%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP043^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:05 AM UTC 2026
% Result : Theorem 2.95s 1.90s
% Output : Refutation 2.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP043^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n021.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:24:24 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP043^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer Descartes_Sign_Rule problem prob_378__5868812_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : Descartes_Sign_Rule/prob_378__5868812_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.00 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 380 ( 123 unt; 50 typ; 0 def)
% 0.29/0.47 % Number of atoms : 858 ( 355 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.47 % Number of connectives : 3700 ( 77 ~; 10 |; 44 &;3118 @)
% 0.29/0.47 % ( 0 <=>; 451 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 34 ( 7 avg)
% 0.29/0.47 % Number of types : 4 ( 3 usr)
% 0.29/0.47 % Number of type conns : 114 ( 114 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 48 ( 47 usr; 6 con; 0-3 aty)
% 0.29/0.47 % Number of variables : 1050 ( 27 ^; 964 !; 20 ?;1050 :)
% 0.29/0.47 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.47 % SPC : TH1_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:53.415
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (4)
% 0.29/0.47 thf(ty_t_List_Olist,type,
% 0.29/0.47 list: $tType > $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Nat_Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_t_Int_Oint,type,
% 0.29/0.47 int: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_tf_a,type,
% 0.29/0.47 a: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (46)
% 0.29/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.47 linordered_idom:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_HOL_Otype,type,
% 0.29/0.47 type:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Nat_Osize,type,
% 0.29/0.47 size:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oplus,type,
% 0.29/0.47 plus:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.47 zero:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.47 ord:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.47 order:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.47 group_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Omonoid__add,type,
% 0.29/0.47 monoid_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.47 linorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.47 preorder:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/0.47 semigroup_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.29/0.47 comm_monoid_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/0.47 ab_semigroup_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/0.47 cancel_semigroup_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.29/0.47 cancel1352612707id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/0.47 linord219039673up_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.29/0.47 ordere216010020id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/0.47 ordere779506340up_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.47 canoni770627133id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 0.29/0.47 ordere1818651114id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 0.29/0.47 strict797366125id_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/0.47 ordere236663937imp_le:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.29/0.47 ordere223160158up_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.29/0.47 strict2144017051up_add:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.29/0.47 ordere516151231imp_le:
% 0.29/0.47 !>[A: $tType] : $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Opsums,type,
% 0.29/0.47 descar1668888542_psums:
% 0.29/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/0.47 plus_plus:
% 0.29/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.47 zero_zero:
% 0.29/0.47 !>[A: $tType] : A ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oappend,type,
% 0.29/0.47 append:
% 0.29/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Olast,type,
% 0.29/0.47 last:
% 0.29/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Olist_OCons,type,
% 0.29/0.47 cons:
% 0.29/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Olist_ONil,type,
% 0.29/0.47 nil:
% 0.29/0.47 !>[A: $tType] : ( list @ A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Onth,type,
% 0.29/0.47 nth:
% 0.29/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Oreplicate,type,
% 0.29/0.47 replicate:
% 0.29/0.47 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_List_Otake,type,
% 0.29/0.47 take:
% 0.29/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.29/0.47 size_size:
% 0.29/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.47 ord_less:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.47 ord_less_eq:
% 0.29/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_p____,type,
% 0.29/0.47 p: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_v,type,
% 0.29/0.47 v: ( list @ a ) > int ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xa____,type,
% 0.29/0.47 xa: a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xs,type,
% 0.29/0.47 xs: list @ a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xs1____,type,
% 0.29/0.47 xs1: list @ a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xs2____,type,
% 0.29/0.47 xs2: list @ a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_xsa____,type,
% 0.29/0.47 xsa: list @ a ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (253)
% 0.29/0.47 thf(fact_0_nonneg_Ohyps_I1_J,axiom,
% 0.29/0.47 ( xsa
% 0.29/0.47 != ( nil @ a ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonneg.hyps(1)
% 0.29/0.47 thf(fact_1_p__nz,axiom,
% 0.29/0.47 ( ( nth @ a @ xsa @ p )
% 0.29/0.47 != ( zero_zero @ a ) ) ).
% 0.29/0.47
% 0.29/0.47 % p_nz
% 0.29/0.47 thf(fact_2_nonneg_Ohyps_I2_J,axiom,
% 0.29/0.47 ( ( last @ a @ xsa )
% 0.29/0.47 != ( zero_zero @ a ) ) ).
% 0.29/0.47
% 0.29/0.47 % nonneg.hyps(2)
% 0.29/0.47 thf(fact_3_psums__0__Cons,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A )
% 0.29/0.47 => ! [Xs: list @ A] :
% 0.29/0.47 ( ( descar1668888542_psums @ A @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) )
% 0.29/0.47 = ( cons @ A @ ( zero_zero @ A ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums_0_Cons
% 0.29/0.47 thf(fact_4_xs1__def,axiom,
% 0.29/0.47 ( xs1
% 0.29/0.47 = ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % xs1_def
% 0.29/0.47 thf(fact_5_xs__decompose,axiom,
% 0.29/0.47 ( xsa
% 0.29/0.47 = ( append @ a @ xs1 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % xs_decompose
% 0.29/0.47 thf(fact_6_psums__replicate__0__append,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A )
% 0.29/0.47 => ! [N: nat,Xs: list @ A] :
% 0.29/0.47 ( ( descar1668888542_psums @ A @ ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ Xs ) )
% 0.29/0.47 = ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums_replicate_0_append
% 0.29/0.47 thf(fact_7_less__p__zero,axiom,
% 0.29/0.47 ! [Q: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ Q @ p )
% 0.29/0.47 => ( ( nth @ a @ xsa @ Q )
% 0.29/0.47 = ( zero_zero @ a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_p_zero
% 0.29/0.47 thf(fact_8__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 0.29/0.47 ( ( take @ a @ p @ xsa )
% 0.29/0.47 = ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>take p xs = replicate p (0::'a)\<close>
% 0.29/0.47 thf(fact_9_replicate__app__Cons__same,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % replicate_app_Cons_same
% 0.29/0.47 thf(fact_10_append_Oassoc,axiom,
% 0.29/0.47 ! [A: $tType,A2: list @ A,B: list @ A,C: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( append @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( append @ A @ A2 @ ( append @ A @ B @ C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append.assoc
% 0.29/0.47 thf(fact_11_append__assoc,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
% 0.29/0.47 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_assoc
% 0.29/0.47 thf(fact_12_append__same__eq,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Ys @ Xs )
% 0.29/0.47 = ( append @ A @ Zs @ Xs ) )
% 0.29/0.47 = ( Ys = Zs ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_same_eq
% 0.29/0.47 thf(fact_13_same__append__eq,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = ( append @ A @ Xs @ Zs ) )
% 0.29/0.47 = ( Ys = Zs ) ) ).
% 0.29/0.47
% 0.29/0.47 % same_append_eq
% 0.29/0.47 thf(fact_14_nonneg_Ohyps_I4_J,axiom,
% 0.29/0.47 ord_less_eq @ a @ ( zero_zero @ a ) @ xa ).
% 0.29/0.47
% 0.29/0.47 % nonneg.hyps(4)
% 0.29/0.47 thf(fact_15_list_Oinject,axiom,
% 0.29/0.47 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 0.29/0.47 ( ( ( cons @ A @ X21 @ X22 )
% 0.29/0.47 = ( cons @ A @ Y21 @ Y22 ) )
% 0.29/0.47 = ( ( X21 = Y21 )
% 0.29/0.47 & ( X22 = Y22 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.inject
% 0.29/0.47 thf(fact_16_append__replicate__commute,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A,K: nat] :
% 0.29/0.47 ( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
% 0.29/0.47 = ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_replicate_commute
% 0.29/0.47 thf(fact_17_Cons__eq__appendI,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 0.29/0.47 ( ( ( cons @ A @ X @ Xs1 )
% 0.29/0.47 = Ys )
% 0.29/0.47 => ( ( Xs
% 0.29/0.47 = ( append @ A @ Xs1 @ Zs ) )
% 0.29/0.47 => ( ( cons @ A @ X @ Xs )
% 0.29/0.47 = ( append @ A @ Ys @ Zs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Cons_eq_appendI
% 0.29/0.47 thf(fact_18_assms_I1_J,axiom,
% 0.29/0.47 ( xs
% 0.29/0.47 != ( nil @ a ) ) ).
% 0.29/0.47
% 0.29/0.47 % assms(1)
% 0.29/0.47 thf(fact_19_assms_I2_J,axiom,
% 0.29/0.47 ( ( last @ a @ xs )
% 0.29/0.47 != ( zero_zero @ a ) ) ).
% 0.29/0.47
% 0.29/0.47 % assms(2)
% 0.29/0.47 thf(fact_20_replicate__eq__replicate,axiom,
% 0.29/0.47 ! [A: $tType,M: nat,X: A,N: nat,Y: A] :
% 0.29/0.47 ( ( ( replicate @ A @ M @ X )
% 0.29/0.47 = ( replicate @ A @ N @ Y ) )
% 0.29/0.47 = ( ( M = N )
% 0.29/0.47 & ( ( M
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 => ( X = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % replicate_eq_replicate
% 0.29/0.47 thf(fact_21_le__zero__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_zero_eq
% 0.29/0.47 thf(fact_22_not__gr__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_gr_zero
% 0.29/0.47 thf(fact_23_append_Oright__neutral,axiom,
% 0.29/0.47 ! [A: $tType,A2: list @ A] :
% 0.29/0.47 ( ( append @ A @ A2 @ ( nil @ A ) )
% 0.29/0.47 = A2 ) ).
% 0.29/0.47
% 0.29/0.47 % append.right_neutral
% 0.29/0.47 thf(fact_24_append__is__Nil__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 = ( ( Xs
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 & ( Ys
% 0.29/0.47 = ( nil @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_is_Nil_conv
% 0.29/0.47 thf(fact_25_Nil__is__append__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( nil @ A )
% 0.29/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( ( Xs
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 & ( Ys
% 0.29/0.47 = ( nil @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Nil_is_append_conv
% 0.29/0.47 thf(fact_26_self__append__conv2,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.29/0.47 ( ( Ys
% 0.29/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( Xs
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % self_append_conv2
% 0.29/0.47 thf(fact_27_append__self__conv2,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = Ys )
% 0.29/0.47 = ( Xs
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_self_conv2
% 0.29/0.47 thf(fact_28_self__append__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( Ys
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % self_append_conv
% 0.29/0.47 thf(fact_29_append__self__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = Xs )
% 0.29/0.47 = ( Ys
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_self_conv
% 0.29/0.47 thf(fact_30_append__Nil2,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( append @ A @ Xs @ ( nil @ A ) )
% 0.29/0.47 = Xs ) ).
% 0.29/0.47
% 0.29/0.47 % append_Nil2
% 0.29/0.47 thf(fact_31_nth__Cons__0,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Xs: list @ A] :
% 0.29/0.47 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % nth_Cons_0
% 0.29/0.47 thf(fact_32_take0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( take @ A @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % take0
% 0.29/0.47 thf(fact_33_take__eq__Nil,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.29/0.47 ( ( ( take @ A @ N @ Xs )
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 = ( ( N
% 0.29/0.47 = ( zero_zero @ nat ) )
% 0.29/0.47 | ( Xs
% 0.29/0.47 = ( nil @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_eq_Nil
% 0.29/0.47 thf(fact_34_empty__replicate,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A] :
% 0.29/0.47 ( ( ( nil @ A )
% 0.29/0.47 = ( replicate @ A @ N @ X ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % empty_replicate
% 0.29/0.47 thf(fact_35_replicate__empty,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A] :
% 0.29/0.47 ( ( ( replicate @ A @ N @ X )
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % replicate_empty
% 0.29/0.47 thf(fact_36_last__replicate,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 => ( ( last @ A @ ( replicate @ A @ N @ X ) )
% 0.29/0.47 = X ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_replicate
% 0.29/0.47 thf(fact_37_append1__eq__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.29/0.47 = ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 0.29/0.47 = ( ( Xs = Ys )
% 0.29/0.47 & ( X = Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append1_eq_conv
% 0.29/0.47 thf(fact_38_nth__take,axiom,
% 0.29/0.47 ! [A: $tType,I: nat,N: nat,Xs: list @ A] :
% 0.29/0.47 ( ( ord_less @ nat @ I @ N )
% 0.29/0.47 => ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
% 0.29/0.47 = ( nth @ A @ Xs @ I ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_take
% 0.29/0.47 thf(fact_39_last__appendL,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.29/0.47 ( ( Ys
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( last @ A @ Xs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_appendL
% 0.29/0.47 thf(fact_40_last__appendR,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.29/0.47 ( ( Ys
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( last @ A @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_appendR
% 0.29/0.47 thf(fact_41_nth__replicate,axiom,
% 0.29/0.47 ! [A: $tType,I: nat,N: nat,X: A] :
% 0.29/0.47 ( ( ord_less @ nat @ I @ N )
% 0.29/0.47 => ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
% 0.29/0.47 = X ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_replicate
% 0.29/0.47 thf(fact_42_last__snoc,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.29/0.47 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % last_snoc
% 0.29/0.47 thf(fact_43_take__0,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( take @ A @ ( zero_zero @ nat ) @ Xs )
% 0.29/0.47 = ( nil @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_0
% 0.29/0.47 thf(fact_44_ext,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 0.29/0.47 ( ! [X2: A] :
% 0.29/0.47 ( ( F @ X2 )
% 0.29/0.47 = ( G @ X2 ) )
% 0.29/0.47 => ( F = G ) ) ).
