
% 0.15/0.43 tree: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_t_Set_Oset,type,
% 0.15/0.43 set: $tType > $tType ).
% 0.15/0.43
% 0.15/0.43 thf(ty_tf_a,type,
% 0.15/0.43 a: $tType ).
% 0.15/0.43
% 0.15/0.43 % Explicit typings (60)
% 0.15/0.43 thf(sy_cl_HOL_Otype,type,
% 0.15/0.43 type:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Oord,type,
% 0.15/0.43 ord:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Oorder,type,
% 0.15/0.43 order:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Olinorder,type,
% 0.15/0.43 linorder:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Orderings_Opreorder,type,
% 0.15/0.43 preorder:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.15/0.43 cancel1352612707id_add:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.15/0.43 linord219039673up_add:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.15/0.43 ordere216010020id_add:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.15/0.43 ordere779506340up_add:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.15/0.43 linord1659791738miring:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.15/0.43 canoni770627133id_add:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.15/0.43 ordere236663937imp_le:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.15/0.43 unique1598680935umeral:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.15/0.43 ordere516151231imp_le:
% 0.15/0.43 !>[A: $tType] : $o ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.15/0.43 bNF_Ca1785829860lChain:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 0.15/0.43 unique455577585es_aux:
% 0.15/0.43 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.15/0.43 plus_plus:
% 0.15/0.43 !>[A: $tType] : ( A > A > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.15/0.43 zero_zero:
% 0.15/0.43 !>[A: $tType] : A ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oheapify,type,
% 0.15/0.43 heapIm818251801eapify:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__is__empty,type,
% 0.15/0.43 heapIm721255937_empty:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__of__list,type,
% 0.15/0.43 heapIm874063447f_list:
% 0.15/0.43 !>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oleft,type,
% 0.15/0.43 heapIm1271749598e_left:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oof__list__tree,type,
% 0.15/0.43 heapIm1912108042t_tree:
% 0.15/0.43 !>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OremoveLeaf,type,
% 0.15/0.43 heapIm970386777veLeaf:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( product_prod @ A @ ( tree @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oright,type,
% 0.15/0.43 heapIm1434396069_right:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OsiftDown,type,
% 0.15/0.43 heapIm748920189ftDown:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_OHeap,type,
% 0.15/0.43 heap:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_OHeap__axioms,type,
% 0.15/0.43 heap_axioms:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_OTree_OE,type,
% 0.15/0.43 e:
% 0.15/0.43 !>[A: $tType] : ( tree @ A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_OTree_OT,type,
% 0.15/0.43 t:
% 0.15/0.43 !>[A: $tType] : ( A > ( tree @ A ) > ( tree @ A ) > ( tree @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_Oin__tree,type,
% 0.15/0.43 in_tree:
% 0.15/0.43 !>[A: $tType] : ( A > ( tree @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_Ois__heap,type,
% 0.15/0.43 is_heap:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_Omultiset,type,
% 0.15/0.43 multiset2:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > ( multiset @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Heap_Oval,type,
% 0.15/0.43 val:
% 0.15/0.43 !>[A: $tType] : ( ( tree @ A ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 0.15/0.43 lattic929149872er_Max:
% 0.15/0.43 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_List_Olist_OCons,type,
% 0.15/0.43 cons:
% 0.15/0.43 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Multiset_Oadd__mset,type,
% 0.15/0.43 add_mset:
% 0.15/0.43 !>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Multiset_Ois__empty,type,
% 0.15/0.43 is_empty:
% 0.15/0.43 !>[A: $tType] : ( ( multiset @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Multiset_Oset__mset,type,
% 0.15/0.43 set_mset:
% 0.15/0.43 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Multiset_Osubseteq__mset,type,
% 0.15/0.43 subseteq_mset:
% 0.15/0.43 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.15/0.43 ord_less_eq:
% 0.15/0.43 !>[A: $tType] : ( A > A > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.15/0.43 order_Greatest:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_OPair,type,
% 0.15/0.43 product_Pair:
% 0.15/0.43 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.15/0.43 produc2004651681e_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.15/0.43 product_rec_prod:
% 0.15/0.43 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_OCollection,type,
% 0.15/0.43 collection:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_OCollection_Oset,type,
% 0.15/0.43 set2:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > ( multiset @ A ) ) > B > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_ORemoveMax,type,
% 0.15/0.43 removeMax:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H,type,
% 0.15/0.43 ssort:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( B > ( product_prod @ A @ B ) ) > B > ( list @ A ) > ( list @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom,type,
% 0.15/0.43 ssort_dom:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( B > ( product_prod @ A @ B ) ) > ( product_prod @ B @ ( list @ A ) ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_RemoveMax_ORemoveMax__axioms,type,
% 0.15/0.43 removeMax_axioms:
% 0.15/0.43 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_Set_OCollect,type,
% 0.15/0.43 collect:
% 0.15/0.43 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_c_member,type,
% 0.15/0.43 member:
% 0.15/0.43 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_l1____,type,
% 0.15/0.43 l1: tree @ a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_l2____,type,
% 0.15/0.43 l2: tree @ a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_r1____,type,
% 0.15/0.43 r1: tree @ a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_r2____,type,
% 0.15/0.43 r2: tree @ a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_v1____,type,
% 0.15/0.43 v1: a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_v2____,type,
% 0.15/0.43 v2: a ).
% 0.15/0.43
% 0.15/0.43 thf(sy_v_v____,type,
% 0.15/0.43 v: a ).
% 0.15/0.43
% 0.15/0.43 % Relevant facts (256)
% 0.15/0.43 thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062,axiom,
% 0.15/0.43 ord_less_eq @ a @ v2 @ v1 ).
% 0.15/0.43
% 0.15/0.43 % \<open>v2 \<le> v1\<close>
% 0.15/0.43 thf(fact_1_True,axiom,
% 0.15/0.43 ord_less_eq @ a @ v1 @ v ).
% 0.15/0.43
% 0.15/0.43 % True
% 0.15/0.43 thf(fact_2__C5__2_Ohyps_C_I2_J,axiom,
% 0.15/0.43 ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
% 0.15/0.43 => ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ v )
% 0.15/0.43 => ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) )
% 0.15/0.43 = ( multiset2 @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % "5_2.hyps"(2)
% 0.15/0.43 thf(fact_3__C5__2_Ohyps_C_I1_J,axiom,
% 0.15/0.43 ( ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
% 0.15/0.43 => ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ v )
% 0.15/0.43 => ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) )
% 0.15/0.43 = ( multiset2 @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % "5_2.hyps"(1)
% 0.15/0.43 thf(fact_4_siftDown__Node,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [T2: tree @ A,V: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( T2
% 0.15/0.43 = ( t @ A @ V @ L @ R ) )
% 0.15/0.43 => ? [L2: tree @ A,V2: A,R2: tree @ A] :
% 0.15/0.43 ( ( ( heapIm748920189ftDown @ A @ T2 )
% 0.15/0.43 = ( t @ A @ V2 @ L2 @ R2 ) )
% 0.15/0.43 & ( ord_less_eq @ A @ V @ V2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown_Node
% 0.15/0.43 thf(fact_5_Tree_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A,Y21: A,Y22: tree @ A,Y23: tree @ A] :
% 0.15/0.43 ( ( ( t @ A @ X21 @ X22 @ X23 )
% 0.15/0.43 = ( t @ A @ Y21 @ Y22 @ Y23 ) )
% 0.15/0.43 = ( ( X21 = Y21 )
% 0.15/0.43 & ( X22 = Y22 )
% 0.15/0.43 & ( X23 = Y23 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Tree.inject
% 0.15/0.43 thf(fact_6_heapify_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( heapIm818251801eapify @ A @ ( t @ A @ V @ L @ R ) )
% 0.15/0.43 = ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm818251801eapify @ A @ L ) @ ( heapIm818251801eapify @ A @ R ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % heapify.simps(2)
% 0.15/0.43 thf(fact_7_siftDown__heap__is__heap,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [L: tree @ A,R: tree @ A,T2: tree @ A,V: A] :
% 0.15/0.43 ( ( is_heap @ A @ L )
% 0.15/0.43 => ( ( is_heap @ A @ R )
% 0.15/0.43 => ( ( T2
% 0.15/0.43 = ( t @ A @ V @ L @ R ) )
% 0.15/0.43 => ( is_heap @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown_heap_is_heap
% 0.15/0.43 thf(fact_8_siftDown__in__tree__set,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( in_tree @ A )
% 0.15/0.43 = ( ^ [V3: A,T3: tree @ A] : ( in_tree @ A @ V3 @ ( heapIm748920189ftDown @ A @ T3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown_in_tree_set
% 0.15/0.43 thf(fact_9_left_Osimps,axiom,
% 0.15/0.43 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( heapIm1271749598e_left @ A @ ( t @ A @ V @ L @ R ) )
% 0.15/0.43 = L ) ).
% 0.15/0.43
% 0.15/0.43 % left.simps
% 0.15/0.43 thf(fact_10_right_Osimps,axiom,
% 0.15/0.43 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( heapIm1434396069_right @ A @ ( t @ A @ V @ L @ R ) )
% 0.15/0.43 = R ) ).
% 0.15/0.43
% 0.15/0.43 % right.simps
% 0.15/0.43 thf(fact_11_siftDown_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A] :
% 0.15/0.43 ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(2)
% 0.15/0.43 thf(fact_12_siftDown_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
% 0.15/0.43 = ( e @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(1)
% 0.15/0.43 thf(fact_13_Heap_Omultiset,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( ( Multiset @ L )
% 0.15/0.43 = ( multiset2 @ A @ ( As_tree @ L ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.multiset
% 0.15/0.43 thf(fact_14_in__tree_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,V: A] :
% 0.15/0.43 ~ ( in_tree @ A @ V @ ( e @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_tree.simps(1)
% 0.15/0.43 thf(fact_15_is__heap_Osimps_I6_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 0.15/0.43 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) )
% 0.15/0.43 & ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(6)
% 0.15/0.43 thf(fact_16_is__heap_Osimps_I5_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
% 0.15/0.43 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) )
% 0.15/0.43 & ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(5)
% 0.15/0.43 thf(fact_17_is__heap_Osimps_I4_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 0.15/0.43 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 0.15/0.43 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(4)
% 0.15/0.43 thf(fact_18_is__heap_Osimps_I3_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 0.15/0.43 ( ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(3)
% 0.15/0.43 thf(fact_19_is__heap_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A] : ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(2)
% 0.15/0.43 thf(fact_20_is__heap_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( is_heap @ A @ ( e @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.simps(1)
% 0.15/0.43 thf(fact_21_heapify_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( heapIm818251801eapify @ A @ ( e @ A ) )
% 0.15/0.43 = ( e @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % heapify.simps(1)
% 0.15/0.43 thf(fact_22_is__heap__max,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,T2: tree @ A] :
% 0.15/0.43 ( ( in_tree @ A @ V @ T2 )
% 0.15/0.43 => ( ( is_heap @ A @ T2 )
% 0.15/0.43 => ( ord_less_eq @ A @ V @ ( val @ A @ T2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap_max
% 0.15/0.43 thf(fact_23_Heap_Oas__tree__empty,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T2: B] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( ( ( As_tree @ T2 )
% 0.15/0.43 = ( e @ A ) )
% 0.15/0.43 = ( Is_empty @ T2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.as_tree_empty
% 0.15/0.43 thf(fact_24_Heap_Ois__heap__of__list,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),I: list @ A] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( is_heap @ A @ ( As_tree @ ( Of_list @ I ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.is_heap_of_list
% 0.15/0.43 thf(fact_25_heapify__heap__is__heap,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [T2: tree @ A] : ( is_heap @ A @ ( heapIm818251801eapify @ A @ T2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % heapify_heap_is_heap
% 0.15/0.43 thf(fact_26_siftDown__in__tree,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [T2: tree @ A] :
% 0.15/0.43 ( ( T2
% 0.15/0.43 != ( e @ A ) )
% 0.15/0.43 => ( in_tree @ A @ ( val @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) @ T2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown_in_tree
% 0.15/0.43 thf(fact_27_is__heap_Ocases,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: tree @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 != ( e @ A ) )
% 0.15/0.43 => ( ! [V4: A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 0.15/0.43 => ~ ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % is_heap.cases
% 0.15/0.43 thf(fact_28_Tree_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,Y: tree @ A] :
% 0.15/0.43 ( ( Y
% 0.15/0.43 != ( e @ A ) )
% 0.15/0.43 => ~ ! [X212: A,X222: tree @ A,X232: tree @ A] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( t @ A @ X212 @ X222 @ X232 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Tree.exhaust
% 0.15/0.43 thf(fact_29_Tree_Oinduct,axiom,
% 0.15/0.43 ! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
% 0.15/0.43 ( ( P @ ( e @ A ) )
% 0.15/0.43 => ( ! [X1: A,X2: tree @ A,X3: tree @ A] :
% 0.15/0.43 ( ( P @ X2 )
% 0.15/0.43 => ( ( P @ X3 )
% 0.15/0.43 => ( P @ ( t @ A @ X1 @ X2 @ X3 ) ) ) )
% 0.15/0.43 => ( P @ Tree ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Tree.induct
% 0.15/0.43 thf(fact_30_Tree_Odistinct_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A] :
% 0.15/0.43 ( ( e @ A )
% 0.15/0.43 != ( t @ A @ X21 @ X22 @ X23 ) ) ).
% 0.15/0.43
% 0.15/0.43 % Tree.distinct(1)
% 0.15/0.43 thf(fact_31_siftDown_Osimps_I3_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( e @ A ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(3)
% 0.15/0.43 thf(fact_32_siftDown_Osimps_I4_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( e @ A ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(4)
% 0.15/0.43 thf(fact_33_val_Osimps,axiom,
% 0.15/0.43 ! [A: $tType,V: A,Uu: tree @ A,Uv: tree @ A] :
% 0.15/0.43 ( ( val @ A @ ( t @ A @ V @ Uu @ Uv ) )
% 0.15/0.43 = V ) ).
% 0.15/0.43
% 0.15/0.43 % val.simps
% 0.15/0.43 thf(fact_34_siftDown_Osimps_I5_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( t @ A @ Va @ Vb @ Vc ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(5)
% 0.15/0.43 thf(fact_35_siftDown_Osimps_I6_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A,V: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 0.15/0.43 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 0.15/0.43 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 0.15/0.43 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( t @ A @ Vd @ Ve @ Vf ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.simps(6)
% 0.15/0.43 thf(fact_36_in__tree_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,V: A,V5: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( in_tree @ A @ V @ ( t @ A @ V5 @ L @ R ) )
% 0.15/0.43 = ( ( V = V5 )
% 0.15/0.43 | ( in_tree @ A @ V @ L )
% 0.15/0.43 | ( in_tree @ A @ V @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % in_tree.simps(2)
% 0.15/0.43 thf(fact_37_removeLeaf_Oinduct,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [P: ( tree @ A ) > $o,A0: tree @ A] :
% 0.15/0.43 ( ! [V4: A] : ( P @ ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( P @ ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( P @ ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 0.15/0.43 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 0.15/0.43 => ( P @ ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 0.15/0.43 => ( ! [V4: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( ( P @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
% 0.15/0.43 => ( ( P @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
% 0.15/0.43 => ( P @ ( t @ A @ V4 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.15/0.43 => ( ( P @ ( e @ A ) )
% 0.15/0.43 => ( P @ A0 ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % removeLeaf.induct
% 0.15/0.43 thf(fact_38_removeLeaf_Ocases,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: tree @ A] :
% 0.15/0.43 ( ! [V4: A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.15/0.43 => ( ! [V4: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.15/0.43 => ( X
% 0.15/0.43 = ( e @ A ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % removeLeaf.cases
% 0.15/0.43 thf(fact_39_siftDown_Ocases,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: tree @ A] :
% 0.15/0.43 ( ( X
% 0.15/0.43 != ( e @ A ) )
% 0.15/0.43 => ( ! [V4: A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 0.15/0.43 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.15/0.43 => ~ ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 0.15/0.43 ( X
% 0.15/0.43 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % siftDown.cases
% 0.15/0.43 thf(fact_40_hs__is__empty__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( heapIm721255937_empty @ A )
% 0.15/0.43 = ( ^ [T3: tree @ A] :
% 0.15/0.43 ( T3
% 0.15/0.43 = ( e @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % hs_is_empty_def
% 0.15/0.43 thf(fact_41_order__refl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_refl
% 0.15/0.43 thf(fact_42_heap__top__geq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [A2: A,T2: tree @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) )
% 0.15/0.43 => ( ( is_heap @ A @ T2 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ ( val @ A @ T2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % heap_top_geq
% 0.15/0.43 thf(fact_43_Heap_Oaxioms_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.axioms(2)
% 0.15/0.43 thf(fact_44_hs__of__list__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( heapIm874063447f_list @ A )
% 0.15/0.43 = ( ^ [L3: list @ A] : ( heapIm818251801eapify @ A @ ( heapIm1912108042t_tree @ A @ L3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % hs_of_list_def
% 0.15/0.43 thf(fact_45_mem__Collect__eq,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,P: A > $o] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.15/0.43 = ( P @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mem_Collect_eq
% 0.15/0.43 thf(fact_46_Collect__mem__eq,axiom,
% 0.15/0.43 ! [A: $tType,A3: set @ A] :
% 0.15/0.43 ( ( collect @ A
% 0.15/0.43 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.15/0.43 = A3 ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_mem_eq
% 0.15/0.43 thf(fact_47_Collect__cong,axiom,
% 0.15/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.15/0.43 ( ! [X5: A] :
% 0.15/0.43 ( ( P @ X5 )
% 0.15/0.43 = ( Q @ X5 ) )
% 0.15/0.43 => ( ( collect @ A @ P )
% 0.15/0.43 = ( collect @ A @ Q ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collect_cong
% 0.15/0.43 thf(fact_48_ext,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.15/0.43 ( ! [X5: A] :
% 0.15/0.43 ( ( F @ X5 )
% 0.15/0.43 = ( G @ X5 ) )
% 0.15/0.43 => ( F = G ) ) ).
% 0.15/0.43
% 0.15/0.43 % ext
% 0.15/0.43 thf(fact_49_le__funD,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F: A > B,G: A > B,X: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funD
% 0.15/0.43 thf(fact_50_le__funE,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F: A > B,G: A > B,X: A] :
% 0.15/0.43 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funE
% 0.15/0.43 thf(fact_51_le__funI,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ! [F: A > B,G: A > B] :
% 0.15/0.43 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.15/0.43 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_funI
% 0.15/0.43 thf(fact_52_le__fun__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ( ( ord_less_eq @ ( A > B ) )
% 0.15/0.43 = ( ^ [F2: A > B,G2: A > B] :
% 0.15/0.43 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_fun_def
% 0.15/0.43 thf(fact_53_order__subst1,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( order @ B )
% 0.15/0.43 & ( order @ A ) )
% 0.15/0.43 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.15/0.43 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.15/0.43 => ( ! [X5: B,Y2: B] :
% 0.15/0.43 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_subst1
% 0.15/0.43 thf(fact_54_order__subst2,axiom,
% 0.15/0.43 ! [A: $tType,C: $tType] :
% 0.15/0.43 ( ( ( order @ C )
% 0.15/0.43 & ( order @ A ) )
% 0.15/0.43 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.15/0.43 => ( ! [X5: A,Y2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.15/0.43 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_subst2
% 0.15/0.43 thf(fact_55_dual__order_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [B2: A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.antisym
% 0.15/0.43 thf(fact_56_dual__order_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [A4: A,B3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.15/0.43 & ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.eq_iff
% 0.15/0.43 thf(fact_57_dual__order_Otrans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [B2: A,A2: A,C2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.trans
% 0.15/0.43 thf(fact_58_linorder__wlog,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.15/0.43 ( ! [A5: A,B4: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.15/0.43 => ( P @ A5 @ B4 ) )
% 0.15/0.43 => ( ! [A5: A,B4: A] :
% 0.15/0.43 ( ( P @ B4 @ A5 )
% 0.15/0.43 => ( P @ A5 @ B4 ) )
% 0.15/0.43 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % linorder_wlog
% 0.15/0.43 thf(fact_59_dual__order_Orefl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % dual_order.refl
% 0.15/0.43 thf(fact_60_order__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X: A,Y: A,Z2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.15/0.43 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_trans
% 0.15/0.43 thf(fact_61_order__class_Oorder_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_class.order.antisym
% 0.15/0.43 thf(fact_62_ord__le__eq__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( B2 = C2 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_le_eq_trans
% 0.15/0.43 thf(fact_63_ord__eq__le__trans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.43 ( ( A2 = B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_eq_le_trans
% 0.15/0.43 thf(fact_64_order__class_Oorder_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [A4: A,B3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.15/0.43 & ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order_class.order.eq_iff
% 0.15/0.43 thf(fact_65_antisym__conv,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [Y: A,X: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ Y @ X )
% 0.15/0.43 => ( ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 = ( X = Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % antisym_conv
% 0.15/0.43 thf(fact_66_le__cases3,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: A,Y: A,Z2: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.15/0.43 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.15/0.43 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.15/0.43 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_cases3
% 0.15/0.43 thf(fact_67_order_Otrans,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % order.trans
% 0.15/0.43 thf(fact_68_le__cases,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_cases
% 0.15/0.43 thf(fact_69_eq__refl,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( preorder @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ( X = Y )
% 0.15/0.43 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_refl
% 0.15/0.43 thf(fact_70_linear,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % linear
% 0.15/0.43 thf(fact_71_antisym,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X @ Y )
% 0.15/0.43 => ( ( ord_less_eq @ A @ Y @ X )
% 0.15/0.43 => ( X = Y ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % antisym
% 0.15/0.43 thf(fact_72_eq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [X4: A,Y4: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.15/0.43 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % eq_iff
% 0.15/0.43 thf(fact_73_ord__le__eq__subst,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ( F @ B2 )
% 0.15/0.43 = C2 )
% 0.15/0.43 => ( ! [X5: A,Y2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.15/0.43 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_le_eq_subst
% 0.15/0.43 thf(fact_74_ord__eq__le__subst,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( ( ord @ B )
% 0.15/0.43 & ( ord @ A ) )
% 0.15/0.43 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( F @ B2 ) )
% 0.15/0.43 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.15/0.43 => ( ! [X5: B,Y2: B] :
% 0.15/0.43 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % ord_eq_le_subst
% 0.15/0.43 thf(fact_75_heap__top__max,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [T2: tree @ A] :
% 0.15/0.43 ( ( T2
% 0.15/0.43 != ( e @ A ) )
% 0.15/0.43 => ( ( is_heap @ A @ T2 )
% 0.15/0.43 => ( ( val @ A @ T2 )
% 0.15/0.43 = ( lattic929149872er_Max @ A @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % heap_top_max
% 0.15/0.43 thf(fact_76_Heap_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.intro
% 0.15/0.43 thf(fact_77_Heap__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( heap @ B @ A )
% 0.15/0.43 = ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
% 0.15/0.43 & ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap_def
% 0.15/0.43 thf(fact_78_of__list__tree_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A,Tail: list @ A] :
% 0.15/0.43 ( ( heapIm1912108042t_tree @ A @ ( cons @ A @ V @ Tail ) )
% 0.15/0.43 = ( t @ A @ V @ ( heapIm1912108042t_tree @ A @ Tail ) @ ( e @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % of_list_tree.simps(2)
% 0.15/0.43 thf(fact_79_Greatest__equality,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [P: A > $o,X: A] :
% 0.15/0.43 ( ( P @ X )
% 0.15/0.43 => ( ! [Y2: A] :
% 0.15/0.43 ( ( P @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.15/0.43 => ( ( order_Greatest @ A @ P )
% 0.15/0.43 = X ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Greatest_equality
% 0.15/0.43 thf(fact_80_GreatestI2__order,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( order @ A )
% 0.15/0.43 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.15/0.43 ( ( P @ X )
% 0.15/0.43 => ( ! [Y2: A] :
% 0.15/0.43 ( ( P @ Y2 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y2 @ X ) )
% 0.15/0.43 => ( ! [X5: A] :
% 0.15/0.43 ( ( P @ X5 )
% 0.15/0.43 => ( ! [Y5: A] :
% 0.15/0.43 ( ( P @ Y5 )
% 0.15/0.43 => ( ord_less_eq @ A @ Y5 @ X5 ) )
% 0.15/0.43 => ( Q @ X5 ) ) )
% 0.15/0.43 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % GreatestI2_order
% 0.15/0.43 thf(fact_81_le__rel__bool__arg__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ord @ A )
% 0.15/0.43 => ( ( ord_less_eq @ ( $o > A ) )
% 0.15/0.43 = ( ^ [X6: $o > A,Y6: $o > A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 0.15/0.43 & ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_rel_bool_arg_iff
% 0.15/0.43 thf(fact_82_verit__la__disequality,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( A2 = B2 )
% 0.15/0.43 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % verit_la_disequality
% 0.15/0.43 thf(fact_83_Heap_Oaxioms_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.axioms(1)
% 0.15/0.43 thf(fact_84_list_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 0.15/0.43 ( ( ( cons @ A @ X21 @ X22 )
% 0.15/0.43 = ( cons @ A @ Y21 @ Y22 ) )
% 0.15/0.43 = ( ( X21 = Y21 )
% 0.15/0.43 & ( X22 = Y22 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % list.inject
% 0.15/0.43 thf(fact_85_Collection_Ois__empty__inj,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E: B] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( Is_empty @ E )
% 0.15/0.43 => ( E = Empty ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.is_empty_inj
% 0.15/0.43 thf(fact_86_Collection_Ois__empty__empty,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( Is_empty @ Empty ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.is_empty_empty
% 0.15/0.43 thf(fact_87_not__Cons__self2,axiom,
% 0.15/0.43 ! [A: $tType,X: A,Xs: list @ A] :
% 0.15/0.43 ( ( cons @ A @ X @ Xs )
% 0.15/0.43 != Xs ) ).
