%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP056^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n002.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 08:55:12 AM UTC 2026
% Result : Theorem 2.67s 1.64s
% Output : Refutation 2.67s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP056^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n002.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:25:03 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP056^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer FLPTheorem problem prob_1320__3306644_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : FLPTheorem/prob_1320__3306644_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 342 ( 70 unt; 49 typ; 0 def)
% 0.19/0.45 % Number of atoms : 841 ( 210 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.45 % Number of connectives : 3320 ( 66 ~; 13 |; 35 &;2740 @)
% 0.19/0.45 % ( 0 <=>; 466 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 17 ( 8 avg)
% 0.19/0.45 % Number of types : 6 ( 5 usr)
% 0.19/0.45 % Number of type conns : 94 ( 94 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 45 ( 44 usr; 2 con; 0-4 aty)
% 0.19/0.45 % Number of variables : 1025 ( 36 ^; 911 !; 39 ?;1025 :)
% 0.19/0.45 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.45 % SPC : TH1_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:54.899
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (7)
% 0.19/0.45 thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext,type,
% 0.19/0.45 configuration_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Product__Type_Ounit,type,
% 0.19/0.45 product_unit: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_List_Olist,type,
% 0.19/0.45 list: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Nat_Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_v,type,
% 0.19/0.45 v: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_s,type,
% 0.19/0.45 s: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_p,type,
% 0.19/0.45 p: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (42)
% 0.19/0.45 thf(sy_cl_HOL_Otype,type,
% 0.19/0.45 type:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Nat_Osize,type,
% 0.19/0.45 size:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oone,type,
% 0.19/0.45 one:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ominus,type,
% 0.19/0.45 minus:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.45 ord:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Num_Oneg__numeral,type,
% 0.19/0.45 neg_numeral:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.45 order:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.45 group_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.19/0.45 no_bot:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.19/0.45 no_top:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.45 linorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.45 preorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oab__group__add,type,
% 0.19/0.45 ab_group_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.19/0.45 wellorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.19/0.45 semigroup_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.19/0.45 dense_order:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.19/0.45 comm_monoid_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.19/0.45 ab_semigroup_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.19/0.45 cancel_semigroup_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.19/0.45 ordered_ab_group_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.19/0.45 cancel1352612707id_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.19/0.45 cancel146912293up_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.19/0.45 ordere779506340up_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.19/0.45 linord1659791738miring:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.45 canoni770627133id_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.19/0.45 ordere236663937imp_le:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.19/0.45 ordere223160158up_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 0.19/0.45 ordere623563068d_diff:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.19/0.45 strict2144017051up_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.45 minus_minus:
% 0.19/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.19/0.45 one_one:
% 0.19/0.45 !>[A: $tType] : A ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.19/0.45 plus_plus:
% 0.19/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_ListUtilities_OprefixList,type,
% 0.19/0.45 prefixList:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Olast,type,
% 0.19/0.45 last:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Olist_ONil,type,
% 0.19/0.45 nil:
% 0.19/0.45 !>[A: $tType] : ( list @ A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_List_Onth,type,
% 0.19/0.45 nth:
% 0.19/0.45 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.19/0.45 size_size:
% 0.19/0.45 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.19/0.45 ord_less:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.45 ord_less_eq:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_fe____,type,
% 0.19/0.45 fe: nat > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_n0____,type,
% 0.19/0.45 n0: nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_n____,type,
% 0.19/0.45 n: nat ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (254)
% 0.19/0.45 thf(fact_0_FeNNotEmpty,axiom,
% 0.19/0.45 ( ( fe @ n )
% 0.19/0.45 != ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % FeNNotEmpty
% 0.19/0.45 thf(fact_1_LastIsLastIndex,axiom,
% 0.19/0.45 ! [A2: $tType,L: list @ A2] :
% 0.19/0.45 ( ( L
% 0.19/0.45 != ( nil @ A2 ) )
% 0.19/0.45 => ( ( last @ A2 @ L )
% 0.19/0.45 = ( nth @ A2 @ L @ ( minus_minus @ nat @ ( size_size @ ( list @ A2 ) @ L ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % LastIsLastIndex
% 0.19/0.45 thf(fact_2_Fou2,axiom,
% 0.19/0.45 ord_less @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% 0.19/0.45
% 0.19/0.45 % Fou2
% 0.19/0.45 thf(fact_3_AssmNoDecided_I1_J,axiom,
% 0.19/0.45 ord_less @ nat @ n0 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% 0.19/0.45
% 0.19/0.45 % AssmNoDecided(1)
% 0.19/0.45 thf(fact_4_last__conv__nth,axiom,
% 0.19/0.45 ! [A: $tType,Xs: list @ A] :
% 0.19/0.45 ( ( Xs
% 0.19/0.45 != ( nil @ A ) )
% 0.19/0.45 => ( ( last @ A @ Xs )
% 0.19/0.45 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % last_conv_nth
% 0.19/0.45 thf(fact_5_minus__apply,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( minus @ B )
% 0.19/0.45 => ( ( minus_minus @ ( A > B ) )
% 0.19/0.45 = ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_apply
% 0.19/0.45 thf(fact_6_Fou,axiom,
% 0.19/0.45 ord_less_eq @ nat @ n0 @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % Fou
% 0.19/0.45 thf(fact_7_one__natural_Orsp,axiom,
% 0.19/0.45 ( ( one_one @ nat )
% 0.19/0.45 = ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_natural.rsp
% 0.19/0.45 thf(fact_8_diff__commute,axiom,
% 0.19/0.45 ! [I: nat,J: nat,K: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.19/0.45 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_commute
% 0.19/0.45 thf(fact_9_Ex__list__of__length,axiom,
% 0.19/0.45 ! [A: $tType,N: nat] :
% 0.19/0.45 ? [Xs2: list @ A] :
% 0.19/0.45 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.19/0.45 = N ) ).
% 0.19/0.45
% 0.19/0.45 % Ex_list_of_length
% 0.19/0.45 thf(fact_10_neq__if__length__neq,axiom,
% 0.19/0.45 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.45 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.19/0.45 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.19/0.45 => ( Xs != Ys ) ) ).
% 0.19/0.45
% 0.19/0.45 % neq_if_length_neq
% 0.19/0.45 thf(fact_11_size__neq__size__imp__neq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( size @ A )
% 0.19/0.45 => ! [X2: A,Y: A] :
% 0.19/0.45 ( ( ( size_size @ A @ X2 )
% 0.19/0.45 != ( size_size @ A @ Y ) )
% 0.19/0.45 => ( X2 != Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % size_neq_size_imp_neq
% 0.19/0.45 thf(fact_12_fun__diff__def,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( minus @ B )
% 0.19/0.45 => ( ( minus_minus @ ( A > B ) )
% 0.19/0.45 = ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fun_diff_def
% 0.19/0.45 thf(fact_13_diff__diff__cancel,axiom,
% 0.19/0.45 ! [I: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ I @ N )
% 0.19/0.45 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.19/0.45 = I ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_diff_cancel
% 0.19/0.45 thf(fact_14_le__refl,axiom,
% 0.19/0.45 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.19/0.45
% 0.19/0.45 % le_refl
% 0.19/0.45 thf(fact_15_le__trans,axiom,
% 0.19/0.45 ! [I: nat,J: nat,K: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ J @ K )
% 0.19/0.45 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_trans
% 0.19/0.45 thf(fact_16_eq__imp__le,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( M = N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % eq_imp_le
% 0.19/0.45 thf(fact_17_le__antisym,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ N @ M )
% 0.19/0.45 => ( M = N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_antisym
% 0.19/0.45 thf(fact_18_nat__less__le,axiom,
% 0.19/0.45 ( ( ord_less @ nat )
% 0.19/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.19/0.45 & ( M2 != N2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_less_le
% 0.19/0.45 thf(fact_19_nat__neq__iff,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( M != N )
% 0.19/0.45 = ( ( ord_less @ nat @ M @ N )
% 0.19/0.45 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_neq_iff
% 0.19/0.45 thf(fact_20_less__diff__iff,axiom,
% 0.19/0.45 ! [K: nat,M: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ K @ M )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.45 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.19/0.45 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_diff_iff
% 0.19/0.45 thf(fact_21_less__not__refl,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl
% 0.19/0.45 thf(fact_22_nat__le__linear,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.45 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_le_linear
% 0.19/0.45 thf(fact_23_diff__less__mono,axiom,
% 0.19/0.45 ! [A4: nat,B3: nat,C: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ A4 @ B3 )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ C @ A4 )
% 0.19/0.45 => ( ord_less @ nat @ ( minus_minus @ nat @ A4 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_less_mono
% 0.19/0.45 thf(fact_24_less__not__refl2,axiom,
% 0.19/0.45 ! [N: nat,M: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ N @ M )
% 0.19/0.45 => ( M != N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl2
% 0.19/0.45 thf(fact_25_less__not__refl3,axiom,
% 0.19/0.45 ! [S: nat,T: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ S @ T )
% 0.19/0.45 => ( S != T ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl3
% 0.19/0.45 thf(fact_26_measure__induct,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( wellorder @ B )
% 0.19/0.45 => ! [F: A > B,P: A > $o,A4: A] :
% 0.19/0.45 ( ! [X3: A] :
% 0.19/0.45 ( ! [Y2: A] :
% 0.19/0.45 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.19/0.45 => ( P @ Y2 ) )
% 0.19/0.45 => ( P @ X3 ) )
% 0.19/0.45 => ( P @ A4 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % measure_induct
% 0.19/0.45 thf(fact_27_less__imp__le__nat,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M @ N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_imp_le_nat
% 0.19/0.45 thf(fact_28_less__irrefl__nat,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.45
% 0.19/0.45 % less_irrefl_nat
% 0.19/0.45 thf(fact_29_nat__less__induct,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ! [N3: nat] :
% 0.19/0.45 ( ! [M3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N3 )
% 0.19/0.45 => ( P @ M3 ) )
% 0.19/0.45 => ( P @ N3 ) )
% 0.19/0.45 => ( P @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_less_induct
% 0.19/0.45 thf(fact_30_infinite__descent,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ! [N3: nat] :
% 0.19/0.45 ( ~ ( P @ N3 )
% 0.19/0.45 => ? [M3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N3 )
% 0.19/0.45 & ~ ( P @ M3 ) ) )
% 0.19/0.45 => ( P @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % infinite_descent
% 0.19/0.45 thf(fact_31_le__eq__less__or__eq,axiom,
% 0.19/0.45 ( ( ord_less_eq @ nat )
% 0.19/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M2 @ N2 )
% 0.19/0.45 | ( M2 = N2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_eq_less_or_eq
% 0.19/0.45 thf(fact_32_less__or__eq__imp__le,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ( ord_less @ nat @ M @ N )
% 0.19/0.45 | ( M = N ) )
% 0.19/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_or_eq_imp_le
% 0.19/0.45 thf(fact_33_linorder__neqE__nat,axiom,
% 0.19/0.45 ! [X2: nat,Y: nat] :
% 0.19/0.45 ( ( X2 != Y )
% 0.19/0.45 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 0.19/0.45 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % linorder_neqE_nat
% 0.19/0.45 thf(fact_34_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.45 ! [P: nat > $o,K: nat,B3: nat] :
% 0.19/0.45 ( ( P @ K )
% 0.19/0.45 => ( ! [Y3: nat] :
% 0.19/0.45 ( ( P @ Y3 )
% 0.19/0.45 => ( ord_less_eq @ nat @ Y3 @ B3 ) )
% 0.19/0.45 => ? [X3: nat] :
% 0.19/0.45 ( ( P @ X3 )
% 0.19/0.45 & ! [Y2: nat] :
% 0.19/0.45 ( ( P @ Y2 )
% 0.19/0.45 => ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Nat.ex_has_greatest_nat
% 0.19/0.45 thf(fact_35_le__neq__implies__less,axiom,
% 0.19/0.45 ! [M: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.45 => ( ( M != N )
% 0.19/0.45 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_neq_implies_less
% 0.19/0.45 thf(fact_36_measure__induct__rule,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( wellorder @ B )
% 0.19/0.45 => ! [F: A > B,P: A > $o,A4: A] :
% 0.19/0.45 ( ! [X3: A] :
% 0.19/0.45 ( ! [Y2: A] :
% 0.19/0.45 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.19/0.46 => ( P @ Y2 ) )
% 0.19/0.46 => ( P @ X3 ) )
% 0.19/0.46 => ( P @ A4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % measure_induct_rule
% 0.19/0.46 thf(fact_37_less__mono__imp__le__mono,axiom,
% 0.19/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.19/0.46 ( ! [I2: nat,J2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ J2 )
% 0.19/0.46 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_mono_imp_le_mono
% 0.19/0.46 thf(fact_38_infinite__descent__measure,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 0.19/0.46 ( ! [X3: A] :
% 0.19/0.46 ( ~ ( P @ X3 )
% 0.19/0.46 => ? [Y2: A] :
% 0.19/0.46 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 0.19/0.46 & ~ ( P @ Y2 ) ) )
% 0.19/0.46 => ( P @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent_measure
% 0.19/0.46 thf(fact_39_diff__strict__right__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_strict_right_mono
% 0.19/0.46 thf(fact_40_diff__strict__left__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.19/0.46 => ( ord_less @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_strict_left_mono
% 0.19/0.46 thf(fact_41_diff__eq__diff__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ( minus_minus @ A @ A4 @ B3 )
% 0.19/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.19/0.46 => ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_diff_less
% 0.19/0.46 thf(fact_42_diff__strict__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,D: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less @ A @ D @ C )
% 0.19/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_strict_mono
% 0.19/0.46 thf(fact_43_ext,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.19/0.46 ( ! [X3: A] :
% 0.19/0.46 ( ( F @ X3 )
% 0.19/0.46 = ( G @ X3 ) )
% 0.19/0.46 => ( F = G ) ) ).
% 0.19/0.46
% 0.19/0.46 % ext
% 0.19/0.46 thf(fact_44_diff__eq__diff__less__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ( minus_minus @ A @ A4 @ B3 )
% 0.19/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_diff_less_eq
% 0.19/0.46 thf(fact_45_diff__right__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_right_mono
% 0.19/0.46 thf(fact_46_diff__left__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_left_mono
% 0.19/0.46 thf(fact_47_diff__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,D: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ D @ C )
% 0.19/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_mono
% 0.19/0.46 thf(fact_48_length__induct,axiom,
% 0.19/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.19/0.46 ( ! [Xs2: list @ A] :
% 0.19/0.46 ( ! [Ys2: list @ A] :
% 0.19/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.19/0.46 => ( P @ Ys2 ) )
% 0.19/0.46 => ( P @ Xs2 ) )
% 0.19/0.46 => ( P @ Xs ) ) ).