% 0.29/0.47
% 0.29/0.47 % ext
% 0.29/0.47 thf(fact_45_last_Osimps,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.29/0.47 ( ( ( Xs
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = X ) )
% 0.29/0.47 & ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = ( last @ A @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last.simps
% 0.29/0.47 thf(fact_46_last__ConsL,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = X ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_ConsL
% 0.29/0.47 thf(fact_47_last__ConsR,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = ( last @ A @ Xs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_ConsR
% 0.29/0.47 thf(fact_48_last__append,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.29/0.47 ( ( ( Ys
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( last @ A @ Xs ) ) )
% 0.29/0.47 & ( ( Ys
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( last @ A @ Ys ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % last_append
% 0.29/0.47 thf(fact_49_take__Nil,axiom,
% 0.29/0.47 ! [A: $tType,N: nat] :
% 0.29/0.47 ( ( take @ A @ N @ ( nil @ A ) )
% 0.29/0.47 = ( nil @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_Nil
% 0.29/0.47 thf(fact_50_take__equalityI,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ! [I2: nat] :
% 0.29/0.47 ( ( take @ A @ I2 @ Xs )
% 0.29/0.47 = ( take @ A @ I2 @ Ys ) )
% 0.29/0.47 => ( Xs = Ys ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_equalityI
% 0.29/0.47 thf(fact_51_transpose_Ocases,axiom,
% 0.29/0.47 ! [A: $tType,X: list @ ( list @ A )] :
% 0.29/0.47 ( ( X
% 0.29/0.47 != ( nil @ ( list @ A ) ) )
% 0.29/0.47 => ( ! [Xss: list @ ( list @ A )] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.29/0.47 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % transpose.cases
% 0.29/0.47 thf(fact_52_longest__common__suffix,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ? [Ss: list @ A,Xs4: list @ A,Ys2: list @ A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 = ( append @ A @ Xs4 @ Ss ) )
% 0.29/0.47 & ( Ys
% 0.29/0.47 = ( append @ A @ Ys2 @ Ss ) )
% 0.29/0.47 & ( ( Xs4
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 | ( Ys2
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 | ( ( last @ A @ Xs4 )
% 0.29/0.47 != ( last @ A @ Ys2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % longest_common_suffix
% 0.29/0.47 thf(fact_53_replicate__0,axiom,
% 0.29/0.47 ! [A: $tType,X: A] :
% 0.29/0.47 ( ( replicate @ A @ ( zero_zero @ nat ) @ X )
% 0.29/0.47 = ( nil @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % replicate_0
% 0.29/0.47 thf(fact_54_zero__less__iff__neq__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.47 = ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_iff_neq_zero
% 0.29/0.47 thf(fact_55_gr__implies__not__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [M: A,N: A] :
% 0.29/0.47 ( ( ord_less @ A @ M @ N )
% 0.29/0.47 => ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not_zero
% 0.29/0.47 thf(fact_56_not__less__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_zero
% 0.29/0.47 thf(fact_57_gr__zeroI,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_zeroI
% 0.29/0.47 thf(fact_58_zero__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le
% 0.29/0.47 thf(fact_59_strict__sorted_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.29/0.47 => ( P @ A0 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_sorted.induct
% 0.29/0.47 thf(fact_60_strict__sorted_Ocases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: list @ A] :
% 0.29/0.47 ( ( X
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ~ ! [X2: A,Ys3: list @ A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_sorted.cases
% 0.29/0.47 thf(fact_61_map__tailrec__rev_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType,P: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
% 0.29/0.47 ( ! [F2: A > B2,X_1: list @ B2] : ( P @ F2 @ ( nil @ A ) @ X_1 )
% 0.29/0.47 => ( ! [F2: A > B2,A3: A,As: list @ A,Bs: list @ B2] :
% 0.29/0.47 ( ( P @ F2 @ As @ ( cons @ B2 @ ( F2 @ A3 ) @ Bs ) )
% 0.29/0.47 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 0.29/0.47 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % map_tailrec_rev.induct
% 0.29/0.47 thf(fact_62_list__nonempty__induct,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( Xs3
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( P @ Xs3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 0.29/0.47 => ( P @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_nonempty_induct
% 0.29/0.47 thf(fact_63_successively_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.29/0.47 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.29/0.47 => ( ! [P2: A > A > $o,X2: A] : ( P @ P2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [P2: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( P @ P2 @ ( cons @ A @ Y2 @ Xs3 ) )
% 0.29/0.47 => ( P @ P2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % successively.induct
% 0.29/0.47 thf(fact_64_arg__min__list_Oinduct,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType] :
% 0.29/0.47 ( ( linorder @ B2 )
% 0.29/0.47 => ! [P: ( A > B2 ) > ( list @ A ) > $o,A0: A > B2,A1: list @ A] :
% 0.29/0.47 ( ! [F2: A > B2,X2: A] : ( P @ F2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [F2: A > B2,X2: A,Y2: A,Zs2: list @ A] :
% 0.29/0.47 ( ( P @ F2 @ ( cons @ A @ Y2 @ Zs2 ) )
% 0.29/0.47 => ( P @ F2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
% 0.29/0.47 => ( ! [A3: A > B2] : ( P @ A3 @ ( nil @ A ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % arg_min_list.induct
% 0.29/0.47 thf(fact_65_remdups__adj_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( ( X2 = Y2 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 0.29/0.47 => ( ( ( X2 != Y2 )
% 0.29/0.47 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 0.29/0.47 => ( P @ A0 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % remdups_adj.induct
% 0.29/0.47 thf(fact_66_sorted__wrt_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.29/0.47 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 0.29/0.47 => ( ! [P2: A > A > $o,X2: A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ P2 @ Ys3 )
% 0.29/0.47 => ( P @ P2 @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sorted_wrt.induct
% 0.29/0.47 thf(fact_67_remdups__adj_Ocases,axiom,
% 0.29/0.47 ! [A: $tType,X: list @ A] :
% 0.29/0.47 ( ( X
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % remdups_adj.cases
% 0.29/0.47 thf(fact_68_shuffles_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.29/0.47 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.29/0.47 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
% 0.29/0.47 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % shuffles.induct
% 0.29/0.47 thf(fact_69_min__list_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.29/0.47 ( ! [X2: A,Xs3: list @ A] :
% 0.29/0.47 ( ! [X212: A,X222: list @ A] :
% 0.29/0.47 ( ( Xs3
% 0.29/0.47 = ( cons @ A @ X212 @ X222 ) )
% 0.29/0.47 => ( P @ Xs3 ) )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 0.29/0.47 => ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( P @ A0 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_list.induct
% 0.29/0.47 thf(fact_70_min__list_Ocases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [X: list @ A] :
% 0.29/0.47 ( ! [X2: A,Xs3: list @ A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ Xs3 ) )
% 0.29/0.47 => ( X
% 0.29/0.47 = ( nil @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % min_list.cases
% 0.29/0.47 thf(fact_71_induct__list012,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [X2: A,Y2: A,Zs2: list @ A] :
% 0.29/0.47 ( ( P @ Zs2 )
% 0.29/0.47 => ( ( P @ ( cons @ A @ Y2 @ Zs2 ) )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
% 0.29/0.47 => ( P @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % induct_list012
% 0.29/0.47 thf(fact_72_splice_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.29/0.47 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ Ys3 @ Xs3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % splice.induct
% 0.29/0.47 thf(fact_73_list__induct2_H,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys: list @ B2] :
% 0.29/0.47 ( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B2 ) )
% 0.29/0.47 => ( ! [Y2: B2,Ys3: list @ B2] : ( P @ ( nil @ A ) @ ( cons @ B2 @ Y2 @ Ys3 ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
% 0.29/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) )
% 0.29/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_induct2'
% 0.29/0.47 thf(fact_74_neq__Nil__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 = ( ? [Y3: A,Ys4: list @ A] :
% 0.29/0.47 ( Xs
% 0.29/0.47 = ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq_Nil_conv
% 0.29/0.47 thf(fact_75_list_Oinducts,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X1: A,X23: list @ A] :
% 0.29/0.47 ( ( P @ X23 )
% 0.29/0.47 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 0.29/0.47 => ( P @ List ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.inducts
% 0.29/0.47 thf(fact_76_list_Oexhaust,axiom,
% 0.29/0.47 ! [A: $tType,Y: list @ A] :
% 0.29/0.47 ( ( Y
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ~ ! [X213: A,X223: list @ A] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( cons @ A @ X213 @ X223 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.exhaust
% 0.29/0.47 thf(fact_77_list_OdiscI,axiom,
% 0.29/0.47 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 0.29/0.47 ( ( List
% 0.29/0.47 = ( cons @ A @ X21 @ X22 ) )
% 0.29/0.47 => ( List
% 0.29/0.47 != ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.discI
% 0.29/0.47 thf(fact_78_list_Odistinct_I1_J,axiom,
% 0.29/0.47 ! [A: $tType,X21: A,X22: list @ A] :
% 0.29/0.47 ( ( nil @ A )
% 0.29/0.47 != ( cons @ A @ X21 @ X22 ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.distinct(1)
% 0.29/0.47 thf(fact_79_append_Oleft__neutral,axiom,
% 0.29/0.47 ! [A: $tType,A2: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( nil @ A ) @ A2 )
% 0.29/0.47 = A2 ) ).
% 0.29/0.47
% 0.29/0.47 % append.left_neutral
% 0.29/0.47 thf(fact_80_append__Nil,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( nil @ A ) @ Ys )
% 0.29/0.47 = Ys ) ).
% 0.29/0.47
% 0.29/0.47 % append_Nil
% 0.29/0.47 thf(fact_81_eq__Nil__appendI,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( Xs = Ys )
% 0.29/0.47 => ( Xs
% 0.29/0.47 = ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_Nil_appendI
% 0.29/0.47 thf(fact_82_psums_Ocases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ! [X: list @ A] :
% 0.29/0.47 ( ( X
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( X
% 0.29/0.47 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums.cases
% 0.29/0.47 thf(fact_83_rev__nonempty__induct,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( Xs3
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ( ( P @ Xs3 )
% 0.29/0.47 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 0.29/0.47 => ( P @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_nonempty_induct
% 0.29/0.47 thf(fact_84_append__eq__Cons__conv,axiom,
% 0.29/0.47 ! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Ys @ Zs )
% 0.29/0.47 = ( cons @ A @ X @ Xs ) )
% 0.29/0.47 = ( ( ( Ys
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 & ( Zs
% 0.29/0.47 = ( cons @ A @ X @ Xs ) ) )
% 0.29/0.47 | ? [Ys5: list @ A] :
% 0.29/0.47 ( ( Ys
% 0.29/0.47 = ( cons @ A @ X @ Ys5 ) )
% 0.29/0.47 & ( ( append @ A @ Ys5 @ Zs )
% 0.29/0.47 = Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_eq_Cons_conv
% 0.29/0.47 thf(fact_85_Cons__eq__append__conv,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 0.29/0.47 ( ( ( cons @ A @ X @ Xs )
% 0.29/0.47 = ( append @ A @ Ys @ Zs ) )
% 0.29/0.47 = ( ( ( Ys
% 0.29/0.47 = ( nil @ A ) )
% 0.29/0.47 & ( ( cons @ A @ X @ Xs )
% 0.29/0.47 = Zs ) )
% 0.29/0.47 | ? [Ys5: list @ A] :
% 0.29/0.47 ( ( ( cons @ A @ X @ Ys5 )
% 0.29/0.47 = Ys )
% 0.29/0.47 & ( Xs
% 0.29/0.47 = ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Cons_eq_append_conv
% 0.29/0.47 thf(fact_86_rev__exhaust,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( Xs
% 0.29/0.47 != ( nil @ A ) )
% 0.29/0.47 => ~ ! [Ys3: list @ A,Y2: A] :
% 0.29/0.47 ( Xs
% 0.29/0.47 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_exhaust
% 0.29/0.47 thf(fact_87_rev__induct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( P @ Xs3 )
% 0.29/0.47 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 0.29/0.47 => ( P @ Xs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_induct
% 0.29/0.47 thf(fact_88_psums_Osimps_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ( ( descar1668888542_psums @ A @ ( nil @ A ) )
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums.simps(1)
% 0.29/0.47 thf(fact_89_replicate__append__same,axiom,
% 0.29/0.47 ! [A: $tType,I: nat,X: A] :
% 0.29/0.47 ( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.29/0.47 = ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % replicate_append_same
% 0.29/0.47 thf(fact_90_psums_Osimps_I2_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ! [X: A] :
% 0.29/0.47 ( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.29/0.47 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums.simps(2)
% 0.29/0.47 thf(fact_91_zero__reorient,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( zero @ A )
% 0.29/0.47 => ! [X: A] :
% 0.29/0.47 ( ( ( zero_zero @ A )
% 0.29/0.47 = X )
% 0.29/0.47 = ( X
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_reorient
% 0.29/0.47 thf(fact_92_not__Cons__self2,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Xs: list @ A] :
% 0.29/0.47 ( ( cons @ A @ X @ Xs )
% 0.29/0.47 != Xs ) ).
% 0.29/0.47
% 0.29/0.47 % not_Cons_self2
% 0.29/0.47 thf(fact_93_append__eq__append__conv2,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = ( append @ A @ Zs @ Ts ) )
% 0.29/0.47 = ( ? [Us: list @ A] :
% 0.29/0.47 ( ( ( Xs
% 0.29/0.47 = ( append @ A @ Zs @ Us ) )
% 0.29/0.47 & ( ( append @ A @ Us @ Ys )
% 0.29/0.47 = Ts ) )
% 0.29/0.47 | ( ( ( append @ A @ Xs @ Us )
% 0.29/0.47 = Zs )
% 0.29/0.47 & ( Ys
% 0.29/0.47 = ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_eq_append_conv2
% 0.29/0.47 thf(fact_94_append__eq__appendI,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
% 0.29/0.47 ( ( ( append @ A @ Xs @ Xs1 )
% 0.29/0.47 = Zs )
% 0.29/0.47 => ( ( Ys
% 0.29/0.47 = ( append @ A @ Xs1 @ Us2 ) )
% 0.29/0.47 => ( ( append @ A @ Xs @ Ys )
% 0.29/0.47 = ( append @ A @ Zs @ Us2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_eq_appendI
% 0.29/0.47 thf(fact_95_append__Cons,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
% 0.29/0.47 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_Cons
% 0.29/0.47 thf(fact_96_v__decompose,axiom,
% 0.29/0.47 ! [Xs5: list @ a] :
% 0.29/0.47 ( ( v @ ( append @ a @ Xs5 @ xsa ) )
% 0.29/0.47 = ( plus_plus @ int @ ( v @ ( append @ a @ Xs5 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ ( nil @ a ) ) ) ) @ ( v @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % v_decompose
% 0.29/0.47 thf(fact_97_p__less__length,axiom,
% 0.29/0.47 ord_less @ nat @ p @ ( size_size @ ( list @ a ) @ xsa ) ).
% 0.29/0.47
% 0.29/0.47 % p_less_length
% 0.29/0.47 thf(fact_98_neq0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq0_conv
% 0.29/0.47 thf(fact_99_less__nat__zero__code,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_nat_zero_code
% 0.29/0.47 thf(fact_100_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 != ( zero_zero @ nat ) )
% 0.29/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.not_eq_extremum
% 0.29/0.47 thf(fact_101_order__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_refl
% 0.29/0.47 thf(fact_102_minus__poly__rev__list_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A )
% 0.29/0.47 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.29/0.47 ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.29/0.47 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 0.29/0.47 => ( ! [Y2: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys3 ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % minus_poly_rev_list.induct
% 0.29/0.47 thf(fact_103_plus__coeffs_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_monoid_add @ A )
% 0.29/0.47 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.29/0.47 ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 0.29/0.47 => ( ! [V: A,Va: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 0.29/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.29/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % plus_coeffs.induct
% 0.29/0.47 thf(fact_104_minf_I8_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Z )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ T @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % minf(8)
% 0.29/0.47 thf(fact_105_minf_I6_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [T: A] :
% 0.29/0.47 ? [Z: A] :
% 0.29/0.47 ! [X3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X3 @ Z )
% 0.29/0.47 => ( ord_less_eq @ A @ X3 @ T ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % minf(6)
% 0.29/0.47 thf(fact_106_add__left__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_cancel
% 0.29/0.47 thf(fact_107_add__right__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_cancel
% 0.29/0.47 thf(fact_108_le0,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.47
% 0.29/0.47 % le0
% 0.29/0.47 thf(fact_109_bot__nat__0_Oextremum,axiom,
% 0.29/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum
% 0.29/0.47 thf(fact_110_add_Oleft__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_neutral
% 0.29/0.47 thf(fact_111_add_Oright__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.right_neutral
% 0.29/0.47 thf(fact_112_double__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_zero
% 0.29/0.47 thf(fact_113_double__zero__sym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( zero_zero @ A )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_zero_sym
% 0.29/0.47 thf(fact_114_add__cancel__left__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = A2 )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_left_left
% 0.29/0.47 thf(fact_115_add__cancel__left__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = A2 )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_left_right
% 0.29/0.47 thf(fact_116_add__cancel__right__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_right_left
% 0.29/0.47 thf(fact_117_add__cancel__right__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_right_right
% 0.29/0.47 thf(fact_118_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( ( X
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 & ( Y
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_eq_0_iff_both_eq_0
% 0.29/0.47 thf(fact_119_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ( zero_zero @ A )
% 0.29/0.47 = ( plus_plus @ A @ X @ Y ) )
% 0.29/0.47 = ( ( X
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 & ( Y
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_eq_add_iff_both_eq_0
% 0.29/0.47 thf(fact_120_add__le__cancel__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_cancel_right
% 0.29/0.47 thf(fact_121_add__le__cancel__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_cancel_left
% 0.29/0.47 thf(fact_122_add__less__cancel__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_cancel_right
% 0.29/0.47 thf(fact_123_add__less__cancel__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_cancel_left
% 0.29/0.47 thf(fact_124_append__eq__append__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
% 0.29/0.47 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 | ( ( size_size @ ( list @ A ) @ Us2 )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 0.29/0.47 => ( ( ( append @ A @ Xs @ Us2 )
% 0.29/0.47 = ( append @ A @ Ys @ Vs ) )
% 0.29/0.47 = ( ( Xs = Ys )
% 0.29/0.47 & ( Us2 = Vs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % append_eq_append_conv
% 0.29/0.47 thf(fact_125_take__all__iff,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.29/0.47 ( ( ( take @ A @ N @ Xs )
% 0.29/0.47 = Xs )
% 0.29/0.47 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_all_iff
% 0.29/0.47 thf(fact_126_take__all,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 0.29/0.47 => ( ( take @ A @ N @ Xs )
% 0.29/0.47 = Xs ) ) ).