% 0.15/0.43
% 0.15/0.43 % not_Cons_self2
% 0.15/0.43 thf(fact_88_Heap__axioms__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( heap_axioms @ B @ A )
% 0.15/0.43 = ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ! [L3: B] :
% 0.15/0.43 ( ( Multiset2 @ L3 )
% 0.15/0.43 = ( multiset2 @ A @ ( As_tree2 @ L3 ) ) )
% 0.15/0.43 & ! [I2: list @ A] : ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I2 ) ) )
% 0.15/0.43 & ! [T3: B] :
% 0.15/0.43 ( ( ( As_tree2 @ T3 )
% 0.15/0.43 = ( e @ A ) )
% 0.15/0.43 = ( Is_empty2 @ T3 ) )
% 0.15/0.43 & ! [L3: B,M: A,L4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ L3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 0.15/0.43 = ( Remove_max2 @ L3 ) )
% 0.15/0.43 => ( ( add_mset @ A @ M @ ( Multiset2 @ L4 ) )
% 0.15/0.43 = ( Multiset2 @ L3 ) ) ) )
% 0.15/0.43 & ! [L3: B,M: A,L4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ L3 )
% 0.15/0.43 => ( ( is_heap @ A @ ( As_tree2 @ L3 ) )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 0.15/0.43 = ( Remove_max2 @ L3 ) )
% 0.15/0.43 => ( is_heap @ A @ ( As_tree2 @ L4 ) ) ) ) )
% 0.15/0.43 & ! [T3: B,M: A,T4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ T3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ T4 )
% 0.15/0.43 = ( Remove_max2 @ T3 ) )
% 0.15/0.43 => ( M
% 0.15/0.43 = ( val @ A @ ( As_tree2 @ T3 ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap_axioms_def
% 0.15/0.43 thf(fact_89_Heap__axioms_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B )] :
% 0.15/0.43 ( ! [L5: B] :
% 0.15/0.43 ( ( Multiset @ L5 )
% 0.15/0.43 = ( multiset2 @ A @ ( As_tree @ L5 ) ) )
% 0.15/0.43 => ( ! [I3: list @ A] : ( is_heap @ A @ ( As_tree @ ( Of_list @ I3 ) ) )
% 0.15/0.43 => ( ! [T5: B] :
% 0.15/0.43 ( ( ( As_tree @ T5 )
% 0.15/0.43 = ( e @ A ) )
% 0.15/0.43 = ( Is_empty @ T5 ) )
% 0.15/0.43 => ( ! [L5: B,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( ( add_mset @ A @ M2 @ ( Multiset @ L2 ) )
% 0.15/0.43 = ( Multiset @ L5 ) ) ) )
% 0.15/0.43 => ( ! [L5: B,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( is_heap @ A @ ( As_tree @ L5 ) )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( is_heap @ A @ ( As_tree @ L2 ) ) ) ) )
% 0.15/0.43 => ( ! [T5: B,M2: A,T6: B] :
% 0.15/0.43 ( ~ ( Is_empty @ T5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ T6 )
% 0.15/0.43 = ( Remove_max @ T5 ) )
% 0.15/0.43 => ( M2
% 0.15/0.43 = ( val @ A @ ( As_tree @ T5 ) ) ) ) )
% 0.15/0.43 => ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap_axioms.intro
% 0.15/0.43 thf(fact_90_Heap_Oremove__max__multiset_H,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M3: A,L6: B] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( ( add_mset @ A @ M3 @ ( Multiset @ L6 ) )
% 0.15/0.43 = ( Multiset @ L ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.remove_max_multiset'
% 0.15/0.43 thf(fact_91_Heap_Oremove__max__val,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T2: B,M3: A,T7: B] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( ~ ( Is_empty @ T2 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ T7 )
% 0.15/0.43 = ( Remove_max @ T2 ) )
% 0.15/0.43 => ( M3
% 0.15/0.43 = ( val @ A @ ( As_tree @ T2 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.remove_max_val
% 0.15/0.43 thf(fact_92_Heap_Oremove__max__is__heap,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M3: A,L6: B] :
% 0.15/0.43 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 0.15/0.43 => ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( is_heap @ A @ ( As_tree @ L ) )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( is_heap @ A @ ( As_tree @ L6 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Heap.remove_max_is_heap
% 0.15/0.43 thf(fact_93_multi__self__add__other__not__self,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,X: A] :
% 0.15/0.43 ( M4
% 0.15/0.43 != ( add_mset @ A @ X @ M4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % multi_self_add_other_not_self
% 0.15/0.43 thf(fact_94_add__mset__add__mset__same__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( ( add_mset @ A @ A2 @ A3 )
% 0.15/0.43 = ( add_mset @ A @ A2 @ B5 ) )
% 0.15/0.43 = ( A3 = B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mset_add_mset_same_iff
% 0.15/0.43 thf(fact_95_old_Oprod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.15/0.43 = ( ( A2 = A6 )
% 0.15/0.43 & ( B2 = B6 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inject
% 0.15/0.43 thf(fact_96_prod_Oinject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y24: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ X12 @ X24 )
% 0.15/0.43 = ( product_Pair @ A @ B @ Y1 @ Y24 ) )
% 0.15/0.43 = ( ( X12 = Y1 )
% 0.15/0.43 & ( X24 = Y24 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod.inject
% 0.15/0.43 thf(fact_97_mset__add,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( set_mset @ A @ A3 ) )
% 0.15/0.43 => ~ ! [B7: multiset @ A] :
% 0.15/0.43 ( A3
% 0.15/0.43 != ( add_mset @ A @ A2 @ B7 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_add
% 0.15/0.43 thf(fact_98_multi__member__split,axiom,
% 0.15/0.43 ! [A: $tType,X: A,M4: multiset @ A] :
% 0.15/0.43 ( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
% 0.15/0.43 => ? [A7: multiset @ A] :
% 0.15/0.43 ( M4
% 0.15/0.43 = ( add_mset @ A @ X @ A7 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multi_member_split
% 0.15/0.43 thf(fact_99_surj__pair,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.15/0.43 ? [X5: A,Y2: B] :
% 0.15/0.43 ( P2
% 0.15/0.43 = ( product_Pair @ A @ B @ X5 @ Y2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % surj_pair
% 0.15/0.43 thf(fact_100_prod__cases,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.15/0.43 => ( P @ P2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases
% 0.15/0.43 thf(fact_101_Pair__inject,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.15/0.43 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.15/0.43 => ~ ( ( A2 = A6 )
% 0.15/0.43 => ( B2 != B6 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Pair_inject
% 0.15/0.43 thf(fact_102_prod__cases3,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ~ ! [A5: A,B4: B,C3: C] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases3
% 0.15/0.43 thf(fact_103_prod__cases4,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ~ ! [A5: A,B4: B,C3: C,D2: D] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases4
% 0.15/0.43 thf(fact_104_prod__cases5,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 0.15/0.43 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases5
% 0.15/0.43 thf(fact_105_prod__cases6,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 0.15/0.43 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D2 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases6
% 0.15/0.43 thf(fact_106_prod__cases7,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.15/0.43 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3,G4: G3] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_cases7
% 0.15/0.43 thf(fact_107_prod__induct3,axiom,
% 0.15/0.43 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.15/0.43 ( ! [A5: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
% 0.15/0.43 => ( P @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct3
% 0.15/0.43 thf(fact_108_prod__induct4,axiom,
% 0.15/0.43 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.15/0.43 ( ! [A5: A,B4: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.15/0.43 => ( P @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct4
% 0.15/0.43 thf(fact_109_prod__induct5,axiom,
% 0.15/0.43 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 0.15/0.43 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) )
% 0.15/0.43 => ( P @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct5
% 0.15/0.43 thf(fact_110_prod__induct6,axiom,
% 0.15/0.43 ! [F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 0.15/0.43 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D2 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) )
% 0.15/0.43 => ( P @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct6
% 0.15/0.43 thf(fact_111_prod__induct7,axiom,
% 0.15/0.43 ! [G3: $tType,F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.15/0.43 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.15/0.43 => ( P @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % prod_induct7
% 0.15/0.43 thf(fact_112_old_Oprod_Oexhaust,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.15/0.43 ~ ! [A5: A,B4: B] :
% 0.15/0.43 ( Y
% 0.15/0.43 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.exhaust
% 0.15/0.43 thf(fact_113_old_Oprod_Oinducts,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.15/0.43 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 0.15/0.43 => ( P @ Prod ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.inducts
% 0.15/0.43 thf(fact_114_add__eq__conv__ex,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,M4: multiset @ A,B2: A,N: multiset @ A] :
% 0.15/0.43 ( ( ( add_mset @ A @ A2 @ M4 )
% 0.15/0.43 = ( add_mset @ A @ B2 @ N ) )
% 0.15/0.43 = ( ( ( M4 = N )
% 0.15/0.43 & ( A2 = B2 ) )
% 0.15/0.43 | ? [K: multiset @ A] :
% 0.15/0.43 ( ( M4
% 0.15/0.43 = ( add_mset @ A @ B2 @ K ) )
% 0.15/0.43 & ( N
% 0.15/0.43 = ( add_mset @ A @ A2 @ K ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_eq_conv_ex
% 0.15/0.43 thf(fact_115_add__mset__commute,axiom,
% 0.15/0.43 ! [A: $tType,X: A,Y: A,M4: multiset @ A] :
% 0.15/0.43 ( ( add_mset @ A @ X @ ( add_mset @ A @ Y @ M4 ) )
% 0.15/0.43 = ( add_mset @ A @ Y @ ( add_mset @ A @ X @ M4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mset_commute
% 0.15/0.43 thf(fact_116_union__single__eq__member,axiom,
% 0.15/0.43 ! [A: $tType,X: A,M4: multiset @ A,N: multiset @ A] :
% 0.15/0.43 ( ( ( add_mset @ A @ X @ M4 )
% 0.15/0.43 = N )
% 0.15/0.43 => ( member @ A @ X @ ( set_mset @ A @ N ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_single_eq_member
% 0.15/0.43 thf(fact_117_insert__noteq__member,axiom,
% 0.15/0.43 ! [A: $tType,B2: A,B5: multiset @ A,C2: A,C4: multiset @ A] :
% 0.15/0.43 ( ( ( add_mset @ A @ B2 @ B5 )
% 0.15/0.43 = ( add_mset @ A @ C2 @ C4 ) )
% 0.15/0.43 => ( ( B2 != C2 )
% 0.15/0.43 => ( member @ A @ C2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % insert_noteq_member
% 0.15/0.43 thf(fact_118_old_Oprod_Orec,axiom,
% 0.15/0.43 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.15/0.43 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.15/0.43 = ( F1 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % old.prod.rec
% 0.15/0.43 thf(fact_119_internal__case__prod__conv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.15/0.43 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.15/0.43 = ( C2 @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % internal_case_prod_conv
% 0.15/0.43 thf(fact_120_Collection_Oset__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),L: B] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( set2 @ B @ A @ Multiset @ L )
% 0.15/0.43 = ( set_mset @ A @ ( Multiset @ L ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.set_def
% 0.15/0.43 thf(fact_121_relChain__def,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( ord @ B )
% 0.15/0.43 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.15/0.43 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.15/0.43 ! [I2: A,J: A] :
% 0.15/0.43 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 0.15/0.43 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % relChain_def
% 0.15/0.43 thf(fact_122_removeLeaf_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [V: A] :
% 0.15/0.43 ( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
% 0.15/0.43 = ( product_Pair @ A @ ( tree @ A ) @ V @ ( e @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % removeLeaf.simps(1)
% 0.15/0.43 thf(fact_123_Collection_Oset_Ocong,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( set2 @ B @ A )
% 0.15/0.43 = ( set2 @ B @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.set.cong
% 0.15/0.43 thf(fact_124_RemoveMax__axioms__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( removeMax_axioms @ B @ A )
% 0.15/0.43 = ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv: B > $o] :
% 0.15/0.43 ( ! [X4: list @ A] : ( Inv @ ( Of_list2 @ X4 ) )
% 0.15/0.43 & ! [L3: B,M: A,L4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ L3 )
% 0.15/0.43 => ( ( Inv @ L3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 0.15/0.43 = ( Remove_max2 @ L3 ) )
% 0.15/0.43 => ( M
% 0.15/0.43 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset2 @ L3 ) ) ) ) ) )
% 0.15/0.43 & ! [L3: B,M: A,L4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ L3 )
% 0.15/0.43 => ( ( Inv @ L3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 0.15/0.43 = ( Remove_max2 @ L3 ) )
% 0.15/0.43 => ( ( add_mset @ A @ M @ ( Multiset2 @ L4 ) )
% 0.15/0.43 = ( Multiset2 @ L3 ) ) ) ) )
% 0.15/0.43 & ! [L3: B,M: A,L4: B] :
% 0.15/0.43 ( ~ ( Is_empty2 @ L3 )
% 0.15/0.43 => ( ( Inv @ L3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 0.15/0.43 = ( Remove_max2 @ L3 ) )
% 0.15/0.43 => ( Inv @ L4 ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax_axioms_def
% 0.15/0.43 thf(fact_125_RemoveMax__axioms_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Inv2: B > $o,Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B ),Multiset: B > ( multiset @ A )] :
% 0.15/0.43 ( ! [X5: list @ A] : ( Inv2 @ ( Of_list @ X5 ) )
% 0.15/0.43 => ( ! [L5: B,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( Inv2 @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( M2
% 0.15/0.43 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L5 ) ) ) ) ) )
% 0.15/0.43 => ( ! [L5: B,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( Inv2 @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( ( add_mset @ A @ M2 @ ( Multiset @ L2 ) )
% 0.15/0.43 = ( Multiset @ L5 ) ) ) ) )
% 0.15/0.43 => ( ! [L5: B,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( Inv2 @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( Inv2 @ L2 ) ) ) )
% 0.15/0.43 => ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax_axioms.intro
% 0.15/0.43 thf(fact_126_RemoveMax_Oremove__max__max,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( Inv2 @ L )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( M3
% 0.15/0.43 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.remove_max_max
% 0.15/0.43 thf(fact_127_multiset__induct__max,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 0.15/0.43 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ! [X5: A,M5: multiset @ A] :
% 0.15/0.43 ( ( P @ M5 )
% 0.15/0.43 => ( ! [Xa: A] :
% 0.15/0.43 ( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
% 0.15/0.43 => ( ord_less_eq @ A @ Xa @ X5 ) )
% 0.15/0.43 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
% 0.15/0.43 => ( P @ M4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_induct_max
% 0.15/0.43 thf(fact_128_add__mset__eq__singleton__iff,axiom,
% 0.15/0.43 ! [A: $tType,X: A,M4: multiset @ A,Y: A] :
% 0.15/0.43 ( ( ( add_mset @ A @ X @ M4 )
% 0.15/0.43 = ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.15/0.43 = ( ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( X = Y ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mset_eq_singleton_iff
% 0.15/0.43 thf(fact_129_single__eq__add__mset,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A,M4: multiset @ A] :
% 0.15/0.43 ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( add_mset @ A @ B2 @ M4 ) )
% 0.15/0.43 = ( ( B2 = A2 )
% 0.15/0.43 & ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % single_eq_add_mset
% 0.15/0.43 thf(fact_130_add__mset__eq__single,axiom,
% 0.15/0.43 ! [A: $tType,B2: A,M4: multiset @ A,A2: A] :
% 0.15/0.43 ( ( ( add_mset @ A @ B2 @ M4 )
% 0.15/0.43 = ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.15/0.43 = ( ( B2 = A2 )
% 0.15/0.43 & ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mset_eq_single
% 0.15/0.43 thf(fact_131_single__eq__single,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,B2: A] :
% 0.15/0.43 ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.15/0.43 = ( A2 = B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % single_eq_single
% 0.15/0.43 thf(fact_132_RemoveMax__def,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( removeMax @ B @ A )
% 0.15/0.43 = ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv: B > $o] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
% 0.15/0.43 & ( removeMax_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax_def
% 0.15/0.43 thf(fact_133_RemoveMax_Ointro,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.intro
% 0.15/0.43 thf(fact_134_multiset__cases,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A] :
% 0.15/0.43 ( ( M4
% 0.15/0.43 != ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ~ ! [X5: A,N2: multiset @ A] :
% 0.15/0.43 ( M4
% 0.15/0.43 != ( add_mset @ A @ X5 @ N2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_cases
% 0.15/0.43 thf(fact_135_multiset__induct,axiom,
% 0.15/0.43 ! [A: $tType,P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 0.15/0.43 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ! [X5: A,M5: multiset @ A] :
% 0.15/0.43 ( ( P @ M5 )
% 0.15/0.43 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) )
% 0.15/0.43 => ( P @ M4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_induct
% 0.15/0.43 thf(fact_136_multiset__induct2,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType,P: ( multiset @ A ) > ( multiset @ B ) > $o,M4: multiset @ A,N: multiset @ B] :
% 0.15/0.43 ( ( P @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
% 0.15/0.43 => ( ! [A5: A,M5: multiset @ A,N2: multiset @ B] :
% 0.15/0.43 ( ( P @ M5 @ N2 )
% 0.15/0.43 => ( P @ ( add_mset @ A @ A5 @ M5 ) @ N2 ) )
% 0.15/0.43 => ( ! [A5: B,M5: multiset @ A,N2: multiset @ B] :
% 0.15/0.43 ( ( P @ M5 @ N2 )
% 0.15/0.43 => ( P @ M5 @ ( add_mset @ B @ A5 @ N2 ) ) )
% 0.15/0.43 => ( P @ M4 @ N ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_induct2
% 0.15/0.43 thf(fact_137_empty__not__add__mset,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A] :
% 0.15/0.43 ( ( zero_zero @ ( multiset @ A ) )
% 0.15/0.43 != ( add_mset @ A @ A2 @ A3 ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_not_add_mset
% 0.15/0.43 thf(fact_138_multi__nonempty__split,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A] :
% 0.15/0.43 ( ( M4
% 0.15/0.43 != ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ? [A7: multiset @ A,A5: A] :
% 0.15/0.43 ( M4
% 0.15/0.43 = ( add_mset @ A @ A5 @ A7 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multi_nonempty_split
% 0.15/0.43 thf(fact_139_multiset__nonemptyE,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A] :
% 0.15/0.43 ( ( A3
% 0.15/0.43 != ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ~ ! [X5: A] :
% 0.15/0.43 ~ ( member @ A @ X5 @ ( set_mset @ A @ A3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_nonemptyE
% 0.15/0.43 thf(fact_140_RemoveMax_Oof__list__inv,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,X: list @ A] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( Inv2 @ ( Of_list @ X ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.of_list_inv
% 0.15/0.43 thf(fact_141_RemoveMax_Oaxioms_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.axioms(2)
% 0.15/0.43 thf(fact_142_RemoveMax_Oremove__max__inv,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( Inv2 @ L )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( Inv2 @ L6 ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.remove_max_inv
% 0.15/0.43 thf(fact_143_Collection_Ois__empty__as__list,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E: B] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( Is_empty @ E )
% 0.15/0.43 => ( ( Multiset @ E )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.is_empty_as_list
% 0.15/0.43 thf(fact_144_Collection_Omultiset__empty,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
% 0.15/0.43 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 0.15/0.43 => ( ( Multiset @ Empty )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Collection.multiset_empty
% 0.15/0.43 thf(fact_145_RemoveMax_Oaxioms_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.axioms(1)
% 0.15/0.43 thf(fact_146_multi__member__last,axiom,
% 0.15/0.43 ! [A: $tType,X: A] : ( member @ A @ X @ ( set_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multi_member_last
% 0.15/0.43 thf(fact_147_RemoveMax_Oremove__max__multiset,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( Inv2 @ L )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( ( add_mset @ A @ M3 @ ( Multiset @ L6 ) )
% 0.15/0.43 = ( Multiset @ L ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.remove_max_multiset
% 0.15/0.43 thf(fact_148_multiset_Osimps_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( multiset2 @ A @ ( e @ A ) )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset.simps(1)
% 0.15/0.43 thf(fact_149_multiset__induct__min,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 0.15/0.43 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ! [X5: A,M5: multiset @ A] :
% 0.15/0.43 ( ( P @ M5 )
% 0.15/0.43 => ( ! [Xa: A] :
% 0.15/0.43 ( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
% 0.15/0.43 => ( ord_less_eq @ A @ X5 @ Xa ) )
% 0.15/0.43 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
% 0.15/0.43 => ( P @ M4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset_induct_min
% 0.15/0.43 thf(fact_150_le__zero__eq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( canoni770627133id_add @ A )
% 0.15/0.43 => ! [N3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ N3 @ ( zero_zero @ A ) )
% 0.15/0.43 = ( N3
% 0.15/0.43 = ( zero_zero @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_zero_eq
% 0.15/0.43 thf(fact_151_Multiset_Ois__empty__def,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( is_empty @ A )
% 0.15/0.43 = ( ^ [A8: multiset @ A] :
% 0.15/0.43 ( A8
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Multiset.is_empty_def
% 0.15/0.43 thf(fact_152_le__numeral__extra_I3_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linord1659791738miring @ A )
% 0.15/0.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_numeral_extra(3)
% 0.15/0.43 thf(fact_153_zero__le,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( canoni770627133id_add @ A )
% 0.15/0.43 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.15/0.43
% 0.15/0.43 % zero_le
% 0.15/0.43 thf(fact_154_RemoveMax_Ossort_HInduct,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,P: B > ( list @ A ) > $o,Sl: list @ A] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ( Inv2 @ L )
% 0.15/0.43 => ( ( P @ L @ Sl )
% 0.15/0.43 => ( ! [L5: B,Sl2: list @ A,M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( Inv2 @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( ( P @ L5 @ Sl2 )
% 0.15/0.43 => ( P @ L2 @ ( cons @ A @ M2 @ Sl2 ) ) ) ) ) )
% 0.15/0.43 => ( P @ Empty @ ( ssort @ B @ A @ Is_empty @ Remove_max @ L @ Sl ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'Induct
% 0.15/0.43 thf(fact_155_RemoveMax_Ossort_H__dom_Ocases,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,A2: product_prod @ B @ ( list @ A )] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ A2 )
% 0.15/0.43 => ~ ! [L5: B,Sl2: list @ A] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( product_Pair @ B @ ( list @ A ) @ L5 @ Sl2 ) )
% 0.15/0.43 => ~ ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ! [M6: A,L7: B] :
% 0.15/0.43 ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'_dom.cases
% 0.15/0.43 thf(fact_156_RemoveMax_Ossort_H__dom_Osimps,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,A2: product_prod @ B @ ( list @ A )] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ A2 )
% 0.15/0.43 = ( ? [L3: B,Sl3: list @ A] :
% 0.15/0.43 ( ( A2
% 0.15/0.43 = ( product_Pair @ B @ ( list @ A ) @ L3 @ Sl3 ) )
% 0.15/0.43 & ! [X4: A,Y4: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L3 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ X4 @ Y4 )
% 0.15/0.43 = ( Remove_max @ L3 ) )
% 0.15/0.43 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ Y4 @ ( cons @ A @ X4 @ Sl3 ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'_dom.simps
% 0.15/0.43 thf(fact_157_RemoveMax_Ossort_H__dom_Ocong,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( ssort_dom @ B @ A )
% 0.15/0.43 = ( ssort_dom @ B @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'_dom.cong
% 0.15/0.43 thf(fact_158_RemoveMax_Ossort_H_Ocong,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ( ssort @ B @ A )
% 0.15/0.43 = ( ssort @ B @ A ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'.cong
% 0.15/0.43 thf(fact_159_RemoveMax_Ossort_H__dom_Oinducts,axiom,
% 0.15/0.43 ! [A: $tType,B: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,X: product_prod @ B @ ( list @ A ),P: ( product_prod @ B @ ( list @ A ) ) > $o] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ X )
% 0.15/0.43 => ( ! [L5: B,Sl2: list @ A] :
% 0.15/0.43 ( ! [M6: A,L7: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) )
% 0.15/0.43 => ( ! [M6: A,L7: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L5 )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 0.15/0.43 = ( Remove_max @ L5 ) )
% 0.15/0.43 => ( P @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) )
% 0.15/0.43 => ( P @ ( product_Pair @ B @ ( list @ A ) @ L5 @ Sl2 ) ) ) )
% 0.15/0.43 => ( P @ X ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'_dom.inducts
% 0.15/0.43 thf(fact_160_RemoveMax_Ossort_H__dom_Ointros,axiom,
% 0.15/0.43 ! [B: $tType,A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,Sl: list @ A] :
% 0.15/0.43 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 0.15/0.43 => ( ! [M2: A,L2: B] :
% 0.15/0.43 ( ~ ( Is_empty @ L )
% 0.15/0.43 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 0.15/0.43 = ( Remove_max @ L ) )
% 0.15/0.43 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L2 @ ( cons @ A @ M2 @ Sl ) ) ) ) )
% 0.15/0.43 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L @ Sl ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % RemoveMax.ssort'_dom.intros
% 0.15/0.43 thf(fact_161_multiset_Osimps_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 0.15/0.43 ( ( multiset2 @ A @ ( t @ A @ V @ L @ R ) )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( multiset2 @ A @ L ) @ ( add_mset @ A @ V @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( multiset2 @ A @ R ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % multiset.simps(2)
% 0.15/0.43 thf(fact_162_divides__aux__eq,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( unique1598680935umeral @ A )
% 0.15/0.43 => ! [Q2: A,R: A] :
% 0.15/0.43 ( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R ) )
% 0.15/0.43 = ( R
% 0.15/0.43 = ( zero_zero @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % divides_aux_eq
% 0.15/0.43 thf(fact_163_single__subset__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,M4: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ M4 )
% 0.15/0.43 = ( member @ A @ A2 @ ( set_mset @ A @ M4 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % single_subset_iff
% 0.15/0.43 thf(fact_164_subset__mset_Oorder__refl,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A] : ( subseteq_mset @ A @ X @ X ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.order_refl
% 0.15/0.43 thf(fact_165_subset__mset_Odual__order_Orefl,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A] : ( subseteq_mset @ A @ A2 @ A2 ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.dual_order.refl
% 0.15/0.43 thf(fact_166_add__le__cancel__left,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere236663937imp_le @ A )
% 0.15/0.43 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.15/0.43 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_cancel_left
% 0.15/0.43 thf(fact_167_add__le__cancel__right,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere236663937imp_le @ A )
% 0.15/0.43 => ! [A2: A,C2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.15/0.43 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_cancel_right
% 0.15/0.43 thf(fact_168_subset__mset_Obot_Oextremum__unique,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( A2
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.bot.extremum_unique
% 0.15/0.43 thf(fact_169_subset__mset_Ole__zero__eq,axiom,
% 0.15/0.43 ! [A: $tType,N3: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ N3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( N3
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.le_zero_eq
% 0.15/0.43 thf(fact_170_subset__mset_Oadd__le__cancel__left,axiom,
% 0.15/0.43 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_cancel_left
% 0.15/0.43 thf(fact_171_subset__mset_Oadd__le__cancel__right,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_cancel_right
% 0.15/0.43 thf(fact_172_mset__subset__eq__mono__add__left__cancel,axiom,
% 0.15/0.43 ! [A: $tType,C4: multiset @ A,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C4 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B5 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_mono_add_left_cancel
% 0.15/0.43 thf(fact_173_mset__subset__eq__mono__add__right__cancel,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,C4: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C4 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_mono_add_right_cancel
% 0.15/0.43 thf(fact_174_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( ( zero_zero @ ( multiset @ A ) )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.zero_eq_add_iff_both_eq_0
% 0.15/0.43 thf(fact_175_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_eq_0_iff_both_eq_0
% 0.15/0.43 thf(fact_176_union__eq__empty,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,N: multiset @ A] :
% 0.15/0.43 ( ( ( plus_plus @ ( multiset @ A ) @ M4 @ N )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( N
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_eq_empty
% 0.15/0.43 thf(fact_177_empty__eq__union,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,N: multiset @ A] :
% 0.15/0.43 ( ( ( zero_zero @ ( multiset @ A ) )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ M4 @ N ) )
% 0.15/0.43 = ( ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( N
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % empty_eq_union
% 0.15/0.43 thf(fact_178_union__mset__add__mset__right,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,A2: A,B5: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ A2 @ B5 ) )
% 0.15/0.43 = ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_mset_add_mset_right
% 0.15/0.43 thf(fact_179_union__mset__add__mset__left,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A3 ) @ B5 )
% 0.15/0.43 = ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_mset_add_mset_left
% 0.15/0.43 thf(fact_180_add__le__same__cancel1,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere516151231imp_le @ A )
% 0.15/0.43 => ! [B2: A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 0.15/0.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_same_cancel1
% 0.15/0.43 thf(fact_181_add__le__same__cancel2,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere516151231imp_le @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 0.15/0.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_same_cancel2
% 0.15/0.43 thf(fact_182_le__add__same__cancel1,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere516151231imp_le @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 0.15/0.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_add_same_cancel1
% 0.15/0.43 thf(fact_183_le__add__same__cancel2,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere516151231imp_le @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 0.15/0.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_add_same_cancel2
% 0.15/0.43 thf(fact_184_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linord219039673up_add @ A )
% 0.15/0.43 => ! [A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.15/0.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % double_add_le_zero_iff_single_add_le_zero
% 0.15/0.43 thf(fact_185_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linord219039673up_add @ A )
% 0.15/0.43 => ! [A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.15/0.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % zero_le_double_add_iff_zero_le_single_add
% 0.15/0.43 thf(fact_186_subset__mset_Oadd__le__same__cancel1,axiom,
% 0.15/0.43 ! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) @ B2 )
% 0.15/0.43 = ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_same_cancel1
% 0.15/0.43 thf(fact_187_subset__mset_Oadd__le__same__cancel2,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ B2 )
% 0.15/0.43 = ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_same_cancel2
% 0.15/0.43 thf(fact_188_subset__mset_Ole__add__same__cancel1,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.le_add_same_cancel1
% 0.15/0.43 thf(fact_189_subset__mset_Ole__add__same__cancel2,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.le_add_same_cancel2
% 0.15/0.43 thf(fact_190_add__mset__subseteq__single__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,M4: multiset @ A,B2: A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ M4 ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.15/0.43 = ( ( M4
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( A2 = B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mset_subseteq_single_iff
% 0.15/0.43 thf(fact_191_union__assoc,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M4 @ N ) @ K2 )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_assoc
% 0.15/0.43 thf(fact_192_union__lcomm,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M4 @ K2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_lcomm
% 0.15/0.43 thf(fact_193_union__commute,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) )
% 0.15/0.43 = ( ^ [M7: multiset @ A,N4: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N4 @ M7 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_commute
% 0.15/0.43 thf(fact_194_union__left__cancel,axiom,
% 0.15/0.43 ! [A: $tType,K2: multiset @ A,M4: multiset @ A,N: multiset @ A] :
% 0.15/0.43 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M4 )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ K2 @ N ) )
% 0.15/0.43 = ( M4 = N ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_left_cancel
% 0.15/0.43 thf(fact_195_subset__mset_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [X4: multiset @ A,Y4: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ X4 @ Y4 )
% 0.15/0.43 & ( subseteq_mset @ A @ Y4 @ X4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.eq_iff
% 0.15/0.43 thf(fact_196_union__right__cancel,axiom,
% 0.15/0.43 ! [A: $tType,M4: multiset @ A,K2: multiset @ A,N: multiset @ A] :
% 0.15/0.43 ( ( ( plus_plus @ ( multiset @ A ) @ M4 @ K2 )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 0.15/0.43 = ( M4 = N ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_right_cancel
% 0.15/0.43 thf(fact_197_subset__mset_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ X @ Y )
% 0.15/0.43 => ( ( subseteq_mset @ A @ Y @ X )
% 0.15/0.43 => ( X = Y ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.antisym
% 0.15/0.43 thf(fact_198_subset__mset_Oeq__refl,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( X = Y )
% 0.15/0.43 => ( subseteq_mset @ A @ X @ Y ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.eq_refl
% 0.15/0.43 thf(fact_199_subset__mset_Oadd__mono,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_mono
% 0.15/0.43 thf(fact_200_subset__mset_Oless__eqE,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ~ ! [C3: multiset @ A] :
% 0.15/0.43 ( B2
% 0.15/0.43 != ( plus_plus @ ( multiset @ A ) @ A2 @ C3 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.less_eqE
% 0.15/0.43 thf(fact_201_subset__mset_Ole__iff__add,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( subseteq_mset @ A )
% 0.15/0.43 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 0.15/0.43 ? [C5: multiset @ A] :
% 0.15/0.43 ( B3
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.le_iff_add
% 0.15/0.43 thf(fact_202_mset__subset__eq__add__left,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_add_left
% 0.15/0.43 thf(fact_203_mset__subset__eq__mono__add,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,B5: multiset @ A,C4: multiset @ A,D4: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A3 @ B5 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ C4 @ D4 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ D4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_mono_add
% 0.15/0.43 thf(fact_204_subset__mset_Oorder_Otrans,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 0.15/0.43 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.order.trans
% 0.15/0.43 thf(fact_205_subset__mset_Oorder__trans,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ X @ Y )
% 0.15/0.43 => ( ( subseteq_mset @ A @ Y @ Z2 )
% 0.15/0.43 => ( subseteq_mset @ A @ X @ Z2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.order_trans
% 0.15/0.43 thf(fact_206_mset__subset__eq__add__right,axiom,
% 0.15/0.43 ! [A: $tType,B5: multiset @ A,A3: multiset @ A] : ( subseteq_mset @ A @ B5 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_add_right
% 0.15/0.43 thf(fact_207_subset__mset_Oantisym__conv,axiom,
% 0.15/0.43 ! [A: $tType,Y: multiset @ A,X: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ Y @ X )
% 0.15/0.43 => ( ( subseteq_mset @ A @ X @ Y )
% 0.15/0.43 = ( X = Y ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.antisym_conv
% 0.15/0.43 thf(fact_208_subset__mset_Oorder_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A4 @ B3 )
% 0.15/0.43 & ( subseteq_mset @ A @ B3 @ A4 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.order.eq_iff
% 0.15/0.43 thf(fact_209_multi__union__self__other__eq,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,X7: multiset @ A,Y7: multiset @ A] :
% 0.15/0.43 ( ( ( plus_plus @ ( multiset @ A ) @ A3 @ X7 )
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ A3 @ Y7 ) )
% 0.15/0.43 => ( X7 = Y7 ) ) ).