% 0.19/0.46
% 0.19/0.46 % length_induct
% 0.19/0.46 thf(fact_49_less__imp__diff__less,axiom,
% 0.19/0.46 ! [J: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J @ K )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_diff_less
% 0.19/0.46 thf(fact_50_diff__less__mono2,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ M @ L )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono2
% 0.19/0.46 thf(fact_51_diff__le__mono2,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono2
% 0.19/0.46 thf(fact_52_le__diff__iff_H,axiom,
% 0.19/0.46 ! [A4: nat,C: nat,B3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ A4 @ C )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ B3 @ C )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A4 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ B3 @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff'
% 0.19/0.46 thf(fact_53_diff__le__self,axiom,
% 0.19/0.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_self
% 0.19/0.46 thf(fact_54_diff__le__mono,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono
% 0.19/0.46 thf(fact_55_Nat_Odiff__diff__eq,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_eq
% 0.19/0.46 thf(fact_56_le__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff
% 0.19/0.46 thf(fact_57_eq__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.46 => ( ( ( minus_minus @ nat @ M @ K )
% 0.19/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.19/0.46 = ( M = N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_diff_iff
% 0.19/0.46 thf(fact_58_list__eq__iff__nth__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.19/0.46 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.19/0.46 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.19/0.46 => ( ( nth @ A @ Xs3 @ I3 )
% 0.19/0.46 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_eq_iff_nth_eq
% 0.19/0.46 thf(fact_59_Skolem__list__nth,axiom,
% 0.19/0.46 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.19/0.46 ( ( ! [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ K )
% 0.19/0.46 => ? [X4: A] : ( P @ I3 @ X4 ) ) )
% 0.19/0.46 = ( ? [Xs3: list @ A] :
% 0.19/0.46 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.19/0.46 = K )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ K )
% 0.19/0.46 => ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Skolem_list_nth
% 0.19/0.46 thf(fact_60_nth__equalityI,axiom,
% 0.19/0.46 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.19/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.19/0.46 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.19/0.46 => ( ! [I2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.19/0.46 => ( ( nth @ A @ Xs @ I2 )
% 0.19/0.46 = ( nth @ A @ Ys @ I2 ) ) )
% 0.19/0.46 => ( Xs = Ys ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nth_equalityI
% 0.19/0.46 thf(fact_61_one__reorient,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( one @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ( ( ( one_one @ A )
% 0.19/0.46 = X2 )
% 0.19/0.46 = ( X2
% 0.19/0.46 = ( one_one @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % one_reorient
% 0.19/0.46 thf(fact_62_diff__right__commute,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 )
% 0.19/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_right_commute
% 0.19/0.46 thf(fact_63_diff__eq__diff__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ( minus_minus @ A @ A4 @ B3 )
% 0.19/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.19/0.46 => ( ( A4 = B3 )
% 0.19/0.46 = ( C = D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_diff_eq
% 0.19/0.46 thf(fact_64_Length,axiom,
% 0.19/0.46 ! [N4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ N4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Length
% 0.19/0.46 thf(fact_65_AllArePrefixesExec,axiom,
% 0.19/0.46 ! [M3: nat,N4: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ N4 )
% 0.19/0.46 => ( prefixList @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ M3 ) @ ( fe @ N4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % AllArePrefixesExec
% 0.19/0.46 thf(fact_66_order__refl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_67_nat__descend__induct,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o,M: nat] :
% 0.19/0.46 ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ N @ K2 )
% 0.19/0.46 => ( P @ K2 ) )
% 0.19/0.46 => ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ! [I4: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ K2 @ I4 )
% 0.19/0.46 => ( P @ I4 ) )
% 0.19/0.46 => ( P @ K2 ) ) )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_descend_induct
% 0.19/0.46 thf(fact_68_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [Y3: A] :
% 0.19/0.46 ( ( P @ Y3 )
% 0.19/0.46 => ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
% 0.19/0.46 => ? [X3: A] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 & ! [Y2: A] :
% 0.19/0.46 ( ( P @ Y2 )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lattices_Big.ex_has_greatest_nat
% 0.19/0.46 thf(fact_69_less__numeral__extra_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linord1659791738miring @ A )
% 0.19/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_numeral_extra(4)
% 0.19/0.46 thf(fact_70_le__numeral__extra_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linord1659791738miring @ A )
% 0.19/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_numeral_extra(4)
% 0.19/0.46 thf(fact_71_minf_I8_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(8)
% 0.19/0.46 thf(fact_72_minf_I6_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z2 )
% 0.19/0.46 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(6)
% 0.19/0.46 thf(fact_73_pinf_I8_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(8)
% 0.19/0.46 thf(fact_74_pinf_I6_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(6)
% 0.19/0.46 thf(fact_75_add__left__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel_semigroup_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ A4 @ B3 )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ C ) )
% 0.19/0.46 = ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_cancel
% 0.19/0.46 thf(fact_76_add__right__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel_semigroup_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ B3 @ A4 )
% 0.19/0.46 = ( plus_plus @ A @ C @ A4 ) )
% 0.19/0.46 = ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_cancel
% 0.19/0.46 thf(fact_77_add__le__cancel__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_right
% 0.19/0.46 thf(fact_78_add__le__cancel__left,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [C: A,A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.19/0.46 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_left
% 0.19/0.46 thf(fact_79_add__less__cancel__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 = ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_right
% 0.19/0.46 thf(fact_80_add__less__cancel__left,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [C: A,A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.19/0.46 = ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_left
% 0.19/0.46 thf(fact_81_add__diff__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
% 0.19/0.46 = A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel
% 0.19/0.46 thf(fact_82_diff__add__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
% 0.19/0.46 = A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_cancel
% 0.19/0.46 thf(fact_83_add__diff__cancel__left,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [C: A,A4: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.19/0.46 = ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left
% 0.19/0.46 thf(fact_84_add__diff__cancel__left_H,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ A4 )
% 0.19/0.46 = B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left'
% 0.19/0.46 thf(fact_85_add__diff__cancel__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 = ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right
% 0.19/0.46 thf(fact_86_add__diff__cancel__right_H,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
% 0.19/0.46 = A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right'
% 0.19/0.46 thf(fact_87_nat__add__left__cancel__less,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.46 = ( ord_less @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_add_left_cancel_less
% 0.19/0.46 thf(fact_88_nat__add__left__cancel__le,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_add_left_cancel_le
% 0.19/0.46 thf(fact_89_diff__diff__left,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.19/0.46 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_left
% 0.19/0.46 thf(fact_90_Nat_Odiff__diff__right,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_right
% 0.19/0.46 thf(fact_91_Nat_Oadd__diff__assoc2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.add_diff_assoc2
% 0.19/0.46 thf(fact_92_Nat_Oadd__diff__assoc,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.add_diff_assoc
% 0.19/0.46 thf(fact_93_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ab_semigroup_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.46 thf(fact_94_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ( plus_plus @ A @ I @ K )
% 0.19/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(4)
% 0.19/0.46 thf(fact_95_group__cancel_Oadd1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( comm_monoid_add @ A )
% 0.19/0.46 => ! [A5: A,K: A,A4: A,B3: A] :
% 0.19/0.46 ( ( A5
% 0.19/0.46 = ( plus_plus @ A @ K @ A4 ) )
% 0.19/0.46 => ( ( plus_plus @ A @ A5 @ B3 )
% 0.19/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add1
% 0.19/0.46 thf(fact_96_group__cancel_Oadd2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( comm_monoid_add @ A )
% 0.19/0.46 => ! [B4: A,K: A,B3: A,A4: A] :
% 0.19/0.46 ( ( B4
% 0.19/0.46 = ( plus_plus @ A @ K @ B3 ) )
% 0.19/0.46 => ( ( plus_plus @ A @ A4 @ B4 )
% 0.19/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add2
% 0.19/0.46 thf(fact_97_add_Oassoc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( semigroup_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.assoc
% 0.19/0.46 thf(fact_98_add_Oleft__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ A4 @ B3 )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ C ) )
% 0.19/0.46 = ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_cancel
% 0.19/0.46 thf(fact_99_add_Oright__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ B3 @ A4 )
% 0.19/0.46 = ( plus_plus @ A @ C @ A4 ) )
% 0.19/0.46 = ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.right_cancel
% 0.19/0.46 thf(fact_100_add_Ocommute,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ab_semigroup_add @ A )
% 0.19/0.46 => ( ( plus_plus @ A )
% 0.19/0.46 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.commute
% 0.19/0.46 thf(fact_101_add_Oleft__commute,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ab_semigroup_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C ) )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_commute
% 0.19/0.46 thf(fact_102_add__left__imp__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel_semigroup_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ A4 @ B3 )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ C ) )
% 0.19/0.46 => ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_imp_eq
% 0.19/0.46 thf(fact_103_add__right__imp__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel_semigroup_add @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ B3 @ A4 )
% 0.19/0.46 = ( plus_plus @ A @ C @ A4 ) )
% 0.19/0.46 => ( B3 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_imp_eq
% 0.19/0.46 thf(fact_104_is__num__normalize_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( neg_numeral @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % is_num_normalize(1)
% 0.19/0.46 thf(fact_105_add__le__imp__le__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_right
% 0.19/0.46 thf(fact_106_add__le__imp__le__left,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [C: A,A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_left
% 0.19/0.46 thf(fact_107_le__iff__add,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( canoni770627133id_add @ A )
% 0.19/0.46 => ( ( ord_less_eq @ A )
% 0.19/0.46 = ( ^ [A6: A,B5: A] :
% 0.19/0.46 ? [C2: A] :
% 0.19/0.46 ( B5
% 0.19/0.46 = ( plus_plus @ A @ A6 @ C2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_iff_add
% 0.19/0.46 thf(fact_108_add__right__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_mono
% 0.19/0.46 thf(fact_109_less__eqE,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( canoni770627133id_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ~ ! [C3: A] :
% 0.19/0.46 ( B3
% 0.19/0.46 != ( plus_plus @ A @ A4 @ C3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eqE
% 0.19/0.46 thf(fact_110_add__left__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_mono
% 0.19/0.46 thf(fact_111_add__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C @ D )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono
% 0.19/0.46 thf(fact_112_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(1)
% 0.19/0.46 thf(fact_113_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(2)
% 0.19/0.46 thf(fact_114_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere779506340up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(3)
% 0.19/0.46 thf(fact_115_add__less__imp__less__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 => ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_imp_less_right
% 0.19/0.46 thf(fact_116_add__less__imp__less__left,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere236663937imp_le @ A )
% 0.19/0.46 => ! [C: A,A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 0.19/0.46 => ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_imp_less_left
% 0.19/0.46 thf(fact_117_add__strict__right__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_right_mono
% 0.19/0.46 thf(fact_118_add__strict__left__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_left_mono
% 0.19/0.46 thf(fact_119_add__strict__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( strict2144017051up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less @ A @ C @ D )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_mono
% 0.19/0.46 thf(fact_120_add__mono__thms__linordered__field_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(1)
% 0.19/0.46 thf(fact_121_add__mono__thms__linordered__field_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_less @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(2)
% 0.19/0.46 thf(fact_122_add__mono__thms__linordered__field_I5_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.46 & ( ord_less @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(5)
% 0.19/0.46 thf(fact_123_group__cancel_Osub1,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ab_group_add @ A )
% 0.19/0.46 => ! [A5: A,K: A,A4: A,B3: A] :
% 0.19/0.46 ( ( A5
% 0.19/0.46 = ( plus_plus @ A @ K @ A4 ) )
% 0.19/0.46 => ( ( minus_minus @ A @ A5 @ B3 )
% 0.19/0.46 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.sub1
% 0.19/0.46 thf(fact_124_diff__eq__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ( minus_minus @ A @ A4 @ B3 )
% 0.19/0.46 = C )
% 0.19/0.46 = ( A4
% 0.19/0.46 = ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_eq_eq
% 0.19/0.46 thf(fact_125_eq__diff__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( A4
% 0.19/0.46 = ( minus_minus @ A @ C @ B3 ) )
% 0.19/0.46 = ( ( plus_plus @ A @ A4 @ B3 )
% 0.19/0.46 = C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_diff_eq
% 0.19/0.46 thf(fact_126_add__diff__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_eq
% 0.19/0.46 thf(fact_127_diff__diff__eq2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_eq2
% 0.19/0.46 thf(fact_128_diff__add__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_eq
% 0.19/0.46 thf(fact_129_diff__add__eq__diff__diff__swap,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) )
% 0.19/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_eq_diff_diff_swap
% 0.19/0.46 thf(fact_130_diff__diff__add,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel146912293up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_add
% 0.19/0.46 thf(fact_131_add__implies__diff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel1352612707id_add @ A )
% 0.19/0.46 => ! [C: A,B3: A,A4: A] :
% 0.19/0.46 ( ( ( plus_plus @ A @ C @ B3 )
% 0.19/0.46 = A4 )
% 0.19/0.46 => ( C
% 0.19/0.46 = ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_implies_diff
% 0.19/0.46 thf(fact_132_less__add__eq__less,axiom,
% 0.19/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ K @ L )
% 0.19/0.46 => ( ( ( plus_plus @ nat @ M @ L )
% 0.19/0.46 = ( plus_plus @ nat @ K @ N ) )
% 0.19/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_eq_less
% 0.19/0.46 thf(fact_133_trans__less__add2,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J )
% 0.19/0.46 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_less_add2
% 0.19/0.46 thf(fact_134_trans__less__add1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J )
% 0.19/0.46 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_less_add1
% 0.19/0.46 thf(fact_135_add__less__mono1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J )
% 0.19/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_mono1
% 0.19/0.46 thf(fact_136_not__add__less2,axiom,
% 0.19/0.46 ! [J: nat,I: nat] :
% 0.19/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 0.19/0.46
% 0.19/0.46 % not_add_less2
% 0.19/0.46 thf(fact_137_not__add__less1,axiom,
% 0.19/0.46 ! [I: nat,J: nat] :
% 0.19/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 0.19/0.46
% 0.19/0.46 % not_add_less1
% 0.19/0.46 thf(fact_138_add__less__mono,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ J )
% 0.19/0.46 => ( ( ord_less @ nat @ K @ L )
% 0.19/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_mono
% 0.19/0.46 thf(fact_139_add__lessD1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.19/0.46 => ( ord_less @ nat @ I @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_lessD1
% 0.19/0.46 thf(fact_140_nat__le__iff__add,axiom,
% 0.19/0.46 ( ( ord_less_eq @ nat )
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ? [K3: nat] :
% 0.19/0.46 ( N2
% 0.19/0.46 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_iff_add
% 0.19/0.46 thf(fact_141_trans__le__add2,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add2
% 0.19/0.46 thf(fact_142_trans__le__add1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add1
% 0.19/0.46 thf(fact_143_add__le__mono1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono1
% 0.19/0.46 thf(fact_144_add__le__mono,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K @ L )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono
% 0.19/0.46 thf(fact_145_le__Suc__ex,axiom,
% 0.19/0.46 ! [K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ L )
% 0.19/0.46 => ? [N3: nat] :
% 0.19/0.46 ( L
% 0.19/0.46 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_Suc_ex
% 0.19/0.46 thf(fact_146_add__leD2,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD2
% 0.19/0.46 thf(fact_147_add__leD1,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD1
% 0.19/0.46 thf(fact_148_le__add2,axiom,
% 0.19/0.46 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add2
% 0.19/0.46 thf(fact_149_le__add1,axiom,
% 0.19/0.46 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add1
% 0.19/0.46 thf(fact_150_add__leE,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.19/0.46 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 0.19/0.46 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leE
% 0.19/0.46 thf(fact_151_Nat_Odiff__cancel,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.46 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_cancel
% 0.19/0.46 thf(fact_152_diff__cancel2,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.19/0.46 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_cancel2
% 0.19/0.46 thf(fact_153_diff__add__inverse,axiom,
% 0.19/0.46 ! [N: nat,M: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_inverse
% 0.19/0.46 thf(fact_154_diff__add__inverse2,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_inverse2
% 0.19/0.46 thf(fact_155_ex__has__greatest__nat__lemma,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [X3: A] :
% 0.19/0.46 ( ( P @ X3 )
% 0.19/0.46 => ? [Y2: A] :
% 0.19/0.46 ( ( P @ Y2 )
% 0.19/0.46 & ~ ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) )
% 0.19/0.46 => ? [Y3: A] :
% 0.19/0.46 ( ( P @ Y3 )
% 0.19/0.46 & ~ ( ord_less @ nat @ ( F @ Y3 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_has_greatest_nat_lemma
% 0.19/0.46 thf(fact_156_add__less__le__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C @ D )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_le_mono
% 0.19/0.46 thf(fact_157_add__le__less__mono,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A,D: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less @ A @ C @ D )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_less_mono
% 0.19/0.46 thf(fact_158_add__mono__thms__linordered__field_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(3)
% 0.19/0.46 thf(fact_159_add__mono__thms__linordered__field_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere223160158up_add @ A )
% 0.19/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.46 & ( ord_less @ A @ K @ L ) )
% 0.19/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(4)
% 0.19/0.46 thf(fact_160_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ( minus_minus @ A @ B3 @ A4 )
% 0.19/0.46 = C )
% 0.19/0.46 = ( B3
% 0.19/0.46 = ( plus_plus @ A @ C @ A4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.19/0.46 thf(fact_161_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ A4 ) )
% 0.19/0.46 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.19/0.46 thf(fact_162_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.19/0.46 thf(fact_163_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 )
% 0.19/0.46 = ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.19/0.46 thf(fact_164_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.19/0.46 thf(fact_165_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 )
% 0.19/0.46 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.19/0.46 thf(fact_166_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 0.19/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.19/0.46 thf(fact_167_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 0.19/0.46 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.19/0.46 thf(fact_168_le__add__diff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_diff
% 0.19/0.46 thf(fact_169_diff__add,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordere623563068d_diff @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ A4 )
% 0.19/0.46 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add
% 0.19/0.46 thf(fact_170_le__diff__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
% 0.19/0.46 = ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_eq
% 0.19/0.46 thf(fact_171_diff__le__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_eq
% 0.19/0.46 thf(fact_172_less__diff__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,C: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
% 0.19/0.46 = ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_eq
% 0.19/0.46 thf(fact_173_diff__less__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 0.19/0.46 = ( ord_less @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_eq
% 0.19/0.46 thf(fact_174_mono__nat__linear__lb,axiom,
% 0.19/0.46 ! [F: nat > nat,M: nat,K: nat] :
% 0.19/0.46 ( ! [M4: nat,N3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M4 @ N3 )
% 0.19/0.46 => ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mono_nat_linear_lb
% 0.19/0.46 thf(fact_175_add__diff__inverse__nat,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ~ ( ord_less @ nat @ M @ N )
% 0.19/0.46 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_inverse_nat
% 0.19/0.46 thf(fact_176_less__diff__conv,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.46 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_conv
% 0.19/0.46 thf(fact_177_Nat_Ole__imp__diff__is__add,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.46 => ( ( ( minus_minus @ nat @ J @ I )
% 0.19/0.46 = K )
% 0.19/0.46 = ( J
% 0.19/0.46 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.le_imp_diff_is_add
% 0.19/0.46 thf(fact_178_Nat_Odiff__add__assoc2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 0.19/0.46 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_add_assoc2
% 0.19/0.46 thf(fact_179_Nat_Odiff__add__assoc,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.19/0.46 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_add_assoc
% 0.19/0.46 thf(fact_180_Nat_Ole__diff__conv2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.le_diff_conv2
% 0.19/0.46 thf(fact_181_le__diff__conv,axiom,
% 0.19/0.46 ! [J: nat,K: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.46 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_conv
% 0.19/0.46 thf(fact_182_less__diff__conv2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.46 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.46 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_conv2
% 0.19/0.46 thf(fact_183_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B3: A,A4: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_184_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A6: A,B5: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B5 @ A6 )
% 0.19/0.46 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_185_dual__order_Otrans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.19/0.46 => ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_186_linorder__wlog,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > A > $o,A4: A,B3: A] :
% 0.19/0.46 ( ! [A7: A,B6: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A7 @ B6 )
% 0.19/0.46 => ( P @ A7 @ B6 ) )
% 0.19/0.46 => ( ! [A7: A,B6: A] :
% 0.19/0.46 ( ( P @ B6 @ A7 )