% 0.29/0.47
% 0.29/0.47 % take_all
% 0.29/0.47 thf(fact_127_length__replicate,axiom,
% 0.29/0.47 ! [A: $tType,N: nat,X: A] :
% 0.29/0.47 ( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 % length_replicate
% 0.29/0.47 thf(fact_128_length__psums,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ! [Xs: list @ A] :
% 0.29/0.47 ( ( size_size @ ( list @ A ) @ ( descar1668888542_psums @ A @ Xs ) )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_psums
% 0.29/0.47 thf(fact_129_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.47 thf(fact_130_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.47 thf(fact_131_le__add__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_add_same_cancel2
% 0.29/0.47 thf(fact_132_le__add__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_add_same_cancel1
% 0.29/0.47 thf(fact_133_add__le__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_same_cancel2
% 0.29/0.47 thf(fact_134_add__le__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_same_cancel1
% 0.29/0.47 thf(fact_135_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.47 thf(fact_136_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.47 thf(fact_137_less__add__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_add_same_cancel2
% 0.29/0.47 thf(fact_138_less__add__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_add_same_cancel1
% 0.29/0.47 thf(fact_139_add__less__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_same_cancel2
% 0.29/0.47 thf(fact_140_add__less__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_same_cancel1
% 0.29/0.47 thf(fact_141_length__0__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( zero_zero @ nat ) )
% 0.29/0.47 = ( Xs
% 0.29/0.47 = ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_0_conv
% 0.29/0.47 thf(fact_142_length__greater__0__conv,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A] :
% 0.29/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.29/0.47 = ( Xs
% 0.29/0.47 != ( nil @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_greater_0_conv
% 0.29/0.47 thf(fact_143_nth__append__length,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
% 0.29/0.47 ( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 % nth_append_length
% 0.29/0.47 thf(fact_144_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.47
% 0.29/0.47 % less_eq_nat.simps(1)
% 0.29/0.47 thf(fact_145_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.47 thf(fact_146_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( I = J )
% 0.29/0.47 & ( K = L ) )
% 0.29/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.29/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(4)
% 0.29/0.47 thf(fact_147_le__0__eq,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_0_eq
% 0.29/0.47 thf(fact_148_impossible__Cons,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 => ( Xs
% 0.29/0.47 != ( cons @ A @ X @ Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % impossible_Cons
% 0.29/0.47 thf(fact_149_Ex__list__of__length,axiom,
% 0.29/0.47 ! [A: $tType,N: nat] :
% 0.29/0.47 ? [Xs3: list @ A] :
% 0.29/0.47 ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 % Ex_list_of_length
% 0.29/0.47 thf(fact_150_neq__if__length__neq,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 => ( Xs != Ys ) ) ).
% 0.29/0.47
% 0.29/0.47 % neq_if_length_neq
% 0.29/0.47 thf(fact_151_group__cancel_Oadd1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_monoid_add @ A )
% 0.29/0.47 => ! [A4: A,K: A,A2: A,B: A] :
% 0.29/0.47 ( ( A4
% 0.29/0.47 = ( plus_plus @ A @ K @ A2 ) )
% 0.29/0.47 => ( ( plus_plus @ A @ A4 @ B )
% 0.29/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % group_cancel.add1
% 0.29/0.47 thf(fact_152_group__cancel_Oadd2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_monoid_add @ A )
% 0.29/0.47 => ! [B3: A,K: A,B: A,A2: A] :
% 0.29/0.47 ( ( B3
% 0.29/0.47 = ( plus_plus @ A @ K @ B ) )
% 0.29/0.47 => ( ( plus_plus @ A @ A2 @ B3 )
% 0.29/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % group_cancel.add2
% 0.29/0.47 thf(fact_153_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_unique
% 0.29/0.47 thf(fact_154_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.47 ! [A2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.47 => ( A2
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_nat_0.extremum_uniqueI
% 0.29/0.47 thf(fact_155_add_Oassoc,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( semigroup_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.assoc
% 0.29/0.47 thf(fact_156_add_Oleft__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_cancel
% 0.29/0.47 thf(fact_157_add_Oright__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.right_cancel
% 0.29/0.47 thf(fact_158_add_Ocommute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A )
% 0.29/0.47 => ( ( plus_plus @ A )
% 0.29/0.47 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.commute
% 0.29/0.47 thf(fact_159_add_Oleft__commute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_commute
% 0.29/0.47 thf(fact_160_add__left__imp__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 => ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_imp_eq
% 0.29/0.47 thf(fact_161_add__right__imp__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 => ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_imp_eq
% 0.29/0.47 thf(fact_162_size__neq__size__imp__neq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( size @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ( size_size @ A @ X )
% 0.29/0.47 != ( size_size @ A @ Y ) )
% 0.29/0.47 => ( X != Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % size_neq_size_imp_neq
% 0.29/0.47 thf(fact_163_less__mono__imp__le__mono,axiom,
% 0.29/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.47 ( ! [I2: nat,J2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/0.47 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.47 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_mono_imp_le_mono
% 0.29/0.47 thf(fact_164_le__neq__implies__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.47 => ( ( M != N )
% 0.29/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_neq_implies_less
% 0.29/0.47 thf(fact_165_less__or__eq__imp__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 | ( M = N ) )
% 0.29/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_or_eq_imp_le
% 0.29/0.47 thf(fact_166_le__eq__less__or__eq,axiom,
% 0.29/0.47 ( ( ord_less_eq @ nat )
% 0.29/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.47 | ( M2 = N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_eq_less_or_eq
% 0.29/0.47 thf(fact_167_less__imp__le__nat,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ M @ N )
% 0.29/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_le_nat
% 0.29/0.47 thf(fact_168_ex__least__nat__le,axiom,
% 0.29/0.47 ! [P: nat > $o,N: nat] :
% 0.29/0.47 ( ( P @ N )
% 0.29/0.47 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.47 => ? [K2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.29/0.47 & ! [I3: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I3 @ K2 )
% 0.29/0.47 => ~ ( P @ I3 ) )
% 0.29/0.47 & ( P @ K2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_least_nat_le
% 0.29/0.47 thf(fact_169_nat__less__le,axiom,
% 0.29/0.47 ( ( ord_less @ nat )
% 0.29/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.47 & ( M2 != N2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_less_le
% 0.29/0.47 thf(fact_170_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_monoid_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % comm_monoid_add_class.add_0
% 0.29/0.47 thf(fact_171_add_Ocomm__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( comm_monoid_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.comm_neutral
% 0.29/0.47 thf(fact_172_add_Ogroup__left__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.group_left_neutral
% 0.29/0.47 thf(fact_173_add__le__imp__le__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_imp_le_right
% 0.29/0.47 thf(fact_174_add__le__imp__le__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_imp_le_left
% 0.29/0.47 thf(fact_175_le__iff__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [A5: A,B4: A] :
% 0.29/0.47 ? [C2: A] :
% 0.29/0.47 ( B4
% 0.29/0.47 = ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_iff_add
% 0.29/0.47 thf(fact_176_add__right__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_mono
% 0.29/0.47 thf(fact_177_less__eqE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ~ ! [C3: A] :
% 0.29/0.47 ( B
% 0.29/0.47 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_eqE
% 0.29/0.47 thf(fact_178_add__left__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_mono
% 0.29/0.47 thf(fact_179_add__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono
% 0.29/0.47 thf(fact_180_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(1)
% 0.29/0.47 thf(fact_181_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( I = J )
% 0.29/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(2)
% 0.29/0.47 thf(fact_182_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.47 & ( K = L ) )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(3)
% 0.29/0.47 thf(fact_183_add__less__imp__less__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_imp_less_right
% 0.29/0.47 thf(fact_184_add__less__imp__less__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_imp_less_left
% 0.29/0.47 thf(fact_185_add__strict__right__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_strict_right_mono
% 0.29/0.47 thf(fact_186_add__strict__left__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_strict_left_mono
% 0.29/0.47 thf(fact_187_add__strict__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( strict2144017051up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ A @ C @ D )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_strict_mono
% 0.29/0.47 thf(fact_188_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.47 & ( K = L ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_field(1)
% 0.29/0.47 thf(fact_189_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( I = J )
% 0.29/0.47 & ( ord_less @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_field(2)
% 0.29/0.47 thf(fact_190_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.47 & ( ord_less @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_field(5)
% 0.29/0.47 thf(fact_191_list_Osize_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 0.29/0.47 = ( zero_zero @ nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % list.size(3)
% 0.29/0.47 thf(fact_192_length__induct,axiom,
% 0.29/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.29/0.47 ( ! [Xs3: list @ A] :
% 0.29/0.47 ( ! [Ys6: list @ A] :
% 0.29/0.47 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.29/0.47 => ( P @ Ys6 ) )
% 0.29/0.47 => ( P @ Xs3 ) )
% 0.29/0.47 => ( P @ Xs ) ) ).
% 0.29/0.47
% 0.29/0.47 % length_induct
% 0.29/0.47 thf(fact_193_nth__take__lemma,axiom,
% 0.29/0.47 ! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.29/0.47 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 => ( ! [I2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.29/0.47 => ( ( nth @ A @ Xs @ I2 )
% 0.29/0.47 = ( nth @ A @ Ys @ I2 ) ) )
% 0.29/0.47 => ( ( take @ A @ K @ Xs )
% 0.29/0.47 = ( take @ A @ K @ Ys ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_take_lemma
% 0.29/0.47 thf(fact_194_add__nonpos__eq__0__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( ( X
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 & ( Y
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonpos_eq_0_iff
% 0.29/0.47 thf(fact_195_add__nonneg__eq__0__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.47 => ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( ( X
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 & ( Y
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonneg_eq_0_iff
% 0.29/0.47 thf(fact_196_add__nonpos__nonpos,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonpos_nonpos
% 0.29/0.47 thf(fact_197_add__nonneg__nonneg,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonneg_nonneg
% 0.29/0.47 thf(fact_198_add__increasing2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [C: A,B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_increasing2
% 0.29/0.47 thf(fact_199_add__decreasing2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_decreasing2
% 0.29/0.47 thf(fact_200_add__increasing,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_increasing
% 0.29/0.47 thf(fact_201_add__decreasing,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_decreasing
% 0.29/0.47 thf(fact_202_pos__add__strict,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( strict797366125id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pos_add_strict
% 0.29/0.47 thf(fact_203_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ~ ! [C3: A] :
% 0.29/0.47 ( ( B
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C3 ) )
% 0.29/0.47 => ( C3
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % canonically_ordered_monoid_add_class.lessE
% 0.29/0.47 thf(fact_204_add__pos__pos,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_pos_pos
% 0.29/0.47 thf(fact_205_add__neg__neg,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_neg_neg
% 0.29/0.47 thf(fact_206_add__less__le__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_le_mono
% 0.29/0.47 thf(fact_207_add__le__less__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less @ A @ C @ D )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_less_mono
% 0.29/0.47 thf(fact_208_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_field(3)
% 0.29/0.47 thf(fact_209_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere223160158up_add @ A )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.47 & ( ord_less @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_field(4)
% 0.29/0.47 thf(fact_210_list__induct4,axiom,
% 0.29/0.47 ! [C4: $tType,A: $tType,B2: $tType,D2: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,Ws: list @ D2,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > ( list @ D2 ) > $o] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ B2 ) @ Ys )
% 0.29/0.47 = ( size_size @ ( list @ C4 ) @ Zs ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ C4 ) @ Zs )
% 0.29/0.47 = ( size_size @ ( list @ D2 ) @ Ws ) )
% 0.29/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) @ ( nil @ D2 ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4,W: D2,Ws2: list @ D2] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
% 0.29/0.47 = ( size_size @ ( list @ C4 ) @ Zs2 ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ C4 ) @ Zs2 )
% 0.29/0.47 = ( size_size @ ( list @ D2 ) @ Ws2 ) )
% 0.29/0.47 => ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) @ ( cons @ D2 @ W @ Ws2 ) ) ) ) ) )
% 0.29/0.47 => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_induct4
% 0.29/0.47 thf(fact_211_list__induct3,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType,C4: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > $o] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ B2 ) @ Ys )
% 0.29/0.47 = ( size_size @ ( list @ C4 ) @ Zs ) )
% 0.29/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 0.29/0.47 => ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
% 0.29/0.47 = ( size_size @ ( list @ C4 ) @ Zs2 ) )
% 0.29/0.47 => ( ( P @ Xs3 @ Ys3 @ Zs2 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) ) ) ) )
% 0.29/0.47 => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_induct3
% 0.29/0.47 thf(fact_212_list__induct2,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType,Xs: list @ A,Ys: list @ B2,P: ( list @ A ) > ( list @ B2 ) > $o] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 0.29/0.47 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
% 0.29/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.29/0.47 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 0.29/0.47 => ( ( P @ Xs3 @ Ys3 )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) ) )
% 0.29/0.47 => ( P @ Xs @ Ys ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_induct2
% 0.29/0.47 thf(fact_213_psums_Oinduct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 0.29/0.47 => ( P @ A0 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums.induct
% 0.29/0.47 thf(fact_214_list__eq__iff__nth__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ^ [Y4: list @ A,Z2: list @ A] : ( Y4 = Z2 ) )
% 0.29/0.47 = ( ^ [Xs2: list @ A,Ys4: list @ A] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Ys4 ) )
% 0.29/0.47 & ! [I4: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.29/0.47 => ( ( nth @ A @ Xs2 @ I4 )
% 0.29/0.47 = ( nth @ A @ Ys4 @ I4 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % list_eq_iff_nth_eq
% 0.29/0.47 thf(fact_215_Skolem__list__nth,axiom,
% 0.29/0.47 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.29/0.47 ( ( ! [I4: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I4 @ K )
% 0.29/0.47 => ? [X4: A] : ( P @ I4 @ X4 ) ) )
% 0.29/0.47 = ( ? [Xs2: list @ A] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.29/0.47 = K )
% 0.29/0.47 & ! [I4: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I4 @ K )
% 0.29/0.47 => ( P @ I4 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Skolem_list_nth
% 0.29/0.47 thf(fact_216_nth__equalityI,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 => ( ! [I2: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.29/0.47 => ( ( nth @ A @ Xs @ I2 )
% 0.29/0.47 = ( nth @ A @ Ys @ I2 ) ) )
% 0.29/0.47 => ( Xs = Ys ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nth_equalityI
% 0.29/0.47 thf(fact_217_psums_Osimps_I3_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( plus @ A )
% 0.29/0.47 => ! [X: A,Y: A,Xs: list @ A] :
% 0.29/0.47 ( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
% 0.29/0.47 = ( cons @ A @ X @ ( descar1668888542_psums @ A @ ( cons @ A @ ( plus_plus @ A @ X @ Y ) @ Xs ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % psums.simps(3)
% 0.29/0.47 thf(fact_218_add__strict__increasing2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere1818651114id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_strict_increasing2
% 0.29/0.47 thf(fact_219_add__strict__increasing,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere1818651114id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_strict_increasing
% 0.29/0.47 thf(fact_220_add__pos__nonneg,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_pos_nonneg
% 0.29/0.47 thf(fact_221_add__nonpos__neg,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonpos_neg
% 0.29/0.47 thf(fact_222_add__nonneg__pos,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_nonneg_pos
% 0.29/0.47 thf(fact_223_add__neg__nonpos,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere216010020id_add @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 0.29/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_neg_nonpos
% 0.29/0.47 thf(fact_224_sign__changes__induct,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ( plus @ A )
% 0.29/0.47 & ( zero @ A ) )
% 0.29/0.47 => ! [P: ( list @ A ) > $o,Xs: list @ A] :
% 0.29/0.47 ( ( P @ ( nil @ A ) )
% 0.29/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.29/0.47 => ( ! [Xs3: list @ A] :
% 0.29/0.47 ( ( P @ Xs3 )
% 0.29/0.47 => ( P @ ( cons @ A @ ( zero_zero @ A ) @ Xs3 ) ) )
% 0.29/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.29/0.47 ( ( X2
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
% 0.29/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 0.29/0.47 => ( P @ Xs ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % sign_changes_induct
% 0.29/0.47 thf(fact_225_same__length__different,axiom,
% 0.29/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.29/0.47 ( ( Xs != Ys )
% 0.29/0.47 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.29/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.29/0.47 => ? [Pre: list @ A,X2: A,Xs4: list @ A,Y2: A,Ys2: list @ A] :
% 0.29/0.47 ( ( X2 != Y2 )
% 0.29/0.47 & ( Xs
% 0.29/0.47 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Xs4 ) ) )
% 0.29/0.47 & ( Ys