% 0.15/0.43
% 0.15/0.43 % multi_union_self_other_eq
% 0.15/0.43 thf(fact_210_subset__mset_Oadd__left__mono,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_left_mono
% 0.15/0.43 thf(fact_211_subset__mset_Oorder_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ A2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.order.antisym
% 0.15/0.43 thf(fact_212_mset__subset__eq__exists__conv,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( subseteq_mset @ A )
% 0.15/0.43 = ( ^ [A8: multiset @ A,B8: multiset @ A] :
% 0.15/0.43 ? [C6: multiset @ A] :
% 0.15/0.43 ( B8
% 0.15/0.43 = ( plus_plus @ ( multiset @ A ) @ A8 @ C6 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_exists_conv
% 0.15/0.43 thf(fact_213_subset__mset_Oadd__right__mono,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_right_mono
% 0.15/0.43 thf(fact_214_subset__mset_Oord__eq__le__trans,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( A2 = B2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 0.15/0.43 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.ord_eq_le_trans
% 0.15/0.43 thf(fact_215_subset__mset_Oord__le__eq__trans,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( B2 = C2 )
% 0.15/0.43 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.ord_le_eq_trans
% 0.15/0.43 thf(fact_216_subset__mset_Odual__order_Otrans,axiom,
% 0.15/0.43 ! [A: $tType,B2: multiset @ A,A2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ C2 @ A2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.dual_order.trans
% 0.15/0.43 thf(fact_217_subset__mset_Odual__order_Oeq__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 0.15/0.43 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ B3 @ A4 )
% 0.15/0.43 & ( subseteq_mset @ A @ A4 @ B3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.dual_order.eq_iff
% 0.15/0.43 thf(fact_218_subset__mset_Oadd__le__imp__le__left,axiom,
% 0.15/0.43 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.15/0.43 => ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_imp_le_left
% 0.15/0.43 thf(fact_219_subset__mset_Odual__order_Oantisym,axiom,
% 0.15/0.43 ! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ B2 @ A2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( A2 = B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.dual_order.antisym
% 0.15/0.43 thf(fact_220_subset__mset_Oadd__le__imp__le__right,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.15/0.43 => ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_le_imp_le_right
% 0.15/0.43 thf(fact_221_union__iff,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) )
% 0.15/0.43 = ( ( member @ A @ A2 @ ( set_mset @ A @ A3 ) )
% 0.15/0.43 | ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % union_iff
% 0.15/0.43 thf(fact_222_mset__subset__eqD,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,B5: multiset @ A,X: A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A3 @ B5 )
% 0.15/0.43 => ( ( member @ A @ X @ ( set_mset @ A @ A3 ) )
% 0.15/0.43 => ( member @ A @ X @ ( set_mset @ A @ B5 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eqD
% 0.15/0.43 thf(fact_223_set__mset__mono,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A3 @ B5 )
% 0.15/0.43 => ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ A3 ) @ ( set_mset @ A @ B5 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % set_mset_mono
% 0.15/0.43 thf(fact_224_mset__subset__eq__add__mset__cancel,axiom,
% 0.15/0.43 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ A3 ) @ ( add_mset @ A @ A2 @ B5 ) )
% 0.15/0.43 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 0.15/0.43
% 0.15/0.43 % mset_subset_eq_add_mset_cancel
% 0.15/0.43 thf(fact_225_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [I: A,J2: A,K3: A,L: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ I @ J2 )
% 0.15/0.43 & ( K3 = L ) )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mono_thms_linordered_semiring(3)
% 0.15/0.43 thf(fact_226_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [I: A,J2: A,K3: A,L: A] :
% 0.15/0.43 ( ( ( I = J2 )
% 0.15/0.43 & ( ord_less_eq @ A @ K3 @ L ) )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mono_thms_linordered_semiring(2)
% 0.15/0.43 thf(fact_227_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [I: A,J2: A,K3: A,L: A] :
% 0.15/0.43 ( ( ( ord_less_eq @ A @ I @ J2 )
% 0.15/0.43 & ( ord_less_eq @ A @ K3 @ L ) )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mono_thms_linordered_semiring(1)
% 0.15/0.43 thf(fact_228_add__mono,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A,D3: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_mono
% 0.15/0.43 thf(fact_229_add__left__mono,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_left_mono
% 0.15/0.43 thf(fact_230_less__eqE,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( canoni770627133id_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ~ ! [C3: A] :
% 0.15/0.43 ( B2
% 0.15/0.43 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % less_eqE
% 0.15/0.43 thf(fact_231_add__right__mono,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere779506340up_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A,C2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_right_mono
% 0.15/0.43 thf(fact_232_le__iff__add,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( canoni770627133id_add @ A )
% 0.15/0.43 => ( ( ord_less_eq @ A )
% 0.15/0.43 = ( ^ [A4: A,B3: A] :
% 0.15/0.43 ? [C5: A] :
% 0.15/0.43 ( B3
% 0.15/0.43 = ( plus_plus @ A @ A4 @ C5 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % le_iff_add
% 0.15/0.43 thf(fact_233_add__le__imp__le__left,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere236663937imp_le @ A )
% 0.15/0.43 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_imp_le_left
% 0.15/0.43 thf(fact_234_add__le__imp__le__right,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere236663937imp_le @ A )
% 0.15/0.43 => ! [A2: A,C2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.15/0.43 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_le_imp_le_right
% 0.15/0.43 thf(fact_235_empty__neutral_I2_J,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = X ) ).
% 0.15/0.43
% 0.15/0.43 % empty_neutral(2)
% 0.15/0.43 thf(fact_236_empty__neutral_I1_J,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A] :
% 0.15/0.43 ( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 0.15/0.43 = X ) ).
% 0.15/0.43
% 0.15/0.43 % empty_neutral(1)
% 0.15/0.43 thf(fact_237_empty__le,axiom,
% 0.15/0.43 ! [A: $tType,A3: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 ) ).
% 0.15/0.43
% 0.15/0.43 % empty_le
% 0.15/0.43 thf(fact_238_subset__mset_Ozero__le,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.zero_le
% 0.15/0.43 thf(fact_239_subset__mset_Obot_Oextremum,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.bot.extremum
% 0.15/0.43 thf(fact_240_subset__mset_Oadd__decreasing,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_decreasing
% 0.15/0.43 thf(fact_241_subset__mset_Oadd__increasing,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 0.15/0.43 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_increasing
% 0.15/0.43 thf(fact_242_subset__mset_Oadd__decreasing2,axiom,
% 0.15/0.43 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ( subseteq_mset @ A @ A2 @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_decreasing2
% 0.15/0.43 thf(fact_243_subset__mset_Oadd__increasing2,axiom,
% 0.15/0.43 ! [A: $tType,C2: multiset @ A,B2: multiset @ A,A2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ A2 )
% 0.15/0.43 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_increasing2
% 0.15/0.43 thf(fact_244_subset__mset_Oadd__nonneg__nonneg,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
% 0.15/0.43 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 0.15/0.43 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_nonneg_nonneg
% 0.15/0.43 thf(fact_245_subset__mset_Oadd__nonpos__nonpos,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_nonpos_nonpos
% 0.15/0.43 thf(fact_246_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 0.15/0.43 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
% 0.15/0.43 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_nonneg_eq_0_iff
% 0.15/0.43 thf(fact_247_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 0.15/0.43 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.add_nonpos_eq_0_iff
% 0.15/0.43 thf(fact_248_subset__mset_Obot_Oextremum__uniqueI,axiom,
% 0.15/0.43 ! [A: $tType,A2: multiset @ A] :
% 0.15/0.43 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.15/0.43 => ( A2
% 0.15/0.43 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % subset_mset.bot.extremum_uniqueI
% 0.15/0.43 thf(fact_249_verit__sum__simplify,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( cancel1352612707id_add @ A )
% 0.15/0.43 => ! [A2: A] :
% 0.15/0.43 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.43 = A2 ) ) ).
% 0.15/0.43
% 0.15/0.43 % verit_sum_simplify
% 0.15/0.43 thf(fact_250_add__nonpos__eq__0__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.15/0.43 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.15/0.43 => ( ( ( plus_plus @ A @ X @ Y )
% 0.15/0.43 = ( zero_zero @ A ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ A ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_nonpos_eq_0_iff
% 0.15/0.43 thf(fact_251_add__nonneg__eq__0__iff,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [X: A,Y: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.15/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.15/0.43 => ( ( ( plus_plus @ A @ X @ Y )
% 0.15/0.43 = ( zero_zero @ A ) )
% 0.15/0.43 = ( ( X
% 0.15/0.43 = ( zero_zero @ A ) )
% 0.15/0.43 & ( Y
% 0.15/0.43 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_nonneg_eq_0_iff
% 0.15/0.43 thf(fact_252_add__nonpos__nonpos,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_nonpos_nonpos
% 0.15/0.43 thf(fact_253_add__nonneg__nonneg,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_nonneg_nonneg
% 0.15/0.43 thf(fact_254_add__increasing2,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [C2: A,B2: A,A2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.15/0.43 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.15/0.43 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_increasing2
% 0.15/0.43 thf(fact_255_add__decreasing2,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( ordere216010020id_add @ A )
% 0.15/0.43 => ! [C2: A,A2: A,B2: A] :
% 0.15/0.43 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.15/0.43 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.15/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % add_decreasing2
% 0.15/0.43
% 0.15/0.43 % Subclasses (4)
% 0.15/0.43 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( type @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( ord @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( order @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 0.15/0.43 ! [A: $tType] :
% 0.15/0.43 ( ( linorder @ A )
% 0.15/0.43 => ( preorder @ A ) ) ).
% 0.15/0.43
% 0.15/0.43 % Type constructors (15)
% 0.15/0.43 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.15/0.43 ! [A9: $tType,A10: $tType] :
% 0.15/0.43 ( ( preorder @ A10 )
% 0.15/0.43 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.15/0.43 ! [A9: $tType,A10: $tType] :
% 0.15/0.43 ( ( order @ A10 )
% 0.15/0.43 => ( order @ ( A9 > A10 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_fun___Orderings_Oord,axiom,
% 0.15/0.43 ! [A9: $tType,A10: $tType] :
% 0.15/0.43 ( ( ord @ A10 )
% 0.15/0.43 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 0.15/0.43 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.15/0.43 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 0.15/0.43 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 0.15/0.43 preorder @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.15/0.43 linorder @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.15/0.43 order @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 0.15/0.43 ord @ $o ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
% 0.15/0.43 ! [A9: $tType] :
% 0.15/0.43 ( ( preorder @ A9 )
% 0.15/0.43 => ( ordere779506340up_add @ ( multiset @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add,axiom,
% 0.15/0.43 ! [A9: $tType] : ( cancel1352612707id_add @ ( multiset @ A9 ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_7,axiom,
% 0.15/0.43 ! [A9: $tType] :
% 0.15/0.43 ( ( preorder @ A9 )
% 0.15/0.43 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Multiset_Omultiset___Orderings_Oorder_8,axiom,
% 0.15/0.43 ! [A9: $tType] :
% 0.15/0.43 ( ( preorder @ A9 )
% 0.15/0.43 => ( order @ ( multiset @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 thf(tcon_Multiset_Omultiset___Orderings_Oord_9,axiom,
% 0.15/0.43 ! [A9: $tType] :
% 0.15/0.43 ( ( preorder @ A9 )
% 0.15/0.43 => ( ord @ ( multiset @ A9 ) ) ) ).
% 0.15/0.43
% 0.15/0.43 % Free types (1)
% 0.15/0.43 thf(tfree_0,hypothesis,
% 0.15/0.43 linorder @ a ).
% 0.15/0.43
% 0.15/0.43 % Conjectures (1)
% 0.15/0.43 thf(conj_0,conjecture,
% 0.15/0.43 ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) )
% 0.15/0.43 = ( multiset2 @ a @ ( t @ a @ v @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ).
% 0.15/0.43
% 0.15/0.43 %------------------------------------------------------------------------------
% 0.15/0.43 ------------------------
% 2.77/1.59 ---- Embedded in THF ---
% 2.77/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.77/1.59 %%% Generated on Tue Feb 24 08:25:37 EST 2026
% 2.77/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.77/1.59 %%% Logic specification used:
% 2.77/1.59 %%% thf(spec, logic, $$dhol).
% 2.77/1.59
% 2.77/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2DTF20638.p
% 2.77/1.59 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.77/1.59 thf(ty_t_Multiset_Omultiset, type, multiset: ($tType > $tType)).
% 2.77/1.59 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.77/1.59 thf(ty_t_Heap_OTree, type, tree: ($tType > $tType)).
% 2.77/1.59 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.77/1.59 thf(ty_tf_a, type, a: $tType).
% 2.77/1.59 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1598680935umeral: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: ($o))).
% 2.77/1.59 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 2.77/1.59 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux, type, unique455577585es_aux: (!> [A:$tType]: ((((product_prod @ A) @ A) > $o)))).
% 2.77/1.59 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.77/1.59 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oheapify, type, heapIm818251801eapify: (!> [A:$tType]: (((tree @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__is__empty, type, heapIm721255937_empty: (!> [A:$tType]: (((tree @ A) > $o)))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__of__list, type, heapIm874063447f_list: (!> [A:$tType]: (((list @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oleft, type, heapIm1271749598e_left: (!> [A:$tType]: (((tree @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oof__list__tree, type, heapIm1912108042t_tree: (!> [A:$tType]: (((list @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OremoveLeaf, type, heapIm970386777veLeaf: (!> [A:$tType]: (((tree @ A) > ((product_prod @ A) @ (tree @ A)))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oright, type, heapIm1434396069_right: (!> [A:$tType]: (((tree @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OsiftDown, type, heapIm748920189ftDown: (!> [A:$tType]: (((tree @ A) > (tree @ A))))).
% 2.77/1.59 thf(sy_c_Heap_OHeap, type, heap: (!> [B:$tType,A:$tType]: ((B > ((B > $o) > (((list @ A) > B) > ((B > (multiset @ A)) > ((B > (tree @ A)) > ((B > ((product_prod @ A) @ B)) > $o))))))))).
% 2.77/1.59 thf(sy_c_Heap_OHeap__axioms, type, heap_axioms: (!> [B:$tType,A:$tType]: (((B > $o) > (((list @ A) > B) > ((B > (multiset @ A)) > ((B > (tree @ A)) > ((B > ((product_prod @ A) @ B)) > $o)))))))).
% 2.77/1.59 thf(sy_c_Heap_OTree_OE, type, e: (!> [A:$tType]: ((tree @ A)))).
% 2.77/1.59 thf(sy_c_Heap_OTree_OT, type, t: (!> [A:$tType]: ((A > ((tree @ A) > ((tree @ A) > (tree @ A))))))).
% 2.77/1.59 thf(sy_c_Heap_Oin__tree, type, in_tree: (!> [A:$tType]: ((A > ((tree @ A) > $o))))).
% 2.77/1.59 thf(sy_c_Heap_Ois__heap, type, is_heap: (!> [A:$tType]: (((tree @ A) > $o)))).
% 2.77/1.59 thf(sy_c_Heap_Omultiset, type, multiset2: (!> [A:$tType]: (((tree @ A) > (multiset @ A))))).
% 2.77/1.59 thf(sy_c_Heap_Oval, type, val: (!> [A:$tType]: (((tree @ A) > A)))).
% 2.77/1.59 thf(sy_c_Lattices__Big_Olinorder__class_OMax, type, lattic929149872er_Max: (!> [A:$tType]: (((set @ A) > A)))).
% 2.77/1.59 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.77/1.59 thf(sy_c_Multiset_Oadd__mset, type, add_mset: (!> [A:$tType]: ((A > ((multiset @ A) > (multiset @ A)))))).
% 2.77/1.59 thf(sy_c_Multiset_Ois__empty, type, is_empty: (!> [A:$tType]: (((multiset @ A) > $o)))).
% 2.77/1.59 thf(sy_c_Multiset_Oset__mset, type, set_mset: (!> [A:$tType]: (((multiset @ A) > (set @ A))))).
% 2.77/1.59 thf(sy_c_Multiset_Osubseteq__mset, type, subseteq_mset: (!> [A:$tType]: (((multiset @ A) > ((multiset @ A) > $o))))).
% 2.77/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.77/1.59 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.77/1.59 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.77/1.59 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.77/1.59 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.77/1.59 thf(sy_c_RemoveMax_OCollection, type, collection: (!> [B:$tType,A:$tType]: ((B > ((B > $o) > (((list @ A) > B) > ((B > (multiset @ A)) > $o))))))).
% 2.77/1.59 thf(sy_c_RemoveMax_OCollection_Oset, type, set2: (!> [B:$tType,A:$tType]: (((B > (multiset @ A)) > (B > (set @ A)))))).
% 2.77/1.59 thf(sy_c_RemoveMax_ORemoveMax, type, removeMax: (!> [B:$tType,A:$tType]: ((B > ((B > $o) > (((list @ A) > B) > ((B > (multiset @ A)) > ((B > ((product_prod @ A) @ B)) > ((B > $o) > $o))))))))).
% 2.77/1.59 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H, type, ssort: (!> [B:$tType,A:$tType]: (((B > $o) > ((B > ((product_prod @ A) @ B)) > (B > ((list @ A) > (list @ A)))))))).
% 2.77/1.59 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom, type, ssort_dom: (!> [B:$tType,A:$tType]: (((B > $o) > ((B > ((product_prod @ A) @ B)) > (((product_prod @ B) @ (list @ A)) > $o)))))).
% 2.77/1.59 thf(sy_c_RemoveMax_ORemoveMax__axioms, type, removeMax_axioms: (!> [B:$tType,A:$tType]: (((B > $o) > (((list @ A) > B) > ((B > (multiset @ A)) > ((B > ((product_prod @ A) @ B)) > ((B > $o) > $o)))))))).
% 2.77/1.59 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.77/1.59 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.77/1.59 thf(sy_v_l1____, type, l1: (tree @ a)).
% 2.77/1.59 thf(sy_v_l2____, type, l2: (tree @ a)).
% 2.77/1.59 thf(sy_v_r1____, type, r1: (tree @ a)).
% 2.77/1.59 thf(sy_v_r2____, type, r2: (tree @ a)).
% 2.77/1.59 thf(sy_v_v1____, type, v1: a).
% 2.77/1.59 thf(sy_v_v2____, type, v2: a).
% 2.77/1.59 thf(sy_v_v____, type, v: a).
% 2.77/1.59 thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062, axiom, (((ord_less_eq @ a) @ v2) @ v1)).
% 2.77/1.59 thf(fact_1_True, axiom, (((ord_less_eq @ a) @ v1) @ v)).
% 2.77/1.59 thf(fact_2__C5__2_Ohyps_C_I2_J, axiom, ((~ ((((ord_less_eq @ a) @ ((val @ a) @ ((((t @ a) @ v2) @ l2) @ r2))) @ ((val @ a) @ ((((t @ a) @ v1) @ l1) @ r1))))) => ((~ ((((ord_less_eq @ a) @ ((val @ a) @ ((((t @ a) @ v2) @ l2) @ r2))) @ v))) => (((multiset2 @ a) @ ((heapIm748920189ftDown @ a) @ ((((t @ a) @ v) @ ((heapIm1271749598e_left @ a) @ ((((t @ a) @ v2) @ l2) @ r2))) @ ((heapIm1434396069_right @ a) @ ((((t @ a) @ v2) @ l2) @ r2))))) = ((multiset2 @ a) @ ((((t @ a) @ v) @ ((heapIm1271749598e_left @ a) @ ((((t @ a) @ v2) @ l2) @ r2))) @ ((heapIm1434396069_right @ a) @ ((((t @ a) @ v2) @ l2) @ r2)))))))).
% 2.77/1.59 thf(fact_3__C5__2_Ohyps_C_I1_J, axiom, ((((ord_less_eq @ a) @ ((val @ a) @ ((((t @ a) @ v2) @ l2) @ r2))) @ ((val @ a) @ ((((t @ a) @ v1) @ l1) @ r1))) => ((~ ((((ord_less_eq @ a) @ ((val @ a) @ ((((t @ a) @ v1) @ l1) @ r1))) @ v))) => (((multiset2 @ a) @ ((heapIm748920189ftDown @ a) @ ((((t @ a) @ v) @ ((heapIm1271749598e_left @ a) @ ((((t @ a) @ v1) @ l1) @ r1))) @ ((heapIm1434396069_right @ a) @ ((((t @ a) @ v1) @ l1) @ r1))))) = ((multiset2 @ a) @ ((((t @ a) @ v) @ ((heapIm1271749598e_left @ a) @ ((((t @ a) @ v1) @ l1) @ r1))) @ ((heapIm1434396069_right @ a) @ ((((t @ a) @ v1) @ l1) @ r1)))))))).
% 2.77/1.59 thf(fact_4_siftDown__Node, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:(tree @ A),V:A,L:(tree @ A),R:(tree @ A)]: (((T2 = ((((t @ A) @ V) @ L) @ R)) => (? [L2:(tree @ A),V2:A,R2:(tree @ A)]: (((((heapIm748920189ftDown @ A) @ T2) = ((((t @ A) @ V2) @ L2) @ R2)) & (((ord_less_eq @ A) @ V) @ V2))))))))))).
% 2.77/1.59 thf(fact_5_Tree_Oinject, axiom, (! [A:$tType,X21:A,X22:(tree @ A),X23:(tree @ A),Y21:A,Y22:(tree @ A),Y23:(tree @ A)]: (((((((t @ A) @ X21) @ X22) @ X23) = ((((t @ A) @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 2.77/1.59 thf(fact_6_heapify_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,L:(tree @ A),R:(tree @ A)]: ((((heapIm818251801eapify @ A) @ ((((t @ A) @ V) @ L) @ R)) = ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm818251801eapify @ A) @ L)) @ ((heapIm818251801eapify @ A) @ R)))))))))).
% 2.77/1.59 thf(fact_7_siftDown__heap__is__heap, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(tree @ A),R:(tree @ A),T2:(tree @ A),V:A]: ((((is_heap @ A) @ L) => (((is_heap @ A) @ R) => ((T2 = ((((t @ A) @ V) @ L) @ R)) => ((is_heap @ A) @ ((heapIm748920189ftDown @ A) @ T2))))))))))).
% 2.77/1.59 thf(fact_8_siftDown__in__tree__set, axiom, (! [A:$tType]: (((linorder @ A) => ((in_tree @ A) = (^ [V3:A,T3:(tree @ A)]: ((((in_tree @ A) @ V3) @ ((heapIm748920189ftDown @ A) @ T3))))))))).
% 2.77/1.59 thf(fact_9_left_Osimps, axiom, (! [A:$tType,V:A,L:(tree @ A),R:(tree @ A)]: ((((heapIm1271749598e_left @ A) @ ((((t @ A) @ V) @ L) @ R)) = L)))).
% 2.77/1.59 thf(fact_10_right_Osimps, axiom, (! [A:$tType,V:A,L:(tree @ A),R:(tree @ A)]: ((((heapIm1434396069_right @ A) @ ((((t @ A) @ V) @ L) @ R)) = R)))).
% 2.77/1.59 thf(fact_11_siftDown_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A]: ((((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ (e @ A)) @ (e @ A))) = ((((t @ A) @ V) @ (e @ A)) @ (e @ A))))))))).
% 2.77/1.59 thf(fact_12_siftDown_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (((heapIm748920189ftDown @ A) @ (e @ A)) = (e @ A)))))).
% 2.77/1.59 thf(fact_13_Heap_Omultiset, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),L:B]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((Multiset @ L) = ((multiset2 @ A) @ (As_tree @ L)))))))))).
% 2.77/1.59 thf(fact_14_in__tree_Osimps_I1_J, axiom, (! [A:$tType,V:A]: ((~ ((((in_tree @ A) @ V) @ (e @ A))))))).
% 2.77/1.59 thf(fact_15_is__heap_Osimps_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Vd:A,Ve:(tree @ A),Vf:(tree @ A),Va:A,Vb:(tree @ A),Vc:(tree @ A)]: ((((is_heap @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) & (((is_heap @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)) & ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V) & ((is_heap @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf)))))))))))).
% 2.77/1.59 thf(fact_16_is__heap_Osimps_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Va:A,Vb:(tree @ A),Vc:(tree @ A),Vd:A,Ve:(tree @ A),Vf:(tree @ A)]: ((((is_heap @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) = ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V) & (((is_heap @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) & ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) & ((is_heap @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))))))))).
% 2.77/1.59 thf(fact_17_is__heap_Osimps_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Va:A,Vb:(tree @ A),Vc:(tree @ A)]: ((((is_heap @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ (e @ A))) = ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) & ((is_heap @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))))))).
% 2.77/1.59 thf(fact_18_is__heap_Osimps_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Va:A,Vb:(tree @ A),Vc:(tree @ A)]: ((((is_heap @ A) @ ((((t @ A) @ V) @ (e @ A)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) & ((is_heap @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))))))).
% 2.77/1.59 thf(fact_19_is__heap_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A]: (((is_heap @ A) @ ((((t @ A) @ V) @ (e @ A)) @ (e @ A))))))))).
% 2.77/1.59 thf(fact_20_is__heap_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => ((is_heap @ A) @ (e @ A)))))).
% 2.77/1.59 thf(fact_21_heapify_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (((heapIm818251801eapify @ A) @ (e @ A)) = (e @ A)))))).
% 2.77/1.59 thf(fact_22_is__heap__max, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,T2:(tree @ A)]: (((((in_tree @ A) @ V) @ T2) => (((is_heap @ A) @ T2) => (((ord_less_eq @ A) @ V) @ ((val @ A) @ T2)))))))))).
% 2.77/1.59 thf(fact_23_Heap_Oas__tree__empty, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),T2:B]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => (((As_tree @ T2) = (e @ A)) = (Is_empty @ T2))))))))).
% 2.77/1.59 thf(fact_24_Heap_Ois__heap__of__list, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),I:(list @ A)]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((is_heap @ A) @ (As_tree @ (Of_list @ I)))))))))).
% 2.77/1.59 thf(fact_25_heapify__heap__is__heap, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:(tree @ A)]: (((is_heap @ A) @ ((heapIm818251801eapify @ A) @ T2)))))))).
% 2.77/1.59 thf(fact_26_siftDown__in__tree, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:(tree @ A)]: (((T2 != (e @ A)) => (((in_tree @ A) @ ((val @ A) @ ((heapIm748920189ftDown @ A) @ T2))) @ T2)))))))).
% 2.77/1.59 thf(fact_27_is__heap_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(tree @ A)]: (((X != (e @ A)) => ((! [V4:A]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ (e @ A))))) => (~ ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A),Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2))))))))))))))))).
% 2.77/1.59 thf(fact_28_Tree_Oexhaust, axiom, (! [A:$tType,Y:(tree @ A)]: (((Y != (e @ A)) => (~ ((! [X212:A,X222:(tree @ A),X232:(tree @ A)]: ((Y != ((((t @ A) @ X212) @ X222) @ X232)))))))))).
% 2.77/1.59 thf(fact_29_Tree_Oinduct, axiom, (! [A:$tType,P:((tree @ A) > $o),Tree:(tree @ A)]: (((P @ (e @ A)) => ((! [X1:A,X2:(tree @ A),X3:(tree @ A)]: (((P @ X2) => ((P @ X3) => (P @ ((((t @ A) @ X1) @ X2) @ X3)))))) => (P @ Tree)))))).
% 2.77/1.59 thf(fact_30_Tree_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(tree @ A),X23:(tree @ A)]: (((e @ A) != ((((t @ A) @ X21) @ X22) @ X23))))).
% 2.77/1.59 thf(fact_31_siftDown_Osimps_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Va:A,Vb:(tree @ A),Vc:(tree @ A),V:A]: ((((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ (e @ A))) = ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ (e @ A)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ (e @ A))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))))) @ (e @ A))))))))))).
% 2.77/1.59 thf(fact_32_siftDown_Osimps_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Va:A,Vb:(tree @ A),Vc:(tree @ A),V:A]: ((((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ (e @ A)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ V) @ (e @ A)) @ ((((t @ A) @ Va) @ Vb) @ Vc)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ (e @ A)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ (e @ A)) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))))))))))).
% 2.77/1.59 thf(fact_33_val_Osimps, axiom, (! [A:$tType,V:A,Uu:(tree @ A),Uv:(tree @ A)]: ((((val @ A) @ ((((t @ A) @ V) @ Uu) @ Uv)) = V)))).
% 2.77/1.59 thf(fact_34_siftDown_Osimps_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Vd:A,Ve:(tree @ A),Vf:(tree @ A),Va:A,Vb:(tree @ A),Vc:(tree @ A),V:A]: ((((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) => (((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) = ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))))) @ ((((t @ A) @ Vd) @ Ve) @ Vf)))))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))))) => (((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) = ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((((t @ A) @ Va) @ Vb) @ Vc)) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf)))))))))))))))).
% 2.77/1.59 thf(fact_35_siftDown_Osimps_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Va:A,Vb:(tree @ A),Vc:(tree @ A),Vd:A,Ve:(tree @ A),Vf:(tree @ A),V:A]: ((((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) => (((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))))) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((val @ A) @ ((((t @ A) @ Vd) @ Ve) @ Vf))))) => (((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc)))) & ((~ ((((ord_less_eq @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ V))) => (((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((((t @ A) @ Va) @ Vb) @ Vc))) = ((((t @ A) @ ((val @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((((t @ A) @ Vd) @ Ve) @ Vf)) @ ((heapIm748920189ftDown @ A) @ ((((t @ A) @ V) @ ((heapIm1271749598e_left @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc))) @ ((heapIm1434396069_right @ A) @ ((((t @ A) @ Va) @ Vb) @ Vc)))))))))))))))).
% 2.77/1.59 thf(fact_36_in__tree_Osimps_I2_J, axiom, (! [A:$tType,V:A,V5:A,L:(tree @ A),R:(tree @ A)]: (((((in_tree @ A) @ V) @ ((((t @ A) @ V5) @ L) @ R)) = ((V = V5) | ((((in_tree @ A) @ V) @ L) | (((in_tree @ A) @ V) @ R))))))).