% 0.19/0.46 => ( P @ A7 @ B6 ) )
% 0.19/0.46 => ( P @ A4 @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_187_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_188_order__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_189_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.19/0.46 => ( A4 = B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_190_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( B3 = C )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_191_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( A4 = B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_192_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A6: A,B5: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A6 @ B5 )
% 0.19/0.46 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_193_antisym__conv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [Y: A,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_194_le__cases3,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z3: A] :
% 0.19/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_195_order_Otrans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_196_le__cases,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_197_eq__refl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_198_linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_199_antisym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_200_eq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [X: A,Y5: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.19/0.46 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_201_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A4: A,B3: A,F: A > B,C: B] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ( F @ B3 )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_202_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A4: A,F: B > A,B3: B,C: B] :
% 0.19/0.46 ( ( A4
% 0.19/0.46 = ( F @ B3 ) )
% 0.19/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.19/0.46 => ( ! [X3: B,Y3: B] :
% 0.19/0.46 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_203_order__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C4: $tType] :
% 0.19/0.46 ( ( ( order @ C4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A4: A,B3: A,F: A > C4,C: C4] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ C4 @ ( F @ B3 ) @ C )
% 0.19/0.46 => ( ! [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_204_order__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( order @ B )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A4: A,F: B > A,B3: B,C: B] :
% 0.19/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 0.19/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.19/0.46 => ( ! [X3: B,Y3: B] :
% 0.19/0.46 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_205_le__fun__def,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.19/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.19/0.46 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_fun_def
% 0.19/0.46 thf(fact_206_le__funI,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B] :
% 0.19/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.19/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funI
% 0.19/0.46 thf(fact_207_le__funE,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funE
% 0.19/0.46 thf(fact_208_le__funD,axiom,
% 0.19/0.46 ! [B: $tType,A: $tType] :
% 0.19/0.46 ( ( ord @ B )
% 0.19/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.19/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.19/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_funD
% 0.19/0.46 thf(fact_209_dual__order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B3: A,A4: A] :
% 0.19/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.19/0.46 => ( A4 != B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_implies_not_eq
% 0.19/0.46 thf(fact_210_order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( A4 != B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_implies_not_eq
% 0.19/0.46 thf(fact_211_not__less__iff__gr__or__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.46 = ( ( ord_less @ A @ Y @ X2 )
% 0.19/0.46 | ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_iff_gr_or_eq
% 0.19/0.46 thf(fact_212_dual__order_Ostrict__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B3: A,A4: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.19/0.46 => ( ( ord_less @ A @ C @ B3 )
% 0.19/0.46 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans
% 0.19/0.46 thf(fact_213_linorder__less__wlog,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > A > $o,A4: A,B3: A] :
% 0.19/0.46 ( ! [A7: A,B6: A] :
% 0.19/0.46 ( ( ord_less @ A @ A7 @ B6 )
% 0.19/0.46 => ( P @ A7 @ B6 ) )
% 0.19/0.46 => ( ! [A7: A] : ( P @ A7 @ A7 )
% 0.19/0.46 => ( ! [A7: A,B6: A] :
% 0.19/0.46 ( ( P @ B6 @ A7 )
% 0.19/0.46 => ( P @ A7 @ B6 ) )
% 0.19/0.46 => ( P @ A4 @ B3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_less_wlog
% 0.19/0.46 thf(fact_214_exists__least__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( wellorder @ A )
% 0.19/0.46 => ( ( ^ [P2: A > $o] :
% 0.19/0.46 ? [X4: A] : ( P2 @ X4 ) )
% 0.19/0.46 = ( ^ [P3: A > $o] :
% 0.19/0.46 ? [N2: A] :
% 0.19/0.46 ( ( P3 @ N2 )
% 0.19/0.46 & ! [M2: A] :
% 0.19/0.46 ( ( ord_less @ A @ M2 @ N2 )
% 0.19/0.46 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % exists_least_iff
% 0.19/0.46 thf(fact_215_less__imp__not__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_less
% 0.19/0.46 thf(fact_216_order_Ostrict__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less @ A @ B3 @ C )
% 0.19/0.46 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans
% 0.19/0.46 thf(fact_217_dual__order_Oirrefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A] :
% 0.19/0.46 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.irrefl
% 0.19/0.46 thf(fact_218_linorder__cases,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ( X2 != Y )
% 0.19/0.46 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_cases
% 0.19/0.46 thf(fact_219_less__imp__triv,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,P: $o] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less @ A @ Y @ X2 )
% 0.19/0.46 => P ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_triv
% 0.19/0.46 thf(fact_220_less__imp__not__eq2,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( Y != X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq2
% 0.19/0.46 thf(fact_221_antisym__conv3,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [Y: A,X2: A] :
% 0.19/0.46 ( ~ ( ord_less @ A @ Y @ X2 )
% 0.19/0.46 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.19/0.46 = ( X2 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv3
% 0.19/0.46 thf(fact_222_less__induct,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( wellorder @ A )
% 0.19/0.46 => ! [P: A > $o,A4: A] :
% 0.19/0.46 ( ! [X3: A] :
% 0.19/0.46 ( ! [Y2: A] :
% 0.19/0.46 ( ( ord_less @ A @ Y2 @ X3 )
% 0.19/0.46 => ( P @ Y2 ) )
% 0.19/0.46 => ( P @ X3 ) )
% 0.19/0.46 => ( P @ A4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_induct
% 0.19/0.46 thf(fact_223_less__not__sym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_sym
% 0.19/0.46 thf(fact_224_less__imp__not__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( X2 != Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq
% 0.19/0.46 thf(fact_225_dual__order_Oasym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [B3: A,A4: A] :
% 0.19/0.46 ( ( ord_less @ A @ B3 @ A4 )
% 0.19/0.46 => ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.asym
% 0.19/0.46 thf(fact_226_ord__less__eq__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( B3 = C )
% 0.19/0.46 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_trans
% 0.19/0.46 thf(fact_227_ord__eq__less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ord @ A )
% 0.19/0.46 => ! [A4: A,B3: A,C: A] :
% 0.19/0.46 ( ( A4 = B3 )
% 0.19/0.46 => ( ( ord_less @ A @ B3 @ C )
% 0.19/0.46 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_trans
% 0.19/0.46 thf(fact_228_less__irrefl,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl
% 0.19/0.46 thf(fact_229_less__linear,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 | ( X2 = Y )
% 0.19/0.46 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_linear
% 0.19/0.46 thf(fact_230_less__trans,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A,Z3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less @ A @ Y @ Z3 )
% 0.19/0.46 => ( ord_less @ A @ X2 @ Z3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_trans
% 0.19/0.46 thf(fact_231_less__asym_H,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym'
% 0.19/0.46 thf(fact_232_less__asym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( preorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym
% 0.19/0.46 thf(fact_233_less__imp__neq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( X2 != Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_neq
% 0.19/0.46 thf(fact_234_dense,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( dense_order @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ? [Z2: A] :
% 0.19/0.46 ( ( ord_less @ A @ X2 @ Z2 )
% 0.19/0.46 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dense
% 0.19/0.46 thf(fact_235_order_Oasym,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [A4: A,B3: A] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.asym
% 0.19/0.46 thf(fact_236_neq__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( X2 != Y )
% 0.19/0.46 = ( ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_iff
% 0.19/0.46 thf(fact_237_neqE,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [X2: A,Y: A] :
% 0.19/0.46 ( ( X2 != Y )
% 0.19/0.46 => ( ~ ( ord_less @ A @ X2 @ Y )
% 0.19/0.46 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neqE
% 0.19/0.46 thf(fact_238_gt__ex,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( no_top @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 0.19/0.46
% 0.19/0.46 % gt_ex
% 0.19/0.46 thf(fact_239_lt__ex,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( no_bot @ A )
% 0.19/0.46 => ! [X2: A] :
% 0.19/0.46 ? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % lt_ex
% 0.19/0.46 thf(fact_240_order__less__subst2,axiom,
% 0.19/0.46 ! [A: $tType,C4: $tType] :
% 0.19/0.46 ( ( ( order @ C4 )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A4: A,B3: A,F: A > C4,C: C4] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ord_less @ C4 @ ( F @ B3 ) @ C )
% 0.19/0.46 => ( ! [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_241_order__less__subst1,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( order @ B )
% 0.19/0.46 & ( order @ A ) )
% 0.19/0.46 => ! [A4: A,F: B > A,B3: B,C: B] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 0.19/0.46 => ( ( ord_less @ B @ B3 @ C )
% 0.19/0.46 => ( ! [X3: B,Y3: B] :
% 0.19/0.46 ( ( ord_less @ B @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_242_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A4: A,B3: A,F: A > B,C: B] :
% 0.19/0.46 ( ( ord_less @ A @ A4 @ B3 )
% 0.19/0.46 => ( ( ( F @ B3 )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X3: A,Y3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_243_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: $tType,B: $tType] :
% 0.19/0.46 ( ( ( ord @ B )
% 0.19/0.46 & ( ord @ A ) )
% 0.19/0.46 => ! [A4: A,F: B > A,B3: B,C: B] :
% 0.19/0.46 ( ( A4
% 0.19/0.46 = ( F @ B3 ) )
% 0.19/0.46 => ( ( ord_less @ B @ B3 @ C )
% 0.19/0.46 => ( ! [X3: B,Y3: B] :
% 0.19/0.46 ( ( ord_less @ B @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.19/0.46 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_244_pinf_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X3 )
% 0.19/0.46 => ( ( P @ X3 )
% 0.19/0.46 = ( P4 @ X3 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X3 )
% 0.19/0.46 => ( ( Q @ X3 )
% 0.19/0.46 = ( Q2 @ X3 ) ) )
% 0.19/0.46 => ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 & ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(1)
% 0.19/0.46 thf(fact_245_pinf_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X3 )
% 0.19/0.46 => ( ( P @ X3 )
% 0.19/0.46 = ( P4 @ X3 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z4 @ X3 )
% 0.19/0.46 => ( ( Q @ X3 )
% 0.19/0.46 = ( Q2 @ X3 ) ) )
% 0.19/0.46 => ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 | ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(2)
% 0.19/0.46 thf(fact_246_pinf_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(3)
% 0.19/0.46 thf(fact_247_pinf_I4_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(4)
% 0.19/0.46 thf(fact_248_pinf_I5_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(5)
% 0.19/0.46 thf(fact_249_pinf_I7_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ Z2 @ X5 )
% 0.19/0.46 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(7)
% 0.19/0.46 thf(fact_250_pinf_I11_J,axiom,
% 0.19/0.46 ! [C4: $tType,D2: $tType] :
% 0.19/0.46 ( ( ord @ C4 )
% 0.19/0.46 => ! [F3: D2] :
% 0.19/0.46 ? [Z2: C4] :
% 0.19/0.46 ! [X5: C4] :
% 0.19/0.46 ( ( ord_less @ C4 @ Z2 @ X5 )
% 0.19/0.46 => ( F3 = F3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(11)
% 0.19/0.46 thf(fact_251_minf_I1_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Z4 )
% 0.19/0.46 => ( ( P @ X3 )
% 0.19/0.46 = ( P4 @ X3 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Z4 )
% 0.19/0.46 => ( ( Q @ X3 )
% 0.19/0.46 = ( Q2 @ X3 ) ) )
% 0.19/0.46 => ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z2 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 & ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(1)
% 0.19/0.46 thf(fact_252_minf_I2_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 0.19/0.46 ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Z4 )
% 0.19/0.46 => ( ( P @ X3 )
% 0.19/0.46 = ( P4 @ X3 ) ) )
% 0.19/0.46 => ( ? [Z4: A] :
% 0.19/0.46 ! [X3: A] :
% 0.19/0.46 ( ( ord_less @ A @ X3 @ Z4 )
% 0.19/0.46 => ( ( Q @ X3 )
% 0.19/0.46 = ( Q2 @ X3 ) ) )
% 0.19/0.46 => ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z2 )
% 0.19/0.46 => ( ( ( P @ X5 )
% 0.19/0.46 | ( Q @ X5 ) )
% 0.19/0.46 = ( ( P4 @ X5 )
% 0.19/0.46 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(2)
% 0.19/0.46 thf(fact_253_minf_I3_J,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( linorder @ A )
% 0.19/0.46 => ! [T: A] :
% 0.19/0.46 ? [Z2: A] :
% 0.19/0.46 ! [X5: A] :
% 0.19/0.46 ( ( ord_less @ A @ X5 @ Z2 )
% 0.19/0.46 => ( X5 != T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(3)
% 0.19/0.46
% 0.19/0.46 % Type constructors (38)
% 0.19/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.19/0.46 ! [A2: $tType,A8: $tType] :
% 0.19/0.46 ( ( preorder @ A8 )
% 0.19/0.46 => ( preorder @ ( A2 > A8 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.46 ! [A2: $tType,A8: $tType] :
% 0.19/0.46 ( ( order @ A8 )
% 0.19/0.46 => ( order @ ( A2 > A8 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.19/0.46 ! [A2: $tType,A8: $tType] :
% 0.19/0.46 ( ( ord @ A8 )
% 0.19/0.46 => ( ord @ ( A2 > A8 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_fun___Groups_Ominus,axiom,
% 0.19/0.46 ! [A2: $tType,A8: $tType] :
% 0.19/0.46 ( ( minus @ A8 )
% 0.19/0.46 => ( minus @ ( A2 > A8 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.19/0.46 strict2144017051up_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 0.19/0.46 ordere623563068d_diff @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.19/0.46 ordere223160158up_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.19/0.46 ordere236663937imp_le @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.19/0.46 canoni770627133id_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.19/0.46 linord1659791738miring @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.19/0.46 ordere779506340up_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.19/0.46 cancel146912293up_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.19/0.46 cancel1352612707id_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.19/0.46 cancel_semigroup_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.19/0.46 ab_semigroup_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.19/0.46 comm_monoid_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.19/0.46 semigroup_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.19/0.46 wellorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.19/0.46 preorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.19/0.46 linorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.19/0.46 no_top @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.19/0.46 order @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.19/0.46 ord @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 0.19/0.46 minus @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.19/0.46 one @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.19/0.46 size @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 0.19/0.46 preorder @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 0.19/0.46 linorder @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 0.19/0.46 order @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.19/0.46 ord @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 0.19/0.46 minus @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_List_Olist___Nat_Osize_10,axiom,
% 0.19/0.46 ! [A2: $tType] : ( size @ ( list @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_11,axiom,
% 0.19/0.46 wellorder @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_12,axiom,
% 0.19/0.46 preorder @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_13,axiom,
% 0.19/0.46 linorder @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_14,axiom,
% 0.19/0.46 order @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_15,axiom,
% 0.19/0.46 ord @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Groups_Ominus_16,axiom,
% 0.19/0.46 minus @ product_unit ).
% 0.19/0.46
% 0.19/0.46 % Conjectures (1)
% 0.19/0.46 thf(conj_0,conjecture,
% 0.19/0.46 ( ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) )
% 0.19/0.46 = ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 2.59/1.52 ---- Embedded in THF ---
% 2.59/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.59/1.52 %%% Generated on Tue Feb 24 08:25:04 EST 2026
% 2.59/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.59/1.52 %%% Logic specification used:
% 2.59/1.52 %%% thf(spec, logic, $$dhol).
% 2.59/1.52
% 2.59/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.UV0DDxgB5h/DTF2DTF6476.p
% 2.59/1.52 thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext, type, configuration_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 2.59/1.52 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.59/1.52 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.59/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.59/1.52 thf(ty_tf_v, type, v: $tType).
% 2.59/1.52 thf(ty_tf_s, type, s: $tType).
% 2.59/1.52 thf(ty_tf_p, type, p: $tType).
% 2.59/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff, type, ordere623563068d_diff: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: ($o))).
% 2.59/1.52 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.59/1.52 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.59/1.52 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.59/1.52 thf(sy_c_ListUtilities_OprefixList, type, prefixList: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 2.59/1.52 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 2.59/1.52 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.59/1.52 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.59/1.52 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.59/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.59/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.59/1.52 thf(sy_v_fe____, type, fe: (nat > (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit)))).
% 2.59/1.52 thf(sy_v_n0____, type, n0: nat).
% 2.59/1.52 thf(sy_v_n____, type, n: nat).
% 2.59/1.52 thf(fact_0_FeNNotEmpty, axiom, ((fe @ n) != (nil @ ((((configuration_ext @ p) @ v) @ s) @ product_unit)))).
% 2.59/1.52 thf(fact_1_LastIsLastIndex, axiom, (! [A2:$tType,L:(list @ A2)]: (((L != (nil @ A2)) => (((last @ A2) @ L) = (((nth @ A2) @ L) @ (((minus_minus @ nat) @ ((size_size @ (list @ A2)) @ L)) @ (one_one @ nat)))))))).
% 2.59/1.52 thf(fact_2_Fou2, axiom, (((ord_less @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ n))) @ (one_one @ nat))) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ n)))).
% 2.59/1.52 thf(fact_3_AssmNoDecided_I1_J, axiom, (((ord_less @ nat) @ n0) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ n)))).
% 2.59/1.52 thf(fact_4_last__conv__nth, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((last @ A) @ Xs) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))))).
% 2.59/1.52 thf(fact_5_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A3:(A > B),B2:(A > B),X:A]: ((((minus_minus @ B) @ (A3 @ X)) @ (B2 @ X))))))))).
% 2.59/1.52 thf(fact_6_Fou, axiom, (((ord_less_eq @ nat) @ n0) @ (((minus_minus @ nat) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ n))) @ (one_one @ nat)))).
% 2.59/1.52 thf(fact_7_one__natural_Orsp, axiom, ((one_one @ nat) = (one_one @ nat))).
% 2.59/1.52 thf(fact_8_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 2.59/1.52 thf(fact_9_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 2.59/1.52 thf(fact_10_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.59/1.52 thf(fact_11_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X2:A,Y:A]: (((((size_size @ A) @ X2) != ((size_size @ A) @ Y)) => (X2 != Y)))))))).
% 2.59/1.52 thf(fact_12_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A3:(A > B),B2:(A > B),X:A]: ((((minus_minus @ B) @ (A3 @ X)) @ (B2 @ X))))))))).
% 2.59/1.52 thf(fact_13_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.59/1.52 thf(fact_14_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.59/1.52 thf(fact_15_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.59/1.52 thf(fact_16_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_17_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.59/1.52 thf(fact_18_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.59/1.52 thf(fact_19_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.59/1.52 thf(fact_20_less__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less @ nat) @ M) @ N))))))).
% 2.59/1.52 thf(fact_21_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.59/1.52 thf(fact_22_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.59/1.52 thf(fact_23_diff__less__mono, axiom, (! [A4:nat,B3:nat,C:nat]: (((((ord_less @ nat) @ A4) @ B3) => ((((ord_less_eq @ nat) @ C) @ A4) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A4) @ C)) @ (((minus_minus @ nat) @ B3) @ C))))))).
% 2.59/1.52 thf(fact_24_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.59/1.52 thf(fact_25_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.59/1.52 thf(fact_26_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A4)))))))).
% 2.59/1.52 thf(fact_27_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_28_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.59/1.52 thf(fact_29_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P @ M3)))) => (P @ N3)))) => (P @ N))))).
% 2.59/1.52 thf(fact_30_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P @ M3))))))))) => (P @ N))))).
% 2.59/1.52 thf(fact_31_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.59/1.52 thf(fact_32_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_33_linorder__neqE__nat, axiom, (! [X2:nat,Y:nat]: (((X2 != Y) => ((~ ((((ord_less @ nat) @ X2) @ Y))) => (((ord_less @ nat) @ Y) @ X2)))))).
% 2.59/1.52 thf(fact_34_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B3:nat]: (((P @ K) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B3)))) => (? [X3:nat]: (((P @ X3) & (! [Y2:nat]: (((P @ Y2) => (((ord_less_eq @ nat) @ Y2) @ X3)))))))))))).
% 2.59/1.52 thf(fact_35_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.59/1.52 thf(fact_36_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A4)))))))).
% 2.59/1.52 thf(fact_37_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 2.59/1.52 thf(fact_38_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X3)) & (~ ((P @ Y2))))))))) => (P @ X2))))).
% 2.59/1.52 thf(fact_39_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => (((ord_less @ A) @ (((minus_minus @ A) @ A4) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_40_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A4:A,C:A]: (((((ord_less @ A) @ B3) @ A4) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A4)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_41_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A4) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A4) @ B3) = (((ord_less @ A) @ C) @ D))))))))).
% 2.59/1.52 thf(fact_42_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,D:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A4) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_43_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.59/1.52 thf(fact_44_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A4) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A4) @ B3) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.59/1.52 thf(fact_45_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_46_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A4:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A4) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A4)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_47_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,D:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_48_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys2:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys2)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.59/1.52 thf(fact_49_less__imp__diff__less, axiom, (! [J:nat,K:nat,N:nat]: (((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ N)) @ K))))).
% 2.59/1.52 thf(fact_50_diff__less__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less @ nat) @ M) @ N) => ((((ord_less @ nat) @ M) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M))))))).