% 0.29/0.47 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % same_length_different
% 0.29/0.47 thf(fact_226_dual__order_Oantisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( A2 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.antisym
% 0.29/0.47 thf(fact_227_dual__order_Oeq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.47 = ( ^ [A5: A,B4: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.29/0.47 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.eq_iff
% 0.29/0.47 thf(fact_228_dual__order_Otrans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.trans
% 0.29/0.47 thf(fact_229_linorder__wlog,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P: A > A > $o,A2: A,B: A] :
% 0.29/0.47 ( ! [A3: A,B5: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A3 @ B5 )
% 0.29/0.47 => ( P @ A3 @ B5 ) )
% 0.29/0.47 => ( ! [A3: A,B5: A] :
% 0.29/0.47 ( ( P @ B5 @ A3 )
% 0.29/0.47 => ( P @ A3 @ B5 ) )
% 0.29/0.47 => ( P @ A2 @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_wlog
% 0.29/0.47 thf(fact_230_dual__order_Orefl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.refl
% 0.29/0.47 thf(fact_231_order__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A,Y: A,Z3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.29/0.47 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_trans
% 0.29/0.47 thf(fact_232_order__class_Oorder_Oantisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.47 => ( A2 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_class.order.antisym
% 0.29/0.47 thf(fact_233_ord__le__eq__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( B = C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_le_eq_trans
% 0.29/0.47 thf(fact_234_ord__eq__le__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( A2 = B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_le_trans
% 0.29/0.47 thf(fact_235_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.47 = ( ^ [A5: A,B4: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.29/0.47 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_class.order.eq_iff
% 0.29/0.47 thf(fact_236_antisym__conv,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [Y: A,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 = ( X = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv
% 0.29/0.47 thf(fact_237_le__cases3,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y: A,Z3: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.29/0.47 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.29/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_cases3
% 0.29/0.47 thf(fact_238_order_Otrans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ B @ C )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.trans
% 0.29/0.47 thf(fact_239_le__cases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_cases
% 0.29/0.47 thf(fact_240_eq__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( X = Y )
% 0.29/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_refl
% 0.29/0.47 thf(fact_241_linear,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linear
% 0.29/0.47 thf(fact_242_antisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X: A,Y: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.47 => ( X = Y ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym
% 0.29/0.47 thf(fact_243_eq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.29/0.47 = ( ^ [X5: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 0.29/0.47 & ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_iff
% 0.29/0.47 thf(fact_244_ord__le__eq__subst,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType] :
% 0.29/0.47 ( ( ( ord @ B2 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ( F @ B )
% 0.29/0.47 = C )
% 0.29/0.47 => ( ! [X2: A,Y2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_le_eq_subst
% 0.29/0.47 thf(fact_245_ord__eq__le__subst,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType] :
% 0.29/0.47 ( ( ( ord @ B2 )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( F @ B ) )
% 0.29/0.47 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.29/0.47 => ( ! [X2: B2,Y2: B2] :
% 0.29/0.47 ( ( ord_less_eq @ B2 @ X2 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_le_subst
% 0.29/0.47 thf(fact_246_order__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C4: $tType] :
% 0.29/0.47 ( ( ( order @ C4 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,F: A > C4,C: C4] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
% 0.29/0.47 => ( ! [X2: A,Y2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ C4 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst2
% 0.29/0.47 thf(fact_247_order__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B2: $tType] :
% 0.29/0.47 ( ( ( order @ B2 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.29/0.47 => ( ( ord_less_eq @ B2 @ B @ C )
% 0.29/0.47 => ( ! [X2: B2,Y2: B2] :
% 0.29/0.47 ( ( ord_less_eq @ B2 @ X2 @ Y2 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst1
% 0.29/0.47 thf(fact_248_le__fun__def,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B2 )
% 0.29/0.47 => ( ( ord_less_eq @ ( A > B2 ) )
% 0.29/0.47 = ( ^ [F3: A > B2,G2: A > B2] :
% 0.29/0.47 ! [X5: A] : ( ord_less_eq @ B2 @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_fun_def
% 0.29/0.47 thf(fact_249_le__funI,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B2 )
% 0.29/0.47 => ! [F: A > B2,G: A > B2] :
% 0.29/0.47 ( ! [X2: A] : ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.29/0.47 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funI
% 0.29/0.47 thf(fact_250_le__funE,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B2 )
% 0.29/0.47 => ! [F: A > B2,G: A > B2,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funE
% 0.29/0.47 thf(fact_251_le__funD,axiom,
% 0.29/0.47 ! [B2: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B2 )
% 0.29/0.47 => ! [F: A > B2,G: A > B2,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funD
% 0.29/0.47 thf(fact_252_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.47 => ( A2 != B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_implies_not_eq
% 0.29/0.47
% 0.29/0.47 % Subclasses (23)
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( type @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oplus,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( plus @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( zero @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ord @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( order @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( group_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( monoid_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( linorder @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( preorder @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( comm_monoid_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ab_semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( cancel_semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( cancel1352612707id_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Olinordered__ab__group__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( linord219039673up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__comm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere216010020id_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere779506340up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere1818651114id_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( strict797366125id_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere236663937imp_le @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere223160158up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( strict2144017051up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere516151231imp_le @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % Type constructors (52)
% 0.29/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.47 ! [A6: $tType,A7: $tType] :
% 0.29/0.47 ( ( preorder @ A7 )
% 0.29/0.47 => ( preorder @ ( A6 > A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.47 ! [A6: $tType,A7: $tType] :
% 0.29/0.47 ( ( order @ A7 )
% 0.29/0.47 => ( order @ ( A6 > A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.47 ! [A6: $tType,A7: $tType] :
% 0.29/0.47 ( ( ord @ A7 )
% 0.29/0.47 => ( ord @ ( A6 > A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.47 ordere516151231imp_le @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.29/0.47 strict2144017051up_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.47 ordere223160158up_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.47 ordere236663937imp_le @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.29/0.47 strict797366125id_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.29/0.47 ordere1818651114id_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.47 ordere779506340up_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
% 0.29/0.47 ordere216010020id_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.29/0.47 linord219039673up_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.47 cancel1352612707id_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.47 cancel_semigroup_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.29/0.47 ab_semigroup_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.47 comm_monoid_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.29/0.47 semigroup_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.29/0.47 preorder @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.29/0.47 linorder @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 0.29/0.47 monoid_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.29/0.47 group_add @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.29/0.47 order @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.29/0.47 ord @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.29/0.47 zero @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oplus,axiom,
% 0.29/0.47 plus @ int ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
% 0.29/0.47 ordere516151231imp_le @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_5,axiom,
% 0.29/0.47 strict2144017051up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_6,axiom,
% 0.29/0.47 ordere223160158up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_7,axiom,
% 0.29/0.47 ordere236663937imp_le @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_8,axiom,
% 0.29/0.47 strict797366125id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_9,axiom,
% 0.29/0.47 ordere1818651114id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.47 canoni770627133id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_10,axiom,
% 0.29/0.47 ordere779506340up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_11,axiom,
% 0.29/0.47 ordere216010020id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_12,axiom,
% 0.29/0.47 cancel1352612707id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_13,axiom,
% 0.29/0.47 cancel_semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_14,axiom,
% 0.29/0.47 ab_semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_15,axiom,
% 0.29/0.47 comm_monoid_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add_16,axiom,
% 0.29/0.47 semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
% 0.29/0.47 preorder @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder_18,axiom,
% 0.29/0.47 linorder @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__add_19,axiom,
% 0.29/0.47 monoid_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_20,axiom,
% 0.29/0.47 order @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Oord_21,axiom,
% 0.29/0.47 ord @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ozero_22,axiom,
% 0.29/0.47 zero @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oplus_23,axiom,
% 0.29/0.47 plus @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.29/0.47 size @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_24,axiom,
% 0.29/0.47 preorder @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_25,axiom,
% 0.29/0.47 linorder @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_26,axiom,
% 0.29/0.47 order @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 0.29/0.47 ord @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_List_Olist___Nat_Osize_28,axiom,
% 0.29/0.47 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Free types (1)
% 0.29/0.47 thf(tfree_0,hypothesis,
% 0.29/0.47 linordered_idom @ a ).
% 0.29/0.47
% 0.29/0.47 % Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ( ( descar1668888542_psums @ a @ xsa )
% 0.29/0.47 = ( append @ a @ ( replicate @ a @ p @ ( zero_zero @ a ) ) @ ( descar1668888542_psums @ a @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.89/1.52 ---- Embedded in THF ---
% 2.89/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.89/1.52 %%% Generated on Tue Feb 24 08:24:25 EST 2026
% 2.89/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.89/1.52 %%% Logic specification used:
% 2.89/1.52 %%% thf(spec, logic, $$dhol).
% 2.89/1.52
% 2.89/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2DTF10522.p
% 2.89/1.52 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.89/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.89/1.52 thf(ty_t_Int_Oint, type, int: $tType).
% 2.89/1.52 thf(ty_tf_a, type, a: $tType).
% 2.89/1.52 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere1818651114id_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict797366125id_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: ($o))).
% 2.89/1.52 thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Opsums, type, descar1668888542_psums: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.89/1.52 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.89/1.52 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.89/1.52 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.89/1.52 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 2.89/1.52 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.89/1.52 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.89/1.52 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 2.89/1.52 thf(sy_c_List_Oreplicate, type, replicate: (!> [A:$tType]: ((nat > (A > (list @ A)))))).
% 2.89/1.52 thf(sy_c_List_Otake, type, take: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 2.89/1.52 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.89/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.89/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.89/1.52 thf(sy_v_p____, type, p: nat).
% 2.89/1.52 thf(sy_v_v, type, v: ((list @ a) > int)).
% 2.89/1.52 thf(sy_v_xa____, type, xa: a).
% 2.89/1.52 thf(sy_v_xs, type, xs: (list @ a)).
% 2.89/1.52 thf(sy_v_xs1____, type, xs1: (list @ a)).
% 2.89/1.52 thf(sy_v_xs2____, type, xs2: (list @ a)).
% 2.89/1.52 thf(sy_v_xsa____, type, xsa: (list @ a)).
% 2.89/1.52 thf(fact_0_nonneg_Ohyps_I1_J, axiom, (xsa != (nil @ a))).
% 2.89/1.52 thf(fact_1_p__nz, axiom, ((((nth @ a) @ xsa) @ p) != (zero_zero @ a))).
% 2.89/1.52 thf(fact_2_nonneg_Ohyps_I2_J, axiom, (((last @ a) @ xsa) != (zero_zero @ a))).
% 2.89/1.52 thf(fact_3_psums__0__Cons, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ A)]: ((((descar1668888542_psums @ A) @ (((cons @ A) @ (zero_zero @ A)) @ Xs)) = (((cons @ A) @ (zero_zero @ A)) @ ((descar1668888542_psums @ A) @ Xs))))))))).
% 2.89/1.52 thf(fact_4_xs1__def, axiom, (xs1 = (((replicate @ a) @ p) @ (zero_zero @ a)))).
% 2.89/1.52 thf(fact_5_xs__decompose, axiom, (xsa = (((append @ a) @ xs1) @ (((cons @ a) @ (((nth @ a) @ xsa) @ p)) @ xs2)))).
% 2.89/1.52 thf(fact_6_psums__replicate__0__append, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [N:nat,Xs:(list @ A)]: ((((descar1668888542_psums @ A) @ (((append @ A) @ (((replicate @ A) @ N) @ (zero_zero @ A))) @ Xs)) = (((append @ A) @ (((replicate @ A) @ N) @ (zero_zero @ A))) @ ((descar1668888542_psums @ A) @ Xs))))))))).
% 2.89/1.52 thf(fact_7_less__p__zero, axiom, (! [Q:nat]: (((((ord_less @ nat) @ Q) @ p) => ((((nth @ a) @ xsa) @ Q) = (zero_zero @ a)))))).
% 2.89/1.52 thf(fact_8__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062, axiom, ((((take @ a) @ p) @ xsa) = (((replicate @ a) @ p) @ (zero_zero @ a)))).
% 2.89/1.52 thf(fact_9_replicate__app__Cons__same, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: (((((append @ A) @ (((replicate @ A) @ N) @ X)) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ (((append @ A) @ (((replicate @ A) @ N) @ X)) @ Xs)))))).
% 2.89/1.52 thf(fact_10_append_Oassoc, axiom, (! [A:$tType,A2:(list @ A),B:(list @ A),C:(list @ A)]: (((((append @ A) @ (((append @ A) @ A2) @ B)) @ C) = (((append @ A) @ A2) @ (((append @ A) @ B) @ C)))))).
% 2.89/1.52 thf(fact_11_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys)) @ Zs) = (((append @ A) @ Xs) @ (((append @ A) @ Ys) @ Zs)))))).
% 2.89/1.52 thf(fact_12_append__same__eq, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Ys) @ Xs) = (((append @ A) @ Zs) @ Xs)) = (Ys = Zs))))).
% 2.89/1.52 thf(fact_13_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Xs) @ Zs)) = (Ys = Zs))))).
% 2.89/1.52 thf(fact_14_nonneg_Ohyps_I4_J, axiom, (((ord_less_eq @ a) @ (zero_zero @ a)) @ xa)).
% 2.89/1.52 thf(fact_15_list_Oinject, axiom, (! [A:$tType,X21:A,X22:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X22) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.89/1.52 thf(fact_16_append__replicate__commute, axiom, (! [A:$tType,N:nat,X:A,K:nat]: (((((append @ A) @ (((replicate @ A) @ N) @ X)) @ (((replicate @ A) @ K) @ X)) = (((append @ A) @ (((replicate @ A) @ K) @ X)) @ (((replicate @ A) @ N) @ X)))))).
% 2.89/1.52 thf(fact_17_Cons__eq__appendI, axiom, (! [A:$tType,X:A,Xs1:(list @ A),Ys:(list @ A),Xs:(list @ A),Zs:(list @ A)]: ((((((cons @ A) @ X) @ Xs1) = Ys) => ((Xs = (((append @ A) @ Xs1) @ Zs)) => ((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys) @ Zs))))))).
% 2.89/1.52 thf(fact_18_assms_I1_J, axiom, (xs != (nil @ a))).
% 2.89/1.52 thf(fact_19_assms_I2_J, axiom, (((last @ a) @ xs) != (zero_zero @ a))).
% 2.89/1.52 thf(fact_20_replicate__eq__replicate, axiom, (! [A:$tType,M:nat,X:A,N:nat,Y:A]: ((((((replicate @ A) @ M) @ X) = (((replicate @ A) @ N) @ Y)) = ((M = N) & ((M != (zero_zero @ nat)) => (X = Y))))))).
% 2.89/1.52 thf(fact_21_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_22_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_23_append_Oright__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ A2) @ (nil @ A)) = A2)))).
% 2.89/1.52 thf(fact_24_append__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))).
% 2.89/1.52 thf(fact_25_Nil__is__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((nil @ A) = (((append @ A) @ Xs) @ Ys)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))).
% 2.89/1.52 thf(fact_26_self__append__conv2, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys = (((append @ A) @ Xs) @ Ys)) = (Xs = (nil @ A)))))).
% 2.89/1.52 thf(fact_27_append__self__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Ys) = (Xs = (nil @ A)))))).
% 2.89/1.52 thf(fact_28_self__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = (((append @ A) @ Xs) @ Ys)) = (Ys = (nil @ A)))))).
% 2.89/1.52 thf(fact_29_append__self__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Xs) = (Ys = (nil @ A)))))).
% 2.89/1.52 thf(fact_30_append__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((append @ A) @ Xs) @ (nil @ A)) = Xs)))).
% 2.89/1.52 thf(fact_31_nth__Cons__0, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ (zero_zero @ nat)) = X)))).
% 2.89/1.52 thf(fact_32_take0, axiom, (! [A:$tType]: ((((take @ A) @ (zero_zero @ nat)) = (^ [Xs2:(list @ A)]: ((nil @ A))))))).
% 2.89/1.52 thf(fact_33_take__eq__Nil, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((take @ A) @ N) @ Xs) = (nil @ A)) = ((N = (zero_zero @ nat)) | (Xs = (nil @ A))))))).
% 2.89/1.52 thf(fact_34_empty__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((nil @ A) = (((replicate @ A) @ N) @ X)) = (N = (zero_zero @ nat)))))).
% 2.89/1.52 thf(fact_35_replicate__empty, axiom, (! [A:$tType,N:nat,X:A]: ((((((replicate @ A) @ N) @ X) = (nil @ A)) = (N = (zero_zero @ nat)))))).