% 2.77/1.59 thf(fact_37_removeLeaf_Oinduct, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((tree @ A) > $o),A0:(tree @ A)]: (((! [V4:A]: ((P @ ((((t @ A) @ V4) @ (e @ A)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: (((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => ((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => (P @ ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ (e @ A))))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: (((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => ((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => (P @ ((((t @ A) @ V4) @ (e @ A)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A),Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A)]: (((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => ((P @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) => (P @ ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2))))))) => ((! [V4:A,Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A),Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: (((P @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2)) => ((P @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2)) => (P @ ((((t @ A) @ V4) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))))) => ((P @ (e @ A)) => (P @ A0))))))))))))).
% 2.77/1.59 thf(fact_38_removeLeaf_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(tree @ A)]: (((! [V4:A]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A),Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2))))) => ((! [V4:A,Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A),Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))) => (X = (e @ A))))))))))))).
% 2.77/1.59 thf(fact_39_siftDown_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(tree @ A)]: (((X != (e @ A)) => ((! [V4:A]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ (e @ A))))) => ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ (e @ A)) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2))))) => (~ ((! [V4:A,Va2:A,Vb2:(tree @ A),Vc2:(tree @ A),Vd2:A,Ve2:(tree @ A),Vf2:(tree @ A)]: ((X != ((((t @ A) @ V4) @ ((((t @ A) @ Va2) @ Vb2) @ Vc2)) @ ((((t @ A) @ Vd2) @ Ve2) @ Vf2))))))))))))))))).
% 2.77/1.59 thf(fact_40_hs__is__empty__def, axiom, (! [A:$tType]: (((heapIm721255937_empty @ A) = (^ [T3:(tree @ A)]: ((T3 = (e @ A)))))))).
% 2.77/1.59 thf(fact_41_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.77/1.59 thf(fact_42_heap__top__geq, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,T2:(tree @ A)]: (((((member @ A) @ A2) @ ((set_mset @ A) @ ((multiset2 @ A) @ T2))) => (((is_heap @ A) @ T2) => (((ord_less_eq @ A) @ A2) @ ((val @ A) @ T2)))))))))).
% 2.77/1.59 thf(fact_43_Heap_Oaxioms_I2_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B))]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => (((((((heap_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max)))))))).
% 2.77/1.59 thf(fact_44_hs__of__list__def, axiom, (! [A:$tType]: (((linorder @ A) => ((heapIm874063447f_list @ A) = (^ [L3:(list @ A)]: (((heapIm818251801eapify @ A) @ ((heapIm1912108042t_tree @ A) @ L3))))))))).
% 2.77/1.59 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.77/1.59 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.77/1.59 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.77/1.59 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 2.77/1.59 thf(fact_49_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.77/1.59 thf(fact_50_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.77/1.59 thf(fact_51_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.77/1.59 thf(fact_52_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F2 @ X4)) @ (G2 @ X4))))))))))).
% 2.77/1.59 thf(fact_53_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less_eq @ B) @ X5) @ Y2) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.77/1.59 thf(fact_54_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => (((ord_less_eq @ C) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A2)) @ C2)))))))))).
% 2.77/1.59 thf(fact_55_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.77/1.59 thf(fact_56_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A4:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A4) & (((ord_less_eq @ A) @ A4) @ B3))))))))).
% 2.77/1.59 thf(fact_57_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.77/1.59 thf(fact_58_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((P @ A5) @ B4)))) => ((! [A5:A,B4:A]: ((((P @ B4) @ A5) => ((P @ A5) @ B4)))) => ((P @ A2) @ B2))))))))).
% 2.77/1.59 thf(fact_59_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.77/1.59 thf(fact_60_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.77/1.59 thf(fact_61_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.77/1.59 thf(fact_62_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.77/1.59 thf(fact_63_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.77/1.59 thf(fact_64_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) & (((ord_less_eq @ A) @ B3) @ A4))))))))).
% 2.77/1.59 thf(fact_65_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.77/1.59 thf(fact_66_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.77/1.59 thf(fact_67_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.77/1.59 thf(fact_68_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.77/1.59 thf(fact_69_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.77/1.59 thf(fact_70_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.77/1.59 thf(fact_71_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.77/1.59 thf(fact_72_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 2.77/1.59 thf(fact_73_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C2) => ((! [X5:A,Y2:A]: (((((ord_less_eq @ A) @ X5) @ Y2) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C2)))))))))).
% 2.77/1.59 thf(fact_74_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C2:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y2:B]: (((((ord_less_eq @ B) @ X5) @ Y2) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C2))))))))))).
% 2.77/1.59 thf(fact_75_heap__top__max, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:(tree @ A)]: (((T2 != (e @ A)) => (((is_heap @ A) @ T2) => (((val @ A) @ T2) = ((lattic929149872er_Max @ A) @ ((set_mset @ A) @ ((multiset2 @ A) @ T2)))))))))))).
% 2.77/1.59 thf(fact_76_Heap_Ointro, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B))]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => ((((((((heap_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max))))))))).
% 2.77/1.59 thf(fact_77_Heap__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((heap @ B) @ A) = (^ [Empty2:B,Is_empty2:(B > $o),Of_list2:((list @ A) > B),Multiset2:(B > (multiset @ A)),As_tree2:(B > (tree @ A)),Remove_max2:(B > ((product_prod @ A) @ B))]: ((((((((collection @ B) @ A) @ Empty2) @ Is_empty2) @ Of_list2) @ Multiset2) & (((((((heap_axioms @ B) @ A) @ Is_empty2) @ Of_list2) @ Multiset2) @ As_tree2) @ Remove_max2))))))))).
% 2.77/1.59 thf(fact_78_of__list__tree_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Tail:(list @ A)]: ((((heapIm1912108042t_tree @ A) @ (((cons @ A) @ V) @ Tail)) = ((((t @ A) @ V) @ ((heapIm1912108042t_tree @ A) @ Tail)) @ (e @ A))))))))).
% 2.77/1.59 thf(fact_79_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y2:A]: (((P @ Y2) => (((ord_less_eq @ A) @ Y2) @ X)))) => (((order_Greatest @ A) @ P) = X))))))))).
% 2.77/1.59 thf(fact_80_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y2:A]: (((P @ Y2) => (((ord_less_eq @ A) @ Y2) @ X)))) => ((! [X5:A]: (((P @ X5) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X5)))) => (Q @ X5))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.77/1.59 thf(fact_81_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X6:($o > A),Y6:($o > A)]: (((((ord_less_eq @ A) @ (X6 @ $false)) @ (Y6 @ $false)) & (((ord_less_eq @ A) @ (X6 @ $true)) @ (Y6 @ $true)))))))))).
% 2.77/1.59 thf(fact_82_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.77/1.59 thf(fact_83_Heap_Oaxioms_I1_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B))]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset)))))))).
% 2.77/1.59 thf(fact_84_list_Oinject, axiom, (! [A:$tType,X21:A,X22:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X22) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.77/1.59 thf(fact_85_Collection_Ois__empty__inj, axiom, (! [A:$tType,B:$tType,Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),E:B]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => ((Is_empty @ E) => (E = Empty)))))).
% 2.77/1.59 thf(fact_86_Collection_Ois__empty__empty, axiom, (! [A:$tType,B:$tType,Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A))]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => (Is_empty @ Empty))))).
% 2.77/1.59 thf(fact_87_not__Cons__self2, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((cons @ A) @ X) @ Xs) != Xs)))).
% 2.77/1.59 thf(fact_88_Heap__axioms__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((heap_axioms @ B) @ A) = (^ [Is_empty2:(B > $o),Of_list2:((list @ A) > B),Multiset2:(B > (multiset @ A)),As_tree2:(B > (tree @ A)),Remove_max2:(B > ((product_prod @ A) @ B))]: (((! [L3:B]: (((Multiset2 @ L3) = ((multiset2 @ A) @ (As_tree2 @ L3))))) & ((! [I2:(list @ A)]: (((is_heap @ A) @ (As_tree2 @ (Of_list2 @ I2))))) & ((! [T3:B]: ((((As_tree2 @ T3) = (e @ A)) = (Is_empty2 @ T3)))) & ((! [L3:B,M:A,L4:B]: (((~ ((Is_empty2 @ L3))) => ((((((product_Pair @ A) @ B) @ M) @ L4) = (Remove_max2 @ L3)) => ((((add_mset @ A) @ M) @ (Multiset2 @ L4)) = (Multiset2 @ L3)))))) & ((! [L3:B,M:A,L4:B]: (((~ ((Is_empty2 @ L3))) => (((is_heap @ A) @ (As_tree2 @ L3)) => ((((((product_Pair @ A) @ B) @ M) @ L4) = (Remove_max2 @ L3)) => ((is_heap @ A) @ (As_tree2 @ L4))))))) & (! [T3:B,M:A,T4:B]: (((~ ((Is_empty2 @ T3))) => ((((((product_Pair @ A) @ B) @ M) @ T4) = (Remove_max2 @ T3)) => (M = ((val @ A) @ (As_tree2 @ T3))))))))))))))))))).
% 2.77/1.59 thf(fact_89_Heap__axioms_Ointro, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Of_list:((list @ A) > B),Is_empty:(B > $o),Remove_max:(B > ((product_prod @ A) @ B))]: (((! [L5:B]: (((Multiset @ L5) = ((multiset2 @ A) @ (As_tree @ L5))))) => ((! [I3:(list @ A)]: (((is_heap @ A) @ (As_tree @ (Of_list @ I3))))) => ((! [T5:B]: ((((As_tree @ T5) = (e @ A)) = (Is_empty @ T5)))) => ((! [L5:B,M2:A,L2:B]: (((~ ((Is_empty @ L5))) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => ((((add_mset @ A) @ M2) @ (Multiset @ L2)) = (Multiset @ L5)))))) => ((! [L5:B,M2:A,L2:B]: (((~ ((Is_empty @ L5))) => (((is_heap @ A) @ (As_tree @ L5)) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => ((is_heap @ A) @ (As_tree @ L2))))))) => ((! [T5:B,M2:A,T6:B]: (((~ ((Is_empty @ T5))) => ((((((product_Pair @ A) @ B) @ M2) @ T6) = (Remove_max @ T5)) => (M2 = ((val @ A) @ (As_tree @ T5))))))) => (((((((heap_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max))))))))))))).
% 2.77/1.59 thf(fact_90_Heap_Oremove__max__multiset_H, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),L:B,M3:A,L6:B]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((~ ((Is_empty @ L))) => ((((((product_Pair @ A) @ B) @ M3) @ L6) = (Remove_max @ L)) => ((((add_mset @ A) @ M3) @ (Multiset @ L6)) = (Multiset @ L))))))))))).
% 2.77/1.59 thf(fact_91_Heap_Oremove__max__val, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),T2:B,M3:A,T7:B]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((~ ((Is_empty @ T2))) => ((((((product_Pair @ A) @ B) @ M3) @ T7) = (Remove_max @ T2)) => (M3 = ((val @ A) @ (As_tree @ T2)))))))))))).
% 2.77/1.59 thf(fact_92_Heap_Oremove__max__is__heap, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),As_tree:(B > (tree @ A)),Remove_max:(B > ((product_prod @ A) @ B)),L:B,M3:A,L6:B]: ((((((((((heap @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ As_tree) @ Remove_max) => ((~ ((Is_empty @ L))) => (((is_heap @ A) @ (As_tree @ L)) => ((((((product_Pair @ A) @ B) @ M3) @ L6) = (Remove_max @ L)) => ((is_heap @ A) @ (As_tree @ L6)))))))))))).
% 2.77/1.59 thf(fact_93_multi__self__add__other__not__self, axiom, (! [A:$tType,M4:(multiset @ A),X:A]: ((M4 != (((add_mset @ A) @ X) @ M4))))).
% 2.77/1.59 thf(fact_94_add__mset__add__mset__same__iff, axiom, (! [A:$tType,A2:A,A3:(multiset @ A),B5:(multiset @ A)]: ((((((add_mset @ A) @ A2) @ A3) = (((add_mset @ A) @ A2) @ B5)) = (A3 = B5))))).
% 2.77/1.59 thf(fact_95_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B6)) = ((A2 = A6) & (B2 = B6)))))).
% 2.77/1.59 thf(fact_96_prod_Oinject, axiom, (! [A:$tType,B:$tType,X12:A,X24:B,Y1:A,Y24:B]: (((((((product_Pair @ A) @ B) @ X12) @ X24) = ((((product_Pair @ A) @ B) @ Y1) @ Y24)) = ((X12 = Y1) & (X24 = Y24)))))).
% 2.77/1.59 thf(fact_97_mset__add, axiom, (! [A:$tType,A2:A,A3:(multiset @ A)]: (((((member @ A) @ A2) @ ((set_mset @ A) @ A3)) => (~ ((! [B7:(multiset @ A)]: ((A3 != (((add_mset @ A) @ A2) @ B7)))))))))).
% 2.77/1.59 thf(fact_98_multi__member__split, axiom, (! [A:$tType,X:A,M4:(multiset @ A)]: (((((member @ A) @ X) @ ((set_mset @ A) @ M4)) => (? [A7:(multiset @ A)]: ((M4 = (((add_mset @ A) @ X) @ A7)))))))).
% 2.77/1.59 thf(fact_99_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X5:A,Y2:B]: ((P2 = ((((product_Pair @ A) @ B) @ X5) @ Y2))))))).
% 2.77/1.59 thf(fact_100_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ P2))))).
% 2.77/1.59 thf(fact_101_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B6)) => (~ (((A2 = A6) => (B2 != B6)))))))).
% 2.77/1.59 thf(fact_102_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B4:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C3)))))))))).
% 2.77/1.59 thf(fact_103_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B4:B,C3:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 2.77/1.59 thf(fact_104_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: ((~ ((! [A5:A,B4:B,C3:C,D2:D,E3:E2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D2) @ E3)))))))))))).
% 2.77/1.59 thf(fact_105_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))))]: ((~ ((! [A5:A,B4:B,C3:C,D2:D,E3:E2,F4:F3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F3)) @ D2) @ ((((product_Pair @ E2) @ F3) @ E3) @ F4))))))))))))).
% 2.77/1.59 thf(fact_106_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F3:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A5:A,B4:B,C3:C,D2:D,E3:E2,F4:F3,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E2) @ ((product_prod @ F3) @ G3)) @ E3) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4)))))))))))))).
% 2.77/1.59 thf(fact_107_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B4:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B4) @ C3))))) => (P @ X))))).
% 2.77/1.59 thf(fact_108_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B4:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X))))).
% 2.77/1.59 thf(fact_109_prod__induct5, axiom, (! [E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: (((! [A5:A,B4:B,C3:C,D2:D,E3:E2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D2) @ E3))))))) => (P @ X))))).
% 2.77/1.59 thf(fact_110_prod__induct6, axiom, (! [F3:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))))]: (((! [A5:A,B4:B,C3:C,D2:D,E3:E2,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F3)) @ D2) @ ((((product_Pair @ E2) @ F3) @ E3) @ F4)))))))) => (P @ X))))).
% 2.77/1.59 thf(fact_111_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))))]: (((! [A5:A,B4:B,C3:C,D2:D,E3:E2,F4:F3,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F3) @ G3))) @ D2) @ ((((product_Pair @ E2) @ ((product_prod @ F3) @ G3)) @ E3) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4))))))))) => (P @ X))))).
% 2.77/1.59 thf(fact_112_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B4:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B4))))))))).
% 2.77/1.59 thf(fact_113_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B4)))) => (P @ Prod))))).
% 2.77/1.59 thf(fact_114_add__eq__conv__ex, axiom, (! [A:$tType,A2:A,M4:(multiset @ A),B2:A,N:(multiset @ A)]: ((((((add_mset @ A) @ A2) @ M4) = (((add_mset @ A) @ B2) @ N)) = (((M4 = N) & (A2 = B2)) | (? [K:(multiset @ A)]: (((M4 = (((add_mset @ A) @ B2) @ K)) & (N = (((add_mset @ A) @ A2) @ K)))))))))).
% 2.77/1.59 thf(fact_115_add__mset__commute, axiom, (! [A:$tType,X:A,Y:A,M4:(multiset @ A)]: (((((add_mset @ A) @ X) @ (((add_mset @ A) @ Y) @ M4)) = (((add_mset @ A) @ Y) @ (((add_mset @ A) @ X) @ M4)))))).
% 2.77/1.59 thf(fact_116_union__single__eq__member, axiom, (! [A:$tType,X:A,M4:(multiset @ A),N:(multiset @ A)]: ((((((add_mset @ A) @ X) @ M4) = N) => (((member @ A) @ X) @ ((set_mset @ A) @ N)))))).
% 2.77/1.59 thf(fact_117_insert__noteq__member, axiom, (! [A:$tType,B2:A,B5:(multiset @ A),C2:A,C4:(multiset @ A)]: ((((((add_mset @ A) @ B2) @ B5) = (((add_mset @ A) @ C2) @ C4)) => ((B2 != C2) => (((member @ A) @ C2) @ ((set_mset @ A) @ B5))))))).
% 2.77/1.59 thf(fact_118_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.77/1.59 thf(fact_119_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 2.77/1.59 thf(fact_120_Collection_Oset__def, axiom, (! [A:$tType,B:$tType,Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),L:B]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => (((((set2 @ B) @ A) @ Multiset) @ L) = ((set_mset @ A) @ (Multiset @ L))))))).
% 2.77/1.59 thf(fact_121_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I2:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I2) @ J)) @ R3) => (((ord_less_eq @ B) @ (As @ I2)) @ (As @ J)))))))))))).
% 2.77/1.59 thf(fact_122_removeLeaf_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A]: ((((heapIm970386777veLeaf @ A) @ ((((t @ A) @ V) @ (e @ A)) @ (e @ A))) = ((((product_Pair @ A) @ (tree @ A)) @ V) @ (e @ A))))))))).
% 2.77/1.59 thf(fact_123_Collection_Oset_Ocong, axiom, (! [A:$tType,B:$tType]: ((((set2 @ B) @ A) = ((set2 @ B) @ A))))).
% 2.77/1.59 thf(fact_124_RemoveMax__axioms__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((removeMax_axioms @ B) @ A) = (^ [Is_empty2:(B > $o),Of_list2:((list @ A) > B),Multiset2:(B > (multiset @ A)),Remove_max2:(B > ((product_prod @ A) @ B)),Inv:(B > $o)]: (((! [X4:(list @ A)]: ((Inv @ (Of_list2 @ X4)))) & ((! [L3:B,M:A,L4:B]: (((~ ((Is_empty2 @ L3))) => ((Inv @ L3) => ((((((product_Pair @ A) @ B) @ M) @ L4) = (Remove_max2 @ L3)) => (M = ((lattic929149872er_Max @ A) @ ((((set2 @ B) @ A) @ Multiset2) @ L3)))))))) & ((! [L3:B,M:A,L4:B]: (((~ ((Is_empty2 @ L3))) => ((Inv @ L3) => ((((((product_Pair @ A) @ B) @ M) @ L4) = (Remove_max2 @ L3)) => ((((add_mset @ A) @ M) @ (Multiset2 @ L4)) = (Multiset2 @ L3))))))) & (! [L3:B,M:A,L4:B]: (((~ ((Is_empty2 @ L3))) => ((Inv @ L3) => ((((((product_Pair @ A) @ B) @ M) @ L4) = (Remove_max2 @ L3)) => (Inv @ L4)))))))))))))))).
% 2.77/1.59 thf(fact_125_RemoveMax__axioms_Ointro, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Inv2:(B > $o),Of_list:((list @ A) > B),Is_empty:(B > $o),Remove_max:(B > ((product_prod @ A) @ B)),Multiset:(B > (multiset @ A))]: (((! [X5:(list @ A)]: ((Inv2 @ (Of_list @ X5)))) => ((! [L5:B,M2:A,L2:B]: (((~ ((Is_empty @ L5))) => ((Inv2 @ L5) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => (M2 = ((lattic929149872er_Max @ A) @ ((((set2 @ B) @ A) @ Multiset) @ L5)))))))) => ((! [L5:B,M2:A,L2:B]: (((~ ((Is_empty @ L5))) => ((Inv2 @ L5) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => ((((add_mset @ A) @ M2) @ (Multiset @ L2)) = (Multiset @ L5))))))) => ((! [L5:B,M2:A,L2:B]: (((~ ((Is_empty @ L5))) => ((Inv2 @ L5) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => (Inv2 @ L2)))))) => (((((((removeMax_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2))))))))))).
% 2.77/1.59 thf(fact_126_RemoveMax_Oremove__max__max, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),L:B,M3:A,L6:B]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((~ ((Is_empty @ L))) => ((Inv2 @ L) => ((((((product_Pair @ A) @ B) @ M3) @ L6) = (Remove_max @ L)) => (M3 = ((lattic929149872er_Max @ A) @ ((((set2 @ B) @ A) @ Multiset) @ L))))))))))))).
% 2.77/1.59 thf(fact_127_multiset__induct__max, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((multiset @ A) > $o),M4:(multiset @ A)]: (((P @ (zero_zero @ (multiset @ A))) => ((! [X5:A,M5:(multiset @ A)]: (((P @ M5) => ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set_mset @ A) @ M5)) => (((ord_less_eq @ A) @ Xa) @ X5)))) => (P @ (((add_mset @ A) @ X5) @ M5)))))) => (P @ M4))))))))).
% 2.77/1.59 thf(fact_128_add__mset__eq__singleton__iff, axiom, (! [A:$tType,X:A,M4:(multiset @ A),Y:A]: ((((((add_mset @ A) @ X) @ M4) = (((add_mset @ A) @ Y) @ (zero_zero @ (multiset @ A)))) = ((M4 = (zero_zero @ (multiset @ A))) & (X = Y)))))).
% 2.77/1.59 thf(fact_129_single__eq__add__mset, axiom, (! [A:$tType,A2:A,B2:A,M4:(multiset @ A)]: ((((((add_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) = (((add_mset @ A) @ B2) @ M4)) = ((B2 = A2) & (M4 = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_130_add__mset__eq__single, axiom, (! [A:$tType,B2:A,M4:(multiset @ A),A2:A]: ((((((add_mset @ A) @ B2) @ M4) = (((add_mset @ A) @ A2) @ (zero_zero @ (multiset @ A)))) = ((B2 = A2) & (M4 = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_131_single__eq__single, axiom, (! [A:$tType,A2:A,B2:A]: ((((((add_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) = (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) = (A2 = B2))))).
% 2.77/1.59 thf(fact_132_RemoveMax__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((removeMax @ B) @ A) = (^ [Empty2:B,Is_empty2:(B > $o),Of_list2:((list @ A) > B),Multiset2:(B > (multiset @ A)),Remove_max2:(B > ((product_prod @ A) @ B)),Inv:(B > $o)]: ((((((((collection @ B) @ A) @ Empty2) @ Is_empty2) @ Of_list2) @ Multiset2) & (((((((removeMax_axioms @ B) @ A) @ Is_empty2) @ Of_list2) @ Multiset2) @ Remove_max2) @ Inv))))))))).
% 2.77/1.59 thf(fact_133_RemoveMax_Ointro, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o)]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => ((((((((removeMax_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2))))))))).
% 2.77/1.59 thf(fact_134_multiset__cases, axiom, (! [A:$tType,M4:(multiset @ A)]: (((M4 != (zero_zero @ (multiset @ A))) => (~ ((! [X5:A,N2:(multiset @ A)]: ((M4 != (((add_mset @ A) @ X5) @ N2)))))))))).
% 2.77/1.59 thf(fact_135_multiset__induct, axiom, (! [A:$tType,P:((multiset @ A) > $o),M4:(multiset @ A)]: (((P @ (zero_zero @ (multiset @ A))) => ((! [X5:A,M5:(multiset @ A)]: (((P @ M5) => (P @ (((add_mset @ A) @ X5) @ M5))))) => (P @ M4)))))).
% 2.77/1.59 thf(fact_136_multiset__induct2, axiom, (! [A:$tType,B:$tType,P:((multiset @ A) > ((multiset @ B) > $o)),M4:(multiset @ A),N:(multiset @ B)]: ((((P @ (zero_zero @ (multiset @ A))) @ (zero_zero @ (multiset @ B))) => ((! [A5:A,M5:(multiset @ A),N2:(multiset @ B)]: ((((P @ M5) @ N2) => ((P @ (((add_mset @ A) @ A5) @ M5)) @ N2)))) => ((! [A5:B,M5:(multiset @ A),N2:(multiset @ B)]: ((((P @ M5) @ N2) => ((P @ M5) @ (((add_mset @ B) @ A5) @ N2))))) => ((P @ M4) @ N))))))).
% 2.77/1.59 thf(fact_137_empty__not__add__mset, axiom, (! [A:$tType,A2:A,A3:(multiset @ A)]: (((zero_zero @ (multiset @ A)) != (((add_mset @ A) @ A2) @ A3))))).
% 2.77/1.59 thf(fact_138_multi__nonempty__split, axiom, (! [A:$tType,M4:(multiset @ A)]: (((M4 != (zero_zero @ (multiset @ A))) => (? [A7:(multiset @ A),A5:A]: ((M4 = (((add_mset @ A) @ A5) @ A7)))))))).
% 2.77/1.59 thf(fact_139_multiset__nonemptyE, axiom, (! [A:$tType,A3:(multiset @ A)]: (((A3 != (zero_zero @ (multiset @ A))) => (~ ((! [X5:A]: ((~ ((((member @ A) @ X5) @ ((set_mset @ A) @ A3)))))))))))).
% 2.77/1.59 thf(fact_140_RemoveMax_Oof__list__inv, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),X:(list @ A)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => (Inv2 @ (Of_list @ X))))))))).
% 2.77/1.59 thf(fact_141_RemoveMax_Oaxioms_I2_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => (((((((removeMax_axioms @ B) @ A) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2)))))))).
% 2.77/1.59 thf(fact_142_RemoveMax_Oremove__max__inv, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),L:B,M3:A,L6:B]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((~ ((Is_empty @ L))) => ((Inv2 @ L) => ((((((product_Pair @ A) @ B) @ M3) @ L6) = (Remove_max @ L)) => (Inv2 @ L6))))))))))).
% 2.77/1.59 thf(fact_143_Collection_Ois__empty__as__list, axiom, (! [B:$tType,A:$tType,Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),E:B]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => ((Is_empty @ E) => ((Multiset @ E) = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_144_Collection_Omultiset__empty, axiom, (! [B:$tType,A:$tType,Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A))]: ((((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) => ((Multiset @ Empty) = (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_145_RemoveMax_Oaxioms_I1_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((((((collection @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset)))))))).
% 2.77/1.59 thf(fact_146_multi__member__last, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ ((set_mset @ A) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_147_RemoveMax_Oremove__max__multiset, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),L:B,M3:A,L6:B]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((~ ((Is_empty @ L))) => ((Inv2 @ L) => ((((((product_Pair @ A) @ B) @ M3) @ L6) = (Remove_max @ L)) => ((((add_mset @ A) @ M3) @ (Multiset @ L6)) = (Multiset @ L)))))))))))).
% 2.77/1.59 thf(fact_148_multiset_Osimps_I1_J, axiom, (! [A:$tType]: ((((multiset2 @ A) @ (e @ A)) = (zero_zero @ (multiset @ A)))))).
% 2.77/1.59 thf(fact_149_multiset__induct__min, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((multiset @ A) > $o),M4:(multiset @ A)]: (((P @ (zero_zero @ (multiset @ A))) => ((! [X5:A,M5:(multiset @ A)]: (((P @ M5) => ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set_mset @ A) @ M5)) => (((ord_less_eq @ A) @ X5) @ Xa)))) => (P @ (((add_mset @ A) @ X5) @ M5)))))) => (P @ M4))))))))).
% 2.77/1.59 thf(fact_150_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N3:A]: (((((ord_less_eq @ A) @ N3) @ (zero_zero @ A)) = (N3 = (zero_zero @ A))))))))).
% 2.77/1.59 thf(fact_151_Multiset_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A8:(multiset @ A)]: ((A8 = (zero_zero @ (multiset @ A))))))))).
% 2.77/1.59 thf(fact_152_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.77/1.59 thf(fact_153_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.77/1.59 thf(fact_154_RemoveMax_Ossort_HInduct, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),L:B,P:(B > ((list @ A) > $o)),Sl:(list @ A)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((Inv2 @ L) => (((P @ L) @ Sl) => ((! [L5:B,Sl2:(list @ A),M2:A,L2:B]: (((~ ((Is_empty @ L5))) => ((Inv2 @ L5) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L5)) => (((P @ L5) @ Sl2) => ((P @ L2) @ (((cons @ A) @ M2) @ Sl2)))))))) => ((P @ Empty) @ ((((((ssort @ B) @ A) @ Is_empty) @ Remove_max) @ L) @ Sl)))))))))))).
% 2.77/1.59 thf(fact_155_RemoveMax_Ossort_H__dom_Ocases, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),A2:((product_prod @ B) @ (list @ A))]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ A2) => (~ ((! [L5:B,Sl2:(list @ A)]: (((A2 = ((((product_Pair @ B) @ (list @ A)) @ L5) @ Sl2)) => (~ (((~ ((Is_empty @ L5))) => (! [M6:A,L7:B]: (((((((product_Pair @ A) @ B) @ M6) @ L7) = (Remove_max @ L5)) => (((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ ((((product_Pair @ B) @ (list @ A)) @ L7) @ (((cons @ A) @ M6) @ Sl2)))))))))))))))))))))).
% 2.77/1.59 thf(fact_156_RemoveMax_Ossort_H__dom_Osimps, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),A2:((product_prod @ B) @ (list @ A))]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ A2) = (? [L3:B,Sl3:(list @ A)]: (((A2 = ((((product_Pair @ B) @ (list @ A)) @ L3) @ Sl3)) & (! [X4:A,Y4:B]: (((~ ((Is_empty @ L3))) => ((((((product_Pair @ A) @ B) @ X4) @ Y4) = (Remove_max @ L3)) => (((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ ((((product_Pair @ B) @ (list @ A)) @ Y4) @ (((cons @ A) @ X4) @ Sl3)))))))))))))))))).
% 2.77/1.59 thf(fact_157_RemoveMax_Ossort_H__dom_Ocong, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((ssort_dom @ B) @ A) = ((ssort_dom @ B) @ A)))))).
% 2.77/1.59 thf(fact_158_RemoveMax_Ossort_H_Ocong, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((ssort @ B) @ A) = ((ssort @ B) @ A)))))).