% 2.59/1.52 thf(fact_51_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.59/1.52 thf(fact_52_le__diff__iff_H, axiom, (! [A4:nat,C:nat,B3:nat]: (((((ord_less_eq @ nat) @ A4) @ C) => ((((ord_less_eq @ nat) @ B3) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A4)) @ (((minus_minus @ nat) @ C) @ B3)) = (((ord_less_eq @ nat) @ B3) @ A4))))))).
% 2.59/1.52 thf(fact_53_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.59/1.52 thf(fact_54_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.59/1.52 thf(fact_55_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.59/1.52 thf(fact_56_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.59/1.52 thf(fact_57_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.59/1.52 thf(fact_58_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y4:(list @ A),Z:(list @ A)]: ((Y4 = Z))) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys3)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs3)) => ((((nth @ A) @ Xs3) @ I3) = (((nth @ A) @ Ys3) @ I3)))))))))))).
% 2.59/1.52 thf(fact_59_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => (? [X4:A]: (((P @ I3) @ X4)))))) = (? [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => ((P @ I3) @ (((nth @ A) @ Xs3) @ I3)))))))))))).
% 2.59/1.52 thf(fact_60_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I2) = (((nth @ A) @ Ys) @ I2))))) => (Xs = Ys)))))).
% 2.59/1.52 thf(fact_61_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X2:A]: ((((one_one @ A) = X2) = (X2 = (one_one @ A))))))))).
% 2.59/1.52 thf(fact_62_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,C:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ C)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ C)))))))).
% 2.59/1.52 thf(fact_63_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A4) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((A4 = B3) = (C = D))))))))).
% 2.59/1.52 thf(fact_64_Length, axiom, (! [N4:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ N4) @ (one_one @ nat))) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ N4)))))).
% 2.59/1.52 thf(fact_65_AllArePrefixesExec, axiom, (! [M3:nat,N4:nat]: (((((ord_less @ nat) @ M3) @ N4) => (((prefixList @ ((((configuration_ext @ p) @ v) @ s) @ product_unit)) @ (fe @ M3)) @ (fe @ N4)))))).
% 2.59/1.52 thf(fact_66_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.59/1.52 thf(fact_67_nat__descend__induct, axiom, (! [N:nat,P:(nat > $o),M:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) => ((! [I4:nat]: (((((ord_less @ nat) @ K2) @ I4) => (P @ I4)))) => (P @ K2))))) => (P @ M)))))).
% 2.59/1.52 thf(fact_68_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),B3:nat]: (((P @ K) => ((! [Y3:A]: (((P @ Y3) => (((ord_less @ nat) @ (F @ Y3)) @ B3)))) => (? [X3:A]: (((P @ X3) & (! [Y2:A]: (((P @ Y2) => (((ord_less_eq @ nat) @ (F @ Y2)) @ (F @ X3))))))))))))).
% 2.59/1.52 thf(fact_69_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.59/1.52 thf(fact_70_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.59/1.52 thf(fact_71_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z2) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.59/1.52 thf(fact_72_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z2) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.59/1.52 thf(fact_73_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.59/1.52 thf(fact_74_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.59/1.52 thf(fact_75_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A4:A,B3:A,C:A]: ((((((plus_plus @ A) @ A4) @ B3) = (((plus_plus @ A) @ A4) @ C)) = (B3 = C)))))))).
% 2.59/1.52 thf(fact_76_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A4:A,C:A]: ((((((plus_plus @ A) @ B3) @ A4) = (((plus_plus @ A) @ C) @ A4)) = (B3 = C)))))))).
% 2.59/1.52 thf(fact_77_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_78_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A4:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3)) = (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_79_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((ord_less @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_80_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A4:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3)) = (((ord_less @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_81_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ B3) = A4))))))).
% 2.59/1.52 thf(fact_82_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ B3) = A4))))))).
% 2.59/1.52 thf(fact_83_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [C:A,A4:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3)) = (((minus_minus @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_84_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ A4) = B3))))))).
% 2.59/1.52 thf(fact_85_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,C:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C)) = (((minus_minus @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_86_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,B3:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ B3) = A4))))))).
% 2.59/1.52 thf(fact_87_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_88_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_89_diff__diff__left, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.59/1.52 thf(fact_90_Nat_Odiff__diff__right, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))).
% 2.59/1.52 thf(fact_91_Nat_Oadd__diff__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K)))))).
% 2.59/1.52 thf(fact_92_Nat_Oadd__diff__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K)))))).
% 2.59/1.52 thf(fact_93_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A4:A,B3:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_94_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_95_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A5:A,K:A,A4:A,B3:A]: (((A5 = (((plus_plus @ A) @ K) @ A4)) => ((((plus_plus @ A) @ A5) @ B3) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A4) @ B3)))))))))).
% 2.59/1.52 thf(fact_96_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B4:A,K:A,B3:A,A4:A]: (((B4 = (((plus_plus @ A) @ K) @ B3)) => ((((plus_plus @ A) @ A4) @ B4) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A4) @ B3)))))))))).
% 2.59/1.52 thf(fact_97_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A4:A,B3:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_98_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A]: ((((((plus_plus @ A) @ A4) @ B3) = (((plus_plus @ A) @ A4) @ C)) = (B3 = C)))))))).
% 2.59/1.52 thf(fact_99_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B3:A,A4:A,C:A]: ((((((plus_plus @ A) @ B3) @ A4) = (((plus_plus @ A) @ C) @ A4)) = (B3 = C)))))))).
% 2.59/1.52 thf(fact_100_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A6:A,B5:A]: ((((plus_plus @ A) @ B5) @ A6)))))))).
% 2.59/1.52 thf(fact_101_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B3:A,A4:A,C:A]: (((((plus_plus @ A) @ B3) @ (((plus_plus @ A) @ A4) @ C)) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_102_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A4:A,B3:A,C:A]: ((((((plus_plus @ A) @ A4) @ B3) = (((plus_plus @ A) @ A4) @ C)) => (B3 = C)))))))).
% 2.59/1.52 thf(fact_103_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B3:A,A4:A,C:A]: ((((((plus_plus @ A) @ B3) @ A4) = (((plus_plus @ A) @ C) @ A4)) => (B3 = C)))))))).
% 2.59/1.52 thf(fact_104_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A4:A,B3:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C) = (((plus_plus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_105_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C)) => (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_106_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A4:A,B3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3)) => (((ord_less_eq @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_107_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A6:A,B5:A]: ((? [C2:A]: ((B5 = (((plus_plus @ A) @ A6) @ C2))))))))))).
% 2.59/1.52 thf(fact_108_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_109_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (~ ((! [C3:A]: ((B3 != (((plus_plus @ A) @ A4) @ C3))))))))))))).
% 2.59/1.52 thf(fact_110_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_111_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A4:A,B3:A,C:A,D:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_112_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_113_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_114_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_115_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C)) => (((ord_less @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_116_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A4:A,B3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3)) => (((ord_less @ A) @ A4) @ B3)))))))).
% 2.59/1.52 thf(fact_117_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => (((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_118_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A4)) @ (((plus_plus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_119_add__strict__mono, axiom, (! [A:$tType]: (((strict2144017051up_add @ A) => (! [A4:A,B3:A,C:A,D:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_120_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_121_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_122_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_123_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A5:A,K:A,A4:A,B3:A]: (((A5 = (((plus_plus @ A) @ K) @ A4)) => ((((minus_minus @ A) @ A5) @ B3) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A4) @ B3)))))))))).
% 2.59/1.52 thf(fact_124_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A]: ((((((minus_minus @ A) @ A4) @ B3) = C) = (A4 = (((plus_plus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_125_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,C:A,B3:A]: (((A4 = (((minus_minus @ A) @ C) @ B3)) = ((((plus_plus @ A) @ A4) @ B3) = C)))))))).
% 2.59/1.52 thf(fact_126_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A]: (((((plus_plus @ A) @ A4) @ (((minus_minus @ A) @ B3) @ C)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C)))))))).
% 2.59/1.52 thf(fact_127_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A]: (((((minus_minus @ A) @ A4) @ (((minus_minus @ A) @ B3) @ C)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ C)) @ B3)))))))).
% 2.59/1.52 thf(fact_128_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,B3:A,C:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ C) = (((minus_minus @ A) @ (((plus_plus @ A) @ A4) @ C)) @ B3)))))))).
% 2.59/1.52 thf(fact_129_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B3:A,C:A]: (((((minus_minus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ C)) @ B3)))))))).
% 2.59/1.52 thf(fact_130_diff__diff__add, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,B3:A,C:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ C) = (((minus_minus @ A) @ A4) @ (((plus_plus @ A) @ B3) @ C))))))))).
% 2.59/1.52 thf(fact_131_add__implies__diff, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [C:A,B3:A,A4:A]: ((((((plus_plus @ A) @ C) @ B3) = A4) => (C = (((minus_minus @ A) @ A4) @ B3))))))))).
% 2.59/1.52 thf(fact_132_less__add__eq__less, axiom, (! [K:nat,L:nat,M:nat,N:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.59/1.52 thf(fact_133_trans__less__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.59/1.52 thf(fact_134_trans__less__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.59/1.52 thf(fact_135_add__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.59/1.52 thf(fact_136_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))).
% 2.59/1.52 thf(fact_137_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))).
% 2.59/1.52 thf(fact_138_add__less__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.59/1.52 thf(fact_139_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))).
% 2.59/1.52 thf(fact_140_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: ((? [K3:nat]: ((N2 = (((plus_plus @ nat) @ M2) @ K3)))))))).
% 2.59/1.52 thf(fact_141_trans__le__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.59/1.52 thf(fact_142_trans__le__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.59/1.52 thf(fact_143_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.59/1.52 thf(fact_144_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.59/1.52 thf(fact_145_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K) @ N3)))))))).
% 2.59/1.52 thf(fact_146_add__leD2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 2.59/1.52 thf(fact_147_add__leD1, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_148_le__add2, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_149_le__add1, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M))))).
% 2.59/1.52 thf(fact_150_add__leE, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 2.59/1.52 thf(fact_151_Nat_Odiff__cancel, axiom, (! [K:nat,M:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_152_diff__cancel2, axiom, (! [M:nat,K:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))).
% 2.59/1.52 thf(fact_153_diff__add__inverse, axiom, (! [N:nat,M:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ M)) @ N) = M)))).
% 2.59/1.52 thf(fact_154_diff__add__inverse2, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N)) @ N) = M)))).
% 2.59/1.52 thf(fact_155_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),N:nat]: (((P @ K) => ((! [X3:A]: (((P @ X3) => (? [Y2:A]: (((P @ Y2) & (~ ((((ord_less_eq @ nat) @ (F @ Y2)) @ (F @ X3)))))))))) => (? [Y3:A]: (((P @ Y3) & (~ ((((ord_less @ nat) @ (F @ Y3)) @ (((plus_plus @ nat) @ (F @ K)) @ N)))))))))))).
% 2.59/1.52 thf(fact_156_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A4:A,B3:A,C:A,D:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_157_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A4:A,B3:A,C:A,D:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A4) @ C)) @ (((plus_plus @ A) @ B3) @ D)))))))))).
% 2.59/1.52 thf(fact_158_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_159_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.59/1.52 thf(fact_160_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ A4) @ B3) => (((((minus_minus @ A) @ B3) @ A4) = C) = (B3 = (((plus_plus @ A) @ C) @ A4))))))))))).
% 2.59/1.52 thf(fact_161_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((plus_plus @ A) @ A4) @ (((minus_minus @ A) @ B3) @ A4)) = B3)))))))).
% 2.59/1.52 thf(fact_162_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((minus_minus @ A) @ C) @ (((minus_minus @ A) @ B3) @ A4)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C) @ A4)) @ B3))))))))).
% 2.59/1.52 thf(fact_163_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((minus_minus @ A) @ (((plus_plus @ A) @ B3) @ C)) @ A4) = (((plus_plus @ A) @ (((minus_minus @ A) @ B3) @ A4)) @ C))))))))).
% 2.59/1.52 thf(fact_164_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B3) @ A4)) @ C) = (((minus_minus @ A) @ (((plus_plus @ A) @ B3) @ C)) @ A4))))))))).
% 2.59/1.52 thf(fact_165_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((minus_minus @ A) @ (((plus_plus @ A) @ C) @ B3)) @ A4) = (((plus_plus @ A) @ C) @ (((minus_minus @ A) @ B3) @ A4)))))))))).
% 2.59/1.52 thf(fact_166_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((plus_plus @ A) @ C) @ (((minus_minus @ A) @ B3) @ A4)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C) @ B3)) @ A4))))))))).
% 2.59/1.52 thf(fact_167_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ C) @ (((minus_minus @ A) @ B3) @ A4)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A4)) @ B3))))))))).
% 2.59/1.52 thf(fact_168_le__add__diff, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => (((ord_less_eq @ A) @ C) @ (((minus_minus @ A) @ (((plus_plus @ A) @ B3) @ C)) @ A4))))))))).
% 2.59/1.52 thf(fact_169_diff__add, axiom, (! [A:$tType]: (((ordere623563068d_diff @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B3) @ A4)) @ A4) = B3)))))))).
% 2.59/1.52 thf(fact_170_le__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ (((minus_minus @ A) @ C) @ B3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C)))))))).
% 2.59/1.52 thf(fact_171_diff__le__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ C) = (((ord_less_eq @ A) @ A4) @ (((plus_plus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_172_less__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,C:A,B3:A]: (((((ord_less @ A) @ A4) @ (((minus_minus @ A) @ C) @ B3)) = (((ord_less @ A) @ (((plus_plus @ A) @ A4) @ B3)) @ C)))))))).
% 2.59/1.52 thf(fact_173_diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ A4) @ B3)) @ C) = (((ord_less @ A) @ A4) @ (((plus_plus @ A) @ C) @ B3))))))))).
% 2.59/1.52 thf(fact_174_mono__nat__linear__lb, axiom, (! [F:(nat > nat),M:nat,K:nat]: (((! [M4:nat,N3:nat]: (((((ord_less @ nat) @ M4) @ N3) => (((ord_less @ nat) @ (F @ M4)) @ (F @ N3))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F @ M)) @ K)) @ (F @ (((plus_plus @ nat) @ M) @ K))))))).
% 2.59/1.52 thf(fact_175_add__diff__inverse__nat, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) => ((((plus_plus @ nat) @ N) @ (((minus_minus @ nat) @ M) @ N)) = M))))).
% 2.59/1.52 thf(fact_176_less__diff__conv, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J))))).
% 2.59/1.52 thf(fact_177_Nat_Ole__imp__diff__is__add, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((minus_minus @ nat) @ J) @ I) = K) = (J = (((plus_plus @ nat) @ K) @ I))))))).
% 2.59/1.52 thf(fact_178_Nat_Odiff__add__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K) = (((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I)))))).
% 2.59/1.52 thf(fact_179_Nat_Odiff__add__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K))))))).
% 2.59/1.52 thf(fact_180_Nat_Ole__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))).
% 2.59/1.52 thf(fact_181_le__diff__conv, axiom, (! [J:nat,K:nat,I:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less_eq @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K)))))).
% 2.59/1.52 thf(fact_182_less__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K))))))).
% 2.59/1.52 thf(fact_183_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ A4) @ B3) => (A4 = B3))))))))).
% 2.59/1.52 thf(fact_184_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A6) & (((ord_less_eq @ A) @ A6) @ B5))))))))).
% 2.59/1.52 thf(fact_185_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A4) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A4))))))))).
% 2.59/1.52 thf(fact_186_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B3:A]: (((! [A7:A,B6:A]: (((((ord_less_eq @ A) @ A7) @ B6) => ((P @ A7) @ B6)))) => ((! [A7:A,B6:A]: ((((P @ B6) @ A7) => ((P @ A7) @ B6)))) => ((P @ A4) @ B3))))))))).
% 2.59/1.52 thf(fact_187_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.59/1.52 thf(fact_188_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X2) @ Z3))))))))).
% 2.59/1.52 thf(fact_189_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ A4) => (A4 = B3))))))))).
% 2.59/1.52 thf(fact_190_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_191_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C:A]: (((A4 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_192_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A6:A,B5:A]: (((((ord_less_eq @ A) @ A6) @ B5) & (((ord_less_eq @ A) @ B5) @ A6))))))))).
% 2.59/1.52 thf(fact_193_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.59/1.52 thf(fact_194_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z3)))) => (((((ord_less_eq @ A) @ X2) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.59/1.52 thf(fact_195_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_196_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.59/1.52 thf(fact_197_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.59/1.52 thf(fact_198_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.59/1.52 thf(fact_199_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.59/1.52 thf(fact_200_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X))))))))).
% 2.59/1.52 thf(fact_201_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A4) @ B3) => (((F @ B3) = C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C)))))))))).
% 2.59/1.52 thf(fact_202_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B3:B,C:B]: (((A4 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A4) @ (F @ C))))))))))).
% 2.59/1.52 thf(fact_203_order__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A4:A,B3:A,F:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A4) @ B3) => ((((ord_less_eq @ C4) @ (F @ B3)) @ C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C4) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ C4) @ (F @ A4)) @ C)))))))))).
% 2.59/1.52 thf(fact_204_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A4) @ (F @ C))))))))))).
% 2.59/1.52 thf(fact_205_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 2.59/1.52 thf(fact_206_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.59/1.52 thf(fact_207_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.59/1.52 thf(fact_208_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.59/1.52 thf(fact_209_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) => (A4 != B3)))))))).
% 2.59/1.52 thf(fact_210_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (A4 != B3)))))))).
% 2.59/1.52 thf(fact_211_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = ((((ord_less @ A) @ Y) @ X2) | (X2 = Y))))))))).
% 2.59/1.52 thf(fact_212_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A,C:A]: (((((ord_less @ A) @ B3) @ A4) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A4))))))))).
% 2.59/1.52 thf(fact_213_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B3:A]: (((! [A7:A,B6:A]: (((((ord_less @ A) @ A7) @ B6) => ((P @ A7) @ B6)))) => ((! [A7:A]: (((P @ A7) @ A7))) => ((! [A7:A,B6:A]: ((((P @ B6) @ A7) => ((P @ A7) @ B6)))) => ((P @ A4) @ B3)))))))))).
% 2.59/1.52 thf(fact_214_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X4:A]: ((P2 @ X4))))) = (^ [P3:(A > $o)]: ((? [N2:A]: (((P3 @ N2) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N2) => (~ ((P3 @ M2)))))))))))))))).
% 2.59/1.52 thf(fact_215_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.59/1.52 thf(fact_216_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_217_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ A4))))))))).
% 2.59/1.52 thf(fact_218_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) => ((X2 != Y) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.59/1.52 thf(fact_219_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,P:$o]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ X2) => P)))))))).