% 2.89/1.52 thf(fact_36_last__replicate, axiom, (! [A:$tType,N:nat,X:A]: (((N != (zero_zero @ nat)) => (((last @ A) @ (((replicate @ A) @ N) @ X)) = X))))).
% 2.89/1.52 thf(fact_37_append1__eq__conv, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys:(list @ A),Y:A]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A))) = (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ (nil @ A)))) = ((Xs = Ys) & (X = Y)))))).
% 2.89/1.52 thf(fact_38_nth__take, axiom, (! [A:$tType,I:nat,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ N) => ((((nth @ A) @ (((take @ A) @ N) @ Xs)) @ I) = (((nth @ A) @ Xs) @ I)))))).
% 2.89/1.52 thf(fact_39_last__appendL, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Xs)))))).
% 2.89/1.52 thf(fact_40_last__appendR, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Ys)))))).
% 2.89/1.52 thf(fact_41_nth__replicate, axiom, (! [A:$tType,I:nat,N:nat,X:A]: (((((ord_less @ nat) @ I) @ N) => ((((nth @ A) @ (((replicate @ A) @ N) @ X)) @ I) = X))))).
% 2.89/1.52 thf(fact_42_last__snoc, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((last @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) = X)))).
% 2.89/1.52 thf(fact_43_take__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((take @ A) @ (zero_zero @ nat)) @ Xs) = (nil @ A))))).
% 2.89/1.52 thf(fact_44_ext, axiom, (! [B2:$tType,A:$tType,F:(A > B2),G:(A > B2)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.89/1.52 thf(fact_45_last_Osimps, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = X)) & ((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = ((last @ A) @ Xs))))))).
% 2.89/1.52 thf(fact_46_last__ConsL, axiom, (! [A:$tType,Xs:(list @ A),X:A]: (((Xs = (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = X))))).
% 2.89/1.52 thf(fact_47_last__ConsR, axiom, (! [A:$tType,Xs:(list @ A),X:A]: (((Xs != (nil @ A)) => (((last @ A) @ (((cons @ A) @ X) @ Xs)) = ((last @ A) @ Xs)))))).
% 2.89/1.52 thf(fact_48_last__append, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: ((((Ys = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Xs))) & ((Ys != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Ys))))))).
% 2.89/1.52 thf(fact_49_take__Nil, axiom, (! [A:$tType,N:nat]: (((((take @ A) @ N) @ (nil @ A)) = (nil @ A))))).
% 2.89/1.52 thf(fact_50_take__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((! [I2:nat]: (((((take @ A) @ I2) @ Xs) = (((take @ A) @ I2) @ Ys)))) => (Xs = Ys))))).
% 2.89/1.52 thf(fact_51_transpose_Ocases, axiom, (! [A:$tType,X:(list @ (list @ A))]: (((X != (nil @ (list @ A))) => ((! [Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (nil @ A)) @ Xss)))) => (~ ((! [X2:A,Xs3:(list @ A),Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs3)) @ Xss))))))))))).
% 2.89/1.52 thf(fact_52_longest__common__suffix, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((? [Ss:(list @ A),Xs4:(list @ A),Ys2:(list @ A)]: (((Xs = (((append @ A) @ Xs4) @ Ss)) & ((Ys = (((append @ A) @ Ys2) @ Ss)) & ((Xs4 = (nil @ A)) | ((Ys2 = (nil @ A)) | (((last @ A) @ Xs4) != ((last @ A) @ Ys2))))))))))).
% 2.89/1.52 thf(fact_53_replicate__0, axiom, (! [A:$tType,X:A]: (((((replicate @ A) @ (zero_zero @ nat)) @ X) = (nil @ A))))).
% 2.89/1.52 thf(fact_54_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_55_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_56_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.89/1.52 thf(fact_57_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.89/1.52 thf(fact_58_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.89/1.52 thf(fact_59_strict__sorted_Oinduct, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A,Ys3:(list @ A)]: (((P @ Ys3) => (P @ (((cons @ A) @ X2) @ Ys3))))) => (P @ A0))))))))).
% 2.89/1.52 thf(fact_60_strict__sorted_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ A)]: (((X != (nil @ A)) => (~ ((! [X2:A,Ys3:(list @ A)]: ((X != (((cons @ A) @ X2) @ Ys3))))))))))))).
% 2.89/1.52 thf(fact_61_map__tailrec__rev_Oinduct, axiom, (! [A:$tType,B2:$tType,P:((A > B2) > ((list @ A) > ((list @ B2) > $o))),A0:(A > B2),A1:(list @ A),A22:(list @ B2)]: (((! [F2:(A > B2),X_1:(list @ B2)]: ((((P @ F2) @ (nil @ A)) @ X_1))) => ((! [F2:(A > B2),A3:A,As:(list @ A),Bs:(list @ B2)]: (((((P @ F2) @ As) @ (((cons @ B2) @ (F2 @ A3)) @ Bs)) => (((P @ F2) @ (((cons @ A) @ A3) @ As)) @ Bs)))) => (((P @ A0) @ A1) @ A22)))))).
% 2.89/1.52 thf(fact_62_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((cons @ A) @ X2) @ Xs3)))))) => (P @ Xs))))))).
% 2.89/1.52 thf(fact_63_successively_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P2:(A > (A > $o))]: (((P @ P2) @ (nil @ A)))) => ((! [P2:(A > (A > $o)),X2:A]: (((P @ P2) @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [P2:(A > (A > $o)),X2:A,Y2:A,Xs3:(list @ A)]: ((((P @ P2) @ (((cons @ A) @ Y2) @ Xs3)) => ((P @ P2) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))) => ((P @ A0) @ A1))))))).
% 2.89/1.52 thf(fact_64_arg__min__list_Oinduct, axiom, (! [B2:$tType,A:$tType]: (((linorder @ B2) => (! [P:((A > B2) > ((list @ A) > $o)),A0:(A > B2),A1:(list @ A)]: (((! [F2:(A > B2),X2:A]: (((P @ F2) @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [F2:(A > B2),X2:A,Y2:A,Zs2:(list @ A)]: ((((P @ F2) @ (((cons @ A) @ Y2) @ Zs2)) => ((P @ F2) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Zs2)))))) => ((! [A3:(A > B2)]: (((P @ A3) @ (nil @ A)))) => ((P @ A0) @ A1)))))))))).
% 2.89/1.52 thf(fact_65_remdups__adj_Oinduct, axiom, (! [A:$tType,P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((((X2 = Y2) => (P @ (((cons @ A) @ X2) @ Xs3))) => (((X2 != Y2) => (P @ (((cons @ A) @ Y2) @ Xs3))) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3))))))) => (P @ A0))))))).
% 2.89/1.52 thf(fact_66_sorted__wrt_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P2:(A > (A > $o))]: (((P @ P2) @ (nil @ A)))) => ((! [P2:(A > (A > $o)),X2:A,Ys3:(list @ A)]: ((((P @ P2) @ Ys3) => ((P @ P2) @ (((cons @ A) @ X2) @ Ys3))))) => ((P @ A0) @ A1)))))).
% 2.89/1.52 thf(fact_67_remdups__adj_Ocases, axiom, (! [A:$tType,X:(list @ A)]: (((X != (nil @ A)) => ((! [X2:A]: ((X != (((cons @ A) @ X2) @ (nil @ A))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((X != (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))))))))).
% 2.89/1.52 thf(fact_68_shuffles_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [Xs3:(list @ A)]: (((P @ Xs3) @ (nil @ A)))) => ((! [X2:A,Xs3:(list @ A),Y2:A,Ys3:(list @ A)]: ((((P @ Xs3) @ (((cons @ A) @ Y2) @ Ys3)) => (((P @ (((cons @ A) @ X2) @ Xs3)) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3)))))) => ((P @ A0) @ A1))))))).
% 2.89/1.52 thf(fact_69_min__list_Oinduct, axiom, (! [A:$tType]: (((ord @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((! [X2:A,Xs3:(list @ A)]: (((! [X212:A,X222:(list @ A)]: (((Xs3 = (((cons @ A) @ X212) @ X222)) => (P @ Xs3)))) => (P @ (((cons @ A) @ X2) @ Xs3))))) => ((P @ (nil @ A)) => (P @ A0))))))))).
% 2.89/1.52 thf(fact_70_min__list_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [X:(list @ A)]: (((! [X2:A,Xs3:(list @ A)]: ((X != (((cons @ A) @ X2) @ Xs3)))) => (X = (nil @ A))))))))).
% 2.89/1.52 thf(fact_71_induct__list012, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Zs2:(list @ A)]: (((P @ Zs2) => ((P @ (((cons @ A) @ Y2) @ Zs2)) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Zs2))))))) => (P @ Xs))))))).
% 2.89/1.52 thf(fact_72_splice_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X2:A,Xs3:(list @ A),Ys3:(list @ A)]: ((((P @ Ys3) @ Xs3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ Ys3)))) => ((P @ A0) @ A1)))))).
% 2.89/1.52 thf(fact_73_list__induct2_H, axiom, (! [A:$tType,B2:$tType,P:((list @ A) > ((list @ B2) > $o)),Xs:(list @ A),Ys:(list @ B2)]: ((((P @ (nil @ A)) @ (nil @ B2)) => ((! [X2:A,Xs3:(list @ A)]: (((P @ (((cons @ A) @ X2) @ Xs3)) @ (nil @ B2)))) => ((! [Y2:B2,Ys3:(list @ B2)]: (((P @ (nil @ A)) @ (((cons @ B2) @ Y2) @ Ys3)))) => ((! [X2:A,Xs3:(list @ A),Y2:B2,Ys3:(list @ B2)]: ((((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B2) @ Y2) @ Ys3))))) => ((P @ Xs) @ Ys)))))))).
% 2.89/1.52 thf(fact_74_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y3:A,Ys4:(list @ A)]: ((Xs = (((cons @ A) @ Y3) @ Ys4)))))))).
% 2.89/1.52 thf(fact_75_list_Oinducts, axiom, (! [A:$tType,P:((list @ A) > $o),List:(list @ A)]: (((P @ (nil @ A)) => ((! [X1:A,X23:(list @ A)]: (((P @ X23) => (P @ (((cons @ A) @ X1) @ X23))))) => (P @ List)))))).
% 2.89/1.52 thf(fact_76_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X213:A,X223:(list @ A)]: ((Y != (((cons @ A) @ X213) @ X223)))))))))).
% 2.89/1.52 thf(fact_77_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X22:(list @ A)]: (((List = (((cons @ A) @ X21) @ X22)) => (List != (nil @ A)))))).
% 2.89/1.52 thf(fact_78_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X22))))).
% 2.89/1.52 thf(fact_79_append_Oleft__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ (nil @ A)) @ A2) = A2)))).
% 2.89/1.52 thf(fact_80_append__Nil, axiom, (! [A:$tType,Ys:(list @ A)]: (((((append @ A) @ (nil @ A)) @ Ys) = Ys)))).
% 2.89/1.52 thf(fact_81_eq__Nil__appendI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = Ys) => (Xs = (((append @ A) @ (nil @ A)) @ Ys)))))).
% 2.89/1.52 thf(fact_82_psums_Ocases, axiom, (! [A:$tType]: (((plus @ A) => (! [X:(list @ A)]: (((X != (nil @ A)) => ((! [X2:A]: ((X != (((cons @ A) @ X2) @ (nil @ A))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((X != (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3))))))))))))))).
% 2.89/1.52 thf(fact_83_rev__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X2) @ (nil @ A)))))))) => (P @ Xs))))))).
% 2.89/1.52 thf(fact_84_append__eq__Cons__conv, axiom, (! [A:$tType,Ys:(list @ A),Zs:(list @ A),X:A,Xs:(list @ A)]: ((((((append @ A) @ Ys) @ Zs) = (((cons @ A) @ X) @ Xs)) = (((Ys = (nil @ A)) & (Zs = (((cons @ A) @ X) @ Xs))) | (? [Ys5:(list @ A)]: (((Ys = (((cons @ A) @ X) @ Ys5)) & ((((append @ A) @ Ys5) @ Zs) = Xs))))))))).
% 2.89/1.52 thf(fact_85_Cons__eq__append__conv, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A)]: ((((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys) @ Zs)) = (((Ys = (nil @ A)) & ((((cons @ A) @ X) @ Xs) = Zs)) | (? [Ys5:(list @ A)]: ((((((cons @ A) @ X) @ Ys5) = Ys) & (Xs = (((append @ A) @ Ys5) @ Zs)))))))))).
% 2.89/1.52 thf(fact_86_rev__exhaust, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (~ ((! [Ys3:(list @ A),Y2:A]: ((Xs != (((append @ A) @ Ys3) @ (((cons @ A) @ Y2) @ (nil @ A)))))))))))).
% 2.89/1.52 thf(fact_87_rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A,Xs3:(list @ A)]: (((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X2) @ (nil @ A))))))) => (P @ Xs)))))).
% 2.89/1.52 thf(fact_88_psums_Osimps_I1_J, axiom, (! [A:$tType]: (((plus @ A) => (((descar1668888542_psums @ A) @ (nil @ A)) = (nil @ A)))))).
% 2.89/1.52 thf(fact_89_replicate__append__same, axiom, (! [A:$tType,I:nat,X:A]: (((((append @ A) @ (((replicate @ A) @ I) @ X)) @ (((cons @ A) @ X) @ (nil @ A))) = (((cons @ A) @ X) @ (((replicate @ A) @ I) @ X)))))).
% 2.89/1.52 thf(fact_90_psums_Osimps_I2_J, axiom, (! [A:$tType]: (((plus @ A) => (! [X:A]: ((((descar1668888542_psums @ A) @ (((cons @ A) @ X) @ (nil @ A))) = (((cons @ A) @ X) @ (nil @ A))))))))).
% 2.89/1.52 thf(fact_91_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_92_not__Cons__self2, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((cons @ A) @ X) @ Xs) != Xs)))).
% 2.89/1.52 thf(fact_93_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs) @ Ts)) = (? [Us:(list @ A)]: ((((Xs = (((append @ A) @ Zs) @ Us)) & ((((append @ A) @ Us) @ Ys) = Ts)) | (((((append @ A) @ Xs) @ Us) = Zs) & (Ys = (((append @ A) @ Us) @ Ts)))))))))).
% 2.89/1.52 thf(fact_94_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs:(list @ A),Ys:(list @ A),Us2:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs) => ((Ys = (((append @ A) @ Xs1) @ Us2)) => ((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs) @ Us2))))))).
% 2.89/1.52 thf(fact_95_append__Cons, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys) = (((cons @ A) @ X) @ (((append @ A) @ Xs) @ Ys)))))).
% 2.89/1.52 thf(fact_96_v__decompose, axiom, (! [Xs5:(list @ a)]: (((v @ (((append @ a) @ Xs5) @ xsa)) = (((plus_plus @ int) @ (v @ (((append @ a) @ Xs5) @ (((cons @ a) @ (((nth @ a) @ xsa) @ p)) @ (nil @ a))))) @ (v @ (((cons @ a) @ (((nth @ a) @ xsa) @ p)) @ xs2))))))).
% 2.89/1.52 thf(fact_97_p__less__length, axiom, (((ord_less @ nat) @ p) @ ((size_size @ (list @ a)) @ xsa))).
% 2.89/1.52 thf(fact_98_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.89/1.52 thf(fact_99_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.89/1.52 thf(fact_100_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.89/1.52 thf(fact_101_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.89/1.52 thf(fact_102_minus__poly__rev__list_Oinduct, axiom, (! [A:$tType]: (((group_add @ A) => (! [P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X2:A,Xs3:(list @ A),Y2:A,Ys3:(list @ A)]: ((((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3))))) => ((! [Xs3:(list @ A)]: (((P @ Xs3) @ (nil @ A)))) => ((! [Y2:A,Ys3:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ Y2) @ Ys3)))) => ((P @ A0) @ A1)))))))))).
% 2.89/1.52 thf(fact_103_plus__coeffs_Oinduct, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [Xs3:(list @ A)]: (((P @ Xs3) @ (nil @ A)))) => ((! [V:A,Va:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ V) @ Va)))) => ((! [X2:A,Xs3:(list @ A),Y2:A,Ys3:(list @ A)]: ((((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3))))) => ((P @ A0) @ A1)))))))))).