% 2.77/1.59 thf(fact_159_RemoveMax_Ossort_H__dom_Oinducts, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),X:((product_prod @ B) @ (list @ A)),P:(((product_prod @ B) @ (list @ A)) > $o)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ X) => ((! [L5:B,Sl2:(list @ A)]: (((! [M6:A,L7:B]: (((~ ((Is_empty @ L5))) => ((((((product_Pair @ A) @ B) @ M6) @ L7) = (Remove_max @ L5)) => (((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ ((((product_Pair @ B) @ (list @ A)) @ L7) @ (((cons @ A) @ M6) @ Sl2))))))) => ((! [M6:A,L7:B]: (((~ ((Is_empty @ L5))) => ((((((product_Pair @ A) @ B) @ M6) @ L7) = (Remove_max @ L5)) => (P @ ((((product_Pair @ B) @ (list @ A)) @ L7) @ (((cons @ A) @ M6) @ Sl2))))))) => (P @ ((((product_Pair @ B) @ (list @ A)) @ L5) @ Sl2)))))) => (P @ X)))))))))).
% 2.77/1.59 thf(fact_160_RemoveMax_Ossort_H__dom_Ointros, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [Empty:B,Is_empty:(B > $o),Of_list:((list @ A) > B),Multiset:(B > (multiset @ A)),Remove_max:(B > ((product_prod @ A) @ B)),Inv2:(B > $o),L:B,Sl:(list @ A)]: ((((((((((removeMax @ B) @ A) @ Empty) @ Is_empty) @ Of_list) @ Multiset) @ Remove_max) @ Inv2) => ((! [M2:A,L2:B]: (((~ ((Is_empty @ L))) => ((((((product_Pair @ A) @ B) @ M2) @ L2) = (Remove_max @ L)) => (((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ ((((product_Pair @ B) @ (list @ A)) @ L2) @ (((cons @ A) @ M2) @ Sl))))))) => (((((ssort_dom @ B) @ A) @ Is_empty) @ Remove_max) @ ((((product_Pair @ B) @ (list @ A)) @ L) @ Sl)))))))))).
% 2.77/1.59 thf(fact_161_multiset_Osimps_I2_J, axiom, (! [A:$tType,V:A,L:(tree @ A),R:(tree @ A)]: ((((multiset2 @ A) @ ((((t @ A) @ V) @ L) @ R)) = (((plus_plus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ ((multiset2 @ A) @ L)) @ (((add_mset @ A) @ V) @ (zero_zero @ (multiset @ A))))) @ ((multiset2 @ A) @ R)))))).
% 2.77/1.59 thf(fact_162_divides__aux__eq, axiom, (! [A:$tType]: (((unique1598680935umeral @ A) => (! [Q2:A,R:A]: ((((unique455577585es_aux @ A) @ ((((product_Pair @ A) @ A) @ Q2) @ R)) = (R = (zero_zero @ A))))))))).
% 2.77/1.59 thf(fact_163_single__subset__iff, axiom, (! [A:$tType,A2:A,M4:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ A2) @ (zero_zero @ (multiset @ A)))) @ M4) = (((member @ A) @ A2) @ ((set_mset @ A) @ M4)))))).
% 2.77/1.59 thf(fact_164_subset__mset_Oorder__refl, axiom, (! [A:$tType,X:(multiset @ A)]: ((((subseteq_mset @ A) @ X) @ X)))).
% 2.77/1.59 thf(fact_165_subset__mset_Odual__order_Orefl, axiom, (! [A:$tType,A2:(multiset @ A)]: ((((subseteq_mset @ A) @ A2) @ A2)))).
% 2.77/1.59 thf(fact_166_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.77/1.59 thf(fact_167_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.77/1.59 thf(fact_168_subset__mset_Obot_Oextremum__unique, axiom, (! [A:$tType,A2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) = (A2 = (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_169_subset__mset_Ole__zero__eq, axiom, (! [A:$tType,N3:(multiset @ A)]: (((((subseteq_mset @ A) @ N3) @ (zero_zero @ (multiset @ A))) = (N3 = (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_170_subset__mset_Oadd__le__cancel__left, axiom, (! [A:$tType,C2:(multiset @ A),A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A2)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) = (((subseteq_mset @ A) @ A2) @ B2))))).
% 2.77/1.59 thf(fact_171_subset__mset_Oadd__le__cancel__right, axiom, (! [A:$tType,A2:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) = (((subseteq_mset @ A) @ A2) @ B2))))).
% 2.77/1.59 thf(fact_172_mset__subset__eq__mono__add__left__cancel, axiom, (! [A:$tType,C4:(multiset @ A),A3:(multiset @ A),B5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C4) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C4) @ B5)) = (((subseteq_mset @ A) @ A3) @ B5))))).
% 2.77/1.59 thf(fact_173_mset__subset__eq__mono__add__right__cancel, axiom, (! [A:$tType,A3:(multiset @ A),C4:(multiset @ A),B5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C4)) @ (((plus_plus @ (multiset @ A)) @ B5) @ C4)) = (((subseteq_mset @ A) @ A3) @ B5))))).
% 2.77/1.59 thf(fact_174_subset__mset_Ozero__eq__add__iff__both__eq__0, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: ((((zero_zero @ (multiset @ A)) = (((plus_plus @ (multiset @ A)) @ X) @ Y)) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_175_subset__mset_Oadd__eq__0__iff__both__eq__0, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_176_union__eq__empty, axiom, (! [A:$tType,M4:(multiset @ A),N:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ M4) @ N) = (zero_zero @ (multiset @ A))) = ((M4 = (zero_zero @ (multiset @ A))) & (N = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_177_empty__eq__union, axiom, (! [A:$tType,M4:(multiset @ A),N:(multiset @ A)]: ((((zero_zero @ (multiset @ A)) = (((plus_plus @ (multiset @ A)) @ M4) @ N)) = ((M4 = (zero_zero @ (multiset @ A))) & (N = (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_178_union__mset__add__mset__right, axiom, (! [A:$tType,A3:(multiset @ A),A2:A,B5:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ A3) @ (((add_mset @ A) @ A2) @ B5)) = (((add_mset @ A) @ A2) @ (((plus_plus @ (multiset @ A)) @ A3) @ B5)))))).
% 2.77/1.59 thf(fact_179_union__mset__add__mset__left, axiom, (! [A:$tType,A2:A,A3:(multiset @ A),B5:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A2) @ A3)) @ B5) = (((add_mset @ A) @ A2) @ (((plus_plus @ (multiset @ A)) @ A3) @ B5)))))).
% 2.77/1.59 thf(fact_180_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.77/1.59 thf(fact_181_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.77/1.59 thf(fact_182_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.77/1.59 thf(fact_183_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 2.77/1.59 thf(fact_184_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.77/1.59 thf(fact_185_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.77/1.59 thf(fact_186_subset__mset_Oadd__le__same__cancel1, axiom, (! [A:$tType,B2:(multiset @ A),A2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ B2) @ A2)) @ B2) = (((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_187_subset__mset_Oadd__le__same__cancel2, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ B2)) @ B2) = (((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_188_subset__mset_Ole__add__same__cancel1, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (((plus_plus @ (multiset @ A)) @ A2) @ B2)) = (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 2.77/1.59 thf(fact_189_subset__mset_Ole__add__same__cancel2, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (((plus_plus @ (multiset @ A)) @ B2) @ A2)) = (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 2.77/1.59 thf(fact_190_add__mset__subseteq__single__iff, axiom, (! [A:$tType,A2:A,M4:(multiset @ A),B2:A]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ A2) @ M4)) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) = ((M4 = (zero_zero @ (multiset @ A))) & (A2 = B2)))))).
% 2.77/1.59 thf(fact_191_union__assoc, axiom, (! [A:$tType,M4:(multiset @ A),N:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ M4) @ N)) @ K2) = (((plus_plus @ (multiset @ A)) @ M4) @ (((plus_plus @ (multiset @ A)) @ N) @ K2)))))).
% 2.77/1.59 thf(fact_192_union__lcomm, axiom, (! [A:$tType,M4:(multiset @ A),N:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ M4) @ (((plus_plus @ (multiset @ A)) @ N) @ K2)) = (((plus_plus @ (multiset @ A)) @ N) @ (((plus_plus @ (multiset @ A)) @ M4) @ K2)))))).
% 2.77/1.59 thf(fact_193_union__commute, axiom, (! [A:$tType]: (((plus_plus @ (multiset @ A)) = (^ [M7:(multiset @ A),N4:(multiset @ A)]: ((((plus_plus @ (multiset @ A)) @ N4) @ M7))))))).
% 2.77/1.59 thf(fact_194_union__left__cancel, axiom, (! [A:$tType,K2:(multiset @ A),M4:(multiset @ A),N:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ K2) @ M4) = (((plus_plus @ (multiset @ A)) @ K2) @ N)) = (M4 = N))))).
% 2.77/1.59 thf(fact_195_subset__mset_Oeq__iff, axiom, (! [A:$tType]: (((^ [Y3:(multiset @ A),Z:(multiset @ A)]: ((Y3 = Z))) = (^ [X4:(multiset @ A),Y4:(multiset @ A)]: (((((subseteq_mset @ A) @ X4) @ Y4) & (((subseteq_mset @ A) @ Y4) @ X4)))))))).
% 2.77/1.59 thf(fact_196_union__right__cancel, axiom, (! [A:$tType,M4:(multiset @ A),K2:(multiset @ A),N:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ M4) @ K2) = (((plus_plus @ (multiset @ A)) @ N) @ K2)) = (M4 = N))))).
% 2.77/1.59 thf(fact_197_subset__mset_Oantisym, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((((subseteq_mset @ A) @ X) @ Y) => ((((subseteq_mset @ A) @ Y) @ X) => (X = Y)))))).
% 2.77/1.59 thf(fact_198_subset__mset_Oeq__refl, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((X = Y) => (((subseteq_mset @ A) @ X) @ Y))))).
% 2.77/1.59 thf(fact_199_subset__mset_Oadd__mono, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A),D3:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => ((((subseteq_mset @ A) @ C2) @ D3) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ D3))))))).
% 2.77/1.59 thf(fact_200_subset__mset_Oless__eqE, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => (~ ((! [C3:(multiset @ A)]: ((B2 != (((plus_plus @ (multiset @ A)) @ A2) @ C3)))))))))).
% 2.77/1.59 thf(fact_201_subset__mset_Ole__iff__add, axiom, (! [A:$tType]: (((subseteq_mset @ A) = (^ [A4:(multiset @ A),B3:(multiset @ A)]: ((? [C5:(multiset @ A)]: ((B3 = (((plus_plus @ (multiset @ A)) @ A4) @ C5)))))))))).
% 2.77/1.59 thf(fact_202_mset__subset__eq__add__left, axiom, (! [A:$tType,A3:(multiset @ A),B5:(multiset @ A)]: ((((subseteq_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ A3) @ B5))))).
% 2.77/1.59 thf(fact_203_mset__subset__eq__mono__add, axiom, (! [A:$tType,A3:(multiset @ A),B5:(multiset @ A),C4:(multiset @ A),D4:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B5) => ((((subseteq_mset @ A) @ C4) @ D4) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C4)) @ (((plus_plus @ (multiset @ A)) @ B5) @ D4))))))).
% 2.77/1.59 thf(fact_204_subset__mset_Oorder_Otrans, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => ((((subseteq_mset @ A) @ B2) @ C2) => (((subseteq_mset @ A) @ A2) @ C2)))))).
% 2.77/1.59 thf(fact_205_subset__mset_Oorder__trans, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A),Z2:(multiset @ A)]: (((((subseteq_mset @ A) @ X) @ Y) => ((((subseteq_mset @ A) @ Y) @ Z2) => (((subseteq_mset @ A) @ X) @ Z2)))))).
% 2.77/1.59 thf(fact_206_mset__subset__eq__add__right, axiom, (! [A:$tType,B5:(multiset @ A),A3:(multiset @ A)]: ((((subseteq_mset @ A) @ B5) @ (((plus_plus @ (multiset @ A)) @ A3) @ B5))))).
% 2.77/1.59 thf(fact_207_subset__mset_Oantisym__conv, axiom, (! [A:$tType,Y:(multiset @ A),X:(multiset @ A)]: (((((subseteq_mset @ A) @ Y) @ X) => ((((subseteq_mset @ A) @ X) @ Y) = (X = Y)))))).
% 2.77/1.59 thf(fact_208_subset__mset_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((^ [Y3:(multiset @ A),Z:(multiset @ A)]: ((Y3 = Z))) = (^ [A4:(multiset @ A),B3:(multiset @ A)]: (((((subseteq_mset @ A) @ A4) @ B3) & (((subseteq_mset @ A) @ B3) @ A4)))))))).
% 2.77/1.59 thf(fact_209_multi__union__self__other__eq, axiom, (! [A:$tType,A3:(multiset @ A),X7:(multiset @ A),Y7:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ A3) @ X7) = (((plus_plus @ (multiset @ A)) @ A3) @ Y7)) => (X7 = Y7))))).
% 2.77/1.59 thf(fact_210_subset__mset_Oadd__left__mono, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A2)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)))))).
% 2.77/1.59 thf(fact_211_subset__mset_Oorder_Oantisym, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => ((((subseteq_mset @ A) @ B2) @ A2) => (A2 = B2)))))).
% 2.77/1.59 thf(fact_212_mset__subset__eq__exists__conv, axiom, (! [A:$tType]: (((subseteq_mset @ A) = (^ [A8:(multiset @ A),B8:(multiset @ A)]: ((? [C6:(multiset @ A)]: ((B8 = (((plus_plus @ (multiset @ A)) @ A8) @ C6)))))))))).
% 2.77/1.59 thf(fact_213_subset__mset_Oadd__right__mono, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)))))).
% 2.77/1.59 thf(fact_214_subset__mset_Oord__eq__le__trans, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((A2 = B2) => ((((subseteq_mset @ A) @ B2) @ C2) => (((subseteq_mset @ A) @ A2) @ C2)))))).
% 2.77/1.59 thf(fact_215_subset__mset_Oord__le__eq__trans, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ B2) => ((B2 = C2) => (((subseteq_mset @ A) @ A2) @ C2)))))).
% 2.77/1.59 thf(fact_216_subset__mset_Odual__order_Otrans, axiom, (! [A:$tType,B2:(multiset @ A),A2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ B2) @ A2) => ((((subseteq_mset @ A) @ C2) @ B2) => (((subseteq_mset @ A) @ C2) @ A2)))))).
% 2.77/1.59 thf(fact_217_subset__mset_Odual__order_Oeq__iff, axiom, (! [A:$tType]: (((^ [Y3:(multiset @ A),Z:(multiset @ A)]: ((Y3 = Z))) = (^ [A4:(multiset @ A),B3:(multiset @ A)]: (((((subseteq_mset @ A) @ B3) @ A4) & (((subseteq_mset @ A) @ A4) @ B3)))))))).
% 2.77/1.59 thf(fact_218_subset__mset_Oadd__le__imp__le__left, axiom, (! [A:$tType,C2:(multiset @ A),A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A2)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) => (((subseteq_mset @ A) @ A2) @ B2))))).
% 2.77/1.59 thf(fact_219_subset__mset_Odual__order_Oantisym, axiom, (! [A:$tType,B2:(multiset @ A),A2:(multiset @ A)]: (((((subseteq_mset @ A) @ B2) @ A2) => ((((subseteq_mset @ A) @ A2) @ B2) => (A2 = B2)))))).
% 2.77/1.59 thf(fact_220_subset__mset_Oadd__le__imp__le__right, axiom, (! [A:$tType,A2:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) => (((subseteq_mset @ A) @ A2) @ B2))))).
% 2.77/1.59 thf(fact_221_union__iff, axiom, (! [A:$tType,A2:A,A3:(multiset @ A),B5:(multiset @ A)]: (((((member @ A) @ A2) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B5))) = ((((member @ A) @ A2) @ ((set_mset @ A) @ A3)) | (((member @ A) @ A2) @ ((set_mset @ A) @ B5))))))).
% 2.77/1.59 thf(fact_222_mset__subset__eqD, axiom, (! [A:$tType,A3:(multiset @ A),B5:(multiset @ A),X:A]: (((((subseteq_mset @ A) @ A3) @ B5) => ((((member @ A) @ X) @ ((set_mset @ A) @ A3)) => (((member @ A) @ X) @ ((set_mset @ A) @ B5))))))).
% 2.77/1.59 thf(fact_223_set__mset__mono, axiom, (! [A:$tType,A3:(multiset @ A),B5:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B5) => (((ord_less_eq @ (set @ A)) @ ((set_mset @ A) @ A3)) @ ((set_mset @ A) @ B5)))))).
% 2.77/1.59 thf(fact_224_mset__subset__eq__add__mset__cancel, axiom, (! [A:$tType,A2:A,A3:(multiset @ A),B5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ A2) @ A3)) @ (((add_mset @ A) @ A2) @ B5)) = (((subseteq_mset @ A) @ A3) @ B5))))).
% 2.77/1.59 thf(fact_225_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J2:A,K3:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J2) & (K3 = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K3)) @ (((plus_plus @ A) @ J2) @ L))))))))).
% 2.77/1.59 thf(fact_226_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J2:A,K3:A,L:A]: ((((I = J2) & (((ord_less_eq @ A) @ K3) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K3)) @ (((plus_plus @ A) @ J2) @ L))))))))).
% 2.77/1.59 thf(fact_227_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J2:A,K3:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J2) & (((ord_less_eq @ A) @ K3) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K3)) @ (((plus_plus @ A) @ J2) @ L))))))))).
% 2.77/1.59 thf(fact_228_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 2.77/1.59 thf(fact_229_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 2.77/1.59 thf(fact_230_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C3:A]: ((B2 != (((plus_plus @ A) @ A2) @ C3))))))))))))).
% 2.77/1.59 thf(fact_231_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 2.77/1.59 thf(fact_232_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((? [C5:A]: ((B3 = (((plus_plus @ A) @ A4) @ C5))))))))))).
% 2.77/1.59 thf(fact_233_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.77/1.59 thf(fact_234_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.77/1.59 thf(fact_235_empty__neutral_I2_J, axiom, (! [A:$tType,X:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ X) @ (zero_zero @ (multiset @ A))) = X)))).
% 2.77/1.59 thf(fact_236_empty__neutral_I1_J, axiom, (! [A:$tType,X:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (zero_zero @ (multiset @ A))) @ X) = X)))).
% 2.77/1.59 thf(fact_237_empty__le, axiom, (! [A:$tType,A3:(multiset @ A)]: ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3)))).
% 2.77/1.59 thf(fact_238_subset__mset_Ozero__le, axiom, (! [A:$tType,X:(multiset @ A)]: ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ X)))).
% 2.77/1.59 thf(fact_239_subset__mset_Obot_Oextremum, axiom, (! [A:$tType,A2:(multiset @ A)]: ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A2)))).
% 2.77/1.59 thf(fact_240_subset__mset_Oadd__decreasing, axiom, (! [A:$tType,A2:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ C2) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ B2)))))).
% 2.77/1.59 thf(fact_241_subset__mset_Oadd__increasing, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A2) => ((((subseteq_mset @ A) @ B2) @ C2) => (((subseteq_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2))))))).
% 2.77/1.59 thf(fact_242_subset__mset_Oadd__decreasing2, axiom, (! [A:$tType,C2:(multiset @ A),A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ C2) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ A2) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2)) @ B2)))))).
% 2.77/1.59 thf(fact_243_subset__mset_Oadd__increasing2, axiom, (! [A:$tType,C2:(multiset @ A),B2:(multiset @ A),A2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ C2) => ((((subseteq_mset @ A) @ B2) @ A2) => (((subseteq_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A2) @ C2))))))).
% 2.77/1.59 thf(fact_244_subset__mset_Oadd__nonneg__nonneg, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A2) => ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (((plus_plus @ (multiset @ A)) @ A2) @ B2))))))).
% 2.77/1.59 thf(fact_245_subset__mset_Oadd__nonpos__nonpos, axiom, (! [A:$tType,A2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A2) @ B2)) @ (zero_zero @ (multiset @ A)))))))).
% 2.77/1.59 thf(fact_246_subset__mset_Oadd__nonneg__eq__0__iff, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ X) => ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ Y) => (((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))))).
% 2.77/1.59 thf(fact_247_subset__mset_Oadd__nonpos__eq__0__iff, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((((subseteq_mset @ A) @ X) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ Y) @ (zero_zero @ (multiset @ A))) => (((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))))).
% 2.77/1.59 thf(fact_248_subset__mset_Obot_Oextremum__uniqueI, axiom, (! [A:$tType,A2:(multiset @ A)]: (((((subseteq_mset @ A) @ A2) @ (zero_zero @ (multiset @ A))) => (A2 = (zero_zero @ (multiset @ A))))))).
% 2.77/1.59 thf(fact_249_verit__sum__simplify, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.77/1.59 thf(fact_250_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.77/1.59 thf(fact_251_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.77/1.59 thf(fact_252_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.77/1.59 thf(fact_253_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.77/1.59 thf(fact_254_add__increasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))).
% 2.77/1.59 thf(fact_255_add__decreasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2))))))))).
% 2.77/1.59 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: (((linorder @ A) => (type @ A))))).
% 2.77/1.59 thf(subcl_Orderings_Olinorder___Orderings_Oord, axiom, (! [A:$tType]: (((linorder @ A) => (ord @ A))))).
% 2.77/1.59 thf(subcl_Orderings_Olinorder___Orderings_Oorder, axiom, (! [A:$tType]: (((linorder @ A) => (order @ A))))).
% 2.77/1.59 thf(subcl_Orderings_Olinorder___Orderings_Opreorder, axiom, (! [A:$tType]: (((linorder @ A) => (preorder @ A))))).
% 2.77/1.59 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 2.77/1.59 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 2.77/1.59 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 2.77/1.59 thf(tcon_Set_Oset___Orderings_Opreorder_1, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 2.77/1.59 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 2.77/1.59 thf(tcon_Set_Oset___Orderings_Oord_3, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 2.77/1.59 thf(tcon_HOL_Obool___Orderings_Opreorder_4, axiom, (preorder @ $o)).
% 2.77/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.77/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, (order @ $o)).
% 2.77/1.59 thf(tcon_HOL_Obool___Orderings_Oord_6, axiom, (ord @ $o)).
% 2.77/1.59 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add, axiom, (! [A9:$tType]: (((preorder @ A9) => (ordere779506340up_add @ (multiset @ A9)))))).
% 2.77/1.59 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add, axiom, (! [A9:$tType]: ((cancel1352612707id_add @ (multiset @ A9))))).
% 2.77/1.59 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_7, axiom, (! [A9:$tType]: (((preorder @ A9) => (preorder @ (multiset @ A9)))))).
% 2.77/1.59 thf(tcon_Multiset_Omultiset___Orderings_Oorder_8, axiom, (! [A9:$tType]: (((preorder @ A9) => (order @ (multiset @ A9)))))).
% 2.77/1.59 thf(tcon_Multiset_Omultiset___Orderings_Oord_9, axiom, (! [A9:$tType]: (((preorder @ A9) => (ord @ (multiset @ A9)))))).
% 2.77/1.59 thf(tfree_0, hypothesis, (linorder @ a)).
% 2.77/1.59 thf(conj_0, conjecture, (((multiset2 @ a) @ ((heapIm748920189ftDown @ a) @ ((((t @ a) @ v) @ ((((t @ a) @ v1) @ l1) @ r1)) @ ((((t @ a) @ v2) @ l2) @ r2)))) = ((multiset2 @ a) @ ((((t @ a) @ v) @ ((((t @ a) @ v1) @ l1) @ r1)) @ ((((t @ a) @ v2) @ l2) @ r2))))).
% 2.77/1.59 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2DTF20638.p
% 2.77/1.59 ------------------------
% 3.10/1.63 ---- Cleaned THF ---
% 3.10/1.63 thf(ty_t_Product__Type_Oprod,type,
% 3.10/1.63 product_prod: $tType > $tType > $tType ).
% 3.10/1.63
% 3.10/1.63 thf(ty_t_Multiset_Omultiset,type,
% 3.10/1.63 multiset: $tType > $tType ).
% 3.10/1.63
% 3.10/1.63 thf(ty_t_List_Olist,type,
% 3.10/1.63 list: $tType > $tType ).
% 3.10/1.63
% 3.10/1.63 thf(ty_t_Heap_OTree,type,
% 3.10/1.63 tree: $tType > $tType ).
% 3.10/1.63
% 3.10/1.63 thf(ty_t_Set_Oset,type,
% 3.10/1.63 set: $tType > $tType ).
% 3.10/1.63
% 3.10/1.63 thf(ty_tf_a,type,
% 3.10/1.63 a: $tType ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_HOL_Otype,type,
% 3.10/1.63 type:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Orderings_Oord,type,
% 3.10/1.63 ord:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Orderings_Oorder,type,
% 3.10/1.63 order:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Orderings_Olinorder,type,
% 3.10/1.63 linorder:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Orderings_Opreorder,type,
% 3.10/1.63 preorder:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 3.10/1.63 cancel1352612707id_add:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 3.10/1.63 linord219039673up_add:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 3.10/1.63 ordere216010020id_add:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 3.10/1.63 ordere779506340up_add:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 3.10/1.63 linord1659791738miring:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.10/1.63 canoni770627133id_add:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 3.10/1.63 ordere236663937imp_le:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 3.10/1.63 unique1598680935umeral:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 3.10/1.63 ordere516151231imp_le:
% 3.10/1.63 !>[A: $tType] : $o ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.10/1.63 bNF_Ca1785829860lChain:
% 3.10/1.63 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 3.10/1.63 unique455577585es_aux:
% 3.10/1.63 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 3.10/1.63 plus_plus:
% 3.10/1.63 !>[A: $tType] : ( A > A > A ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.10/1.63 zero_zero:
% 3.10/1.63 !>[A: $tType] : A ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oheapify,type,
% 3.10/1.63 heapIm818251801eapify:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__is__empty,type,
% 3.10/1.63 heapIm721255937_empty:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__of__list,type,
% 3.10/1.63 heapIm874063447f_list:
% 3.10/1.63 !>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oleft,type,
% 3.10/1.63 heapIm1271749598e_left:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oof__list__tree,type,
% 3.10/1.63 heapIm1912108042t_tree:
% 3.10/1.63 !>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OremoveLeaf,type,
% 3.10/1.63 heapIm970386777veLeaf:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( product_prod @ A @ ( tree @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oright,type,
% 3.10/1.63 heapIm1434396069_right:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OsiftDown,type,
% 3.10/1.63 heapIm748920189ftDown:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_OHeap,type,
% 3.10/1.63 heap:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_OHeap__axioms,type,
% 3.10/1.63 heap_axioms:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_OTree_OE,type,
% 3.10/1.63 e:
% 3.10/1.63 !>[A: $tType] : ( tree @ A ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_OTree_OT,type,
% 3.10/1.63 t:
% 3.10/1.63 !>[A: $tType] : ( A > ( tree @ A ) > ( tree @ A ) > ( tree @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_Oin__tree,type,
% 3.10/1.63 in_tree:
% 3.10/1.63 !>[A: $tType] : ( A > ( tree @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_Ois__heap,type,
% 3.10/1.63 is_heap:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_Omultiset,type,
% 3.10/1.63 multiset2:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > ( multiset @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Heap_Oval,type,
% 3.10/1.63 val:
% 3.10/1.63 !>[A: $tType] : ( ( tree @ A ) > A ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 3.10/1.63 lattic929149872er_Max:
% 3.10/1.63 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_List_Olist_OCons,type,
% 3.10/1.63 cons:
% 3.10/1.63 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Multiset_Oadd__mset,type,
% 3.10/1.63 add_mset:
% 3.10/1.63 !>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Multiset_Ois__empty,type,
% 3.10/1.63 is_empty:
% 3.10/1.63 !>[A: $tType] : ( ( multiset @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Multiset_Oset__mset,type,
% 3.10/1.63 set_mset:
% 3.10/1.63 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Multiset_Osubseteq__mset,type,
% 3.10/1.63 subseteq_mset:
% 3.10/1.63 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.10/1.63 ord_less_eq:
% 3.10/1.63 !>[A: $tType] : ( A > A > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.10/1.63 order_Greatest:
% 3.10/1.63 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Product__Type_OPair,type,
% 3.10/1.63 product_Pair:
% 3.10/1.63 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.10/1.63 produc2004651681e_prod:
% 3.10/1.63 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.10/1.63 product_rec_prod:
% 3.10/1.63 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_OCollection,type,
% 3.10/1.63 collection:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_OCollection_Oset,type,
% 3.10/1.63 set2:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( ( B > ( multiset @ A ) ) > B > ( set @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_ORemoveMax,type,
% 3.10/1.63 removeMax:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H,type,
% 3.10/1.63 ssort:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( B > ( product_prod @ A @ B ) ) > B > ( list @ A ) > ( list @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom,type,
% 3.10/1.63 ssort_dom:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( B > ( product_prod @ A @ B ) ) > ( product_prod @ B @ ( list @ A ) ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_RemoveMax_ORemoveMax__axioms,type,
% 3.10/1.63 removeMax_axioms:
% 3.10/1.63 !>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_Set_OCollect,type,
% 3.10/1.63 collect:
% 3.10/1.63 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_c_member,type,
% 3.10/1.63 member:
% 3.10/1.63 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_l1____,type,
% 3.10/1.63 l1: tree @ a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_l2____,type,
% 3.10/1.63 l2: tree @ a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_r1____,type,
% 3.10/1.63 r1: tree @ a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_r2____,type,
% 3.10/1.63 r2: tree @ a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_v1____,type,
% 3.10/1.63 v1: a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_v2____,type,
% 3.10/1.63 v2: a ).
% 3.10/1.63
% 3.10/1.63 thf(sy_v_v____,type,
% 3.10/1.63 v: a ).
% 3.10/1.63
% 3.10/1.63 thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062,axiom,
% 3.10/1.63 ord_less_eq @ a @ v2 @ v1 ).
% 3.10/1.63
% 3.10/1.63 thf(fact_1_True,axiom,
% 3.10/1.63 ord_less_eq @ a @ v1 @ v ).