% 2.59/1.52 thf(fact_220_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (Y != X2)))))))).
% 2.59/1.52 thf(fact_221_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X2:A]: (((~ ((((ord_less @ A) @ Y) @ X2))) => ((~ ((((ord_less @ A) @ X2) @ Y))) = (X2 = Y))))))))).
% 2.59/1.52 thf(fact_222_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ A) @ Y2) @ X3) => (P @ Y2)))) => (P @ X3)))) => (P @ A4)))))))).
% 2.59/1.52 thf(fact_223_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.59/1.52 thf(fact_224_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.59/1.52 thf(fact_225_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) => (~ ((((ord_less @ A) @ A4) @ B3)))))))))).
% 2.59/1.52 thf(fact_226_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C:A]: (((((ord_less @ A) @ A4) @ B3) => ((B3 = C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_227_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B3:A,C:A]: (((A4 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A4) @ C))))))))).
% 2.59/1.52 thf(fact_228_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((~ ((((ord_less @ A) @ X2) @ X2))))))))).
% 2.59/1.52 thf(fact_229_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) | ((X2 = Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.59/1.52 thf(fact_230_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z3:A]: (((((ord_less @ A) @ X2) @ Y) => ((((ord_less @ A) @ Y) @ Z3) => (((ord_less @ A) @ X2) @ Z3))))))))).
% 2.59/1.52 thf(fact_231_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (~ ((((ord_less @ A) @ B3) @ A4)))))))))).
% 2.59/1.52 thf(fact_232_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (~ ((((ord_less @ A) @ Y) @ X2)))))))))).
% 2.59/1.52 thf(fact_233_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (X2 != Y)))))))).
% 2.59/1.52 thf(fact_234_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (? [Z2:A]: (((((ord_less @ A) @ X2) @ Z2) & (((ord_less @ A) @ Z2) @ Y))))))))))).
% 2.59/1.52 thf(fact_235_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B3:A]: (((((ord_less @ A) @ A4) @ B3) => (~ ((((ord_less @ A) @ B3) @ A4)))))))))).
% 2.59/1.52 thf(fact_236_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) = ((((ord_less @ A) @ X2) @ Y) | (((ord_less @ A) @ Y) @ X2))))))))).
% 2.59/1.52 thf(fact_237_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((X2 != Y) => ((~ ((((ord_less @ A) @ X2) @ Y))) => (((ord_less @ A) @ Y) @ X2))))))))).
% 2.59/1.52 thf(fact_238_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))).
% 2.59/1.52 thf(fact_239_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X2:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X2))))))))).
% 2.59/1.52 thf(fact_240_order__less__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A4:A,B3:A,F:(A > C4),C:C4]: (((((ord_less @ A) @ A4) @ B3) => ((((ord_less @ C4) @ (F @ B3)) @ C) => ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ C4) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ C4) @ (F @ A4)) @ C)))))))))).
% 2.59/1.52 thf(fact_241_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A4) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X3:B,Y3:B]: (((((ord_less @ B) @ X3) @ Y3) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.59/1.52 thf(fact_242_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A4) @ B3) => (((F @ B3) = C) => ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A4)) @ C)))))))))).
% 2.59/1.52 thf(fact_243_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B3:B,C:B]: (((A4 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X3:B,Y3:B]: (((((ord_less @ B) @ X3) @ Y3) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ A) @ A4) @ (F @ C))))))))))).
% 2.59/1.52 thf(fact_244_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.59/1.52 thf(fact_245_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.59/1.52 thf(fact_246_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (X5 != T)))))))))))).
% 2.59/1.52 thf(fact_247_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (X5 != T)))))))))))).
% 2.59/1.52 thf(fact_248_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.59/1.52 thf(fact_249_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.59/1.52 thf(fact_250_pinf_I11_J, axiom, (! [C4:$tType,D2:$tType]: (((ord @ C4) => (! [F3:D2]: ((? [Z2:C4]: ((! [X5:C4]: (((((ord_less @ C4) @ Z2) @ X5) => (F3 = F3)))))))))))).
% 2.59/1.52 thf(fact_251_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z2) => (((P @ X5) & (Q @ X5)) = ((P4 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.59/1.52 thf(fact_252_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P4:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P4 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q2 @ X3))))))) => (? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z2) => (((P @ X5) | (Q @ X5)) = ((P4 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.59/1.52 thf(fact_253_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z2:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z2) => (X5 != T)))))))))))).
% 2.59/1.52 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A2:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A2 > A8)))))).
% 2.59/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A2:$tType,A8:$tType]: (((order @ A8) => (order @ (A2 > A8)))))).
% 2.59/1.52 thf(tcon_fun___Orderings_Oord, axiom, (! [A2:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A2 > A8)))))).
% 2.59/1.52 thf(tcon_fun___Groups_Ominus, axiom, (! [A2:$tType,A8:$tType]: (((minus @ A8) => (minus @ (A2 > A8)))))).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict2144017051up_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff, axiom, (ordere623563068d_diff @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Ominus_4, axiom, (minus @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 2.59/1.52 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.59/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_5, axiom, (preorder @ $o)).
% 2.59/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_6, axiom, (linorder @ $o)).
% 2.59/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_7, axiom, (order @ $o)).
% 2.59/1.52 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, (ord @ $o)).
% 2.59/1.52 thf(tcon_HOL_Obool___Groups_Ominus_9, axiom, (minus @ $o)).
% 2.59/1.52 thf(tcon_List_Olist___Nat_Osize_10, axiom, (! [A2:$tType]: ((size @ (list @ A2))))).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_11, axiom, (wellorder @ product_unit)).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_12, axiom, (preorder @ product_unit)).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_13, axiom, (linorder @ product_unit)).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Orderings_Oorder_14, axiom, (order @ product_unit)).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Orderings_Oord_15, axiom, (ord @ product_unit)).
% 2.59/1.52 thf(tcon_Product__Type_Ounit___Groups_Ominus_16, axiom, (minus @ product_unit)).
% 2.59/1.52 thf(conj_0, conjecture, (((last @ ((((configuration_ext @ p) @ v) @ s) @ product_unit)) @ (fe @ n)) = (((nth @ ((((configuration_ext @ p) @ v) @ s) @ product_unit)) @ (fe @ n)) @ (((minus_minus @ nat) @ ((size_size @ (list @ ((((configuration_ext @ p) @ v) @ s) @ product_unit))) @ (fe @ n))) @ (one_one @ nat))))).
% 2.59/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.UV0DDxgB5h/DTF2DTF6476.p
% 2.59/1.52 ------------------------
% 2.59/1.56 ---- Cleaned THF ---
% 2.59/1.56 thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext,type,
% 2.59/1.56 configuration_ext: $tType > $tType > $tType > $tType > $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_t_Product__Type_Ounit,type,
% 2.59/1.56 product_unit: $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_t_List_Olist,type,
% 2.59/1.56 list: $tType > $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_t_Nat_Onat,type,
% 2.59/1.56 nat: $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_tf_v,type,
% 2.59/1.56 v: $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_tf_s,type,
% 2.59/1.56 s: $tType ).
% 2.59/1.56
% 2.59/1.56 thf(ty_tf_p,type,
% 2.59/1.56 p: $tType ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_HOL_Otype,type,
% 2.59/1.56 type:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Nat_Osize,type,
% 2.59/1.56 size:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oone,type,
% 2.59/1.56 one:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ominus,type,
% 2.59/1.56 minus:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.59/1.56 ord:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Num_Oneg__numeral,type,
% 2.59/1.56 neg_numeral:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.59/1.56 order:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.59/1.56 group_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Ono__bot,type,
% 2.59/1.56 no_bot:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Ono__top,type,
% 2.59/1.56 no_top:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.59/1.56 linorder:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.59/1.56 preorder:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oab__group__add,type,
% 2.59/1.56 ab_group_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Owellorder,type,
% 2.59/1.56 wellorder:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.59/1.56 semigroup_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Orderings_Odense__order,type,
% 2.59/1.56 dense_order:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.59/1.56 comm_monoid_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.59/1.56 ab_semigroup_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.59/1.56 cancel_semigroup_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.59/1.56 ordered_ab_group_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.59/1.56 cancel1352612707id_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.59/1.56 cancel146912293up_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.59/1.56 ordere779506340up_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.59/1.56 linord1659791738miring:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.59/1.56 canoni770627133id_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.59/1.56 ordere236663937imp_le:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.59/1.56 ordere223160158up_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 2.59/1.56 ordere623563068d_diff:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.59/1.56 strict2144017051up_add:
% 2.59/1.56 !>[A: $tType] : $o ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.59/1.56 minus_minus:
% 2.59/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.59/1.56 one_one:
% 2.59/1.56 !>[A: $tType] : A ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.59/1.56 plus_plus:
% 2.59/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_ListUtilities_OprefixList,type,
% 2.59/1.56 prefixList:
% 2.59/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_List_Olast,type,
% 2.59/1.56 last:
% 2.59/1.56 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_List_Olist_ONil,type,
% 2.59/1.56 nil:
% 2.59/1.56 !>[A: $tType] : ( list @ A ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_List_Onth,type,
% 2.59/1.56 nth:
% 2.59/1.56 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.59/1.56 size_size:
% 2.59/1.56 !>[A: $tType] : ( A > nat ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.59/1.56 ord_less:
% 2.59/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.59/1.56 ord_less_eq:
% 2.59/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_v_fe____,type,
% 2.59/1.56 fe: nat > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(sy_v_n0____,type,
% 2.59/1.56 n0: nat ).
% 2.59/1.56
% 2.59/1.56 thf(sy_v_n____,type,
% 2.59/1.56 n: nat ).
% 2.59/1.56
% 2.59/1.56 thf(fact_0_FeNNotEmpty,axiom,
% 2.59/1.56 ( ( fe @ n )
% 2.59/1.56 != ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_1_LastIsLastIndex,axiom,
% 2.59/1.56 ! [A2: $tType,L: list @ A2] :
% 2.59/1.56 ( ( L
% 2.59/1.56 != ( nil @ A2 ) )
% 2.59/1.56 => ( ( last @ A2 @ L )
% 2.59/1.56 = ( nth @ A2 @ L @ ( minus_minus @ nat @ ( size_size @ ( list @ A2 ) @ L ) @ ( one_one @ nat ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_2_Fou2,axiom,
% 2.59/1.56 ord_less @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_3_AssmNoDecided_I1_J,axiom,
% 2.59/1.56 ord_less @ nat @ n0 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_4_last__conv__nth,axiom,
% 2.59/1.56 ! [A: $tType,Xs: list @ A] :
% 2.59/1.56 ( ( Xs
% 2.59/1.56 != ( nil @ A ) )
% 2.59/1.56 => ( ( last @ A @ Xs )
% 2.59/1.56 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_5_minus__apply,axiom,
% 2.59/1.56 ! [B: $tType,A: $tType] :
% 2.59/1.56 ( ( minus @ B )
% 2.59/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.59/1.56 = ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_6_Fou,axiom,
% 2.59/1.56 ord_less_eq @ nat @ n0 @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_7_one__natural_Orsp,axiom,
% 2.59/1.56 ( ( one_one @ nat )
% 2.59/1.56 = ( one_one @ nat ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_8_diff__commute,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.59/1.56 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_9_Ex__list__of__length,axiom,
% 2.59/1.56 ! [A: $tType,N: nat] :
% 2.59/1.56 ? [Xs2: list @ A] :
% 2.59/1.56 ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.59/1.56 = N ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_10_neq__if__length__neq,axiom,
% 2.59/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.59/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.56 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.56 => ( Xs != Ys ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_11_size__neq__size__imp__neq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( size @ A )
% 2.59/1.56 => ! [X2: A,Y: A] :
% 2.59/1.56 ( ( ( size_size @ A @ X2 )
% 2.59/1.56 != ( size_size @ A @ Y ) )
% 2.59/1.56 => ( X2 != Y ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_12_fun__diff__def,axiom,
% 2.59/1.56 ! [B: $tType,A: $tType] :
% 2.59/1.56 ( ( minus @ B )
% 2.59/1.56 => ( ( minus_minus @ ( A > B ) )
% 2.59/1.56 = ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_13_diff__diff__cancel,axiom,
% 2.59/1.56 ! [I: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ N )
% 2.59/1.56 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 2.59/1.56 = I ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_14_le__refl,axiom,
% 2.59/1.56 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_15_le__trans,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ J @ K )
% 2.59/1.56 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_16_eq__imp__le,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( M = N )
% 2.59/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_17_le__antisym,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ N @ M )
% 2.59/1.56 => ( M = N ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_18_nat__less__le,axiom,
% 2.59/1.56 ( ( ord_less @ nat )
% 2.59/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.59/1.56 & ( M2 != N2 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_19_nat__neq__iff,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( M != N )
% 2.59/1.56 = ( ( ord_less @ nat @ M @ N )
% 2.59/1.56 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_20_less__diff__iff,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.56 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.56 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_21_less__not__refl,axiom,
% 2.59/1.56 ! [N: nat] :
% 2.59/1.56 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_22_nat__le__linear,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_23_diff__less__mono,axiom,
% 2.59/1.56 ! [A4: nat,B3: nat,C: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ C @ A4 )
% 2.59/1.56 => ( ord_less @ nat @ ( minus_minus @ nat @ A4 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_24_less__not__refl2,axiom,
% 2.59/1.56 ! [N: nat,M: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ N @ M )
% 2.59/1.56 => ( M != N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_25_less__not__refl3,axiom,
% 2.59/1.56 ! [S: nat,T: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ S @ T )
% 2.59/1.56 => ( S != T ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_26_measure__induct,axiom,
% 2.59/1.56 ! [B: $tType,A: $tType] :
% 2.59/1.56 ( ( wellorder @ B )
% 2.59/1.56 => ! [F: A > B,P: A > $o,A4: A] :
% 2.59/1.56 ( ! [X3: A] :
% 2.59/1.56 ( ! [Y2: A] :
% 2.59/1.56 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 2.59/1.56 => ( P @ Y2 ) )
% 2.59/1.56 => ( P @ X3 ) )
% 2.59/1.56 => ( P @ A4 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_27_less__imp__le__nat,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M @ N )
% 2.59/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_28_less__irrefl__nat,axiom,
% 2.59/1.56 ! [N: nat] :
% 2.59/1.56 ~ ( ord_less @ nat @ N @ N ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_29_nat__less__induct,axiom,
% 2.59/1.56 ! [P: nat > $o,N: nat] :
% 2.59/1.56 ( ! [N3: nat] :
% 2.59/1.56 ( ! [M3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M3 @ N3 )
% 2.59/1.56 => ( P @ M3 ) )
% 2.59/1.56 => ( P @ N3 ) )
% 2.59/1.56 => ( P @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_30_infinite__descent,axiom,
% 2.59/1.56 ! [P: nat > $o,N: nat] :
% 2.59/1.56 ( ! [N3: nat] :
% 2.59/1.56 ( ~ ( P @ N3 )
% 2.59/1.56 => ? [M3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M3 @ N3 )
% 2.59/1.56 & ~ ( P @ M3 ) ) )
% 2.59/1.56 => ( P @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_31_le__eq__less__or__eq,axiom,
% 2.59/1.56 ( ( ord_less_eq @ nat )
% 2.59/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M2 @ N2 )
% 2.59/1.56 | ( M2 = N2 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_32_less__or__eq__imp__le,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( ( ord_less @ nat @ M @ N )
% 2.59/1.56 | ( M = N ) )
% 2.59/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_33_linorder__neqE__nat,axiom,
% 2.59/1.56 ! [X2: nat,Y: nat] :
% 2.59/1.56 ( ( X2 != Y )
% 2.59/1.56 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 2.59/1.56 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_34_Nat_Oex__has__greatest__nat,axiom,
% 2.59/1.56 ! [P: nat > $o,K: nat,B3: nat] :
% 2.59/1.56 ( ( P @ K )
% 2.59/1.56 => ( ! [Y3: nat] :
% 2.59/1.56 ( ( P @ Y3 )
% 2.59/1.56 => ( ord_less_eq @ nat @ Y3 @ B3 ) )
% 2.59/1.56 => ? [X3: nat] :
% 2.59/1.56 ( ( P @ X3 )
% 2.59/1.56 & ! [Y2: nat] :
% 2.59/1.56 ( ( P @ Y2 )
% 2.59/1.56 => ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_35_le__neq__implies__less,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 => ( ( M != N )
% 2.59/1.56 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_36_measure__induct__rule,axiom,
% 2.59/1.56 ! [B: $tType,A: $tType] :
% 2.59/1.56 ( ( wellorder @ B )
% 2.59/1.56 => ! [F: A > B,P: A > $o,A4: A] :
% 2.59/1.56 ( ! [X3: A] :
% 2.59/1.56 ( ! [Y2: A] :
% 2.59/1.56 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 2.59/1.56 => ( P @ Y2 ) )
% 2.59/1.56 => ( P @ X3 ) )
% 2.59/1.56 => ( P @ A4 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_37_less__mono__imp__le__mono,axiom,
% 2.59/1.56 ! [F: nat > nat,I: nat,J: nat] :
% 2.59/1.56 ( ! [I2: nat,J2: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I2 @ J2 )
% 2.59/1.56 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_38_infinite__descent__measure,axiom,
% 2.59/1.56 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 2.59/1.56 ( ! [X3: A] :
% 2.59/1.56 ( ~ ( P @ X3 )
% 2.59/1.56 => ? [Y2: A] :
% 2.59/1.56 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 2.59/1.56 & ~ ( P @ Y2 ) ) )
% 2.59/1.56 => ( P @ X2 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_39_diff__strict__right__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_40_diff__strict__left__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ B3 @ A4 )
% 2.59/1.56 => ( ord_less @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_41_diff__eq__diff__less,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ( minus_minus @ A @ A4 @ B3 )
% 2.59/1.56 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.56 => ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_42_diff__strict__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,D: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less @ A @ D @ C )
% 2.59/1.56 => ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_43_ext,axiom,
% 2.59/1.56 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.59/1.56 ( ! [X3: A] :
% 2.59/1.56 ( ( F @ X3 )
% 2.59/1.56 = ( G @ X3 ) )
% 2.59/1.56 => ( F = G ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_44_diff__eq__diff__less__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ( minus_minus @ A @ A4 @ B3 )
% 2.59/1.56 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.56 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_45_diff__right__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_46_diff__left__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.59/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_47_diff__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,D: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ D @ C )
% 2.59/1.56 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_48_length__induct,axiom,
% 2.59/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.59/1.56 ( ! [Xs2: list @ A] :
% 2.59/1.56 ( ! [Ys2: list @ A] :
% 2.59/1.56 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.59/1.56 => ( P @ Ys2 ) )
% 2.59/1.56 => ( P @ Xs2 ) )
% 2.59/1.56 => ( P @ Xs ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_49_less__imp__diff__less,axiom,
% 2.59/1.56 ! [J: nat,K: nat,N: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ J @ K )
% 2.59/1.56 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_50_diff__less__mono2,axiom,
% 2.59/1.56 ! [M: nat,N: nat,L: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M @ N )
% 2.59/1.56 => ( ( ord_less @ nat @ M @ L )
% 2.59/1.56 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_51_diff__le__mono2,axiom,
% 2.59/1.56 ! [M: nat,N: nat,L: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_52_le__diff__iff_H,axiom,
% 2.59/1.56 ! [A4: nat,C: nat,B3: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ A4 @ C )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ B3 @ C )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A4 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 2.59/1.56 = ( ord_less_eq @ nat @ B3 @ A4 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_53_diff__le__self,axiom,
% 2.59/1.56 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_54_diff__le__mono,axiom,
% 2.59/1.56 ! [M: nat,N: nat,L: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_55_Nat_Odiff__diff__eq,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.56 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.56 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_56_le__diff__iff,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.59/1.56 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_57_eq__diff__iff,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ M )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ K @ N )
% 2.59/1.56 => ( ( ( minus_minus @ nat @ M @ K )
% 2.59/1.56 = ( minus_minus @ nat @ N @ K ) )
% 2.59/1.56 = ( M = N ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_58_list__eq__iff__nth__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ^ [Y4: list @ A,Z: list @ A] : ( Y4 = Z ) )