% 2.89/1.52 thf(fact_104_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (~ ((((ord_less_eq @ A) @ T) @ X3)))))))))))))).
% 2.89/1.52 thf(fact_105_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (((ord_less_eq @ A) @ X3) @ T)))))))))))).
% 2.89/1.52 thf(fact_106_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 2.89/1.52 thf(fact_107_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 2.89/1.52 thf(fact_108_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.89/1.52 thf(fact_109_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.89/1.52 thf(fact_110_add_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.89/1.52 thf(fact_111_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.89/1.52 thf(fact_112_double__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_113_double__zero__sym, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_114_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [B:A,A2:A]: ((((((plus_plus @ A) @ B) @ A2) = A2) = (B = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_115_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B:A]: ((((((plus_plus @ A) @ A2) @ B) = A2) = (B = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_116_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ B) @ A2)) = (B = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_117_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ A2) @ B)) = (B = (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_118_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.89/1.52 thf(fact_119_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X) @ Y)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.89/1.52 thf(fact_120_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_121_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_122_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_123_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_124_append__eq__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Us2:(list @ A),Vs:(list @ A)]: ((((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) | (((size_size @ (list @ A)) @ Us2) = ((size_size @ (list @ A)) @ Vs))) => (((((append @ A) @ Xs) @ Us2) = (((append @ A) @ Ys) @ Vs)) = ((Xs = Ys) & (Us2 = Vs))))))).
% 2.89/1.52 thf(fact_125_take__all__iff, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((take @ A) @ N) @ Xs) = Xs) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N))))).
% 2.89/1.52 thf(fact_126_take__all, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N) => ((((take @ A) @ N) @ Xs) = Xs))))).
% 2.89/1.52 thf(fact_127_length__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((size_size @ (list @ A)) @ (((replicate @ A) @ N) @ X)) = N)))).
% 2.89/1.52 thf(fact_128_length__psums, axiom, (! [A:$tType]: (((plus @ A) => (! [Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((descar1668888542_psums @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs)))))))).
% 2.89/1.52 thf(fact_129_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.89/1.52 thf(fact_130_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_131_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.89/1.52 thf(fact_132_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.89/1.52 thf(fact_133_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_134_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_135_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.89/1.52 thf(fact_136_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_137_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.89/1.52 thf(fact_138_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 2.89/1.52 thf(fact_139_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_140_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.89/1.52 thf(fact_141_length__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = (zero_zero @ nat)) = (Xs = (nil @ A)))))).
% 2.89/1.52 thf(fact_142_length__greater__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)) = (Xs != (nil @ A)))))).
% 2.89/1.52 thf(fact_143_nth__append__length, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys:(list @ A)]: (((((nth @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ Ys))) @ ((size_size @ (list @ A)) @ Xs)) = X)))).
% 2.89/1.52 thf(fact_144_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.89/1.52 thf(fact_145_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.89/1.52 thf(fact_146_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.52 thf(fact_147_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.89/1.52 thf(fact_148_impossible__Cons, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),X:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys)) => (Xs != (((cons @ A) @ X) @ Ys)))))).
% 2.89/1.52 thf(fact_149_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs3:(list @ A)]: ((((size_size @ (list @ A)) @ Xs3) = N)))))).
% 2.89/1.52 thf(fact_150_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))).
% 2.89/1.52 thf(fact_151_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A2:A,B:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A4) @ B) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.52 thf(fact_152_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B3:A,K:A,B:A,A2:A]: (((B3 = (((plus_plus @ A) @ K) @ B)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.52 thf(fact_153_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.89/1.52 thf(fact_154_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.89/1.52 thf(fact_155_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.89/1.52 thf(fact_156_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 2.89/1.52 thf(fact_157_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 2.89/1.52 thf(fact_158_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A5:A,B4:A]: ((((plus_plus @ A) @ B4) @ A5)))))))).
% 2.89/1.52 thf(fact_159_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B:A,A2:A,C:A]: (((((plus_plus @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.89/1.52 thf(fact_160_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) => (B = C)))))))).
% 2.89/1.52 thf(fact_161_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B = C)))))))).
% 2.89/1.52 thf(fact_162_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X:A,Y:A]: (((((size_size @ A) @ X) != ((size_size @ A) @ Y)) => (X != Y)))))))).
% 2.89/1.52 thf(fact_163_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))))))).
% 2.89/1.52 thf(fact_164_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.89/1.52 thf(fact_165_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.89/1.52 thf(fact_166_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.89/1.52 thf(fact_167_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.89/1.52 thf(fact_168_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ K2)))))))))).
% 2.89/1.52 thf(fact_169_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.89/1.52 thf(fact_170_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.89/1.52 thf(fact_171_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.89/1.52 thf(fact_172_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 2.89/1.52 thf(fact_173_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_174_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_175_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A5:A,B4:A]: ((? [C2:A]: ((B4 = (((plus_plus @ A) @ A5) @ C2))))))))))).
% 2.89/1.52 thf(fact_176_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.89/1.52 thf(fact_177_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => (~ ((! [C3:A]: ((B != (((plus_plus @ A) @ A2) @ C3))))))))))))).
% 2.89/1.52 thf(fact_178_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B))))))))).
% 2.89/1.52 thf(fact_179_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.89/1.52 thf(fact_180_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$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))))))))).
% 2.89/1.52 thf(fact_181_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.52 thf(fact_182_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$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))))))))).
% 2.89/1.52 thf(fact_183_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) => (((ord_less @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_184_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) => (((ord_less @ A) @ A2) @ B)))))))).
% 2.89/1.52 thf(fact_185_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C))))))))).
% 2.89/1.52 thf(fact_186_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B))))))))).
% 2.89/1.52 thf(fact_187_add__strict__mono, axiom, (! [A:$tType]: (((strict2144017051up_add @ A) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.89/1.52 thf(fact_188_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$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))))))))).
% 2.89/1.52 thf(fact_189_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.52 thf(fact_190_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$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))))))))).
% 2.89/1.52 thf(fact_191_list_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (list @ A)) @ (nil @ A)) = (zero_zero @ nat))))).
% 2.89/1.52 thf(fact_192_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs3:(list @ A)]: (((! [Ys6:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys6)) @ ((size_size @ (list @ A)) @ Xs3)) => (P @ Ys6)))) => (P @ Xs3)))) => (P @ Xs))))).
% 2.89/1.52 thf(fact_193_nth__take__lemma, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Ys:(list @ A)]: (((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Ys)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((((nth @ A) @ Xs) @ I2) = (((nth @ A) @ Ys) @ I2))))) => ((((take @ A) @ K) @ Xs) = (((take @ A) @ K) @ Ys)))))))).
% 2.89/1.52 thf(fact_194_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.89/1.52 thf(fact_195_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.89/1.52 thf(fact_196_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.89/1.52 thf(fact_197_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.52 thf(fact_198_add__increasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,B:A,A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => ((((ord_less_eq @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.89/1.52 thf(fact_199_add__decreasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B))))))))).
% 2.89/1.52 thf(fact_200_add__increasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.89/1.52 thf(fact_201_add__decreasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ B))))))))).
% 2.89/1.52 thf(fact_202_pos__add__strict, axiom, (! [A:$tType]: (((strict797366125id_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.89/1.52 thf(fact_203_canonically__ordered__monoid__add__class_OlessE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((! [C3:A]: (((B = (((plus_plus @ A) @ A2) @ C3)) => (C3 = (zero_zero @ A)))))))))))))).
% 2.89/1.52 thf(fact_204_add__pos__pos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.52 thf(fact_205_add__neg__neg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.89/1.52 thf(fact_206_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.89/1.52 thf(fact_207_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 2.89/1.52 thf(fact_208_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.89/1.52 thf(fact_209_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$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))))))))).
% 2.89/1.52 thf(fact_210_list__induct4, axiom, (! [C4:$tType,A:$tType,B2:$tType,D2:$tType,Xs:(list @ A),Ys:(list @ B2),Zs:(list @ C4),Ws:(list @ D2),P:((list @ A) > ((list @ B2) > ((list @ C4) > ((list @ D2) > $o))))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B2)) @ Ys)) => ((((size_size @ (list @ B2)) @ Ys) = ((size_size @ (list @ C4)) @ Zs)) => ((((size_size @ (list @ C4)) @ Zs) = ((size_size @ (list @ D2)) @ Ws)) => (((((P @ (nil @ A)) @ (nil @ B2)) @ (nil @ C4)) @ (nil @ D2)) => ((! [X2:A,Xs3:(list @ A),Y2:B2,Ys3:(list @ B2),Z:C4,Zs2:(list @ C4),W:D2,Ws2:(list @ D2)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B2)) @ Ys3)) => ((((size_size @ (list @ B2)) @ Ys3) = ((size_size @ (list @ C4)) @ Zs2)) => ((((size_size @ (list @ C4)) @ Zs2) = ((size_size @ (list @ D2)) @ Ws2)) => (((((P @ Xs3) @ Ys3) @ Zs2) @ Ws2) => ((((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B2) @ Y2) @ Ys3)) @ (((cons @ C4) @ Z) @ Zs2)) @ (((cons @ D2) @ W) @ Ws2)))))))) => ((((P @ Xs) @ Ys) @ Zs) @ Ws))))))))).
% 2.89/1.52 thf(fact_211_list__induct3, axiom, (! [B2:$tType,A:$tType,C4:$tType,Xs:(list @ A),Ys:(list @ B2),Zs:(list @ C4),P:((list @ A) > ((list @ B2) > ((list @ C4) > $o)))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B2)) @ Ys)) => ((((size_size @ (list @ B2)) @ Ys) = ((size_size @ (list @ C4)) @ Zs)) => ((((P @ (nil @ A)) @ (nil @ B2)) @ (nil @ C4)) => ((! [X2:A,Xs3:(list @ A),Y2:B2,Ys3:(list @ B2),Z:C4,Zs2:(list @ C4)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B2)) @ Ys3)) => ((((size_size @ (list @ B2)) @ Ys3) = ((size_size @ (list @ C4)) @ Zs2)) => ((((P @ Xs3) @ Ys3) @ Zs2) => (((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B2) @ Y2) @ Ys3)) @ (((cons @ C4) @ Z) @ Zs2))))))) => (((P @ Xs) @ Ys) @ Zs)))))))).
% 2.89/1.52 thf(fact_212_list__induct2, axiom, (! [A:$tType,B2:$tType,Xs:(list @ A),Ys:(list @ B2),P:((list @ A) > ((list @ B2) > $o))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B2)) @ Ys)) => (((P @ (nil @ A)) @ (nil @ B2)) => ((! [X2:A,Xs3:(list @ A),Y2:B2,Ys3:(list @ B2)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B2)) @ Ys3)) => (((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B2) @ Y2) @ Ys3)))))) => ((P @ Xs) @ Ys))))))).
% 2.89/1.52 thf(fact_213_psums_Oinduct, axiom, (! [A:$tType]: (((plus @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Xs3:(list @ A)]: (((P @ (((cons @ A) @ (((plus_plus @ A) @ X2) @ Y2)) @ Xs3)) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))) => (P @ A0)))))))))).
% 2.89/1.52 thf(fact_214_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y4:(list @ A),Z2:(list @ A)]: ((Y4 = Z2))) = (^ [Xs2:(list @ A),Ys4:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys4)) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs2)) => ((((nth @ A) @ Xs2) @ I4) = (((nth @ A) @ Ys4) @ I4)))))))))))).
% 2.89/1.52 thf(fact_215_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => (? [X4:A]: (((P @ I4) @ X4)))))) = (? [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => ((P @ I4) @ (((nth @ A) @ Xs2) @ I4)))))))))))).
% 2.89/1.52 thf(fact_216_nth__equalityI, axiom, (! [A:$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)))))).
% 2.89/1.52 thf(fact_217_psums_Osimps_I3_J, axiom, (! [A:$tType]: (((plus @ A) => (! [X:A,Y:A,Xs:(list @ A)]: ((((descar1668888542_psums @ A) @ (((cons @ A) @ X) @ (((cons @ A) @ Y) @ Xs))) = (((cons @ A) @ X) @ ((descar1668888542_psums @ A) @ (((cons @ A) @ (((plus_plus @ A) @ X) @ Y)) @ Xs)))))))))).
% 2.89/1.53 thf(fact_218_add__strict__increasing2, axiom, (! [A:$tType]: (((ordere1818651114id_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.89/1.53 thf(fact_219_add__strict__increasing, axiom, (! [A:$tType]: (((ordere1818651114id_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)))))))))).
% 2.89/1.53 thf(fact_220_add__pos__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.53 thf(fact_221_add__nonpos__neg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.89/1.53 thf(fact_222_add__nonneg__pos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B)))))))))).
% 2.89/1.53 thf(fact_223_add__neg__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ (zero_zero @ A)))))))))).
% 2.89/1.53 thf(fact_224_sign__changes__induct, axiom, (! [A:$tType]: ((((plus @ A) & (zero @ A)) => (! [P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [Xs3:(list @ A)]: (((P @ Xs3) => (P @ (((cons @ A) @ (zero_zero @ A)) @ Xs3))))) => ((! [X2:A,Y2:A,Xs3:(list @ A)]: (((X2 != (zero_zero @ A)) => ((P @ (((cons @ A) @ (((plus_plus @ A) @ X2) @ Y2)) @ Xs3)) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3))))))) => (P @ Xs))))))))))).
% 2.89/1.53 thf(fact_225_same__length__different, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs != Ys) => ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => (? [Pre:(list @ A),X2:A,Xs4:(list @ A),Y2:A,Ys2:(list @ A)]: (((X2 != Y2) & ((Xs = (((append @ A) @ Pre) @ (((append @ A) @ (((cons @ A) @ X2) @ (nil @ A))) @ Xs4))) & (Ys = (((append @ A) @ Pre) @ (((append @ A) @ (((cons @ A) @ Y2) @ (nil @ A))) @ Ys2)))))))))))).
% 2.89/1.53 thf(fact_226_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 2.89/1.53 thf(fact_227_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (((ord_less_eq @ A) @ A5) @ B4))))))))).
% 2.89/1.53 thf(fact_228_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.89/1.53 thf(fact_229_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B:A]: (((! [A3:A,B5:A]: (((((ord_less_eq @ A) @ A3) @ B5) => ((P @ A3) @ B5)))) => ((! [A3:A,B5:A]: ((((P @ B5) @ A3) => ((P @ A3) @ B5)))) => ((P @ A2) @ B))))))))).
% 2.89/1.53 thf(fact_230_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.89/1.53 thf(fact_231_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 2.89/1.53 thf(fact_232_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 2.89/1.53 thf(fact_233_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.89/1.53 thf(fact_234_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.89/1.53 thf(fact_235_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (((ord_less_eq @ A) @ B4) @ A5))))))))).
% 2.89/1.53 thf(fact_236_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.89/1.53 thf(fact_237_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.89/1.53 thf(fact_238_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.89/1.53 thf(fact_239_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.89/1.53 thf(fact_240_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.89/1.53 thf(fact_241_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.89/1.53 thf(fact_242_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.89/1.53 thf(fact_243_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X5:A,Y3:A]: (((((ord_less_eq @ A) @ X5) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X5))))))))).
% 2.89/1.53 thf(fact_244_ord__le__eq__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,B:A,F:(A > B2),C:B2]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B2) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ B2) @ (F @ A2)) @ C)))))))))).
% 2.89/1.53 thf(fact_245_ord__eq__le__subst, axiom, (! [A:$tType,B2:$tType]: ((((ord @ B2) & (ord @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((A2 = (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X2:B2,Y2:B2]: (((((ord_less_eq @ B2) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.89/1.53 thf(fact_246_order__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A2:A,B:A,F:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C4) @ (F @ B)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C4) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ C4) @ (F @ A2)) @ C)))))))))).
% 2.89/1.53 thf(fact_247_order__subst1, axiom, (! [A:$tType,B2:$tType]: ((((order @ B2) & (order @ A)) => (! [A2:A,F:(B2 > A),B:B2,C:B2]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B2) @ B) @ C) => ((! [X2:B2,Y2:B2]: (((((ord_less_eq @ B2) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.89/1.53 thf(fact_248_le__fun__def, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => ((ord_less_eq @ (A > B2)) = (^ [F3:(A > B2),G2:(A > B2)]: ((! [X5:A]: ((((ord_less_eq @ B2) @ (F3 @ X5)) @ (G2 @ X5))))))))))).