% 3.10/1.63
% 3.10/1.63 thf(fact_2__C5__2_Ohyps_C_I2_J,axiom,
% 3.10/1.63 ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
% 3.10/1.63 => ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ v )
% 3.10/1.63 => ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) )
% 3.10/1.63 = ( multiset2 @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_3__C5__2_Ohyps_C_I1_J,axiom,
% 3.10/1.63 ( ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
% 3.10/1.63 => ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ v )
% 3.10/1.63 => ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) )
% 3.10/1.63 = ( multiset2 @ a @ ( t @ a @ v @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_4_siftDown__Node,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [T2: tree @ A,V: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( T2
% 3.10/1.63 = ( t @ A @ V @ L @ R ) )
% 3.10/1.63 => ? [L2: tree @ A,V2: A,R2: tree @ A] :
% 3.10/1.63 ( ( ( heapIm748920189ftDown @ A @ T2 )
% 3.10/1.63 = ( t @ A @ V2 @ L2 @ R2 ) )
% 3.10/1.63 & ( ord_less_eq @ A @ V @ V2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_5_Tree_Oinject,axiom,
% 3.10/1.63 ! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A,Y21: A,Y22: tree @ A,Y23: tree @ A] :
% 3.10/1.63 ( ( ( t @ A @ X21 @ X22 @ X23 )
% 3.10/1.63 = ( t @ A @ Y21 @ Y22 @ Y23 ) )
% 3.10/1.63 = ( ( X21 = Y21 )
% 3.10/1.63 & ( X22 = Y22 )
% 3.10/1.63 & ( X23 = Y23 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_6_heapify_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( heapIm818251801eapify @ A @ ( t @ A @ V @ L @ R ) )
% 3.10/1.63 = ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm818251801eapify @ A @ L ) @ ( heapIm818251801eapify @ A @ R ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_7_siftDown__heap__is__heap,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [L: tree @ A,R: tree @ A,T2: tree @ A,V: A] :
% 3.10/1.63 ( ( is_heap @ A @ L )
% 3.10/1.63 => ( ( is_heap @ A @ R )
% 3.10/1.63 => ( ( T2
% 3.10/1.63 = ( t @ A @ V @ L @ R ) )
% 3.10/1.63 => ( is_heap @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_8_siftDown__in__tree__set,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( in_tree @ A )
% 3.10/1.63 = ( ^ [V3: A,T3: tree @ A] : ( in_tree @ A @ V3 @ ( heapIm748920189ftDown @ A @ T3 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_9_left_Osimps,axiom,
% 3.10/1.63 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( heapIm1271749598e_left @ A @ ( t @ A @ V @ L @ R ) )
% 3.10/1.63 = L ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_10_right_Osimps,axiom,
% 3.10/1.63 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( heapIm1434396069_right @ A @ ( t @ A @ V @ L @ R ) )
% 3.10/1.63 = R ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_11_siftDown_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A] :
% 3.10/1.63 ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_12_siftDown_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
% 3.10/1.63 = ( e @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_13_Heap_Omultiset,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( ( Multiset @ L )
% 3.10/1.63 = ( multiset2 @ A @ ( As_tree @ L ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_14_in__tree_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType,V: A] :
% 3.10/1.63 ~ ( in_tree @ A @ V @ ( e @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_15_is__heap_Osimps_I6_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 3.10/1.63 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) )
% 3.10/1.63 & ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_16_is__heap_Osimps_I5_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
% 3.10/1.63 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) )
% 3.10/1.63 & ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_17_is__heap_Osimps_I4_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 3.10/1.63 ( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 3.10/1.63 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_18_is__heap_Osimps_I3_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
% 3.10/1.63 ( ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 & ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_19_is__heap_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A] : ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_20_is__heap_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( is_heap @ A @ ( e @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_21_heapify_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( heapIm818251801eapify @ A @ ( e @ A ) )
% 3.10/1.63 = ( e @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_22_is__heap__max,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,T2: tree @ A] :
% 3.10/1.63 ( ( in_tree @ A @ V @ T2 )
% 3.10/1.63 => ( ( is_heap @ A @ T2 )
% 3.10/1.63 => ( ord_less_eq @ A @ V @ ( val @ A @ T2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_23_Heap_Oas__tree__empty,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T2: B] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( ( ( As_tree @ T2 )
% 3.10/1.63 = ( e @ A ) )
% 3.10/1.63 = ( Is_empty @ T2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_24_Heap_Ois__heap__of__list,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),I: list @ A] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( is_heap @ A @ ( As_tree @ ( Of_list @ I ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_25_heapify__heap__is__heap,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [T2: tree @ A] : ( is_heap @ A @ ( heapIm818251801eapify @ A @ T2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_26_siftDown__in__tree,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [T2: tree @ A] :
% 3.10/1.63 ( ( T2
% 3.10/1.63 != ( e @ A ) )
% 3.10/1.63 => ( in_tree @ A @ ( val @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) @ T2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_27_is__heap_Ocases,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: tree @ A] :
% 3.10/1.63 ( ( X
% 3.10/1.63 != ( e @ A ) )
% 3.10/1.63 => ( ! [V4: A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 3.10/1.63 => ~ ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_28_Tree_Oexhaust,axiom,
% 3.10/1.63 ! [A: $tType,Y: tree @ A] :
% 3.10/1.63 ( ( Y
% 3.10/1.63 != ( e @ A ) )
% 3.10/1.63 => ~ ! [X212: A,X222: tree @ A,X232: tree @ A] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( t @ A @ X212 @ X222 @ X232 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_29_Tree_Oinduct,axiom,
% 3.10/1.63 ! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
% 3.10/1.63 ( ( P @ ( e @ A ) )
% 3.10/1.63 => ( ! [X1: A,X2: tree @ A,X3: tree @ A] :
% 3.10/1.63 ( ( P @ X2 )
% 3.10/1.63 => ( ( P @ X3 )
% 3.10/1.63 => ( P @ ( t @ A @ X1 @ X2 @ X3 ) ) ) )
% 3.10/1.63 => ( P @ Tree ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_30_Tree_Odistinct_I1_J,axiom,
% 3.10/1.63 ! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A] :
% 3.10/1.63 ( ( e @ A )
% 3.10/1.63 != ( t @ A @ X21 @ X22 @ X23 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_31_siftDown_Osimps_I3_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( e @ A ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_32_siftDown_Osimps_I4_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( e @ A ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_33_val_Osimps,axiom,
% 3.10/1.63 ! [A: $tType,V: A,Uu: tree @ A,Uv: tree @ A] :
% 3.10/1.63 ( ( val @ A @ ( t @ A @ V @ Uu @ Uv ) )
% 3.10/1.63 = V ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_34_siftDown_Osimps_I5_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( t @ A @ Va @ Vb @ Vc ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_35_siftDown_Osimps_I6_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A,V: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
% 3.10/1.63 & ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
% 3.10/1.63 => ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
% 3.10/1.63 = ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( t @ A @ Vd @ Ve @ Vf ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_36_in__tree_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType,V: A,V5: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( in_tree @ A @ V @ ( t @ A @ V5 @ L @ R ) )
% 3.10/1.63 = ( ( V = V5 )
% 3.10/1.63 | ( in_tree @ A @ V @ L )
% 3.10/1.63 | ( in_tree @ A @ V @ R ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_37_removeLeaf_Oinduct,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [P: ( tree @ A ) > $o,A0: tree @ A] :
% 3.10/1.63 ( ! [V4: A] : ( P @ ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( P @ ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( P @ ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 3.10/1.63 ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( ( P @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
% 3.10/1.63 => ( P @ ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 3.10/1.63 => ( ! [V4: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( ( P @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
% 3.10/1.63 => ( ( P @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
% 3.10/1.63 => ( P @ ( t @ A @ V4 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 3.10/1.63 => ( ( P @ ( e @ A ) )
% 3.10/1.63 => ( P @ A0 ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_38_removeLeaf_Ocases,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: tree @ A] :
% 3.10/1.63 ( ! [V4: A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
% 3.10/1.63 => ( ! [V4: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 3.10/1.63 => ( X
% 3.10/1.63 = ( e @ A ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_39_siftDown_Ocases,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: tree @ A] :
% 3.10/1.63 ( ( X
% 3.10/1.63 != ( e @ A ) )
% 3.10/1.63 => ( ! [V4: A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
% 3.10/1.63 => ( ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
% 3.10/1.63 => ~ ! [V4: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
% 3.10/1.63 ( X
% 3.10/1.63 != ( t @ A @ V4 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_40_hs__is__empty__def,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( heapIm721255937_empty @ A )
% 3.10/1.63 = ( ^ [T3: tree @ A] :
% 3.10/1.63 ( T3
% 3.10/1.63 = ( e @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_41_order__refl,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( preorder @ A )
% 3.10/1.63 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_42_heap__top__geq,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [A2: A,T2: tree @ A] :
% 3.10/1.63 ( ( member @ A @ A2 @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) )
% 3.10/1.63 => ( ( is_heap @ A @ T2 )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ ( val @ A @ T2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_43_Heap_Oaxioms_I2_J,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_44_hs__of__list__def,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( heapIm874063447f_list @ A )
% 3.10/1.63 = ( ^ [L3: list @ A] : ( heapIm818251801eapify @ A @ ( heapIm1912108042t_tree @ A @ L3 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_45_mem__Collect__eq,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,P: A > $o] :
% 3.10/1.63 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.10/1.63 = ( P @ A2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_46_Collect__mem__eq,axiom,
% 3.10/1.63 ! [A: $tType,A3: set @ A] :
% 3.10/1.63 ( ( collect @ A
% 3.10/1.63 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 3.10/1.63 = A3 ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_47_Collect__cong,axiom,
% 3.10/1.63 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.10/1.63 ( ! [X5: A] :
% 3.10/1.63 ( ( P @ X5 )
% 3.10/1.63 = ( Q @ X5 ) )
% 3.10/1.63 => ( ( collect @ A @ P )
% 3.10/1.63 = ( collect @ A @ Q ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_48_ext,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.10/1.63 ( ! [X5: A] :
% 3.10/1.63 ( ( F @ X5 )
% 3.10/1.63 = ( G @ X5 ) )
% 3.10/1.63 => ( F = G ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_49_le__funD,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( ord @ B )
% 3.10/1.63 => ! [F: A > B,G: A > B,X: A] :
% 3.10/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.10/1.63 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_50_le__funE,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( ord @ B )
% 3.10/1.63 => ! [F: A > B,G: A > B,X: A] :
% 3.10/1.63 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.10/1.63 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_51_le__funI,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( ord @ B )
% 3.10/1.63 => ! [F: A > B,G: A > B] :
% 3.10/1.63 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 3.10/1.63 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_52_le__fun__def,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( ord @ B )
% 3.10/1.63 => ( ( ord_less_eq @ ( A > B ) )
% 3.10/1.63 = ( ^ [F2: A > B,G2: A > B] :
% 3.10/1.63 ! [X4: A] : ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_53_order__subst1,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( ( order @ B )
% 3.10/1.63 & ( order @ A ) )
% 3.10/1.63 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.10/1.63 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.10/1.63 => ( ! [X5: B,Y2: B] :
% 3.10/1.63 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_54_order__subst2,axiom,
% 3.10/1.63 ! [A: $tType,C: $tType] :
% 3.10/1.63 ( ( ( order @ C )
% 3.10/1.63 & ( order @ A ) )
% 3.10/1.63 => ! [A2: A,B2: A,F: A > C,C2: C] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 3.10/1.63 => ( ! [X5: A,Y2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 3.10/1.63 => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_55_dual__order_Oantisym,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [B2: A,A2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( A2 = B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_56_dual__order_Oeq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [A4: A,B3: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ B3 @ A4 )
% 3.10/1.63 & ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_57_dual__order_Otrans,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [B2: A,A2: A,C2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.10/1.63 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_58_linorder__wlog,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.10/1.63 ( ! [A5: A,B4: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A5 @ B4 )
% 3.10/1.63 => ( P @ A5 @ B4 ) )
% 3.10/1.63 => ( ! [A5: A,B4: A] :
% 3.10/1.63 ( ( P @ B4 @ A5 )
% 3.10/1.63 => ( P @ A5 @ B4 ) )
% 3.10/1.63 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_59_dual__order_Orefl,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_60_order__trans,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( preorder @ A )
% 3.10/1.63 => ! [X: A,Y: A,Z2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.10/1.63 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_61_order__class_Oorder_Oantisym,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.10/1.63 => ( A2 = B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_62_ord__le__eq__trans,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ord @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( B2 = C2 )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_63_ord__eq__le__trans,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ord @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.10/1.63 ( ( A2 = B2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_64_order__class_Oorder_Oeq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [A4: A,B3: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A4 @ B3 )
% 3.10/1.63 & ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_65_antisym__conv,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [Y: A,X: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ Y @ X )
% 3.10/1.63 => ( ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 = ( X = Y ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_66_le__cases3,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: A,Y: A,Z2: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.10/1.63 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 3.10/1.63 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 3.10/1.63 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_67_order_Otrans,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_68_le__cases,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_69_eq__refl,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( preorder @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ( X = Y )
% 3.10/1.63 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_70_linear,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_71_antisym,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X @ Y )
% 3.10/1.63 => ( ( ord_less_eq @ A @ Y @ X )
% 3.10/1.63 => ( X = Y ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_72_eq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [X4: A,Y4: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.10/1.63 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_73_ord__le__eq__subst,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( ( ord @ B )
% 3.10/1.63 & ( ord @ A ) )
% 3.10/1.63 => ! [A2: A,B2: A,F: A > B,C2: B] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( ( F @ B2 )
% 3.10/1.63 = C2 )
% 3.10/1.63 => ( ! [X5: A,Y2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X5 @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 3.10/1.63 => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_74_ord__eq__le__subst,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( ( ord @ B )
% 3.10/1.63 & ( ord @ A ) )
% 3.10/1.63 => ! [A2: A,F: B > A,B2: B,C2: B] :
% 3.10/1.63 ( ( A2
% 3.10/1.63 = ( F @ B2 ) )
% 3.10/1.63 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.10/1.63 => ( ! [X5: B,Y2: B] :
% 3.10/1.63 ( ( ord_less_eq @ B @ X5 @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_75_heap__top__max,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [T2: tree @ A] :
% 3.10/1.63 ( ( T2
% 3.10/1.63 != ( e @ A ) )
% 3.10/1.63 => ( ( is_heap @ A @ T2 )
% 3.10/1.63 => ( ( val @ A @ T2 )
% 3.10/1.63 = ( lattic929149872er_Max @ A @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_76_Heap_Ointro,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_77_Heap__def,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( heap @ B @ A )
% 3.10/1.63 = ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
% 3.10/1.63 & ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_78_of__list__tree_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A,Tail: list @ A] :
% 3.10/1.63 ( ( heapIm1912108042t_tree @ A @ ( cons @ A @ V @ Tail ) )
% 3.10/1.63 = ( t @ A @ V @ ( heapIm1912108042t_tree @ A @ Tail ) @ ( e @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_79_Greatest__equality,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [P: A > $o,X: A] :
% 3.10/1.63 ( ( P @ X )
% 3.10/1.63 => ( ! [Y2: A] :
% 3.10/1.63 ( ( P @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ A @ Y2 @ X ) )
% 3.10/1.63 => ( ( order_Greatest @ A @ P )
% 3.10/1.63 = X ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_80_GreatestI2__order,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( order @ A )
% 3.10/1.63 => ! [P: A > $o,X: A,Q: A > $o] :
% 3.10/1.63 ( ( P @ X )
% 3.10/1.63 => ( ! [Y2: A] :
% 3.10/1.63 ( ( P @ Y2 )
% 3.10/1.63 => ( ord_less_eq @ A @ Y2 @ X ) )
% 3.10/1.63 => ( ! [X5: A] :
% 3.10/1.63 ( ( P @ X5 )
% 3.10/1.63 => ( ! [Y5: A] :
% 3.10/1.63 ( ( P @ Y5 )
% 3.10/1.63 => ( ord_less_eq @ A @ Y5 @ X5 ) )
% 3.10/1.63 => ( Q @ X5 ) ) )
% 3.10/1.63 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_81_le__rel__bool__arg__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ord @ A )
% 3.10/1.63 => ( ( ord_less_eq @ ( $o > A ) )
% 3.10/1.63 = ( ^ [X6: $o > A,Y6: $o > A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 3.10/1.63 & ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_82_verit__la__disequality,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( A2 = B2 )
% 3.10/1.63 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_83_Heap_Oaxioms_I1_J,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_84_list_Oinject,axiom,
% 3.10/1.63 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 3.10/1.63 ( ( ( cons @ A @ X21 @ X22 )
% 3.10/1.63 = ( cons @ A @ Y21 @ Y22 ) )
% 3.10/1.63 = ( ( X21 = Y21 )
% 3.10/1.63 & ( X22 = Y22 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_85_Collection_Ois__empty__inj,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E: B] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( Is_empty @ E )
% 3.10/1.63 => ( E = Empty ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_86_Collection_Ois__empty__empty,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( Is_empty @ Empty ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_87_not__Cons__self2,axiom,
% 3.10/1.63 ! [A: $tType,X: A,Xs: list @ A] :
% 3.10/1.63 ( ( cons @ A @ X @ Xs )
% 3.10/1.63 != Xs ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_88_Heap__axioms__def,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( heap_axioms @ B @ A )
% 3.10/1.63 = ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ! [L3: B] :
% 3.10/1.63 ( ( Multiset2 @ L3 )
% 3.10/1.63 = ( multiset2 @ A @ ( As_tree2 @ L3 ) ) )
% 3.10/1.63 & ! [I2: list @ A] : ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I2 ) ) )
% 3.10/1.63 & ! [T3: B] :
% 3.10/1.63 ( ( ( As_tree2 @ T3 )
% 3.10/1.63 = ( e @ A ) )
% 3.10/1.63 = ( Is_empty2 @ T3 ) )
% 3.10/1.63 & ! [L3: B,M: A,L4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ L3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 3.10/1.63 = ( Remove_max2 @ L3 ) )
% 3.10/1.63 => ( ( add_mset @ A @ M @ ( Multiset2 @ L4 ) )
% 3.10/1.63 = ( Multiset2 @ L3 ) ) ) )
% 3.10/1.63 & ! [L3: B,M: A,L4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ L3 )
% 3.10/1.63 => ( ( is_heap @ A @ ( As_tree2 @ L3 ) )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 3.10/1.63 = ( Remove_max2 @ L3 ) )
% 3.10/1.63 => ( is_heap @ A @ ( As_tree2 @ L4 ) ) ) ) )
% 3.10/1.63 & ! [T3: B,M: A,T4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ T3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ T4 )
% 3.10/1.63 = ( Remove_max2 @ T3 ) )
% 3.10/1.63 => ( M
% 3.10/1.63 = ( val @ A @ ( As_tree2 @ T3 ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_89_Heap__axioms_Ointro,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B )] :
% 3.10/1.63 ( ! [L5: B] :
% 3.10/1.63 ( ( Multiset @ L5 )
% 3.10/1.63 = ( multiset2 @ A @ ( As_tree @ L5 ) ) )
% 3.10/1.63 => ( ! [I3: list @ A] : ( is_heap @ A @ ( As_tree @ ( Of_list @ I3 ) ) )
% 3.10/1.63 => ( ! [T5: B] :
% 3.10/1.63 ( ( ( As_tree @ T5 )
% 3.10/1.63 = ( e @ A ) )
% 3.10/1.63 = ( Is_empty @ T5 ) )
% 3.10/1.63 => ( ! [L5: B,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( ( add_mset @ A @ M2 @ ( Multiset @ L2 ) )
% 3.10/1.63 = ( Multiset @ L5 ) ) ) )
% 3.10/1.63 => ( ! [L5: B,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( is_heap @ A @ ( As_tree @ L5 ) )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( is_heap @ A @ ( As_tree @ L2 ) ) ) ) )
% 3.10/1.63 => ( ! [T5: B,M2: A,T6: B] :
% 3.10/1.63 ( ~ ( Is_empty @ T5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ T6 )
% 3.10/1.63 = ( Remove_max @ T5 ) )
% 3.10/1.63 => ( M2
% 3.10/1.63 = ( val @ A @ ( As_tree @ T5 ) ) ) ) )
% 3.10/1.63 => ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_90_Heap_Oremove__max__multiset_H,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M3: A,L6: B] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( ( add_mset @ A @ M3 @ ( Multiset @ L6 ) )
% 3.10/1.63 = ( Multiset @ L ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_91_Heap_Oremove__max__val,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T2: B,M3: A,T7: B] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( ~ ( Is_empty @ T2 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ T7 )
% 3.10/1.63 = ( Remove_max @ T2 ) )
% 3.10/1.63 => ( M3
% 3.10/1.63 = ( val @ A @ ( As_tree @ T2 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_92_Heap_Oremove__max__is__heap,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M3: A,L6: B] :
% 3.10/1.63 ( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
% 3.10/1.63 => ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( is_heap @ A @ ( As_tree @ L ) )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( is_heap @ A @ ( As_tree @ L6 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_93_multi__self__add__other__not__self,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,X: A] :
% 3.10/1.63 ( M4
% 3.10/1.63 != ( add_mset @ A @ X @ M4 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_94_add__mset__add__mset__same__iff,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( ( add_mset @ A @ A2 @ A3 )
% 3.10/1.63 = ( add_mset @ A @ A2 @ B5 ) )
% 3.10/1.63 = ( A3 = B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_95_old_Oprod_Oinject,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 3.10/1.63 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.10/1.63 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 3.10/1.63 = ( ( A2 = A6 )
% 3.10/1.63 & ( B2 = B6 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_96_prod_Oinject,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y24: B] :
% 3.10/1.63 ( ( ( product_Pair @ A @ B @ X12 @ X24 )
% 3.10/1.63 = ( product_Pair @ A @ B @ Y1 @ Y24 ) )
% 3.10/1.63 = ( ( X12 = Y1 )
% 3.10/1.63 & ( X24 = Y24 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_97_mset__add,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A] :
% 3.10/1.63 ( ( member @ A @ A2 @ ( set_mset @ A @ A3 ) )
% 3.10/1.63 => ~ ! [B7: multiset @ A] :
% 3.10/1.63 ( A3
% 3.10/1.63 != ( add_mset @ A @ A2 @ B7 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_98_multi__member__split,axiom,
% 3.10/1.63 ! [A: $tType,X: A,M4: multiset @ A] :
% 3.10/1.63 ( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
% 3.10/1.63 => ? [A7: multiset @ A] :
% 3.10/1.63 ( M4
% 3.10/1.63 = ( add_mset @ A @ X @ A7 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_99_surj__pair,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.10/1.63 ? [X5: A,Y2: B] :
% 3.10/1.63 ( P2
% 3.10/1.63 = ( product_Pair @ A @ B @ X5 @ Y2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_100_prod__cases,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.10/1.63 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 3.10/1.63 => ( P @ P2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_101_Pair__inject,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 3.10/1.63 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.10/1.63 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 3.10/1.63 => ~ ( ( A2 = A6 )
% 3.10/1.63 => ( B2 != B6 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_102_prod__cases3,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.10/1.63 ~ ! [A5: A,B4: B,C3: C] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_103_prod__cases4,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.10/1.63 ~ ! [A5: A,B4: B,C3: C,D2: D] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_104_prod__cases5,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 3.10/1.63 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_105_prod__cases6,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 3.10/1.63 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D2 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_106_prod__cases7,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 3.10/1.63 ~ ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3,G4: G3] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_107_prod__induct3,axiom,
% 3.10/1.63 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.10/1.63 ( ! [A5: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
% 3.10/1.63 => ( P @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_108_prod__induct4,axiom,
% 3.10/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.10/1.63 ( ! [A5: A,B4: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 3.10/1.63 => ( P @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_109_prod__induct5,axiom,
% 3.10/1.63 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
% 3.10/1.63 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) )
% 3.10/1.63 => ( P @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_110_prod__induct6,axiom,
% 3.10/1.63 ! [F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) )] :
% 3.10/1.63 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F3 ) @ D2 @ ( product_Pair @ E2 @ F3 @ E3 @ F4 ) ) ) ) ) )
% 3.10/1.63 => ( P @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_111_prod__induct7,axiom,
% 3.10/1.63 ! [G3: $tType,F3: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 3.10/1.63 ( ! [A5: A,B4: B,C3: C,D2: D,E3: E2,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 3.10/1.63 => ( P @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_112_old_Oprod_Oexhaust,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.10/1.63 ~ ! [A5: A,B4: B] :
% 3.10/1.63 ( Y
% 3.10/1.63 != ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_113_old_Oprod_Oinducts,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.10/1.63 ( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
% 3.10/1.63 => ( P @ Prod ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_114_add__eq__conv__ex,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,M4: multiset @ A,B2: A,N: multiset @ A] :
% 3.10/1.63 ( ( ( add_mset @ A @ A2 @ M4 )
% 3.10/1.63 = ( add_mset @ A @ B2 @ N ) )
% 3.10/1.63 = ( ( ( M4 = N )
% 3.10/1.63 & ( A2 = B2 ) )
% 3.10/1.63 | ? [K: multiset @ A] :
% 3.10/1.63 ( ( M4
% 3.10/1.63 = ( add_mset @ A @ B2 @ K ) )
% 3.10/1.63 & ( N
% 3.10/1.63 = ( add_mset @ A @ A2 @ K ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_115_add__mset__commute,axiom,
% 3.10/1.63 ! [A: $tType,X: A,Y: A,M4: multiset @ A] :
% 3.10/1.63 ( ( add_mset @ A @ X @ ( add_mset @ A @ Y @ M4 ) )
% 3.10/1.63 = ( add_mset @ A @ Y @ ( add_mset @ A @ X @ M4 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_116_union__single__eq__member,axiom,
% 3.10/1.63 ! [A: $tType,X: A,M4: multiset @ A,N: multiset @ A] :
% 3.10/1.63 ( ( ( add_mset @ A @ X @ M4 )
% 3.10/1.63 = N )
% 3.10/1.63 => ( member @ A @ X @ ( set_mset @ A @ N ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_117_insert__noteq__member,axiom,
% 3.10/1.63 ! [A: $tType,B2: A,B5: multiset @ A,C2: A,C4: multiset @ A] :
% 3.10/1.63 ( ( ( add_mset @ A @ B2 @ B5 )
% 3.10/1.63 = ( add_mset @ A @ C2 @ C4 ) )
% 3.10/1.63 => ( ( B2 != C2 )
% 3.10/1.63 => ( member @ A @ C2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_118_old_Oprod_Orec,axiom,
% 3.10/1.63 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.10/1.63 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.10/1.63 = ( F1 @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_119_internal__case__prod__conv,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 3.10/1.63 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.10/1.63 = ( C2 @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_120_Collection_Oset__def,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),L: B] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( set2 @ B @ A @ Multiset @ L )
% 3.10/1.63 = ( set_mset @ A @ ( Multiset @ L ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_121_relChain__def,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( ord @ B )
% 3.10/1.63 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.10/1.63 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.10/1.63 ! [I2: A,J: A] :
% 3.10/1.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
% 3.10/1.63 => ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_122_removeLeaf_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [V: A] :
% 3.10/1.63 ( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
% 3.10/1.63 = ( product_Pair @ A @ ( tree @ A ) @ V @ ( e @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_123_Collection_Oset_Ocong,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( set2 @ B @ A )
% 3.10/1.63 = ( set2 @ B @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_124_RemoveMax__axioms__def,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( removeMax_axioms @ B @ A )
% 3.10/1.63 = ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv: B > $o] :
% 3.10/1.63 ( ! [X4: list @ A] : ( Inv @ ( Of_list2 @ X4 ) )
% 3.10/1.63 & ! [L3: B,M: A,L4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ L3 )
% 3.10/1.63 => ( ( Inv @ L3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 3.10/1.63 = ( Remove_max2 @ L3 ) )
% 3.10/1.63 => ( M
% 3.10/1.63 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset2 @ L3 ) ) ) ) ) )
% 3.10/1.63 & ! [L3: B,M: A,L4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ L3 )
% 3.10/1.63 => ( ( Inv @ L3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 3.10/1.63 = ( Remove_max2 @ L3 ) )
% 3.10/1.63 => ( ( add_mset @ A @ M @ ( Multiset2 @ L4 ) )
% 3.10/1.63 = ( Multiset2 @ L3 ) ) ) ) )
% 3.10/1.63 & ! [L3: B,M: A,L4: B] :
% 3.10/1.63 ( ~ ( Is_empty2 @ L3 )
% 3.10/1.63 => ( ( Inv @ L3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M @ L4 )
% 3.10/1.63 = ( Remove_max2 @ L3 ) )
% 3.10/1.63 => ( Inv @ L4 ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_125_RemoveMax__axioms_Ointro,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Inv2: B > $o,Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B ),Multiset: B > ( multiset @ A )] :
% 3.10/1.63 ( ! [X5: list @ A] : ( Inv2 @ ( Of_list @ X5 ) )
% 3.10/1.63 => ( ! [L5: B,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( Inv2 @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( M2
% 3.10/1.63 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L5 ) ) ) ) ) )
% 3.10/1.63 => ( ! [L5: B,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( Inv2 @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( ( add_mset @ A @ M2 @ ( Multiset @ L2 ) )
% 3.10/1.63 = ( Multiset @ L5 ) ) ) ) )
% 3.10/1.63 => ( ! [L5: B,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( Inv2 @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( Inv2 @ L2 ) ) ) )
% 3.10/1.63 => ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_126_RemoveMax_Oremove__max__max,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( Inv2 @ L )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( M3
% 3.10/1.63 = ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_127_multiset__induct__max,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 3.10/1.63 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ! [X5: A,M5: multiset @ A] :
% 3.10/1.63 ( ( P @ M5 )
% 3.10/1.63 => ( ! [Xa: A] :
% 3.10/1.63 ( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
% 3.10/1.63 => ( ord_less_eq @ A @ Xa @ X5 ) )
% 3.10/1.63 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
% 3.10/1.63 => ( P @ M4 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_128_add__mset__eq__singleton__iff,axiom,
% 3.10/1.63 ! [A: $tType,X: A,M4: multiset @ A,Y: A] :
% 3.10/1.63 ( ( ( add_mset @ A @ X @ M4 )
% 3.10/1.63 = ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
% 3.10/1.63 = ( ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( X = Y ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_129_single__eq__add__mset,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,B2: A,M4: multiset @ A] :
% 3.10/1.63 ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( add_mset @ A @ B2 @ M4 ) )
% 3.10/1.63 = ( ( B2 = A2 )
% 3.10/1.63 & ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_130_add__mset__eq__single,axiom,
% 3.10/1.63 ! [A: $tType,B2: A,M4: multiset @ A,A2: A] :
% 3.10/1.63 ( ( ( add_mset @ A @ B2 @ M4 )
% 3.10/1.63 = ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 3.10/1.63 = ( ( B2 = A2 )
% 3.10/1.63 & ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_131_single__eq__single,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,B2: A] :
% 3.10/1.63 ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 3.10/1.63 = ( A2 = B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_132_RemoveMax__def,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( removeMax @ B @ A )