% 2.59/1.56 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 2.59/1.56 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.59/1.56 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 2.59/1.56 & ! [I3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.59/1.56 => ( ( nth @ A @ Xs3 @ I3 )
% 2.59/1.56 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_59_Skolem__list__nth,axiom,
% 2.59/1.56 ! [A: $tType,K: nat,P: nat > A > $o] :
% 2.59/1.56 ( ( ! [I3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I3 @ K )
% 2.59/1.56 => ? [X4: A] : ( P @ I3 @ X4 ) ) )
% 2.59/1.56 = ( ? [Xs3: list @ A] :
% 2.59/1.56 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.59/1.56 = K )
% 2.59/1.56 & ! [I3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I3 @ K )
% 2.59/1.56 => ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_60_nth__equalityI,axiom,
% 2.59/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.59/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.59/1.56 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.59/1.56 => ( ! [I2: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.59/1.56 => ( ( nth @ A @ Xs @ I2 )
% 2.59/1.56 = ( nth @ A @ Ys @ I2 ) ) )
% 2.59/1.56 => ( Xs = Ys ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_61_one__reorient,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( one @ A )
% 2.59/1.56 => ! [X2: A] :
% 2.59/1.56 ( ( ( one_one @ A )
% 2.59/1.56 = X2 )
% 2.59/1.56 = ( X2
% 2.59/1.56 = ( one_one @ A ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_62_diff__right__commute,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 )
% 2.59/1.56 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_63_diff__eq__diff__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ( minus_minus @ A @ A4 @ B3 )
% 2.59/1.56 = ( minus_minus @ A @ C @ D ) )
% 2.59/1.56 => ( ( A4 = B3 )
% 2.59/1.56 = ( C = D ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_64_Length,axiom,
% 2.59/1.56 ! [N4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ N4 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_65_AllArePrefixesExec,axiom,
% 2.59/1.56 ! [M3: nat,N4: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M3 @ N4 )
% 2.59/1.56 => ( prefixList @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ M3 ) @ ( fe @ N4 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_66_order__refl,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( preorder @ A )
% 2.59/1.56 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_67_nat__descend__induct,axiom,
% 2.59/1.56 ! [N: nat,P: nat > $o,M: nat] :
% 2.59/1.56 ( ! [K2: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ N @ K2 )
% 2.59/1.56 => ( P @ K2 ) )
% 2.59/1.56 => ( ! [K2: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.59/1.56 => ( ! [I4: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ K2 @ I4 )
% 2.59/1.56 => ( P @ I4 ) )
% 2.59/1.56 => ( P @ K2 ) ) )
% 2.59/1.56 => ( P @ M ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_68_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.59/1.56 ! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
% 2.59/1.56 ( ( P @ K )
% 2.59/1.56 => ( ! [Y3: A] :
% 2.59/1.56 ( ( P @ Y3 )
% 2.59/1.56 => ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
% 2.59/1.56 => ? [X3: A] :
% 2.59/1.56 ( ( P @ X3 )
% 2.59/1.56 & ! [Y2: A] :
% 2.59/1.56 ( ( P @ Y2 )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_69_less__numeral__extra_I4_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linord1659791738miring @ A )
% 2.59/1.56 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_70_le__numeral__extra_I4_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linord1659791738miring @ A )
% 2.59/1.56 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_71_minf_I8_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linorder @ A )
% 2.59/1.56 => ! [T: A] :
% 2.59/1.56 ? [Z2: A] :
% 2.59/1.56 ! [X5: A] :
% 2.59/1.56 ( ( ord_less @ A @ X5 @ Z2 )
% 2.59/1.56 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_72_minf_I6_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linorder @ A )
% 2.59/1.56 => ! [T: A] :
% 2.59/1.56 ? [Z2: A] :
% 2.59/1.56 ! [X5: A] :
% 2.59/1.56 ( ( ord_less @ A @ X5 @ Z2 )
% 2.59/1.56 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_73_pinf_I8_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linorder @ A )
% 2.59/1.56 => ! [T: A] :
% 2.59/1.56 ? [Z2: A] :
% 2.59/1.56 ! [X5: A] :
% 2.59/1.56 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.56 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_74_pinf_I6_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( linorder @ A )
% 2.59/1.56 => ! [T: A] :
% 2.59/1.56 ? [Z2: A] :
% 2.59/1.56 ! [X5: A] :
% 2.59/1.56 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.56 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_75_add__left__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel_semigroup_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ A4 @ B3 )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ C ) )
% 2.59/1.56 = ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_76_add__right__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel_semigroup_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ B3 @ A4 )
% 2.59/1.56 = ( plus_plus @ A @ C @ A4 ) )
% 2.59/1.56 = ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_77_add__le__cancel__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_78_add__le__cancel__left,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [C: A,A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.56 = ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_79_add__less__cancel__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 = ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_80_add__less__cancel__left,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [C: A,A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.56 = ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_81_add__diff__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
% 2.59/1.56 = A4 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_82_diff__add__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
% 2.59/1.56 = A4 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_83_add__diff__cancel__left,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [C: A,A4: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.56 = ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_84_add__diff__cancel__left_H,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ A4 )
% 2.59/1.56 = B3 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_85_add__diff__cancel__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 = ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_86_add__diff__cancel__right_H,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
% 2.59/1.56 = A4 ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_87_nat__add__left__cancel__less,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.59/1.56 = ( ord_less @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_88_nat__add__left__cancel__le,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.59/1.56 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_89_diff__diff__left,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.59/1.56 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_90_Nat_Odiff__diff__right,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 2.59/1.56 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_91_Nat_Oadd__diff__assoc2,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 2.59/1.56 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_92_Nat_Oadd__diff__assoc,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 2.59/1.56 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_93_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ab_semigroup_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_94_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( I = J )
% 2.59/1.56 & ( K = L ) )
% 2.59/1.56 => ( ( plus_plus @ A @ I @ K )
% 2.59/1.56 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_95_group__cancel_Oadd1,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( comm_monoid_add @ A )
% 2.59/1.56 => ! [A5: A,K: A,A4: A,B3: A] :
% 2.59/1.56 ( ( A5
% 2.59/1.56 = ( plus_plus @ A @ K @ A4 ) )
% 2.59/1.56 => ( ( plus_plus @ A @ A5 @ B3 )
% 2.59/1.56 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_96_group__cancel_Oadd2,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( comm_monoid_add @ A )
% 2.59/1.56 => ! [B4: A,K: A,B3: A,A4: A] :
% 2.59/1.56 ( ( B4
% 2.59/1.56 = ( plus_plus @ A @ K @ B3 ) )
% 2.59/1.56 => ( ( plus_plus @ A @ A4 @ B4 )
% 2.59/1.56 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_97_add_Oassoc,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( semigroup_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_98_add_Oleft__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ A4 @ B3 )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ C ) )
% 2.59/1.56 = ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_99_add_Oright__cancel,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ B3 @ A4 )
% 2.59/1.56 = ( plus_plus @ A @ C @ A4 ) )
% 2.59/1.56 = ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_100_add_Ocommute,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ab_semigroup_add @ A )
% 2.59/1.56 => ( ( plus_plus @ A )
% 2.59/1.56 = ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_101_add_Oleft__commute,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ab_semigroup_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C ) )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_102_add__left__imp__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel_semigroup_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ A4 @ B3 )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ C ) )
% 2.59/1.56 => ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_103_add__right__imp__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel_semigroup_add @ A )
% 2.59/1.56 => ! [B3: A,A4: A,C: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ B3 @ A4 )
% 2.59/1.56 = ( plus_plus @ A @ C @ A4 ) )
% 2.59/1.56 => ( B3 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_104_is__num__normalize_I1_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( neg_numeral @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_105_add__le__imp__le__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_106_add__le__imp__le__left,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [C: A,A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.56 => ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_107_le__iff__add,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( canoni770627133id_add @ A )
% 2.59/1.56 => ( ( ord_less_eq @ A )
% 2.59/1.56 = ( ^ [A6: A,B5: A] :
% 2.59/1.56 ? [C2: A] :
% 2.59/1.56 ( B5
% 2.59/1.56 = ( plus_plus @ A @ A6 @ C2 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_108_add__right__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_109_less__eqE,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( canoni770627133id_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ~ ! [C3: A] :
% 2.59/1.56 ( B3
% 2.59/1.56 != ( plus_plus @ A @ A4 @ C3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_110_add__left__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_111_add__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ C @ D )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_112_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.59/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_113_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( I = J )
% 2.59/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_114_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere779506340up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.59/1.56 & ( K = L ) )
% 2.59/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_115_add__less__imp__less__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 => ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_116_add__less__imp__less__left,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere236663937imp_le @ A )
% 2.59/1.56 => ! [C: A,A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
% 2.59/1.56 => ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_117_add__strict__right__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_118_add__strict__left__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_119_add__strict__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( strict2144017051up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less @ A @ C @ D )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_120_add__mono__thms__linordered__field_I1_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.59/1.56 & ( K = L ) )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_121_add__mono__thms__linordered__field_I2_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( I = J )
% 2.59/1.56 & ( ord_less @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_122_add__mono__thms__linordered__field_I5_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.59/1.56 & ( ord_less @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_123_group__cancel_Osub1,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ab_group_add @ A )
% 2.59/1.56 => ! [A5: A,K: A,A4: A,B3: A] :
% 2.59/1.56 ( ( A5
% 2.59/1.56 = ( plus_plus @ A @ K @ A4 ) )
% 2.59/1.56 => ( ( minus_minus @ A @ A5 @ B3 )
% 2.59/1.56 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_124_diff__eq__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ( minus_minus @ A @ A4 @ B3 )
% 2.59/1.56 = C )
% 2.59/1.56 = ( A4
% 2.59/1.56 = ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_125_eq__diff__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( A4
% 2.59/1.56 = ( minus_minus @ A @ C @ B3 ) )
% 2.59/1.56 = ( ( plus_plus @ A @ A4 @ B3 )
% 2.59/1.56 = C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_126_add__diff__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_127_diff__diff__eq2,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_128_diff__add__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_129_diff__add__eq__diff__diff__swap,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) )
% 2.59/1.56 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_130_diff__diff__add,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel146912293up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_131_add__implies__diff,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( cancel1352612707id_add @ A )
% 2.59/1.56 => ! [C: A,B3: A,A4: A] :
% 2.59/1.56 ( ( ( plus_plus @ A @ C @ B3 )
% 2.59/1.56 = A4 )
% 2.59/1.56 => ( C
% 2.59/1.56 = ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_132_less__add__eq__less,axiom,
% 2.59/1.56 ! [K: nat,L: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ K @ L )
% 2.59/1.56 => ( ( ( plus_plus @ nat @ M @ L )
% 2.59/1.56 = ( plus_plus @ nat @ K @ N ) )
% 2.59/1.56 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_133_trans__less__add2,axiom,
% 2.59/1.56 ! [I: nat,J: nat,M: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I @ J )
% 2.59/1.56 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_134_trans__less__add1,axiom,
% 2.59/1.56 ! [I: nat,J: nat,M: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I @ J )
% 2.59/1.56 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_135_add__less__mono1,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I @ J )
% 2.59/1.56 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_136_not__add__less2,axiom,
% 2.59/1.56 ! [J: nat,I: nat] :
% 2.59/1.56 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_137_not__add__less1,axiom,
% 2.59/1.56 ! [I: nat,J: nat] :
% 2.59/1.56 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_138_add__less__mono,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I @ J )
% 2.59/1.56 => ( ( ord_less @ nat @ K @ L )
% 2.59/1.56 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_139_add__lessD1,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 2.59/1.56 => ( ord_less @ nat @ I @ K ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_140_nat__le__iff__add,axiom,
% 2.59/1.56 ( ( ord_less_eq @ nat )
% 2.59/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.59/1.56 ? [K3: nat] :
% 2.59/1.56 ( N2
% 2.59/1.56 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_141_trans__le__add2,axiom,
% 2.59/1.56 ! [I: nat,J: nat,M: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_142_trans__le__add1,axiom,
% 2.59/1.56 ! [I: nat,J: nat,M: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_143_add__le__mono1,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_144_add__le__mono,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ K @ L )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_145_le__Suc__ex,axiom,
% 2.59/1.56 ! [K: nat,L: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ L )
% 2.59/1.56 => ? [N3: nat] :
% 2.59/1.56 ( L
% 2.59/1.56 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_146_add__leD2,axiom,
% 2.59/1.56 ! [M: nat,K: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.59/1.56 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_147_add__leD1,axiom,
% 2.59/1.56 ! [M: nat,K: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.59/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_148_le__add2,axiom,
% 2.59/1.56 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_149_le__add1,axiom,
% 2.59/1.56 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_150_add__leE,axiom,
% 2.59/1.56 ! [M: nat,K: nat,N: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.59/1.56 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 2.59/1.56 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_151_Nat_Odiff__cancel,axiom,
% 2.59/1.56 ! [K: nat,M: nat,N: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.59/1.56 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_152_diff__cancel2,axiom,
% 2.59/1.56 ! [M: nat,K: nat,N: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 2.59/1.56 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_153_diff__add__inverse,axiom,
% 2.59/1.56 ! [N: nat,M: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 2.59/1.56 = M ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_154_diff__add__inverse2,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 2.59/1.56 = M ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_155_ex__has__greatest__nat__lemma,axiom,
% 2.59/1.56 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 2.59/1.56 ( ( P @ K )
% 2.59/1.56 => ( ! [X3: A] :
% 2.59/1.56 ( ( P @ X3 )
% 2.59/1.56 => ? [Y2: A] :
% 2.59/1.56 ( ( P @ Y2 )
% 2.59/1.56 & ~ ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) )
% 2.59/1.56 => ? [Y3: A] :
% 2.59/1.56 ( ( P @ Y3 )
% 2.59/1.56 & ~ ( ord_less @ nat @ ( F @ Y3 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_156_add__less__le__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ C @ D )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_157_add__le__less__mono,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A,D: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less @ A @ C @ D )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_158_add__mono__thms__linordered__field_I3_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.59/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_159_add__mono__thms__linordered__field_I4_J,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere223160158up_add @ A )
% 2.59/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.59/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.59/1.56 & ( ord_less @ A @ K @ L ) )
% 2.59/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_160_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ( minus_minus @ A @ B3 @ A4 )
% 2.59/1.56 = C )
% 2.59/1.56 = ( B3
% 2.59/1.56 = ( plus_plus @ A @ C @ A4 ) ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_161_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ A4 ) )
% 2.59/1.56 = B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_162_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_163_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 )
% 2.59/1.56 = ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_164_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_165_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 )
% 2.59/1.56 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_166_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 2.59/1.56 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_167_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
% 2.59/1.56 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_168_le__add__diff,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_169_diff__add,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordere623563068d_diff @ A )
% 2.59/1.56 => ! [A4: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ A4 )
% 2.59/1.56 = B3 ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_170_le__diff__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
% 2.59/1.56 = ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_171_diff__le__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_172_less__diff__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,C: A,B3: A] :
% 2.59/1.56 ( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
% 2.59/1.56 = ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_173_diff__less__eq,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( ordered_ab_group_add @ A )
% 2.59/1.56 => ! [A4: A,B3: A,C: A] :
% 2.59/1.56 ( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
% 2.59/1.56 = ( ord_less @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_174_mono__nat__linear__lb,axiom,
% 2.59/1.56 ! [F: nat > nat,M: nat,K: nat] :
% 2.59/1.56 ( ! [M4: nat,N3: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ M4 @ N3 )
% 2.59/1.56 => ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 2.59/1.56 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_175_add__diff__inverse__nat,axiom,
% 2.59/1.56 ! [M: nat,N: nat] :
% 2.59/1.56 ( ~ ( ord_less @ nat @ M @ N )
% 2.59/1.56 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 2.59/1.56 = M ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_176_less__diff__conv,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 2.59/1.56 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_177_Nat_Ole__imp__diff__is__add,axiom,
% 2.59/1.56 ! [I: nat,J: nat,K: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ I @ J )
% 2.59/1.56 => ( ( ( minus_minus @ nat @ J @ I )
% 2.59/1.56 = K )
% 2.59/1.56 = ( J
% 2.59/1.56 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_178_Nat_Odiff__add__assoc2,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 2.59/1.56 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_179_Nat_Odiff__add__assoc,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 2.59/1.56 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_180_Nat_Ole__diff__conv2,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 2.59/1.56 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_181_le__diff__conv,axiom,
% 2.59/1.56 ! [J: nat,K: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 2.59/1.56 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_182_less__diff__conv2,axiom,
% 2.59/1.56 ! [K: nat,J: nat,I: nat] :
% 2.59/1.56 ( ( ord_less_eq @ nat @ K @ J )
% 2.59/1.56 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 2.59/1.56 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_183_dual__order_Oantisym,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( order @ A )
% 2.59/1.56 => ! [B3: A,A4: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.59/1.56 => ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.56 => ( A4 = B3 ) ) ) ) ).