% 2.89/1.53 thf(fact_249_le__funI, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2)]: (((! [X2:A]: ((((ord_less_eq @ B2) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B2)) @ F) @ G)))))))).
% 2.89/1.53 thf(fact_250_le__funE, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 2.89/1.53 thf(fact_251_le__funD, axiom, (! [B2:$tType,A:$tType]: (((ord @ B2) => (! [F:(A > B2),G:(A > B2),X:A]: (((((ord_less_eq @ (A > B2)) @ F) @ G) => (((ord_less_eq @ B2) @ (F @ X)) @ (G @ X))))))))).
% 2.89/1.53 thf(fact_252_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___HOL_Otype, axiom, (! [A:$tType]: (((linordered_idom @ A) => (type @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oplus, axiom, (! [A:$tType]: (((linordered_idom @ A) => (plus @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ozero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (zero @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Orderings_Oord, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ord @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (order @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (group_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (monoid_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linorder @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (preorder @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (semigroup_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (comm_monoid_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ab_semigroup_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (cancel_semigroup_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__comm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (cancel1352612707id_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Olinordered__ab__group__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linord219039673up_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__comm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere216010020id_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere779506340up_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__comm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere1818651114id_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__comm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (strict797366125id_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere236663937imp_le @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere223160158up_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (strict2144017051up_add @ A))))).
% 2.89/1.53 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere516151231imp_le @ A))))).
% 2.89/1.53 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: (((preorder @ A7) => (preorder @ (A6 > A7)))))).
% 2.89/1.53 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: (((order @ A7) => (order @ (A6 > A7)))))).
% 2.89/1.53 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: (((ord @ A7) => (ord @ (A6 > A7)))))).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere516151231imp_le @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict2144017051up_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add, axiom, (strict797366125id_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add, axiom, (ordere1818651114id_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add, axiom, (ordere216010020id_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, (preorder @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, (monoid_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, (order @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, (ord @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.89/1.53 thf(tcon_Int_Oint___Groups_Oplus, axiom, (plus @ int)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4, axiom, (ordere516151231imp_le @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_5, axiom, (strict2144017051up_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_6, axiom, (ordere223160158up_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_7, axiom, (ordere236663937imp_le @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_8, axiom, (strict797366125id_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_9, axiom, (ordere1818651114id_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_10, axiom, (ordere779506340up_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_11, axiom, (ordere216010020id_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_12, axiom, (cancel1352612707id_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_13, axiom, (cancel_semigroup_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_14, axiom, (ab_semigroup_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_15, axiom, (comm_monoid_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add_16, axiom, (semigroup_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_17, axiom, (preorder @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder_18, axiom, (linorder @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Omonoid__add_19, axiom, (monoid_add @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_20, axiom, (order @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Oord_21, axiom, (ord @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ozero_22, axiom, (zero @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oplus_23, axiom, (plus @ nat)).
% 2.89/1.53 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_24, axiom, (preorder @ $o)).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_25, axiom, (linorder @ $o)).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_26, axiom, (order @ $o)).
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Oord_27, axiom, (ord @ $o)).
% 2.89/1.53 thf(tcon_List_Olist___Nat_Osize_28, axiom, (! [A6:$tType]: ((size @ (list @ A6))))).
% 2.89/1.53 thf(tfree_0, hypothesis, (linordered_idom @ a)).
% 2.89/1.53 thf(conj_0, conjecture, (((descar1668888542_psums @ a) @ xsa) = (((append @ a) @ (((replicate @ a) @ p) @ (zero_zero @ a))) @ ((descar1668888542_psums @ a) @ (((cons @ a) @ (((nth @ a) @ xsa) @ p)) @ xs2))))).
% 2.89/1.53 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2DTF10522.p
% 2.89/1.53 ------------------------
% 2.89/1.55 ---- Cleaned THF ---
% 2.89/1.56 thf(ty_t_List_Olist,type,
% 2.89/1.56 list: $tType > $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_t_Nat_Onat,type,
% 2.89/1.56 nat: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_t_Int_Oint,type,
% 2.89/1.56 int: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(ty_tf_a,type,
% 2.89/1.56 a: $tType ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.89/1.56 linordered_idom:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_HOL_Otype,type,
% 2.89/1.56 type:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Nat_Osize,type,
% 2.89/1.56 size:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oplus,type,
% 2.89/1.56 plus:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ozero,type,
% 2.89/1.56 zero:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.89/1.56 ord:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.89/1.56 order:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.89/1.56 group_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Omonoid__add,type,
% 2.89/1.56 monoid_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.89/1.56 linorder:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.89/1.56 preorder:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.89/1.56 semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.89/1.56 comm_monoid_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.89/1.56 ab_semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.89/1.56 cancel_semigroup_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.89/1.56 cancel1352612707id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.89/1.56 linord219039673up_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 2.89/1.56 ordere216010020id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.89/1.56 ordere779506340up_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.89/1.56 canoni770627133id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 2.89/1.56 ordere1818651114id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 2.89/1.56 strict797366125id_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.89/1.56 ordere236663937imp_le:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.89/1.56 ordere223160158up_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.89/1.56 strict2144017051up_add:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.89/1.56 ordere516151231imp_le:
% 2.89/1.56 !>[A: $tType] : $o ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Opsums,type,
% 2.89/1.56 descar1668888542_psums:
% 2.89/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.89/1.56 plus_plus:
% 2.89/1.56 !>[A: $tType] : ( A > A > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.89/1.56 zero_zero:
% 2.89/1.56 !>[A: $tType] : A ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Oappend,type,
% 2.89/1.56 append:
% 2.89/1.56 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Olast,type,
% 2.89/1.56 last:
% 2.89/1.56 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Olist_OCons,type,
% 2.89/1.56 cons:
% 2.89/1.56 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Olist_ONil,type,
% 2.89/1.56 nil:
% 2.89/1.56 !>[A: $tType] : ( list @ A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Onth,type,
% 2.89/1.56 nth:
% 2.89/1.56 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Oreplicate,type,
% 2.89/1.56 replicate:
% 2.89/1.56 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_List_Otake,type,
% 2.89/1.56 take:
% 2.89/1.56 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.89/1.56 size_size:
% 2.89/1.56 !>[A: $tType] : ( A > nat ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.89/1.56 ord_less:
% 2.89/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.89/1.56 ord_less_eq:
% 2.89/1.56 !>[A: $tType] : ( A > A > $o ) ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_p____,type,
% 2.89/1.56 p: nat ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_v,type,
% 2.89/1.56 v: ( list @ a ) > int ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_xa____,type,
% 2.89/1.56 xa: a ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_xs,type,
% 2.89/1.56 xs: list @ a ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_xs1____,type,
% 2.89/1.56 xs1: list @ a ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_xs2____,type,
% 2.89/1.56 xs2: list @ a ).
% 2.89/1.56
% 2.89/1.56 thf(sy_v_xsa____,type,
% 2.89/1.56 xsa: list @ a ).
% 2.89/1.56
% 2.89/1.56 thf(fact_0_nonneg_Ohyps_I1_J,axiom,
% 2.89/1.56 ( xsa
% 2.89/1.56 != ( nil @ a ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_1_p__nz,axiom,
% 2.89/1.56 ( ( nth @ a @ xsa @ p )
% 2.89/1.56 != ( zero_zero @ a ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_2_nonneg_Ohyps_I2_J,axiom,
% 2.89/1.56 ( ( last @ a @ xsa )
% 2.89/1.56 != ( zero_zero @ a ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_3_psums__0__Cons,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( monoid_add @ A )
% 2.89/1.56 => ! [Xs: list @ A] :
% 2.89/1.56 ( ( descar1668888542_psums @ A @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) )
% 2.89/1.56 = ( cons @ A @ ( zero_zero @ A ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_4_xs1__def,axiom,
% 2.89/1.56 ( xs1
% 2.89/1.56 = ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_5_xs__decompose,axiom,
% 2.89/1.56 ( xsa
% 2.89/1.56 = ( append @ a @ xs1 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_6_psums__replicate__0__append,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( monoid_add @ A )
% 2.89/1.56 => ! [N: nat,Xs: list @ A] :
% 2.89/1.56 ( ( descar1668888542_psums @ A @ ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ Xs ) )
% 2.89/1.56 = ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_7_less__p__zero,axiom,
% 2.89/1.56 ! [Q: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ Q @ p )
% 2.89/1.56 => ( ( nth @ a @ xsa @ Q )
% 2.89/1.56 = ( zero_zero @ a ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_8__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062,axiom,
% 2.89/1.56 ( ( take @ a @ p @ xsa )
% 2.89/1.56 = ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_9_replicate__app__Cons__same,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_10_append_Oassoc,axiom,
% 2.89/1.56 ! [A: $tType,A2: list @ A,B: list @ A,C: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( append @ A @ A2 @ B ) @ C )
% 2.89/1.56 = ( append @ A @ A2 @ ( append @ A @ B @ C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_11_append__assoc,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
% 2.89/1.56 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_12_append__same__eq,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Ys @ Xs )
% 2.89/1.56 = ( append @ A @ Zs @ Xs ) )
% 2.89/1.56 = ( Ys = Zs ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_13_same__append__eq,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = ( append @ A @ Xs @ Zs ) )
% 2.89/1.56 = ( Ys = Zs ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_14_nonneg_Ohyps_I4_J,axiom,
% 2.89/1.56 ord_less_eq @ a @ ( zero_zero @ a ) @ xa ).
% 2.89/1.56
% 2.89/1.56 thf(fact_15_list_Oinject,axiom,
% 2.89/1.56 ! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
% 2.89/1.56 ( ( ( cons @ A @ X21 @ X22 )
% 2.89/1.56 = ( cons @ A @ Y21 @ Y22 ) )
% 2.89/1.56 = ( ( X21 = Y21 )
% 2.89/1.56 & ( X22 = Y22 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_16_append__replicate__commute,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A,K: nat] :
% 2.89/1.56 ( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
% 2.89/1.56 = ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_17_Cons__eq__appendI,axiom,
% 2.89/1.56 ! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
% 2.89/1.56 ( ( ( cons @ A @ X @ Xs1 )
% 2.89/1.56 = Ys )
% 2.89/1.56 => ( ( Xs
% 2.89/1.56 = ( append @ A @ Xs1 @ Zs ) )
% 2.89/1.56 => ( ( cons @ A @ X @ Xs )
% 2.89/1.56 = ( append @ A @ Ys @ Zs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_18_assms_I1_J,axiom,
% 2.89/1.56 ( xs
% 2.89/1.56 != ( nil @ a ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_19_assms_I2_J,axiom,
% 2.89/1.56 ( ( last @ a @ xs )
% 2.89/1.56 != ( zero_zero @ a ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_20_replicate__eq__replicate,axiom,
% 2.89/1.56 ! [A: $tType,M: nat,X: A,N: nat,Y: A] :
% 2.89/1.56 ( ( ( replicate @ A @ M @ X )
% 2.89/1.56 = ( replicate @ A @ N @ Y ) )
% 2.89/1.56 = ( ( M = N )
% 2.89/1.56 & ( ( M
% 2.89/1.56 != ( zero_zero @ nat ) )
% 2.89/1.56 => ( X = Y ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_21_le__zero__eq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [N: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.89/1.56 = ( N
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_22_not__gr__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [N: A] :
% 2.89/1.56 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.89/1.56 = ( N
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_23_append_Oright__neutral,axiom,
% 2.89/1.56 ! [A: $tType,A2: list @ A] :
% 2.89/1.56 ( ( append @ A @ A2 @ ( nil @ A ) )
% 2.89/1.56 = A2 ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_24_append__is__Nil__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 = ( ( Xs
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 & ( Ys
% 2.89/1.56 = ( nil @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_25_Nil__is__append__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( nil @ A )
% 2.89/1.56 = ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( ( Xs
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 & ( Ys
% 2.89/1.56 = ( nil @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_26_self__append__conv2,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.89/1.56 ( ( Ys
% 2.89/1.56 = ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( Xs
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_27_append__self__conv2,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = Ys )
% 2.89/1.56 = ( Xs
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_28_self__append__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 = ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( Ys
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_29_append__self__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = Xs )
% 2.89/1.56 = ( Ys
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_30_append__Nil2,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( append @ A @ Xs @ ( nil @ A ) )
% 2.89/1.56 = Xs ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_31_nth__Cons__0,axiom,
% 2.89/1.56 ! [A: $tType,X: A,Xs: list @ A] :
% 2.89/1.56 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
% 2.89/1.56 = X ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_32_take0,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( take @ A @ ( zero_zero @ nat ) )
% 2.89/1.56 = ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_33_take__eq__Nil,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.89/1.56 ( ( ( take @ A @ N @ Xs )
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 = ( ( N
% 2.89/1.56 = ( zero_zero @ nat ) )
% 2.89/1.56 | ( Xs
% 2.89/1.56 = ( nil @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_34_empty__replicate,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A] :
% 2.89/1.56 ( ( ( nil @ A )
% 2.89/1.56 = ( replicate @ A @ N @ X ) )
% 2.89/1.56 = ( N
% 2.89/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_35_replicate__empty,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A] :
% 2.89/1.56 ( ( ( replicate @ A @ N @ X )
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 = ( N
% 2.89/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_36_last__replicate,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A] :
% 2.89/1.56 ( ( N
% 2.89/1.56 != ( zero_zero @ nat ) )
% 2.89/1.56 => ( ( last @ A @ ( replicate @ A @ N @ X ) )
% 2.89/1.56 = X ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_37_append1__eq__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.89/1.56 = ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 2.89/1.56 = ( ( Xs = Ys )
% 2.89/1.56 & ( X = Y ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_38_nth__take,axiom,
% 2.89/1.56 ! [A: $tType,I: nat,N: nat,Xs: list @ A] :
% 2.89/1.56 ( ( ord_less @ nat @ I @ N )
% 2.89/1.56 => ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
% 2.89/1.56 = ( nth @ A @ Xs @ I ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_39_last__appendL,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.89/1.56 ( ( Ys
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( last @ A @ Xs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_40_last__appendR,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.89/1.56 ( ( Ys
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( last @ A @ Ys ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_41_nth__replicate,axiom,
% 2.89/1.56 ! [A: $tType,I: nat,N: nat,X: A] :
% 2.89/1.56 ( ( ord_less @ nat @ I @ N )
% 2.89/1.56 => ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
% 2.89/1.56 = X ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_42_last__snoc,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A] :
% 2.89/1.56 ( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 2.89/1.56 = X ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_43_take__0,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( take @ A @ ( zero_zero @ nat ) @ Xs )
% 2.89/1.56 = ( nil @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_44_ext,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
% 2.89/1.56 ( ! [X2: A] :
% 2.89/1.56 ( ( F @ X2 )
% 2.89/1.56 = ( G @ X2 ) )
% 2.89/1.56 => ( F = G ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_45_last_Osimps,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A] :
% 2.89/1.56 ( ( ( Xs
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = X ) )
% 2.89/1.56 & ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = ( last @ A @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_46_last__ConsL,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = X ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_47_last__ConsR,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = ( last @ A @ Xs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_48_last__append,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.89/1.56 ( ( ( Ys
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( last @ A @ Xs ) ) )
% 2.89/1.56 & ( ( Ys
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.89/1.56 = ( last @ A @ Ys ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_49_take__Nil,axiom,
% 2.89/1.56 ! [A: $tType,N: nat] :
% 2.89/1.56 ( ( take @ A @ N @ ( nil @ A ) )
% 2.89/1.56 = ( nil @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_50_take__equalityI,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ! [I2: nat] :