% 3.10/1.63 = ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv: B > $o] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
% 3.10/1.63 & ( removeMax_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_133_RemoveMax_Ointro,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_134_multiset__cases,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A] :
% 3.10/1.63 ( ( M4
% 3.10/1.63 != ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ~ ! [X5: A,N2: multiset @ A] :
% 3.10/1.63 ( M4
% 3.10/1.63 != ( add_mset @ A @ X5 @ N2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_135_multiset__induct,axiom,
% 3.10/1.63 ! [A: $tType,P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 3.10/1.63 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ! [X5: A,M5: multiset @ A] :
% 3.10/1.63 ( ( P @ M5 )
% 3.10/1.63 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) )
% 3.10/1.63 => ( P @ M4 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_136_multiset__induct2,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType,P: ( multiset @ A ) > ( multiset @ B ) > $o,M4: multiset @ A,N: multiset @ B] :
% 3.10/1.63 ( ( P @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
% 3.10/1.63 => ( ! [A5: A,M5: multiset @ A,N2: multiset @ B] :
% 3.10/1.63 ( ( P @ M5 @ N2 )
% 3.10/1.63 => ( P @ ( add_mset @ A @ A5 @ M5 ) @ N2 ) )
% 3.10/1.63 => ( ! [A5: B,M5: multiset @ A,N2: multiset @ B] :
% 3.10/1.63 ( ( P @ M5 @ N2 )
% 3.10/1.63 => ( P @ M5 @ ( add_mset @ B @ A5 @ N2 ) ) )
% 3.10/1.63 => ( P @ M4 @ N ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_137_empty__not__add__mset,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A] :
% 3.10/1.63 ( ( zero_zero @ ( multiset @ A ) )
% 3.10/1.63 != ( add_mset @ A @ A2 @ A3 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_138_multi__nonempty__split,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A] :
% 3.10/1.63 ( ( M4
% 3.10/1.63 != ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ? [A7: multiset @ A,A5: A] :
% 3.10/1.63 ( M4
% 3.10/1.63 = ( add_mset @ A @ A5 @ A7 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_139_multiset__nonemptyE,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A] :
% 3.10/1.63 ( ( A3
% 3.10/1.63 != ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ~ ! [X5: A] :
% 3.10/1.63 ~ ( member @ A @ X5 @ ( set_mset @ A @ A3 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_140_RemoveMax_Oof__list__inv,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,X: list @ A] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( Inv2 @ ( Of_list @ X ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_141_RemoveMax_Oaxioms_I2_J,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_142_RemoveMax_Oremove__max__inv,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( Inv2 @ L )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( Inv2 @ L6 ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_143_Collection_Ois__empty__as__list,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E: B] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( Is_empty @ E )
% 3.10/1.63 => ( ( Multiset @ E )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_144_Collection_Omultiset__empty,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
% 3.10/1.63 ( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
% 3.10/1.63 => ( ( Multiset @ Empty )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_145_RemoveMax_Oaxioms_I1_J,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_146_multi__member__last,axiom,
% 3.10/1.63 ! [A: $tType,X: A] : ( member @ A @ X @ ( set_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_147_RemoveMax_Oremove__max__multiset,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,M3: A,L6: B] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( Inv2 @ L )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M3 @ L6 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( ( add_mset @ A @ M3 @ ( Multiset @ L6 ) )
% 3.10/1.63 = ( Multiset @ L ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_148_multiset_Osimps_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( multiset2 @ A @ ( e @ A ) )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_149_multiset__induct__min,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
% 3.10/1.63 ( ( P @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ! [X5: A,M5: multiset @ A] :
% 3.10/1.63 ( ( P @ M5 )
% 3.10/1.63 => ( ! [Xa: A] :
% 3.10/1.63 ( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
% 3.10/1.63 => ( ord_less_eq @ A @ X5 @ Xa ) )
% 3.10/1.63 => ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
% 3.10/1.63 => ( P @ M4 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_150_le__zero__eq,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( canoni770627133id_add @ A )
% 3.10/1.63 => ! [N3: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ N3 @ ( zero_zero @ A ) )
% 3.10/1.63 = ( N3
% 3.10/1.63 = ( zero_zero @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_151_Multiset_Ois__empty__def,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( is_empty @ A )
% 3.10/1.63 = ( ^ [A8: multiset @ A] :
% 3.10/1.63 ( A8
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_152_le__numeral__extra_I3_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linord1659791738miring @ A )
% 3.10/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_153_zero__le,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( canoni770627133id_add @ A )
% 3.10/1.63 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_154_RemoveMax_Ossort_HInduct,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,P: B > ( list @ A ) > $o,Sl: list @ A] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ( Inv2 @ L )
% 3.10/1.63 => ( ( P @ L @ Sl )
% 3.10/1.63 => ( ! [L5: B,Sl2: list @ A,M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( Inv2 @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( ( P @ L5 @ Sl2 )
% 3.10/1.63 => ( P @ L2 @ ( cons @ A @ M2 @ Sl2 ) ) ) ) ) )
% 3.10/1.63 => ( P @ Empty @ ( ssort @ B @ A @ Is_empty @ Remove_max @ L @ Sl ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_155_RemoveMax_Ossort_H__dom_Ocases,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,A2: product_prod @ B @ ( list @ A )] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ A2 )
% 3.10/1.63 => ~ ! [L5: B,Sl2: list @ A] :
% 3.10/1.63 ( ( A2
% 3.10/1.63 = ( product_Pair @ B @ ( list @ A ) @ L5 @ Sl2 ) )
% 3.10/1.63 => ~ ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ! [M6: A,L7: B] :
% 3.10/1.63 ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_156_RemoveMax_Ossort_H__dom_Osimps,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,A2: product_prod @ B @ ( list @ A )] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ A2 )
% 3.10/1.63 = ( ? [L3: B,Sl3: list @ A] :
% 3.10/1.63 ( ( A2
% 3.10/1.63 = ( product_Pair @ B @ ( list @ A ) @ L3 @ Sl3 ) )
% 3.10/1.63 & ! [X4: A,Y4: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L3 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ X4 @ Y4 )
% 3.10/1.63 = ( Remove_max @ L3 ) )
% 3.10/1.63 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ Y4 @ ( cons @ A @ X4 @ Sl3 ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_157_RemoveMax_Ossort_H__dom_Ocong,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( ssort_dom @ B @ A )
% 3.10/1.63 = ( ssort_dom @ B @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_158_RemoveMax_Ossort_H_Ocong,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ( ssort @ B @ A )
% 3.10/1.63 = ( ssort @ B @ A ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_159_RemoveMax_Ossort_H__dom_Oinducts,axiom,
% 3.10/1.63 ! [A: $tType,B: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,X: product_prod @ B @ ( list @ A ),P: ( product_prod @ B @ ( list @ A ) ) > $o] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ X )
% 3.10/1.63 => ( ! [L5: B,Sl2: list @ A] :
% 3.10/1.63 ( ! [M6: A,L7: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) )
% 3.10/1.63 => ( ! [M6: A,L7: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L5 )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M6 @ L7 )
% 3.10/1.63 = ( Remove_max @ L5 ) )
% 3.10/1.63 => ( P @ ( product_Pair @ B @ ( list @ A ) @ L7 @ ( cons @ A @ M6 @ Sl2 ) ) ) ) )
% 3.10/1.63 => ( P @ ( product_Pair @ B @ ( list @ A ) @ L5 @ Sl2 ) ) ) )
% 3.10/1.63 => ( P @ X ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_160_RemoveMax_Ossort_H__dom_Ointros,axiom,
% 3.10/1.63 ! [B: $tType,A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv2: B > $o,L: B,Sl: list @ A] :
% 3.10/1.63 ( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv2 )
% 3.10/1.63 => ( ! [M2: A,L2: B] :
% 3.10/1.63 ( ~ ( Is_empty @ L )
% 3.10/1.63 => ( ( ( product_Pair @ A @ B @ M2 @ L2 )
% 3.10/1.63 = ( Remove_max @ L ) )
% 3.10/1.63 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L2 @ ( cons @ A @ M2 @ Sl ) ) ) ) )
% 3.10/1.63 => ( ssort_dom @ B @ A @ Is_empty @ Remove_max @ ( product_Pair @ B @ ( list @ A ) @ L @ Sl ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_161_multiset_Osimps_I2_J,axiom,
% 3.10/1.63 ! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
% 3.10/1.63 ( ( multiset2 @ A @ ( t @ A @ V @ L @ R ) )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( multiset2 @ A @ L ) @ ( add_mset @ A @ V @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( multiset2 @ A @ R ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_162_divides__aux__eq,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( unique1598680935umeral @ A )
% 3.10/1.63 => ! [Q2: A,R: A] :
% 3.10/1.63 ( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R ) )
% 3.10/1.63 = ( R
% 3.10/1.63 = ( zero_zero @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_163_single__subset__iff,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,M4: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ M4 )
% 3.10/1.63 = ( member @ A @ A2 @ ( set_mset @ A @ M4 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_164_subset__mset_Oorder__refl,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A] : ( subseteq_mset @ A @ X @ X ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_165_subset__mset_Odual__order_Orefl,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A] : ( subseteq_mset @ A @ A2 @ A2 ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_166_add__le__cancel__left,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere236663937imp_le @ A )
% 3.10/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 3.10/1.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_167_add__le__cancel__right,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere236663937imp_le @ A )
% 3.10/1.63 => ! [A2: A,C2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 3.10/1.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_168_subset__mset_Obot_Oextremum__unique,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( A2
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_169_subset__mset_Ole__zero__eq,axiom,
% 3.10/1.63 ! [A: $tType,N3: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ N3 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( N3
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_170_subset__mset_Oadd__le__cancel__left,axiom,
% 3.10/1.63 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_171_subset__mset_Oadd__le__cancel__right,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_172_mset__subset__eq__mono__add__left__cancel,axiom,
% 3.10/1.63 ! [A: $tType,C4: multiset @ A,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C4 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B5 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_173_mset__subset__eq__mono__add__right__cancel,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,C4: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C4 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_174_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( ( zero_zero @ ( multiset @ A ) )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_175_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_176_union__eq__empty,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,N: multiset @ A] :
% 3.10/1.63 ( ( ( plus_plus @ ( multiset @ A ) @ M4 @ N )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( N
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_177_empty__eq__union,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,N: multiset @ A] :
% 3.10/1.63 ( ( ( zero_zero @ ( multiset @ A ) )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ M4 @ N ) )
% 3.10/1.63 = ( ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( N
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_178_union__mset__add__mset__right,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,A2: A,B5: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ A2 @ B5 ) )
% 3.10/1.63 = ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_179_union__mset__add__mset__left,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A3 ) @ B5 )
% 3.10/1.63 = ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_180_add__le__same__cancel1,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere516151231imp_le @ A )
% 3.10/1.63 => ! [B2: A,A2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
% 3.10/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_181_add__le__same__cancel2,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere516151231imp_le @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
% 3.10/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_182_le__add__same__cancel1,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere516151231imp_le @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
% 3.10/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_183_le__add__same__cancel2,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere516151231imp_le @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
% 3.10/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_184_double__add__le__zero__iff__single__add__le__zero,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linord219039673up_add @ A )
% 3.10/1.63 => ! [A2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 3.10/1.63 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_185_zero__le__double__add__iff__zero__le__single__add,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linord219039673up_add @ A )
% 3.10/1.63 => ! [A2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 3.10/1.63 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_186_subset__mset_Oadd__le__same__cancel1,axiom,
% 3.10/1.63 ! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) @ B2 )
% 3.10/1.63 = ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_187_subset__mset_Oadd__le__same__cancel2,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ B2 )
% 3.10/1.63 = ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_188_subset__mset_Ole__add__same__cancel1,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_189_subset__mset_Ole__add__same__cancel2,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_190_add__mset__subseteq__single__iff,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,M4: multiset @ A,B2: A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ M4 ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 3.10/1.63 = ( ( M4
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( A2 = B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_191_union__assoc,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M4 @ N ) @ K2 )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_192_union__lcomm,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,N: multiset @ A,K2: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M4 @ K2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_193_union__commute,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) )
% 3.10/1.63 = ( ^ [M7: multiset @ A,N4: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N4 @ M7 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_194_union__left__cancel,axiom,
% 3.10/1.63 ! [A: $tType,K2: multiset @ A,M4: multiset @ A,N: multiset @ A] :
% 3.10/1.63 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M4 )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ K2 @ N ) )
% 3.10/1.63 = ( M4 = N ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_195_subset__mset_Oeq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [X4: multiset @ A,Y4: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ X4 @ Y4 )
% 3.10/1.63 & ( subseteq_mset @ A @ Y4 @ X4 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_196_union__right__cancel,axiom,
% 3.10/1.63 ! [A: $tType,M4: multiset @ A,K2: multiset @ A,N: multiset @ A] :
% 3.10/1.63 ( ( ( plus_plus @ ( multiset @ A ) @ M4 @ K2 )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
% 3.10/1.63 = ( M4 = N ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_197_subset__mset_Oantisym,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ X @ Y )
% 3.10/1.63 => ( ( subseteq_mset @ A @ Y @ X )
% 3.10/1.63 => ( X = Y ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_198_subset__mset_Oeq__refl,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( X = Y )
% 3.10/1.63 => ( subseteq_mset @ A @ X @ Y ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_199_subset__mset_Oadd__mono,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_200_subset__mset_Oless__eqE,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ~ ! [C3: multiset @ A] :
% 3.10/1.63 ( B2
% 3.10/1.63 != ( plus_plus @ ( multiset @ A ) @ A2 @ C3 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_201_subset__mset_Ole__iff__add,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( subseteq_mset @ A )
% 3.10/1.63 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 3.10/1.63 ? [C5: multiset @ A] :
% 3.10/1.63 ( B3
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_202_mset__subset__eq__add__left,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_203_mset__subset__eq__mono__add,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,B5: multiset @ A,C4: multiset @ A,D4: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A3 @ B5 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ C4 @ D4 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ D4 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_204_subset__mset_Oorder_Otrans,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 3.10/1.63 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_205_subset__mset_Oorder__trans,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ X @ Y )
% 3.10/1.63 => ( ( subseteq_mset @ A @ Y @ Z2 )
% 3.10/1.63 => ( subseteq_mset @ A @ X @ Z2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_206_mset__subset__eq__add__right,axiom,
% 3.10/1.63 ! [A: $tType,B5: multiset @ A,A3: multiset @ A] : ( subseteq_mset @ A @ B5 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_207_subset__mset_Oantisym__conv,axiom,
% 3.10/1.63 ! [A: $tType,Y: multiset @ A,X: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ Y @ X )
% 3.10/1.63 => ( ( subseteq_mset @ A @ X @ Y )
% 3.10/1.63 = ( X = Y ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_208_subset__mset_Oorder_Oeq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A4 @ B3 )
% 3.10/1.63 & ( subseteq_mset @ A @ B3 @ A4 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_209_multi__union__self__other__eq,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,X7: multiset @ A,Y7: multiset @ A] :
% 3.10/1.63 ( ( ( plus_plus @ ( multiset @ A ) @ A3 @ X7 )
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ A3 @ Y7 ) )
% 3.10/1.63 => ( X7 = Y7 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_210_subset__mset_Oadd__left__mono,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_211_subset__mset_Oorder_Oantisym,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ A2 )
% 3.10/1.63 => ( A2 = B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_212_mset__subset__eq__exists__conv,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( subseteq_mset @ A )
% 3.10/1.63 = ( ^ [A8: multiset @ A,B8: multiset @ A] :
% 3.10/1.63 ? [C6: multiset @ A] :
% 3.10/1.63 ( B8
% 3.10/1.63 = ( plus_plus @ ( multiset @ A ) @ A8 @ C6 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_213_subset__mset_Oadd__right__mono,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_214_subset__mset_Oord__eq__le__trans,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( A2 = B2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 3.10/1.63 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_215_subset__mset_Oord__le__eq__trans,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( B2 = C2 )
% 3.10/1.63 => ( subseteq_mset @ A @ A2 @ C2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_216_subset__mset_Odual__order_Otrans,axiom,
% 3.10/1.63 ! [A: $tType,B2: multiset @ A,A2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ B2 @ A2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ C2 @ A2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_217_subset__mset_Odual__order_Oeq__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ^ [Y3: multiset @ A,Z: multiset @ A] : ( Y3 = Z ) )
% 3.10/1.63 = ( ^ [A4: multiset @ A,B3: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ B3 @ A4 )
% 3.10/1.63 & ( subseteq_mset @ A @ A4 @ B3 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_218_subset__mset_Oadd__le__imp__le__left,axiom,
% 3.10/1.63 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 3.10/1.63 => ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_219_subset__mset_Odual__order_Oantisym,axiom,
% 3.10/1.63 ! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ B2 @ A2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( A2 = B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_220_subset__mset_Oadd__le__imp__le__right,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 3.10/1.63 => ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_221_union__iff,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B5 ) ) )
% 3.10/1.63 = ( ( member @ A @ A2 @ ( set_mset @ A @ A3 ) )
% 3.10/1.63 | ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_222_mset__subset__eqD,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,B5: multiset @ A,X: A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A3 @ B5 )
% 3.10/1.63 => ( ( member @ A @ X @ ( set_mset @ A @ A3 ) )
% 3.10/1.63 => ( member @ A @ X @ ( set_mset @ A @ B5 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_223_set__mset__mono,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A3 @ B5 )
% 3.10/1.63 => ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ A3 ) @ ( set_mset @ A @ B5 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_224_mset__subset__eq__add__mset__cancel,axiom,
% 3.10/1.63 ! [A: $tType,A2: A,A3: multiset @ A,B5: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( add_mset @ A @ A2 @ A3 ) @ ( add_mset @ A @ A2 @ B5 ) )
% 3.10/1.63 = ( subseteq_mset @ A @ A3 @ B5 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_225_add__mono__thms__linordered__semiring_I3_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [I: A,J2: A,K3: A,L: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ I @ J2 )
% 3.10/1.63 & ( K3 = L ) )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_226_add__mono__thms__linordered__semiring_I2_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [I: A,J2: A,K3: A,L: A] :
% 3.10/1.63 ( ( ( I = J2 )
% 3.10/1.63 & ( ord_less_eq @ A @ K3 @ L ) )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_227_add__mono__thms__linordered__semiring_I1_J,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [I: A,J2: A,K3: A,L: A] :
% 3.10/1.63 ( ( ( ord_less_eq @ A @ I @ J2 )
% 3.10/1.63 & ( ord_less_eq @ A @ K3 @ L ) )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_228_add__mono,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A,D3: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_229_add__left__mono,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_230_less__eqE,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( canoni770627133id_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ~ ! [C3: A] :
% 3.10/1.63 ( B2
% 3.10/1.63 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_231_add__right__mono,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere779506340up_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A,C2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_232_le__iff__add,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( canoni770627133id_add @ A )
% 3.10/1.63 => ( ( ord_less_eq @ A )
% 3.10/1.63 = ( ^ [A4: A,B3: A] :
% 3.10/1.63 ? [C5: A] :
% 3.10/1.63 ( B3
% 3.10/1.63 = ( plus_plus @ A @ A4 @ C5 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_233_add__le__imp__le__left,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere236663937imp_le @ A )
% 3.10/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_234_add__le__imp__le__right,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere236663937imp_le @ A )
% 3.10/1.63 => ! [A2: A,C2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 3.10/1.63 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_235_empty__neutral_I2_J,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = X ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_236_empty__neutral_I1_J,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A] :
% 3.10/1.63 ( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 3.10/1.63 = X ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_237_empty__le,axiom,
% 3.10/1.63 ! [A: $tType,A3: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_238_subset__mset_Ozero__le,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_239_subset__mset_Obot_Oextremum,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A] : ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_240_subset__mset_Oadd__decreasing,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_241_subset__mset_Oadd__increasing,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 3.10/1.63 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_242_subset__mset_Oadd__decreasing2,axiom,
% 3.10/1.63 ! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ( subseteq_mset @ A @ A2 @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_243_subset__mset_Oadd__increasing2,axiom,
% 3.10/1.63 ! [A: $tType,C2: multiset @ A,B2: multiset @ A,A2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ A2 )
% 3.10/1.63 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_244_subset__mset_Oadd__nonneg__nonneg,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
% 3.10/1.63 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 3.10/1.63 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_245_subset__mset_Oadd__nonpos__nonpos,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_246_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 3.10/1.63 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
% 3.10/1.63 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_247_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 3.10/1.63 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_248_subset__mset_Obot_Oextremum__uniqueI,axiom,
% 3.10/1.63 ! [A: $tType,A2: multiset @ A] :
% 3.10/1.63 ( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
% 3.10/1.63 => ( A2
% 3.10/1.63 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_249_verit__sum__simplify,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( cancel1352612707id_add @ A )
% 3.10/1.63 => ! [A2: A] :
% 3.10/1.63 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 3.10/1.63 = A2 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_250_add__nonpos__eq__0__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 3.10/1.63 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 3.10/1.63 => ( ( ( plus_plus @ A @ X @ Y )
% 3.10/1.63 = ( zero_zero @ A ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ A ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_251_add__nonneg__eq__0__iff,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [X: A,Y: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 3.10/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 3.10/1.63 => ( ( ( plus_plus @ A @ X @ Y )
% 3.10/1.63 = ( zero_zero @ A ) )
% 3.10/1.63 = ( ( X
% 3.10/1.63 = ( zero_zero @ A ) )
% 3.10/1.63 & ( Y
% 3.10/1.63 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_252_add__nonpos__nonpos,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 3.10/1.63 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_253_add__nonneg__nonneg,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_254_add__increasing2,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [C2: A,B2: A,A2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.10/1.63 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.10/1.63 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(fact_255_add__decreasing2,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( ordere216010020id_add @ A )
% 3.10/1.63 => ! [C2: A,A2: A,B2: A] :
% 3.10/1.63 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 3.10/1.63 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.10/1.63 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( type @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( ord @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( order @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 3.10/1.63 ! [A: $tType] :
% 3.10/1.63 ( ( linorder @ A )
% 3.10/1.63 => ( preorder @ A ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.10/1.63 ! [A9: $tType,A10: $tType] :
% 3.10/1.63 ( ( preorder @ A10 )
% 3.10/1.63 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.10/1.63 ! [A9: $tType,A10: $tType] :
% 3.10/1.63 ( ( order @ A10 )
% 3.10/1.63 => ( order @ ( A9 > A10 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_fun___Orderings_Oord,axiom,
% 3.10/1.63 ! [A9: $tType,A10: $tType] :
% 3.10/1.63 ( ( ord @ A10 )
% 3.10/1.63 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 3.10/1.63 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 3.10/1.63 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 3.10/1.63 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 3.10/1.63 preorder @ $o ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.10/1.63 linorder @ $o ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 3.10/1.63 order @ $o ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 3.10/1.63 ord @ $o ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
% 3.10/1.63 ! [A9: $tType] :
% 3.10/1.63 ( ( preorder @ A9 )
% 3.10/1.63 => ( ordere779506340up_add @ ( multiset @ A9 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add,axiom,
% 3.10/1.63 ! [A9: $tType] : ( cancel1352612707id_add @ ( multiset @ A9 ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_7,axiom,
% 3.10/1.63 ! [A9: $tType] :
% 3.10/1.63 ( ( preorder @ A9 )
% 3.10/1.63 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Multiset_Omultiset___Orderings_Oorder_8,axiom,
% 3.10/1.63 ! [A9: $tType] :
% 3.10/1.63 ( ( preorder @ A9 )
% 3.10/1.63 => ( order @ ( multiset @ A9 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tcon_Multiset_Omultiset___Orderings_Oord_9,axiom,
% 3.10/1.63 ! [A9: $tType] :
% 3.10/1.63 ( ( preorder @ A9 )
% 3.10/1.63 => ( ord @ ( multiset @ A9 ) ) ) ).
% 3.10/1.63
% 3.10/1.63 thf(tfree_0,hypothesis,
% 3.10/1.63 linorder @ a ).
% 3.10/1.63
% 3.10/1.63 thf(conj_0,conjecture,
% 3.10/1.63 ( ( multiset2 @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) )
% 3.10/1.63 = ( multiset2 @ a @ ( t @ a @ v @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ).
% 3.10/1.63 ------------------------
% 3.10/1.65 % (20829)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/183Mi)
% 3.10/1.66 % (20833)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_20638 for (2999ds/2Mi)
% 3.10/1.66 % (20831)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_20638 for (2999ds/27Mi)
% 3.10/1.66 % (20832)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/2Mi)
% 3.10/1.66 % (20830)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_20638 for (2999ds/4Mi)
% 3.10/1.66 % (20833)Instruction limit reached!
% 3.10/1.66 % (20833)------------------------------
% 3.10/1.66 % (20833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.66 % (20833)Termination reason: Unknown
% 3.10/1.66 % (20833)Termination phase: shuffling
% 3.10/1.66
% 3.10/1.66 % (20833)Memory used [KB]: 1407
% 3.10/1.66 % (20833)Time elapsed: 0.002 s
% 3.10/1.66 % (20833)Instructions burned: 2 (million)
% 3.10/1.66 % (20833)------------------------------
% 3.10/1.66 % (20833)------------------------------
% 3.10/1.66 % (20832)Instruction limit reached!
% 3.10/1.66 % (20832)------------------------------
% 3.10/1.66 % (20832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.66 % (20832)Termination reason: Unknown
% 3.10/1.66 % (20832)Termination phase: shuffling
% 3.10/1.66
% 3.10/1.66 % (20832)Memory used [KB]: 1407
% 3.10/1.66 % (20835)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/18Mi)
% 3.10/1.66 % (20832)Time elapsed: 0.003 s
% 3.10/1.66 % (20832)Instructions burned: 3 (million)
% 3.10/1.66 % (20832)------------------------------
% 3.10/1.66 % (20832)------------------------------
% 3.10/1.66 % (20834)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_20638 for (2999ds/275Mi)
% 3.10/1.66 % (20830)Instruction limit reached!
% 3.10/1.66 % (20830)------------------------------
% 3.10/1.66 % (20830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.66 % (20830)Termination reason: Unknown
% 3.10/1.66 % (20830)Termination phase: shuffling
% 3.10/1.66
% 3.10/1.66 % (20830)Memory used [KB]: 1407
% 3.10/1.66 % (20830)Time elapsed: 0.004 s
% 3.10/1.66 % (20830)Instructions burned: 5 (million)
% 3.10/1.66 % (20830)------------------------------
% 3.10/1.66 % (20830)------------------------------
% 3.10/1.67 % (20835)Instruction limit reached!
% 3.10/1.67 % (20835)------------------------------
% 3.10/1.67 % (20835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.67 % (20835)Termination reason: Unknown
% 3.10/1.67 % (20835)Termination phase: shuffling
% 3.10/1.67
% 3.10/1.67 % (20835)Memory used [KB]: 1791
% 3.10/1.67 % (20835)Time elapsed: 0.012 s
% 3.10/1.67 % (20835)Instructions burned: 19 (million)
% 3.10/1.67 % (20835)------------------------------
% 3.10/1.67 % (20835)------------------------------
% 3.10/1.67 % (20831)Instruction limit reached!
% 3.10/1.67 % (20831)------------------------------
% 3.10/1.67 % (20831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.67 % (20831)Termination reason: Unknown
% 3.10/1.67 % (20831)Termination phase: shuffling
% 3.10/1.67
% 3.10/1.67 % (20831)Memory used [KB]: 1918
% 3.10/1.67 % (20831)Time elapsed: 0.015 s
% 3.10/1.67 % (20831)Instructions burned: 28 (million)
% 3.10/1.67 % (20831)------------------------------
% 3.10/1.67 % (20831)------------------------------
% 3.10/1.68 % (20837)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/3Mi)
% 3.10/1.68 % (20838)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_20638 for (2999ds/37Mi)
% 3.10/1.68 % (20837)Instruction limit reached!
% 3.10/1.68 % (20837)------------------------------
% 3.10/1.68 % (20837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.68 % (20837)Termination reason: Unknown
% 3.10/1.68 % (20837)Termination phase: shuffling
% 3.10/1.68
% 3.10/1.68 % (20837)Memory used [KB]: 1407
% 3.10/1.68 % (20837)Time elapsed: 0.003 s
% 3.10/1.68 % (20837)Instructions burned: 3 (million)
% 3.10/1.68 % (20837)------------------------------
% 3.10/1.68 % (20837)------------------------------
% 3.10/1.68 % (20839)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_20638 for (2999ds/15Mi)
% 3.10/1.69 % (20839)Instruction limit reached!
% 3.10/1.69 % (20839)------------------------------
% 3.10/1.69 % (20839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (20839)Termination reason: Unknown
% 3.10/1.69 % (20839)Termination phase: shuffling
% 3.10/1.69
% 3.10/1.69 % (20839)Memory used [KB]: 1791
% 3.10/1.69 % (20839)Time elapsed: 0.009 s
% 3.10/1.69 % (20839)Instructions burned: 15 (million)
% 3.10/1.69 % (20839)------------------------------
% 3.10/1.69 % (20839)------------------------------
% 3.10/1.69 % (20841)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_20638 for (2999ds/1041Mi)
% 3.10/1.69 % (20840)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_20638 for (2999ds/3Mi)
% 3.10/1.69 % (20838)Instruction limit reached!