% 2.59/1.56
% 2.59/1.56 thf(fact_184_dual__order_Oeq__iff,axiom,
% 2.59/1.56 ! [A: $tType] :
% 2.59/1.56 ( ( order @ A )
% 2.59/1.56 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.56 = ( ^ [A6: A,B5: A] :
% 2.59/1.56 ( ( ord_less_eq @ A @ B5 @ A6 )
% 2.59/1.56 & ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_185_dual__order_Otrans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [B3: A,A4: A,C: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.59/1.57 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.59/1.57 => ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_186_linorder__wlog,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > A > $o,A4: A,B3: A] :
% 2.59/1.57 ( ! [A7: A,B6: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A7 @ B6 )
% 2.59/1.57 => ( P @ A7 @ B6 ) )
% 2.59/1.57 => ( ! [A7: A,B6: A] :
% 2.59/1.57 ( ( P @ B6 @ A7 )
% 2.59/1.57 => ( P @ A7 @ B6 ) )
% 2.59/1.57 => ( P @ A4 @ B3 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_187_dual__order_Orefl,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_188_order__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A,Z3: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.59/1.57 => ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_189_order__class_Oorder_Oantisym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A,B3: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ord_less_eq @ A @ B3 @ A4 )
% 2.59/1.57 => ( A4 = B3 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_190_ord__le__eq__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( ord @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( B3 = C )
% 2.59/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_191_ord__eq__le__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( ord @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( A4 = B3 )
% 2.59/1.57 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.59/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_192_order__class_Oorder_Oeq__iff,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.57 = ( ^ [A6: A,B5: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A6 @ B5 )
% 2.59/1.57 & ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_193_antisym__conv,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [Y: A,X2: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.59/1.57 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 = ( X2 = Y ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_194_le__cases3,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A,Z3: A] :
% 2.59/1.57 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.59/1.57 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
% 2.59/1.57 => ( ( ( ord_less_eq @ A @ X2 @ Z3 )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.59/1.57 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.59/1.57 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
% 2.59/1.57 => ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
% 2.59/1.57 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_195_order_Otrans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.59/1.57 => ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_196_le__cases,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_197_eq__refl,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( X2 = Y )
% 2.59/1.57 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_198_linear,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_199_antisym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.59/1.57 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.59/1.57 => ( X2 = Y ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_200_eq__iff,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.59/1.57 = ( ^ [X: A,Y5: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.59/1.57 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_201_ord__le__eq__subst,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( ord @ B )
% 2.59/1.57 & ( ord @ A ) )
% 2.59/1.57 => ! [A4: A,B3: A,F: A > B,C: B] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ( F @ B3 )
% 2.59/1.57 = C )
% 2.59/1.57 => ( ! [X3: A,Y3: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_202_ord__eq__le__subst,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( ord @ B )
% 2.59/1.57 & ( ord @ A ) )
% 2.59/1.57 => ! [A4: A,F: B > A,B3: B,C: B] :
% 2.59/1.57 ( ( A4
% 2.59/1.57 = ( F @ B3 ) )
% 2.59/1.57 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.59/1.57 => ( ! [X3: B,Y3: B] :
% 2.59/1.57 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_203_order__subst2,axiom,
% 2.59/1.57 ! [A: $tType,C4: $tType] :
% 2.59/1.57 ( ( ( order @ C4 )
% 2.59/1.57 & ( order @ A ) )
% 2.59/1.57 => ! [A4: A,B3: A,F: A > C4,C: C4] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ord_less_eq @ C4 @ ( F @ B3 ) @ C )
% 2.59/1.57 => ( ! [X3: A,Y3: A] :
% 2.59/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less_eq @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_204_order__subst1,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( order @ B )
% 2.59/1.57 & ( order @ A ) )
% 2.59/1.57 => ! [A4: A,F: B > A,B3: B,C: B] :
% 2.59/1.57 ( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
% 2.59/1.57 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.59/1.57 => ( ! [X3: B,Y3: B] :
% 2.59/1.57 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_205_le__fun__def,axiom,
% 2.59/1.57 ! [B: $tType,A: $tType] :
% 2.59/1.57 ( ( ord @ B )
% 2.59/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.59/1.57 = ( ^ [F2: A > B,G2: A > B] :
% 2.59/1.57 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_206_le__funI,axiom,
% 2.59/1.57 ! [B: $tType,A: $tType] :
% 2.59/1.57 ( ( ord @ B )
% 2.59/1.57 => ! [F: A > B,G: A > B] :
% 2.59/1.57 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.59/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_207_le__funE,axiom,
% 2.59/1.57 ! [B: $tType,A: $tType] :
% 2.59/1.57 ( ( ord @ B )
% 2.59/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.59/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_208_le__funD,axiom,
% 2.59/1.57 ! [B: $tType,A: $tType] :
% 2.59/1.57 ( ( ord @ B )
% 2.59/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.59/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.59/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_209_dual__order_Ostrict__implies__not__eq,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [B3: A,A4: A] :
% 2.59/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.59/1.57 => ( A4 != B3 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_210_order_Ostrict__implies__not__eq,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A,B3: A] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ( A4 != B3 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_211_not__less__iff__gr__or__eq,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.59/1.57 = ( ( ord_less @ A @ Y @ X2 )
% 2.59/1.57 | ( X2 = Y ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_212_dual__order_Ostrict__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [B3: A,A4: A,C: A] :
% 2.59/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.59/1.57 => ( ( ord_less @ A @ C @ B3 )
% 2.59/1.57 => ( ord_less @ A @ C @ A4 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_213_linorder__less__wlog,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > A > $o,A4: A,B3: A] :
% 2.59/1.57 ( ! [A7: A,B6: A] :
% 2.59/1.57 ( ( ord_less @ A @ A7 @ B6 )
% 2.59/1.57 => ( P @ A7 @ B6 ) )
% 2.59/1.57 => ( ! [A7: A] : ( P @ A7 @ A7 )
% 2.59/1.57 => ( ! [A7: A,B6: A] :
% 2.59/1.57 ( ( P @ B6 @ A7 )
% 2.59/1.57 => ( P @ A7 @ B6 ) )
% 2.59/1.57 => ( P @ A4 @ B3 ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_214_exists__least__iff,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( wellorder @ A )
% 2.59/1.57 => ( ( ^ [P2: A > $o] :
% 2.59/1.57 ? [X4: A] : ( P2 @ X4 ) )
% 2.59/1.57 = ( ^ [P3: A > $o] :
% 2.59/1.57 ? [N2: A] :
% 2.59/1.57 ( ( P3 @ N2 )
% 2.59/1.57 & ! [M2: A] :
% 2.59/1.57 ( ( ord_less @ A @ M2 @ N2 )
% 2.59/1.57 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_215_less__imp__not__less,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_216_order_Ostrict__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ord_less @ A @ B3 @ C )
% 2.59/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_217_dual__order_Oirrefl,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A] :
% 2.59/1.57 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_218_linorder__cases,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ~ ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( ( X2 != Y )
% 2.59/1.57 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_219_less__imp__triv,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A,P: $o] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( ( ord_less @ A @ Y @ X2 )
% 2.59/1.57 => P ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_220_less__imp__not__eq2,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( Y != X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_221_antisym__conv3,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [Y: A,X2: A] :
% 2.59/1.57 ( ~ ( ord_less @ A @ Y @ X2 )
% 2.59/1.57 => ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.59/1.57 = ( X2 = Y ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_222_less__induct,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( wellorder @ A )
% 2.59/1.57 => ! [P: A > $o,A4: A] :
% 2.59/1.57 ( ! [X3: A] :
% 2.59/1.57 ( ! [Y2: A] :
% 2.59/1.57 ( ( ord_less @ A @ Y2 @ X3 )
% 2.59/1.57 => ( P @ Y2 ) )
% 2.59/1.57 => ( P @ X3 ) )
% 2.59/1.57 => ( P @ A4 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_223_less__not__sym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_224_less__imp__not__eq,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( X2 != Y ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_225_dual__order_Oasym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [B3: A,A4: A] :
% 2.59/1.57 ( ( ord_less @ A @ B3 @ A4 )
% 2.59/1.57 => ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_226_ord__less__eq__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( ord @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( B3 = C )
% 2.59/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_227_ord__eq__less__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( ord @ A )
% 2.59/1.57 => ! [A4: A,B3: A,C: A] :
% 2.59/1.57 ( ( A4 = B3 )
% 2.59/1.57 => ( ( ord_less @ A @ B3 @ C )
% 2.59/1.57 => ( ord_less @ A @ A4 @ C ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_228_less__irrefl,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A] :
% 2.59/1.57 ~ ( ord_less @ A @ X2 @ X2 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_229_less__linear,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 | ( X2 = Y )
% 2.59/1.57 | ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_230_less__trans,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A,Z3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( ( ord_less @ A @ Y @ Z3 )
% 2.59/1.57 => ( ord_less @ A @ X2 @ Z3 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_231_less__asym_H,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [A4: A,B3: A] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_232_less__asym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( preorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_233_less__imp__neq,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( X2 != Y ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_234_dense,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( dense_order @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ? [Z2: A] :
% 2.59/1.57 ( ( ord_less @ A @ X2 @ Z2 )
% 2.59/1.57 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_235_order_Oasym,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( order @ A )
% 2.59/1.57 => ! [A4: A,B3: A] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_236_neq__iff,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( X2 != Y )
% 2.59/1.57 = ( ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 | ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_237_neqE,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [X2: A,Y: A] :
% 2.59/1.57 ( ( X2 != Y )
% 2.59/1.57 => ( ~ ( ord_less @ A @ X2 @ Y )
% 2.59/1.57 => ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_238_gt__ex,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( no_top @ A )
% 2.59/1.57 => ! [X2: A] :
% 2.59/1.57 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_239_lt__ex,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( no_bot @ A )
% 2.59/1.57 => ! [X2: A] :
% 2.59/1.57 ? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_240_order__less__subst2,axiom,
% 2.59/1.57 ! [A: $tType,C4: $tType] :
% 2.59/1.57 ( ( ( order @ C4 )
% 2.59/1.57 & ( order @ A ) )
% 2.59/1.57 => ! [A4: A,B3: A,F: A > C4,C: C4] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ord_less @ C4 @ ( F @ B3 ) @ C )
% 2.59/1.57 => ( ! [X3: A,Y3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_241_order__less__subst1,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( order @ B )
% 2.59/1.57 & ( order @ A ) )
% 2.59/1.57 => ! [A4: A,F: B > A,B3: B,C: B] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ ( F @ B3 ) )
% 2.59/1.57 => ( ( ord_less @ B @ B3 @ C )
% 2.59/1.57 => ( ! [X3: B,Y3: B] :
% 2.59/1.57 ( ( ord_less @ B @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_242_ord__less__eq__subst,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( ord @ B )
% 2.59/1.57 & ( ord @ A ) )
% 2.59/1.57 => ! [A4: A,B3: A,F: A > B,C: B] :
% 2.59/1.57 ( ( ord_less @ A @ A4 @ B3 )
% 2.59/1.57 => ( ( ( F @ B3 )
% 2.59/1.57 = C )
% 2.59/1.57 => ( ! [X3: A,Y3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_243_ord__eq__less__subst,axiom,
% 2.59/1.57 ! [A: $tType,B: $tType] :
% 2.59/1.57 ( ( ( ord @ B )
% 2.59/1.57 & ( ord @ A ) )
% 2.59/1.57 => ! [A4: A,F: B > A,B3: B,C: B] :
% 2.59/1.57 ( ( A4
% 2.59/1.57 = ( F @ B3 ) )
% 2.59/1.57 => ( ( ord_less @ B @ B3 @ C )
% 2.59/1.57 => ( ! [X3: B,Y3: B] :
% 2.59/1.57 ( ( ord_less @ B @ X3 @ Y3 )
% 2.59/1.57 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.59/1.57 => ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_244_pinf_I1_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.57 ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.59/1.57 => ( ( P @ X3 )
% 2.59/1.57 = ( P4 @ X3 ) ) )
% 2.59/1.57 => ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.59/1.57 => ( ( Q @ X3 )
% 2.59/1.57 = ( Q2 @ X3 ) ) )
% 2.59/1.57 => ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ( ( ( P @ X5 )
% 2.59/1.57 & ( Q @ X5 ) )
% 2.59/1.57 = ( ( P4 @ X5 )
% 2.59/1.57 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_245_pinf_I2_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.57 ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.59/1.57 => ( ( P @ X3 )
% 2.59/1.57 = ( P4 @ X3 ) ) )
% 2.59/1.57 => ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z4 @ X3 )
% 2.59/1.57 => ( ( Q @ X3 )
% 2.59/1.57 = ( Q2 @ X3 ) ) )
% 2.59/1.57 => ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ( ( ( P @ X5 )
% 2.59/1.57 | ( Q @ X5 ) )
% 2.59/1.57 = ( ( P4 @ X5 )
% 2.59/1.57 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_246_pinf_I3_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [T: A] :
% 2.59/1.57 ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ( X5 != T ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_247_pinf_I4_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [T: A] :
% 2.59/1.57 ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ( X5 != T ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_248_pinf_I5_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [T: A] :
% 2.59/1.57 ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_249_pinf_I7_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [T: A] :
% 2.59/1.57 ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ Z2 @ X5 )
% 2.59/1.57 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_250_pinf_I11_J,axiom,
% 2.59/1.57 ! [C4: $tType,D2: $tType] :
% 2.59/1.57 ( ( ord @ C4 )
% 2.59/1.57 => ! [F3: D2] :
% 2.59/1.57 ? [Z2: C4] :
% 2.59/1.57 ! [X5: C4] :
% 2.59/1.57 ( ( ord_less @ C4 @ Z2 @ X5 )
% 2.59/1.57 => ( F3 = F3 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_251_minf_I1_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.57 ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.59/1.57 => ( ( P @ X3 )
% 2.59/1.57 = ( P4 @ X3 ) ) )
% 2.59/1.57 => ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.59/1.57 => ( ( Q @ X3 )
% 2.59/1.57 = ( Q2 @ X3 ) ) )
% 2.59/1.57 => ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ X5 @ Z2 )
% 2.59/1.57 => ( ( ( P @ X5 )
% 2.59/1.57 & ( Q @ X5 ) )
% 2.59/1.57 = ( ( P4 @ X5 )
% 2.59/1.57 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_252_minf_I2_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
% 2.59/1.57 ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.59/1.57 => ( ( P @ X3 )
% 2.59/1.57 = ( P4 @ X3 ) ) )
% 2.59/1.57 => ( ? [Z4: A] :
% 2.59/1.57 ! [X3: A] :
% 2.59/1.57 ( ( ord_less @ A @ X3 @ Z4 )
% 2.59/1.57 => ( ( Q @ X3 )
% 2.59/1.57 = ( Q2 @ X3 ) ) )
% 2.59/1.57 => ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ X5 @ Z2 )
% 2.59/1.57 => ( ( ( P @ X5 )
% 2.59/1.57 | ( Q @ X5 ) )
% 2.59/1.57 = ( ( P4 @ X5 )
% 2.59/1.57 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(fact_253_minf_I3_J,axiom,
% 2.59/1.57 ! [A: $tType] :
% 2.59/1.57 ( ( linorder @ A )
% 2.59/1.57 => ! [T: A] :
% 2.59/1.57 ? [Z2: A] :
% 2.59/1.57 ! [X5: A] :
% 2.59/1.57 ( ( ord_less @ A @ X5 @ Z2 )
% 2.59/1.57 => ( X5 != T ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.59/1.57 ! [A2: $tType,A8: $tType] :
% 2.59/1.57 ( ( preorder @ A8 )
% 2.59/1.57 => ( preorder @ ( A2 > A8 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.59/1.57 ! [A2: $tType,A8: $tType] :
% 2.59/1.57 ( ( order @ A8 )
% 2.59/1.57 => ( order @ ( A2 > A8 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.59/1.57 ! [A2: $tType,A8: $tType] :
% 2.59/1.57 ( ( ord @ A8 )
% 2.59/1.57 => ( ord @ ( A2 > A8 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_fun___Groups_Ominus,axiom,
% 2.59/1.57 ! [A2: $tType,A8: $tType] :
% 2.59/1.57 ( ( minus @ A8 )
% 2.59/1.57 => ( minus @ ( A2 > A8 ) ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.59/1.57 strict2144017051up_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 2.59/1.57 ordere623563068d_diff @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.59/1.57 ordere223160158up_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.59/1.57 ordere236663937imp_le @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.59/1.57 canoni770627133id_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.59/1.57 linord1659791738miring @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.59/1.57 ordere779506340up_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.59/1.57 cancel146912293up_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.59/1.57 cancel1352612707id_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.59/1.57 cancel_semigroup_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.59/1.57 ab_semigroup_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.59/1.57 comm_monoid_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.59/1.57 semigroup_add @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.59/1.57 wellorder @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.59/1.57 preorder @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.59/1.57 linorder @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.59/1.57 no_top @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.59/1.57 order @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.59/1.57 ord @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 2.59/1.57 minus @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.59/1.57 one @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.59/1.57 size @ nat ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 2.59/1.57 preorder @ $o ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 2.59/1.57 linorder @ $o ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 2.59/1.57 order @ $o ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 2.59/1.57 ord @ $o ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
% 2.59/1.57 minus @ $o ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_List_Olist___Nat_Osize_10,axiom,
% 2.59/1.57 ! [A2: $tType] : ( size @ ( list @ A2 ) ) ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_11,axiom,
% 2.59/1.57 wellorder @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_12,axiom,
% 2.59/1.57 preorder @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_13,axiom,
% 2.59/1.57 linorder @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder_14,axiom,
% 2.59/1.57 order @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oord_15,axiom,
% 2.59/1.57 ord @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(tcon_Product__Type_Ounit___Groups_Ominus_16,axiom,
% 2.59/1.57 minus @ product_unit ).
% 2.59/1.57
% 2.59/1.57 thf(conj_0,conjecture,
% 2.59/1.57 ( ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) )
% 2.59/1.57 = ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ) ) ).
% 2.59/1.57 ------------------------
% 2.59/1.58 % (6813)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/18Mi)
% 2.59/1.59 % (6807)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/183Mi)
% 2.59/1.59 % (6813)Instruction limit reached!
% 2.59/1.59 % (6813)------------------------------
% 2.59/1.59 % (6813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.59 % (6813)Termination reason: Unknown
% 2.59/1.59 % (6813)Termination phase: shuffling
% 2.59/1.59
% 2.59/1.59 % (6813)Memory used [KB]: 1663
% 2.59/1.59 % (6813)Time elapsed: 0.008 s
% 2.59/1.59 % (6813)Instructions burned: 19 (million)
% 2.59/1.59 % (6813)------------------------------
% 2.59/1.59 % (6813)------------------------------
% 2.59/1.59 % (6808)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_6476 for (2999ds/4Mi)
% 2.59/1.59 % (6809)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_6476 for (2999ds/27Mi)
% 2.59/1.59 % (6810)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/2Mi)
% 2.59/1.59 % (6811)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_6476 for (2999ds/2Mi)
% 2.59/1.59 % (6812)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_6476 for (2999ds/275Mi)
% 2.59/1.59 % (6811)Instruction limit reached!
% 2.59/1.59 % (6811)------------------------------
% 2.59/1.59 % (6811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.59 % (6811)Termination reason: Unknown
% 2.59/1.59 % (6811)Termination phase: shuffling
% 2.59/1.59
% 2.59/1.59 % (6811)Memory used [KB]: 1279
% 2.59/1.59 % (6811)Time elapsed: 0.003 s
% 2.59/1.59 % (6811)Instructions burned: 2 (million)
% 2.59/1.59 % (6811)------------------------------
% 2.59/1.59 % (6811)------------------------------
% 2.59/1.59 % (6810)Instruction limit reached!
% 2.59/1.59 % (6810)------------------------------
% 2.59/1.59 % (6810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.59 % (6810)Termination reason: Unknown
% 2.59/1.59 % (6810)Termination phase: shuffling
% 2.59/1.59
% 2.59/1.59 % (6810)Memory used [KB]: 1279
% 2.59/1.59 % (6810)Time elapsed: 0.004 s
% 2.59/1.59 % (6810)Instructions burned: 3 (million)
% 2.59/1.59 % (6810)------------------------------
% 2.59/1.59 % (6810)------------------------------
% 2.59/1.59 % (6808)Instruction limit reached!
% 2.59/1.59 % (6808)------------------------------
% 2.59/1.59 % (6808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.59/1.59 % (6808)Termination reason: Unknown
% 2.59/1.59 % (6808)Termination phase: shuffling
% 2.59/1.59
% 2.59/1.59 % (6808)Memory used [KB]: 1407
% 2.59/1.59 % (6808)Time elapsed: 0.005 s
% 2.59/1.59 % (6808)Instructions burned: 5 (million)
% 2.59/1.59 % (6808)------------------------------
% 2.59/1.59 % (6808)------------------------------
% 2.67/1.60 % (6814)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/3Mi)
% 2.67/1.60 % (6814)Instruction limit reached!
% 2.67/1.60 % (6814)------------------------------
% 2.67/1.60 % (6814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.60 % (6814)Termination reason: Unknown
% 2.67/1.60 % (6814)Termination phase: shuffling
% 2.67/1.60
% 2.67/1.60 % (6814)Memory used [KB]: 1407
% 2.67/1.60 % (6814)Time elapsed: 0.003 s
% 2.67/1.60 % (6814)Instructions burned: 4 (million)
% 2.67/1.60 % (6814)------------------------------
% 2.67/1.60 % (6814)------------------------------
% 2.67/1.60 % (6809)Instruction limit reached!
% 2.67/1.60 % (6809)------------------------------
% 2.67/1.60 % (6809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.60 % (6809)Termination reason: Unknown
% 2.67/1.60 % (6809)Termination phase: Property scanning
% 2.67/1.60
% 2.67/1.60 % (6809)Memory used [KB]: 1791
% 2.67/1.60 % (6809)Time elapsed: 0.018 s
% 2.67/1.60 % (6809)Instructions burned: 28 (million)
% 2.67/1.60 % (6809)------------------------------
% 2.67/1.60 % (6809)------------------------------
% 2.67/1.61 % (6815)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_6476 for (2999ds/37Mi)
% 2.67/1.61 % (6816)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_6476 for (2999ds/15Mi)
% 2.67/1.61 % (6817)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_6476 for (2999ds/3Mi)
% 2.67/1.61 % (6818)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_6476 for (2999ds/1041Mi)
% 2.67/1.61 % (6817)Instruction limit reached!
% 2.67/1.61 % (6817)------------------------------
% 2.67/1.61 % (6817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (6817)Termination reason: Unknown
% 2.67/1.61 % (6817)Termination phase: shuffling
% 2.67/1.61
% 2.67/1.61 % (6817)Memory used [KB]: 1407
% 2.67/1.61 % (6817)Time elapsed: 0.004 s
% 2.67/1.61 % (6817)Instructions burned: 4 (million)
% 2.67/1.61 % (6817)------------------------------
% 2.67/1.61 % (6817)------------------------------
% 2.67/1.62 % (6816)Instruction limit reached!
% 2.67/1.62 % (6816)------------------------------
% 2.67/1.62 % (6816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (6816)Termination reason: Unknown
% 2.67/1.62 % (6816)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (6816)Memory used [KB]: 1663
% 2.67/1.62 % (6816)Time elapsed: 0.012 s
% 2.67/1.62 % (6816)Instructions burned: 16 (million)
% 2.67/1.62 % (6816)------------------------------
% 2.67/1.62 % (6816)------------------------------
% 2.67/1.62 % (6819)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/7Mi)
% 2.67/1.63 % (6819)Instruction limit reached!
% 2.67/1.63 % (6819)------------------------------
% 2.67/1.63 % (6819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.63 % (6819)Termination reason: Unknown
% 2.67/1.63 % (6819)Termination phase: shuffling
% 2.67/1.63
% 2.67/1.63 % (6819)Memory used [KB]: 1407
% 2.67/1.63 % (6819)Time elapsed: 0.005 s
% 2.67/1.63 % (6819)Instructions burned: 7 (million)
% 2.67/1.63 % (6819)------------------------------
% 2.67/1.63 % (6819)------------------------------
% 2.67/1.63 % (6815)Instruction limit reached!
% 2.67/1.63 % (6815)------------------------------
% 2.67/1.63 % (6815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.63 % (6815)Termination reason: Unknown
% 2.67/1.63 % (6815)Termination phase: shuffling
% 2.67/1.63
% 2.67/1.63 % (6815)Memory used [KB]: 2046
% 2.67/1.63 % (6815)Time elapsed: 0.021 s
% 2.67/1.63 % (6815)Instructions burned: 37 (million)
% 2.67/1.63 % (6815)------------------------------
% 2.67/1.63 % (6815)------------------------------
% 2.67/1.63 % (6820)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_6476 for (2999ds/16Mi)
% 2.67/1.64 % (6812)First to succeed.
% 2.67/1.64 % (6821)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_6476 for (2999ds/3Mi)
% 2.67/1.64 % (6822)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_6476 for (2999ds/3Mi)
% 2.67/1.64 % (6820)Instruction limit reached!
% 2.67/1.64 % (6820)------------------------------
% 2.67/1.64 % (6820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (6820)Termination reason: Unknown
% 2.67/1.64 % (6820)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (6820)Memory used [KB]: 1663
% 2.67/1.64 % (6820)Time elapsed: 0.010 s
% 2.67/1.64 % (6820)Instructions burned: 16 (million)
% 2.67/1.64 % (6820)------------------------------
% 2.67/1.64 % (6820)------------------------------
% 2.67/1.64 % (6822)Instruction limit reached!
% 2.67/1.64 % (6822)------------------------------
% 2.67/1.64 % (6822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (6822)Termination reason: Unknown
% 2.67/1.64 % (6822)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (6822)Memory used [KB]: 1407
% 2.67/1.64 % (6822)Time elapsed: 0.002 s
% 2.67/1.64 % (6822)Instructions burned: 3 (million)
% 2.67/1.64 % (6822)------------------------------
% 2.67/1.64 % (6822)------------------------------
% 2.67/1.64 % (6812)Refutation found. Thanks to Tanya!
% 2.67/1.64 % SZS status Theorem for DTF2THF_6476
% 2.67/1.64 % SZS output start Proof for DTF2THF_6476
% 2.67/1.64 thf(type_def_6, type, list: ($tType) > $tType).
% 2.67/1.64 thf(type_def_7, type, nat: $tType).
% 2.67/1.64 thf(type_def_8, type, p: $tType).
% 2.67/1.64 thf(type_def_9, type, v: $tType).
% 2.67/1.64 thf(type_def_10, type, s: $tType).
% 2.67/1.64 thf(type_def_11, type, product_unit: $tType).
% 2.67/1.64 thf(type_def_12, type, configuration_ext: ($tType * $tType * $tType * $tType) > $tType).
% 2.67/1.64 thf(func_def_0, type, configuration_ext: ($tType * $tType * $tType * $tType) > $tType).
% 2.67/1.64 thf(func_def_1, type, product_unit: $tType).
% 2.67/1.64 thf(func_def_2, type, list: ($tType) > $tType).
% 2.67/1.64 thf(func_def_3, type, nat: $tType).
% 2.67/1.64 thf(func_def_4, type, v: $tType).
% 2.67/1.64 thf(func_def_5, type, s: $tType).
% 2.67/1.64 thf(func_def_6, type, p: $tType).
% 2.67/1.64 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_8, type, size: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_9, type, one: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_10, type, minus: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_11, type, ord: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_12, type, neg_numeral: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_13, type, order: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_14, type, group_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_15, type, no_bot: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_16, type, no_top: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_17, type, linorder: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_18, type, preorder: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_19, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_20, type, wellorder: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_21, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_22, type, dense_order: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_23, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_24, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_25, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_26, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_27, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_28, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_29, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_30, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_31, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_32, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_33, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_34, type, ordere623563068d_diff: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_35, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 2.67/1.64 thf(func_def_36, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.67/1.64 thf(func_def_37, type, one_one: !>[X0: $tType]:(X0)).
% 2.67/1.64 thf(func_def_38, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.67/1.64 thf(func_def_39, type, prefixList: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 2.67/1.64 thf(func_def_40, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.67/1.64 thf(func_def_41, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.67/1.64 thf(func_def_42, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.67/1.64 thf(func_def_43, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.67/1.64 thf(func_def_44, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.67/1.64 thf(func_def_45, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.67/1.64 thf(func_def_46, type, fe: nat > list @ configuration_ext @ p @ v @ s @ product_unit).
% 2.67/1.64 thf(func_def_47, type, n0: nat).
% 2.67/1.64 thf(func_def_48, type, n: nat).
% 2.67/1.64 thf(func_def_63, type, sP0: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.67/1.64 thf(func_def_64, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.67/1.64 thf(func_def_65, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.67/1.64 thf(func_def_66, type, sP3: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.67/1.64 thf(func_def_67, type, sP4: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.67/1.64 thf(func_def_68, type, sP5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.67/1.64 thf(func_def_69, type, sP6: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 2.67/1.64 thf(func_def_70, type, sK7: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.67/1.64 thf(func_def_71, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_72, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_73, type, sK10: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_74, type, sK11: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_75, type, sK12: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_76, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_77, type, sK14: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_78, type, sK15: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_79, type, sK16: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_80, type, sK17: nat > (nat > $o) > nat).
% 2.67/1.64 thf(func_def_81, type, sK18: (nat > $o) > nat).
% 2.67/1.64 thf(func_def_82, type, sK19: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_83, type, sK20: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_84, type, sK21: (nat > nat) > nat).
% 2.67/1.64 thf(func_def_85, type, sK22: (nat > nat) > nat).
% 2.67/1.64 thf(func_def_86, type, sK23: !>[X0: $tType]:((nat > X0 > $o) > nat > list @ X0)).
% 2.67/1.64 thf(func_def_87, type, sK24: !>[X0: $tType]:((nat > X0 > $o) > nat > nat)).
% 2.67/1.64 thf(func_def_88, type, sK25: !>[X0: $tType]:(nat > (nat > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_89, type, sK26: !>[X0: $tType]:(list @ X0 > (nat > X0 > $o) > nat > nat)).
% 2.67/1.64 thf(func_def_90, type, sK27: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_91, type, sK28: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_92, type, sK29: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_93, type, sK30: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_94, type, sK31: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_95, type, sK32: !>[X0: $tType, X1: $tType]:(X1 > X0)).
% 2.67/1.64 thf(func_def_96, type, sK33: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_97, type, sK34: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_98, type, sK35: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_99, type, sK36: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_100, type, sK37: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_101, type, sK38: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_102, type, sK39: (nat > nat) > nat).
% 2.67/1.64 thf(func_def_103, type, sK40: (nat > nat) > nat).
% 2.67/1.64 thf(func_def_104, type, sK41: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_105, type, sK42: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_106, type, sK43: (nat > $o) > nat).
% 2.67/1.64 thf(func_def_107, type, sK44: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_108, type, sK45: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_109, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_110, type, sK47: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_111, type, sK48: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_112, type, sK49: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_113, type, sK50: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_114, type, sK51: nat > nat > nat).
% 2.67/1.64 thf(func_def_115, type, sK52: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_116, type, sK53: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_117, type, sK54: !>[X0: $tType]:(nat > list @ X0)).
% 2.67/1.64 thf(func_def_118, type, sK55: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_119, type, sK56: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_120, type, sK57: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_121, type, sK58: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_122, type, sK59: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_123, type, sK60: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_124, type, sK61: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.67/1.64 thf(func_def_125, type, sK62: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_126, type, sK63: !>[X0: $tType]:(nat > (X0 > nat) > X0 > (X0 > $o) > X0)).
% 2.67/1.64 thf(func_def_127, type, sK64: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_128, type, sK65: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_129, type, sK66: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_130, type, sK67: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_131, type, sK68: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.67/1.64 thf(func_def_132, type, sK69: (nat > $o) > nat).
% 2.67/1.64 thf(func_def_133, type, sK70: (nat > $o) > nat > nat).
% 2.67/1.64 thf(func_def_134, type, sK71: (nat > $o) > nat > nat).
% 2.67/1.64 thf(func_def_135, type, sK72: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_136, type, sK73: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_137, type, sK74: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_138, type, sK75: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_139, type, sK76: !>[X0: $tType]:(X0 > X0)).
% 2.67/1.64 thf(func_def_140, type, sK77: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_141, type, sK78: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.67/1.64 thf(func_def_143, type, ph80: !>[X0: $tType]:(X0)).
% 2.67/1.64 thf(f1522,plain,(
% 2.67/1.64 $false),
% 2.67/1.64 inference(subsumption_resolution,[status(thm)],[f1521,f1307])).
% 2.67/1.64 thf(f1307,plain,(
% 2.67/1.64 ((fe @ n) != nil @ configuration_ext @ p @ v @ s @ product_unit)),
% 2.67/1.64 inference(cnf_transformation,[status(thm)],[f1])).
% 2.67/1.64 thf(f1,axiom,(
% 2.67/1.64 ((fe @ n) != nil @ configuration_ext @ p @ v @ s @ product_unit)),
% 2.67/1.64 file('/export/starexec/sandbox/tmp/tmp.UV0DDxgB5h/DTF2THF_6476.p',fact_0_FeNNotEmpty)).
% 2.67/1.64 thf(f1521,plain,(
% 2.67/1.64 ((fe @ n) = nil @ configuration_ext @ p @ v @ s @ product_unit)),
% 2.67/1.64 inference(trivial_inequality_removal,[status(thm)],[f1519])).
% 2.67/1.64 thf(f1519,plain,(
% 2.67/1.64 ((fe @ n) = nil @ configuration_ext @ p @ v @ s @ product_unit) | ((last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) != (last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)))),
% 2.67/1.64 inference(superposition,[status(thm)],[f1257,f1467])).
% 2.67/1.64 thf(f1467,plain,(
% 2.67/1.64 ( ! [X0 : $tType,X1 : list @ X0] : (((last @ X0 @ X1) = (nth @ X0 @ X1 @ (minus_minus @ nat @ (size_size @ list @ X0 @ X1) @ one_one @ nat))) | (nil @ X0 = X1)) )),
% 2.67/1.64 inference(cnf_transformation,[status(thm)],[f965])).
% 2.67/1.64 thf(f965,plain,(
% 2.67/1.64 ! [X0 : $tType,X1 : list @ X0] : ((nil @ X0 = X1) | ((last @ X0 @ X1) = (nth @ X0 @ X1 @ (minus_minus @ nat @ (size_size @ list @ X0 @ X1) @ one_one @ nat))))),
% 2.67/1.64 inference(ennf_transformation,[status(thm)],[f862])).
% 2.67/1.64 thf(f862,plain,(
% 2.67/1.64 ! [X0 : $tType,X1 : list @ X0] : ((nil @ X0 != X1) => ((last @ X0 @ X1) = (nth @ X0 @ X1 @ (minus_minus @ nat @ (size_size @ list @ X0 @ X1) @ one_one @ nat))))),
% 2.67/1.64 inference(rectify,[status(thm)],[f5])).
% 2.67/1.64 thf(f5,axiom,(
% 2.67/1.64 ! [X0 : $tType,X3 : list @ X0] : ((nil @ X0 != X3) => ((last @ X0 @ X3) = (nth @ X0 @ X3 @ (minus_minus @ nat @ (size_size @ list @ X0 @ X3) @ one_one @ nat))))),
% 2.67/1.64 file('/export/starexec/sandbox/tmp/tmp.UV0DDxgB5h/DTF2THF_6476.p',fact_4_last__conv__nth)).
% 2.67/1.64 thf(f1257,plain,(
% 2.67/1.64 ((last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) != (nth @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n) @ (minus_minus @ nat @ (size_size @ list @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) @ one_one @ nat)))),
% 2.67/1.64 inference(cnf_transformation,[status(thm)],[f846])).
% 2.67/1.64 thf(f846,plain,(
% 2.67/1.64 ((last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) != (nth @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n) @ (minus_minus @ nat @ (size_size @ list @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) @ one_one @ nat)))),
% 2.67/1.64 inference(flattening,[status(thm)],[f294])).
% 2.67/1.64 thf(f294,negated_conjecture,(
% 2.67/1.64 ~((last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) = (nth @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n) @ (minus_minus @ nat @ (size_size @ list @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) @ one_one @ nat)))),
% 2.67/1.64 inference(negated_conjecture,[status(cth)],[f293])).
% 2.67/1.64 thf(f293,conjecture,(
% 2.67/1.64 ((last @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) = (nth @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n) @ (minus_minus @ nat @ (size_size @ list @ configuration_ext @ p @ v @ s @ product_unit @ (fe @ n)) @ one_one @ nat)))),
% 2.67/1.64 file('/export/starexec/sandbox/tmp/tmp.UV0DDxgB5h/DTF2THF_6476.p',conj_0)).
% 2.67/1.64 % SZS output end Proof for DTF2THF_6476
% 2.67/1.64 % (6812)------------------------------
% 2.67/1.64 % (6812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (6812)Termination reason: Refutation
% 2.67/1.64
% 2.67/1.64 % (6812)Memory used [KB]: 6908
% 2.67/1.64 % (6812)Time elapsed: 0.049 s
% 2.67/1.64 % (6812)Instructions burned: 88 (million)
% 2.67/1.64 % (6812)------------------------------
% 2.67/1.64 % (6812)------------------------------
% 2.67/1.64 % (6806)Success in time 0.067 s
%------------------------------------------------------------------------------