% 3.10/1.69 % (20838)------------------------------
% 3.10/1.69 % (20838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (20838)Termination reason: Unknown
% 3.10/1.69 % (20838)Termination phase: shuffling
% 3.10/1.69
% 3.10/1.69 % (20838)Memory used [KB]: 2174
% 3.10/1.69 % (20838)Time elapsed: 0.014 s
% 3.10/1.69 % (20838)Instructions burned: 40 (million)
% 3.10/1.69 % (20838)------------------------------
% 3.10/1.69 % (20838)------------------------------
% 3.10/1.69 % (20840)Instruction limit reached!
% 3.10/1.69 % (20840)------------------------------
% 3.10/1.69 % (20840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.69 % (20840)Termination reason: Unknown
% 3.10/1.69 % (20840)Termination phase: shuffling
% 3.10/1.69
% 3.10/1.69 % (20840)Memory used [KB]: 1407
% 3.10/1.69 % (20840)Time elapsed: 0.004 s
% 3.10/1.69 % (20840)Instructions burned: 3 (million)
% 3.10/1.69 % (20840)------------------------------
% 3.10/1.69 % (20840)------------------------------
% 3.10/1.69 % (20842)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/7Mi)
% 3.14/1.70 % (20842)Instruction limit reached!
% 3.14/1.70 % (20842)------------------------------
% 3.14/1.70 % (20842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.70 % (20842)Termination reason: Unknown
% 3.14/1.70 % (20842)Termination phase: shuffling
% 3.14/1.70
% 3.14/1.70 % (20842)Memory used [KB]: 1535
% 3.14/1.70 % (20842)Time elapsed: 0.005 s
% 3.14/1.70 % (20842)Instructions burned: 8 (million)
% 3.14/1.70 % (20842)------------------------------
% 3.14/1.70 % (20842)------------------------------
% 3.14/1.70 % (20845)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_20638 for (2999ds/3Mi)
% 3.14/1.70 % (20845)Instruction limit reached!
% 3.14/1.70 % (20845)------------------------------
% 3.14/1.70 % (20845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.70 % (20845)Termination reason: Unknown
% 3.14/1.70 % (20845)Termination phase: shuffling
% 3.14/1.70
% 3.14/1.70 % (20845)Memory used [KB]: 1407
% 3.14/1.70 % (20845)Time elapsed: 0.002 s
% 3.14/1.70 % (20845)Instructions burned: 3 (million)
% 3.14/1.70 % (20845)------------------------------
% 3.14/1.70 % (20845)------------------------------
% 3.14/1.70 % (20844)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/16Mi)
% 3.14/1.71 % (20846)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_20638 for (2999ds/3Mi)
% 3.14/1.71 % (20846)Instruction limit reached!
% 3.14/1.71 % (20846)------------------------------
% 3.14/1.71 % (20846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.71 % (20846)Termination reason: Unknown
% 3.14/1.71 % (20846)Termination phase: shuffling
% 3.14/1.71
% 3.14/1.71 % (20846)Memory used [KB]: 1407
% 3.14/1.71 % (20846)Time elapsed: 0.003 s
% 3.14/1.71 % (20846)Instructions burned: 3 (million)
% 3.14/1.71 % (20846)------------------------------
% 3.14/1.71 % (20846)------------------------------
% 3.14/1.71 % (20848)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/3Mi)
% 3.14/1.71 % (20844)Instruction limit reached!
% 3.14/1.71 % (20844)------------------------------
% 3.14/1.71 % (20844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.71 % (20844)Termination reason: Unknown
% 3.14/1.71 % (20844)Termination phase: shuffling
% 3.14/1.71
% 3.14/1.71 % (20844)Memory used [KB]: 1791
% 3.14/1.71 % (20844)Time elapsed: 0.010 s
% 3.14/1.71 % (20844)Instructions burned: 17 (million)
% 3.14/1.71 % (20844)------------------------------
% 3.14/1.71 % (20844)------------------------------
% 3.14/1.71 % (20848)Instruction limit reached!
% 3.14/1.71 % (20848)------------------------------
% 3.14/1.71 % (20848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.71 % (20848)Termination reason: Unknown
% 3.14/1.71 % (20848)Termination phase: shuffling
% 3.14/1.71
% 3.14/1.71 % (20848)Memory used [KB]: 1407
% 3.14/1.71 % (20848)Time elapsed: 0.002 s
% 3.14/1.71 % (20848)Instructions burned: 4 (million)
% 3.14/1.71 % (20848)------------------------------
% 3.14/1.71 % (20848)------------------------------
% 3.14/1.72 % (20847)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/7Mi)
% 3.14/1.72 % (20851)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/710Mi)
% 3.14/1.72 % (20847)Instruction limit reached!
% 3.14/1.72 % (20847)------------------------------
% 3.14/1.72 % (20847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.72 % (20847)Termination reason: Unknown
% 3.14/1.72 % (20847)Termination phase: shuffling
% 3.14/1.72
% 3.14/1.72 % (20847)Memory used [KB]: 1535
% 3.14/1.72 % (20847)Time elapsed: 0.006 s
% 3.14/1.72 % (20847)Instructions burned: 8 (million)
% 3.14/1.72 % (20847)------------------------------
% 3.14/1.72 % (20847)------------------------------
% 3.14/1.72 % (20829)Instruction limit reached!
% 3.14/1.72 % (20829)------------------------------
% 3.14/1.72 % (20829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.72 % (20829)Termination reason: Unknown
% 3.14/1.72 % (20829)Termination phase: Saturation
% 3.14/1.72
% 3.14/1.72 % (20829)Memory used [KB]: 7419
% 3.14/1.72 % (20829)Time elapsed: 0.071 s
% 3.14/1.72 % (20829)Instructions burned: 183 (million)
% 3.14/1.72 % (20829)------------------------------
% 3.14/1.72 % (20829)------------------------------
% 3.14/1.72 % (20849)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/4Mi)
% 3.14/1.73 % (20850)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20638 for (2999ds/18Mi)
% 3.14/1.73 % (20834)First to succeed.
% 3.14/1.73 % (20849)Instruction limit reached!
% 3.14/1.73 % (20849)------------------------------
% 3.14/1.73 % (20849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.73 % (20849)Termination reason: Unknown
% 3.14/1.73 % (20849)Termination phase: shuffling
% 3.14/1.73
% 3.14/1.73 % (20849)Memory used [KB]: 1407
% 3.14/1.73 % (20849)Time elapsed: 0.004 s
% 3.14/1.73 % (20849)Instructions burned: 5 (million)
% 3.14/1.73 % (20849)------------------------------
% 3.14/1.73 % (20849)------------------------------
% 3.14/1.73 % (20834)Refutation found. Thanks to Tanya!
% 3.14/1.73 % SZS status Theorem for DTF2THF_20638
% 3.14/1.73 % SZS output start Proof for DTF2THF_20638
% 3.14/1.73 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.14/1.73 thf(type_def_6, type, set: ($tType) > $tType).
% 3.14/1.73 thf(type_def_8, type, tree: ($tType) > $tType).
% 3.14/1.73 thf(type_def_9, type, list: ($tType) > $tType).
% 3.14/1.73 thf(type_def_10, type, multiset: ($tType) > $tType).
% 3.14/1.73 thf(type_def_11, type, a: $tType).
% 3.14/1.73 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 3.14/1.73 thf(func_def_1, type, multiset: ($tType) > $tType).
% 3.14/1.73 thf(func_def_2, type, list: ($tType) > $tType).
% 3.14/1.73 thf(func_def_3, type, tree: ($tType) > $tType).
% 3.14/1.73 thf(func_def_4, type, set: ($tType) > $tType).
% 3.14/1.73 thf(func_def_5, type, a: $tType).
% 3.14/1.73 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_8, type, order: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_9, type, linorder: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_10, type, preorder: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_11, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_12, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_13, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_14, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_15, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_16, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_17, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_18, type, unique1598680935umeral: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_19, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 3.14/1.73 thf(func_def_20, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.14/1.73 thf(func_def_21, type, unique455577585es_aux: !>[X0: $tType]:(product_prod @ X0 @ X0 > $o)).
% 3.14/1.73 thf(func_def_22, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.14/1.73 thf(func_def_23, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.14/1.73 thf(func_def_24, type, heapIm818251801eapify: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_25, type, heapIm721255937_empty: !>[X0: $tType]:(tree @ X0 > $o)).
% 3.14/1.73 thf(func_def_26, type, heapIm874063447f_list: !>[X0: $tType]:(list @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_27, type, heapIm1271749598e_left: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_28, type, heapIm1912108042t_tree: !>[X0: $tType]:(list @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_29, type, heapIm970386777veLeaf: !>[X0: $tType]:(tree @ X0 > product_prod @ X0 @ tree @ X0)).
% 3.14/1.73 thf(func_def_30, type, heapIm1434396069_right: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_31, type, heapIm748920189ftDown: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_32, type, heap: !>[X0: $tType, X1: $tType]:(X0 > (X0 > $o) > (list @ X1 > X0) > (X0 > multiset @ X1) > (X0 > tree @ X1) > (X0 > product_prod @ X1 @ X0) > $o)).
% 3.14/1.73 thf(func_def_33, type, heap_axioms: !>[X0: $tType, X1: $tType]:((X0 > $o) > (list @ X1 > X0) > (X0 > multiset @ X1) > (X0 > tree @ X1) > (X0 > product_prod @ X1 @ X0) > $o)).
% 3.14/1.73 thf(func_def_34, type, e: !>[X0: $tType]:(tree @ X0)).
% 3.14/1.73 thf(func_def_35, type, t: !>[X0: $tType]:(X0 > tree @ X0 > tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_36, type, in_tree: !>[X0: $tType]:(X0 > tree @ X0 > $o)).
% 3.14/1.73 thf(func_def_37, type, is_heap: !>[X0: $tType]:(tree @ X0 > $o)).
% 3.14/1.73 thf(func_def_38, type, multiset2: !>[X0: $tType]:(tree @ X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_39, type, val: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_40, type, lattic929149872er_Max: !>[X0: $tType]:(set @ X0 > X0)).
% 3.14/1.73 thf(func_def_41, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 3.14/1.73 thf(func_def_42, type, add_mset: !>[X0: $tType]:(X0 > multiset @ X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_43, type, is_empty: !>[X0: $tType]:(multiset @ X0 > $o)).
% 3.14/1.73 thf(func_def_44, type, set_mset: !>[X0: $tType]:(multiset @ X0 > set @ X0)).
% 3.14/1.73 thf(func_def_45, type, subseteq_mset: !>[X0: $tType]:(multiset @ X0 > multiset @ X0 > $o)).
% 3.14/1.73 thf(func_def_46, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.14/1.73 thf(func_def_47, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_48, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.14/1.73 thf(func_def_49, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.14/1.73 thf(func_def_50, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.14/1.73 thf(func_def_51, type, collection: !>[X0: $tType, X1: $tType]:(X0 > (X0 > $o) > (list @ X1 > X0) > (X0 > multiset @ X1) > $o)).
% 3.14/1.73 thf(func_def_52, type, set2: !>[X0: $tType, X1: $tType]:((X0 > multiset @ X1) > X0 > set @ X1)).
% 3.14/1.73 thf(func_def_53, type, removeMax: !>[X0: $tType, X1: $tType]:(X0 > (X0 > $o) > (list @ X1 > X0) > (X0 > multiset @ X1) > (X0 > product_prod @ X1 @ X0) > (X0 > $o) > $o)).
% 3.14/1.73 thf(func_def_54, type, ssort: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > product_prod @ X1 @ X0) > X0 > list @ X1 > list @ X1)).
% 3.14/1.73 thf(func_def_55, type, ssort_dom: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > product_prod @ X1 @ X0) > product_prod @ X0 @ list @ X1 > $o)).
% 3.14/1.73 thf(func_def_56, type, removeMax_axioms: !>[X0: $tType, X1: $tType]:((X0 > $o) > (list @ X1 > X0) > (X0 > multiset @ X1) > (X0 > product_prod @ X1 @ X0) > (X0 > $o) > $o)).
% 3.14/1.73 thf(func_def_57, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.14/1.73 thf(func_def_58, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.14/1.73 thf(func_def_59, type, l1: tree @ a).
% 3.14/1.73 thf(func_def_60, type, l2: tree @ a).
% 3.14/1.73 thf(func_def_61, type, r1: tree @ a).
% 3.14/1.73 thf(func_def_62, type, r2: tree @ a).
% 3.14/1.73 thf(func_def_63, type, v1: a).
% 3.14/1.73 thf(func_def_64, type, v2: a).
% 3.14/1.73 thf(func_def_65, type, v: a).
% 3.14/1.73 thf(func_def_83, type, sP0: !>[X0: $tType]:((tree @ X0 > $o) > $o)).
% 3.14/1.73 thf(func_def_84, type, sP1: !>[X0: $tType]:((tree @ X0 > $o) > $o)).
% 3.14/1.73 thf(func_def_85, type, sP2: !>[X0: $tType]:((tree @ X0 > $o) > $o)).
% 3.14/1.73 thf(func_def_86, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.14/1.73 thf(func_def_87, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.14/1.73 thf(func_def_88, type, sP5: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.14/1.73 thf(func_def_89, type, sK6: !>[X0: $tType]:(multiset @ X0 > X0)).
% 3.14/1.73 thf(func_def_90, type, sK7: !>[X0: $tType]:(multiset @ X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_91, type, sK8: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_92, type, sK9: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_93, type, sK10: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_94, type, sK11: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_95, type, sK12: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_96, type, sK13: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_97, type, sK14: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_98, type, sK15: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_99, type, sK16: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_100, type, sK17: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_101, type, sK18: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_102, type, sK19: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_103, type, sK20: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_104, type, sK21: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_105, type, sK22: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_106, type, sK23: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_107, type, sK24: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_108, type, sK25: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_109, type, sK26: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_110, type, sK27: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_111, type, sK28: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_112, type, sK29: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_113, type, sK30: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_114, type, sK31: !>[X0: $tType]:((multiset @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_115, type, sK32: !>[X0: $tType]:((multiset @ X0 > $o) > multiset @ X0)).
% 3.14/1.73 thf(func_def_116, type, sK33: !>[X0: $tType]:(multiset @ X0 > X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_117, type, sK34: !>[X0: $tType]:(multiset @ X0 > X0 > multiset @ X0 > X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_118, type, sK35: !>[X0: $tType]:((multiset @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_119, type, sK36: !>[X0: $tType]:((multiset @ X0 > $o) > multiset @ X0)).
% 3.14/1.73 thf(func_def_120, type, sK37: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_121, type, sK38: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_122, type, sK39: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_123, type, sK40: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_124, type, sK41: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_125, type, sK42: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_126, type, sK43: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_127, type, sK44: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_128, type, sK45: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_129, type, sK46: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_130, type, sK47: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_131, type, sK48: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_132, type, sK49: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_133, type, sK50: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_134, type, sK51: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_135, type, sK52: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_136, type, sK53: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_137, type, sK54: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_138, type, sK55: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_139, type, sK56: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_140, type, sK57: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_141, type, sK58: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_142, type, sK59: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_143, type, sK60: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_144, type, sK61: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_145, type, sK62: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_146, type, sK63: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_147, type, sK64: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_148, type, sK65: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_149, type, sK66: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_150, type, sK67: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_151, type, sK68: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_152, type, sK69: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_153, type, sK70: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_154, type, sK71: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_155, type, sK72: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_156, type, sK73: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_157, type, sK74: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_158, type, sK75: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_159, type, sK76: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_160, type, sK77: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_161, type, sK78: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_162, type, sK79: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_163, type, sK80: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_164, type, sK81: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_165, type, sK82: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_166, type, sK83: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_167, type, sK84: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_168, type, sK85: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_169, type, sK86: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_170, type, sK87: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_171, type, sK88: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_172, type, sK89: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_173, type, sK90: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_174, type, sK91: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_175, type, sK92: !>[X0: $tType]:(tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_176, type, sK93: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_177, type, sK94: !>[X0: $tType]:(tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_178, type, sK95: !>[X0: $tType]:(X0 > tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_179, type, sK96: !>[X0: $tType]:(X0 > tree @ X0 > X0)).
% 3.14/1.73 thf(func_def_180, type, sK97: !>[X0: $tType]:(X0 > tree @ X0 > tree @ X0)).
% 3.14/1.73 thf(func_def_181, type, sK98: !>[X0: $tType]:((multiset @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_182, type, sK99: !>[X0: $tType]:((multiset @ X0 > $o) > multiset @ X0)).
% 3.14/1.73 thf(func_def_183, type, sK100: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > X0)).
% 3.14/1.73 thf(func_def_184, type, sK101: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > multiset @ X0)).
% 3.14/1.73 thf(func_def_185, type, sK102: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > multiset @ X1)).
% 3.14/1.73 thf(func_def_186, type, sK103: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > X1)).
% 3.14/1.73 thf(func_def_187, type, sK104: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > multiset @ X0)).
% 3.14/1.73 thf(func_def_188, type, sK105: !>[X0: $tType, X1: $tType]:((multiset @ X0 > multiset @ X1 > $o) > multiset @ X1)).
% 3.14/1.73 thf(func_def_189, type, sK106: !>[X0: $tType]:(multiset @ X0 > X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_190, type, sK107: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.14/1.73 thf(func_def_191, type, sK108: !>[X0: $tType]:(multiset @ X0 > multiset @ X0)).
% 3.14/1.73 thf(func_def_192, type, sK109: !>[X0: $tType]:(multiset @ X0 > X0)).
% 3.14/1.73 thf(func_def_193, type, sK110: !>[X0: $tType]:(multiset @ X0 > X0)).
% 3.14/1.73 thf(func_def_194, type, sK111: !>[X0: $tType]:((tree @ X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_195, type, sK112: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_196, type, sK113: !>[X0: $tType]:((tree @ X0 > $o) > tree @ X0)).
% 3.14/1.73 thf(func_def_197, type, sK114: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_198, type, sK115: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_199, type, sK116: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_200, type, sK117: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.14/1.73 thf(func_def_202, type, ph119: !>[X0: $tType]:(X0)).
% 3.14/1.73 thf(f1189,plain,(
% 3.14/1.73 $false),
% 3.14/1.73 inference(subsumption_resolution,[status(thm)],[f1188,f1046])).
% 3.14/1.73 thf(f1046,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v2 @ v1) = $true)),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f497])).
% 3.14/1.73 thf(f497,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v2 @ v1) = $true)),
% 3.14/1.73 inference(fool_elimination,[status(thm)],[f496])).
% 3.14/1.73 thf(f496,plain,(
% 3.14/1.73 (ord_less_eq @ a @ v2 @ v1)),
% 3.14/1.73 inference(rectify,[status(thm)],[f1])).
% 3.14/1.73 thf(f1,axiom,(
% 3.14/1.73 (ord_less_eq @ a @ v2 @ v1)),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062)).
% 3.14/1.73 thf(f1188,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v2 @ v1) != $true)),
% 3.14/1.73 inference(forward_demodulation,[status(thm)],[f1187,f966])).
% 3.14/1.73 thf(f966,plain,(
% 3.14/1.73 ( ! [X0 : $tType,X2 : tree @ X0,X3 : tree @ X0,X1 : X0] : (((val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) = X1)) )),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f767])).
% 3.14/1.73 thf(f767,plain,(
% 3.14/1.73 ! [X0 : $tType,X1 : X0,X2 : tree @ X0,X3 : tree @ X0] : ((val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) = X1)),
% 3.14/1.73 inference(rectify,[status(thm)],[f34])).
% 3.14/1.73 thf(f34,axiom,(
% 3.14/1.73 ! [X0 : $tType,X5 : X0,X49 : tree @ X0,X50 : tree @ X0] : ((val @ X0 @ (t @ X0 @ X5 @ X49 @ X50)) = X5)),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',fact_33_val_Osimps)).
% 3.14/1.73 thf(f1187,plain,(
% 3.14/1.73 ($true != (ord_less_eq @ a @ v2 @ (val @ a @ (t @ a @ v1 @ l1 @ r1))))),
% 3.14/1.73 inference(forward_demodulation,[status(thm)],[f1186,f966])).
% 3.14/1.73 thf(f1186,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ (val @ a @ (t @ a @ v2 @ l2 @ r2)) @ (val @ a @ (t @ a @ v1 @ l1 @ r1))) != $true)),
% 3.14/1.73 inference(subsumption_resolution,[status(thm)],[f1185,f1137])).
% 3.14/1.73 thf(f1137,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v1 @ v) = $true)),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f485])).
% 3.14/1.73 thf(f485,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v1 @ v) = $true)),
% 3.14/1.73 inference(fool_elimination,[status(thm)],[f484])).
% 3.14/1.73 thf(f484,plain,(
% 3.14/1.73 (ord_less_eq @ a @ v1 @ v)),
% 3.14/1.73 inference(rectify,[status(thm)],[f2])).
% 3.14/1.73 thf(f2,axiom,(
% 3.14/1.73 (ord_less_eq @ a @ v1 @ v)),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',fact_1_True)).
% 3.14/1.73 thf(f1185,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ v1 @ v) != $true) | ((ord_less_eq @ a @ (val @ a @ (t @ a @ v2 @ l2 @ r2)) @ (val @ a @ (t @ a @ v1 @ l1 @ r1))) != $true)),
% 3.14/1.73 inference(forward_demodulation,[status(thm)],[f1184,f966])).
% 3.14/1.73 thf(f1184,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ (val @ a @ (t @ a @ v1 @ l1 @ r1)) @ v) != $true) | ((ord_less_eq @ a @ (val @ a @ (t @ a @ v2 @ l2 @ r2)) @ (val @ a @ (t @ a @ v1 @ l1 @ r1))) != $true)),
% 3.14/1.73 inference(subsumption_resolution,[status(thm)],[f1183,f981])).
% 3.14/1.73 thf(f981,plain,(
% 3.14/1.73 (linorder @ a = $true)),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f527])).
% 3.14/1.73 thf(f527,plain,(
% 3.14/1.73 (linorder @ a = $true)),
% 3.14/1.73 inference(fool_elimination,[status(thm)],[f526])).
% 3.14/1.73 thf(f526,plain,(
% 3.14/1.73 linorder @ a),
% 3.14/1.73 inference(rectify,[status(thm)],[f276])).
% 3.14/1.73 thf(f276,axiom,(
% 3.14/1.73 linorder @ a),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',tfree_0)).
% 3.14/1.73 thf(f1183,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ (val @ a @ (t @ a @ v1 @ l1 @ r1)) @ v) != $true) | (linorder @ a != $true) | ((ord_less_eq @ a @ (val @ a @ (t @ a @ v2 @ l2 @ r2)) @ (val @ a @ (t @ a @ v1 @ l1 @ r1))) != $true)),
% 3.14/1.73 inference(trivial_inequality_removal,[status(thm)],[f1179])).
% 3.14/1.73 thf(f1179,plain,(
% 3.14/1.73 ((ord_less_eq @ a @ (val @ a @ (t @ a @ v2 @ l2 @ r2)) @ (val @ a @ (t @ a @ v1 @ l1 @ r1))) != $true) | ((multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2))) != (multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2)))) | (linorder @ a != $true) | ((ord_less_eq @ a @ (val @ a @ (t @ a @ v1 @ l1 @ r1)) @ v) != $true)),
% 3.14/1.73 inference(superposition,[status(thm)],[f1109,f1052])).
% 3.14/1.73 thf(f1052,plain,(
% 3.14/1.73 ( ! [X0 : $tType,X2 : tree @ X0,X3 : tree @ X0,X1 : X0,X6 : tree @ X0,X7 : X0,X4 : X0,X5 : tree @ X0] : (((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) | (linorder @ X0 != $true) | ($true != (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) | ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) != $true)) )),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f824])).
% 3.14/1.73 thf(f824,plain,(
% 3.14/1.73 ! [X0 : $tType] : ((linorder @ X0 != $true) | ! [X1 : X0,X2 : tree @ X0,X3 : tree @ X0,X4 : X0,X5 : tree @ X0,X6 : tree @ X0,X7 : X0] : ((((($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) | ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (t @ X0 @ X4 @ X5 @ X6) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X1 @ X2 @ X3))))))) & (($true != (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) | ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))))) | ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) = $true)) & (((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) != $true) | ((($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) | ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X4 @ X5 @ X6)))) @ (t @ X0 @ X1 @ X2 @ X3)))) & (($true != (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) | ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))))))))),
% 3.14/1.73 inference(ennf_transformation,[status(thm)],[f776])).
% 3.14/1.73 thf(f776,plain,(
% 3.14/1.73 ! [X0 : $tType] : ((linorder @ X0 = $true) => ! [X1 : X0,X2 : tree @ X0,X3 : tree @ X0,X4 : X0,X5 : tree @ X0,X6 : tree @ X0,X7 : X0] : ((((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) != $true) => ((($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3)))) & (($true != (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (t @ X0 @ X4 @ X5 @ X6) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X1 @ X2 @ X3))))))))) & (((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) = $true) => ((($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3)))) & (($true != (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X4 @ X5 @ X6)))) @ (t @ X0 @ X1 @ X2 @ X3))))))))),
% 3.14/1.73 inference(flattening,[status(thm)],[f321])).
% 3.14/1.73 thf(f321,plain,(
% 3.14/1.73 ! [X0 : $tType] : ((linorder @ X0 = $true) => ! [X1 : X0,X2 : tree @ X0,X3 : tree @ X0,X4 : X0,X5 : tree @ X0,X6 : tree @ X0,X7 : X0] : ((((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) = $true) => ((~($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X4 @ X5 @ X6)))) @ (t @ X0 @ X1 @ X2 @ X3)))) & (($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3)))))) & (~((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) = $true) => ((~($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (t @ X0 @ X4 @ X5 @ X6) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X1 @ X2 @ X3))))))) & (($true = (ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7)) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))))))))),
% 3.14/1.73 inference(fool_elimination,[status(thm)],[f320])).
% 3.14/1.73 thf(f320,plain,(
% 3.14/1.73 ! [X0 : $tType] : (linorder @ X0 => ! [X1 : X0,X2 : tree @ X0,X3 : tree @ X0,X4 : X0,X5 : tree @ X0,X6 : tree @ X0,X7 : X0] : (((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) => ((~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X4 @ X5 @ X6)))) @ (t @ X0 @ X1 @ X2 @ X3)))) & ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6)) @ X7) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3)))))) & (~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (val @ X0 @ (t @ X0 @ X4 @ X5 @ X6))) => ((~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (t @ X0 @ X4 @ X5 @ X6) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X1 @ X2 @ X3))))))) & ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X1 @ X2 @ X3)) @ X7) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))) = (t @ X0 @ X7 @ (t @ X0 @ X4 @ X5 @ X6) @ (t @ X0 @ X1 @ X2 @ X3))))))))),
% 3.14/1.73 inference(rectify,[status(thm)],[f35])).
% 3.14/1.73 thf(f35,axiom,(
% 3.14/1.73 ! [X0 : $tType] : (linorder @ X0 => ! [X25 : X0,X26 : tree @ X0,X27 : tree @ X0,X28 : X0,X29 : tree @ X0,X30 : tree @ X0,X5 : X0] : (((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ (val @ X0 @ (t @ X0 @ X28 @ X29 @ X30))) => ((~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X28 @ X29 @ X30)) @ X5) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X28 @ X29 @ X30)) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X28 @ X29 @ X30)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X28 @ X29 @ X30)))) @ (t @ X0 @ X25 @ X26 @ X27)))) & ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X28 @ X29 @ X30)) @ X5) => ((t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27)) = (heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27))))))) & (~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ (val @ X0 @ (t @ X0 @ X28 @ X29 @ X30))) => ((~(ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ X5) => ((heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27))) = (t @ X0 @ (val @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ (t @ X0 @ X28 @ X29 @ X30) @ (heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (heapIm1271749598e_left @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ (heapIm1434396069_right @ X0 @ (t @ X0 @ X25 @ X26 @ X27))))))) & ((ord_less_eq @ X0 @ (val @ X0 @ (t @ X0 @ X25 @ X26 @ X27)) @ X5) => ((t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27)) = (heapIm748920189ftDown @ X0 @ (t @ X0 @ X5 @ (t @ X0 @ X28 @ X29 @ X30) @ (t @ X0 @ X25 @ X26 @ X27)))))))))),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',fact_34_siftDown_Osimps_I5_J)).
% 3.14/1.73 thf(f1109,plain,(
% 3.14/1.73 ((multiset2 @ a @ (heapIm748920189ftDown @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2)))) != (multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2))))),
% 3.14/1.73 inference(cnf_transformation,[status(thm)],[f744])).
% 3.14/1.73 thf(f744,plain,(
% 3.14/1.73 ((multiset2 @ a @ (heapIm748920189ftDown @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2)))) != (multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2))))),
% 3.14/1.73 inference(flattening,[status(thm)],[f278])).
% 3.14/1.73 thf(f278,negated_conjecture,(
% 3.14/1.73 ~((multiset2 @ a @ (heapIm748920189ftDown @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2)))) = (multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2))))),
% 3.14/1.73 inference(negated_conjecture,[status(cth)],[f277])).
% 3.14/1.73 thf(f277,conjecture,(
% 3.14/1.73 ((multiset2 @ a @ (heapIm748920189ftDown @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2)))) = (multiset2 @ a @ (t @ a @ v @ (t @ a @ v1 @ l1 @ r1) @ (t @ a @ v2 @ l2 @ r2))))),
% 3.14/1.73 file('/export/starexec/sandbox2/tmp/tmp.jk9cZwfdt6/DTF2THF_20638.p',conj_0)).
% 3.14/1.73 % SZS output end Proof for DTF2THF_20638
% 3.14/1.73 % (20834)------------------------------
% 3.14/1.73 % (20834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.14/1.73 % (20834)Termination reason: Refutation
% 3.14/1.73
% 3.14/1.73 % (20834)Memory used [KB]: 7291
% 3.14/1.73 % (20834)Time elapsed: 0.071 s
% 3.14/1.73 % (20834)Instructions burned: 125 (million)
% 3.14/1.73 % (20834)------------------------------
% 3.14/1.73 % (20834)------------------------------
% 3.14/1.73 % (20827)Success in time 0.09 s
%------------------------------------------------------------------------------