% 2.89/1.56 ( ( take @ A @ I2 @ Xs )
% 2.89/1.56 = ( take @ A @ I2 @ Ys ) )
% 2.89/1.56 => ( Xs = Ys ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_51_transpose_Ocases,axiom,
% 2.89/1.56 ! [A: $tType,X: list @ ( list @ A )] :
% 2.89/1.56 ( ( X
% 2.89/1.56 != ( nil @ ( list @ A ) ) )
% 2.89/1.56 => ( ! [Xss: list @ ( list @ A )] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 2.89/1.56 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_52_longest__common__suffix,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ? [Ss: list @ A,Xs4: list @ A,Ys2: list @ A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 = ( append @ A @ Xs4 @ Ss ) )
% 2.89/1.56 & ( Ys
% 2.89/1.56 = ( append @ A @ Ys2 @ Ss ) )
% 2.89/1.56 & ( ( Xs4
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 | ( Ys2
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 | ( ( last @ A @ Xs4 )
% 2.89/1.56 != ( last @ A @ Ys2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_53_replicate__0,axiom,
% 2.89/1.56 ! [A: $tType,X: A] :
% 2.89/1.56 ( ( replicate @ A @ ( zero_zero @ nat ) @ X )
% 2.89/1.56 = ( nil @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_54_zero__less__iff__neq__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [N: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.89/1.56 = ( N
% 2.89/1.56 != ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_55_gr__implies__not__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [M: A,N: A] :
% 2.89/1.56 ( ( ord_less @ A @ M @ N )
% 2.89/1.56 => ( N
% 2.89/1.56 != ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_56_not__less__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [N: A] :
% 2.89/1.56 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_57_gr__zeroI,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [N: A] :
% 2.89/1.56 ( ( N
% 2.89/1.56 != ( zero_zero @ A ) )
% 2.89/1.56 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_58_zero__le,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_59_strict__sorted_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
% 2.89/1.56 => ( P @ A0 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_60_strict__sorted_Ocases,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [X: list @ A] :
% 2.89/1.56 ( ( X
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ~ ! [X2: A,Ys3: list @ A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_61_map__tailrec__rev_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType,P: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
% 2.89/1.56 ( ! [F2: A > B2,X_1: list @ B2] : ( P @ F2 @ ( nil @ A ) @ X_1 )
% 2.89/1.56 => ( ! [F2: A > B2,A3: A,As: list @ A,Bs: list @ B2] :
% 2.89/1.56 ( ( P @ F2 @ As @ ( cons @ B2 @ ( F2 @ A3 ) @ Bs ) )
% 2.89/1.56 => ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 2.89/1.56 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_62_list__nonempty__induct,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( Xs3
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( P @ Xs3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 2.89/1.56 => ( P @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_63_successively_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 2.89/1.56 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 2.89/1.56 => ( ! [P2: A > A > $o,X2: A] : ( P @ P2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [P2: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( P @ P2 @ ( cons @ A @ Y2 @ Xs3 ) )
% 2.89/1.56 => ( P @ P2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_64_arg__min__list_Oinduct,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType] :
% 2.89/1.56 ( ( linorder @ B2 )
% 2.89/1.56 => ! [P: ( A > B2 ) > ( list @ A ) > $o,A0: A > B2,A1: list @ A] :
% 2.89/1.56 ( ! [F2: A > B2,X2: A] : ( P @ F2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [F2: A > B2,X2: A,Y2: A,Zs2: list @ A] :
% 2.89/1.56 ( ( P @ F2 @ ( cons @ A @ Y2 @ Zs2 ) )
% 2.89/1.56 => ( P @ F2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
% 2.89/1.56 => ( ! [A3: A > B2] : ( P @ A3 @ ( nil @ A ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_65_remdups__adj_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( ( X2 = Y2 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 2.89/1.56 => ( ( ( X2 != Y2 )
% 2.89/1.56 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 2.89/1.56 => ( P @ A0 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_66_sorted__wrt_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 2.89/1.56 ( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
% 2.89/1.56 => ( ! [P2: A > A > $o,X2: A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ P2 @ Ys3 )
% 2.89/1.56 => ( P @ P2 @ ( cons @ A @ X2 @ Ys3 ) ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_67_remdups__adj_Ocases,axiom,
% 2.89/1.56 ! [A: $tType,X: list @ A] :
% 2.89/1.56 ( ( X
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_68_shuffles_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.89/1.56 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 2.89/1.56 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
% 2.89/1.56 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_69_min__list_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ord @ A )
% 2.89/1.56 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 2.89/1.56 ( ! [X2: A,Xs3: list @ A] :
% 2.89/1.56 ( ! [X212: A,X222: list @ A] :
% 2.89/1.56 ( ( Xs3
% 2.89/1.56 = ( cons @ A @ X212 @ X222 ) )
% 2.89/1.56 => ( P @ Xs3 ) )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 2.89/1.56 => ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( P @ A0 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_70_min__list_Ocases,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ord @ A )
% 2.89/1.56 => ! [X: list @ A] :
% 2.89/1.56 ( ! [X2: A,Xs3: list @ A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ Xs3 ) )
% 2.89/1.56 => ( X
% 2.89/1.56 = ( nil @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_71_induct__list012,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [X2: A,Y2: A,Zs2: list @ A] :
% 2.89/1.56 ( ( P @ Zs2 )
% 2.89/1.56 => ( ( P @ ( cons @ A @ Y2 @ Zs2 ) )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
% 2.89/1.56 => ( P @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_72_splice_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.89/1.56 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ Ys3 @ Xs3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_73_list__induct2_H,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys: list @ B2] :
% 2.89/1.56 ( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B2 ) )
% 2.89/1.56 => ( ! [Y2: B2,Ys3: list @ B2] : ( P @ ( nil @ A ) @ ( cons @ B2 @ Y2 @ Ys3 ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
% 2.89/1.56 ( ( P @ Xs3 @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) )
% 2.89/1.56 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_74_neq__Nil__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 = ( ? [Y3: A,Ys4: list @ A] :
% 2.89/1.56 ( Xs
% 2.89/1.56 = ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_75_list_Oinducts,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X1: A,X23: list @ A] :
% 2.89/1.56 ( ( P @ X23 )
% 2.89/1.56 => ( P @ ( cons @ A @ X1 @ X23 ) ) )
% 2.89/1.56 => ( P @ List ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_76_list_Oexhaust,axiom,
% 2.89/1.56 ! [A: $tType,Y: list @ A] :
% 2.89/1.56 ( ( Y
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ~ ! [X213: A,X223: list @ A] :
% 2.89/1.56 ( Y
% 2.89/1.56 != ( cons @ A @ X213 @ X223 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_77_list_OdiscI,axiom,
% 2.89/1.56 ! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
% 2.89/1.56 ( ( List
% 2.89/1.56 = ( cons @ A @ X21 @ X22 ) )
% 2.89/1.56 => ( List
% 2.89/1.56 != ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_78_list_Odistinct_I1_J,axiom,
% 2.89/1.56 ! [A: $tType,X21: A,X22: list @ A] :
% 2.89/1.56 ( ( nil @ A )
% 2.89/1.56 != ( cons @ A @ X21 @ X22 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_79_append_Oleft__neutral,axiom,
% 2.89/1.56 ! [A: $tType,A2: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( nil @ A ) @ A2 )
% 2.89/1.56 = A2 ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_80_append__Nil,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( nil @ A ) @ Ys )
% 2.89/1.56 = Ys ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_81_eq__Nil__appendI,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( Xs = Ys )
% 2.89/1.56 => ( Xs
% 2.89/1.56 = ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_82_psums_Ocases,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ! [X: list @ A] :
% 2.89/1.56 ( ( X
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( X
% 2.89/1.56 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_83_rev__nonempty__induct,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( Xs3
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ( ( P @ Xs3 )
% 2.89/1.56 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 2.89/1.56 => ( P @ Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_84_append__eq__Cons__conv,axiom,
% 2.89/1.56 ! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Ys @ Zs )
% 2.89/1.56 = ( cons @ A @ X @ Xs ) )
% 2.89/1.56 = ( ( ( Ys
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 & ( Zs
% 2.89/1.56 = ( cons @ A @ X @ Xs ) ) )
% 2.89/1.56 | ? [Ys5: list @ A] :
% 2.89/1.56 ( ( Ys
% 2.89/1.56 = ( cons @ A @ X @ Ys5 ) )
% 2.89/1.56 & ( ( append @ A @ Ys5 @ Zs )
% 2.89/1.56 = Xs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_85_Cons__eq__append__conv,axiom,
% 2.89/1.56 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
% 2.89/1.56 ( ( ( cons @ A @ X @ Xs )
% 2.89/1.56 = ( append @ A @ Ys @ Zs ) )
% 2.89/1.56 = ( ( ( Ys
% 2.89/1.56 = ( nil @ A ) )
% 2.89/1.56 & ( ( cons @ A @ X @ Xs )
% 2.89/1.56 = Zs ) )
% 2.89/1.56 | ? [Ys5: list @ A] :
% 2.89/1.56 ( ( ( cons @ A @ X @ Ys5 )
% 2.89/1.56 = Ys )
% 2.89/1.56 & ( Xs
% 2.89/1.56 = ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_86_rev__exhaust,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( Xs
% 2.89/1.56 != ( nil @ A ) )
% 2.89/1.56 => ~ ! [Ys3: list @ A,Y2: A] :
% 2.89/1.56 ( Xs
% 2.89/1.56 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_87_rev__induct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( P @ Xs3 )
% 2.89/1.56 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 2.89/1.56 => ( P @ Xs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_88_psums_Osimps_I1_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ( ( descar1668888542_psums @ A @ ( nil @ A ) )
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_89_replicate__append__same,axiom,
% 2.89/1.56 ! [A: $tType,I: nat,X: A] :
% 2.89/1.56 ( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.89/1.56 = ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_90_psums_Osimps_I2_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ! [X: A] :
% 2.89/1.56 ( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.89/1.56 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_91_zero__reorient,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( zero @ A )
% 2.89/1.56 => ! [X: A] :
% 2.89/1.56 ( ( ( zero_zero @ A )
% 2.89/1.56 = X )
% 2.89/1.56 = ( X
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_92_not__Cons__self2,axiom,
% 2.89/1.56 ! [A: $tType,X: A,Xs: list @ A] :
% 2.89/1.56 ( ( cons @ A @ X @ Xs )
% 2.89/1.56 != Xs ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_93_append__eq__append__conv2,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = ( append @ A @ Zs @ Ts ) )
% 2.89/1.56 = ( ? [Us: list @ A] :
% 2.89/1.56 ( ( ( Xs
% 2.89/1.56 = ( append @ A @ Zs @ Us ) )
% 2.89/1.56 & ( ( append @ A @ Us @ Ys )
% 2.89/1.56 = Ts ) )
% 2.89/1.56 | ( ( ( append @ A @ Xs @ Us )
% 2.89/1.56 = Zs )
% 2.89/1.56 & ( Ys
% 2.89/1.56 = ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_94_append__eq__appendI,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
% 2.89/1.56 ( ( ( append @ A @ Xs @ Xs1 )
% 2.89/1.56 = Zs )
% 2.89/1.56 => ( ( Ys
% 2.89/1.56 = ( append @ A @ Xs1 @ Us2 ) )
% 2.89/1.56 => ( ( append @ A @ Xs @ Ys )
% 2.89/1.56 = ( append @ A @ Zs @ Us2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_95_append__Cons,axiom,
% 2.89/1.56 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
% 2.89/1.56 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_96_v__decompose,axiom,
% 2.89/1.56 ! [Xs5: list @ a] :
% 2.89/1.56 ( ( v @ ( append @ a @ Xs5 @ xsa ) )
% 2.89/1.56 = ( plus_plus @ int @ ( v @ ( append @ a @ Xs5 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ ( nil @ a ) ) ) ) @ ( v @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_97_p__less__length,axiom,
% 2.89/1.56 ord_less @ nat @ p @ ( size_size @ ( list @ a ) @ xsa ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_98_neq0__conv,axiom,
% 2.89/1.56 ! [N: nat] :
% 2.89/1.56 ( ( N
% 2.89/1.56 != ( zero_zero @ nat ) )
% 2.89/1.56 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_99_less__nat__zero__code,axiom,
% 2.89/1.56 ! [N: nat] :
% 2.89/1.56 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_100_bot__nat__0_Onot__eq__extremum,axiom,
% 2.89/1.56 ! [A2: nat] :
% 2.89/1.56 ( ( A2
% 2.89/1.56 != ( zero_zero @ nat ) )
% 2.89/1.56 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_101_order__refl,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( preorder @ A )
% 2.89/1.56 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_102_minus__poly__rev__list_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( group_add @ A )
% 2.89/1.56 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.89/1.56 ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ Xs3 @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.89/1.56 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 2.89/1.56 => ( ! [Y2: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys3 ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_103_plus__coeffs_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( comm_monoid_add @ A )
% 2.89/1.56 => ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.89/1.56 ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 2.89/1.56 => ( ! [V: A,Va: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 2.89/1.56 ( ( P @ Xs3 @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.89/1.56 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_104_minf_I8_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [T: A] :
% 2.89/1.56 ? [Z: A] :
% 2.89/1.56 ! [X3: A] :
% 2.89/1.56 ( ( ord_less @ A @ X3 @ Z )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ T @ X3 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_105_minf_I6_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [T: A] :
% 2.89/1.56 ? [Z: A] :
% 2.89/1.56 ! [X3: A] :
% 2.89/1.56 ( ( ord_less @ A @ X3 @ Z )
% 2.89/1.56 => ( ord_less_eq @ A @ X3 @ T ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_106_add__left__cancel,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel_semigroup_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ A2 @ B )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ C ) )
% 2.89/1.56 = ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_107_add__right__cancel,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel_semigroup_add @ A )
% 2.89/1.56 => ! [B: A,A2: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ B @ A2 )
% 2.89/1.56 = ( plus_plus @ A @ C @ A2 ) )
% 2.89/1.56 = ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_108_le0,axiom,
% 2.89/1.56 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_109_bot__nat__0_Oextremum,axiom,
% 2.89/1.56 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_110_add_Oleft__neutral,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( monoid_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 = A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_111_add_Oright__neutral,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( monoid_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 = A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_112_double__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 = ( A2
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_113_double__zero__sym,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ( zero_zero @ A )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.89/1.56 = ( A2
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_114_add__cancel__left__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel1352612707id_add @ A )
% 2.89/1.56 => ! [B: A,A2: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ B @ A2 )
% 2.89/1.56 = A2 )
% 2.89/1.56 = ( B
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_115_add__cancel__left__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel1352612707id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ A2 @ B )
% 2.89/1.56 = A2 )
% 2.89/1.56 = ( B
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_116_add__cancel__right__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel1352612707id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( A2
% 2.89/1.56 = ( plus_plus @ A @ B @ A2 ) )
% 2.89/1.56 = ( B
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_117_add__cancel__right__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel1352612707id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( A2
% 2.89/1.56 = ( plus_plus @ A @ A2 @ B ) )
% 2.89/1.56 = ( B
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_118_add__eq__0__iff__both__eq__0,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ X @ Y )
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 = ( ( X
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 & ( Y
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_119_zero__eq__add__iff__both__eq__0,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ( zero_zero @ A )
% 2.89/1.56 = ( plus_plus @ A @ X @ Y ) )
% 2.89/1.56 = ( ( X
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 & ( Y
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_120_add__le__cancel__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [A2: A,C: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.89/1.56 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_121_add__le__cancel__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [C: A,A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.89/1.56 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_122_add__less__cancel__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [A2: A,C: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.89/1.56 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_123_add__less__cancel__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [C: A,A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.89/1.56 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_124_append__eq__append__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
% 2.89/1.56 ( ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 | ( ( size_size @ ( list @ A ) @ Us2 )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Vs ) ) )
% 2.89/1.56 => ( ( ( append @ A @ Xs @ Us2 )
% 2.89/1.56 = ( append @ A @ Ys @ Vs ) )
% 2.89/1.56 = ( ( Xs = Ys )
% 2.89/1.56 & ( Us2 = Vs ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_125_take__all__iff,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.89/1.56 ( ( ( take @ A @ N @ Xs )
% 2.89/1.56 = Xs )
% 2.89/1.56 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_126_take__all,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,N: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
% 2.89/1.56 => ( ( take @ A @ N @ Xs )
% 2.89/1.56 = Xs ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_127_length__replicate,axiom,
% 2.89/1.56 ! [A: $tType,N: nat,X: A] :
% 2.89/1.56 ( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
% 2.89/1.56 = N ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_128_length__psums,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ! [Xs: list @ A] :
% 2.89/1.56 ( ( size_size @ ( list @ A ) @ ( descar1668888542_psums @ A @ Xs ) )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_129_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.89/1.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_130_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.89/1.56 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_131_le__add__same__cancel2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.89/1.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_132_le__add__same__cancel1,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.89/1.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_133_add__le__same__cancel2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.89/1.56 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_134_add__le__same__cancel1,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [B: A,A2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.89/1.56 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_135_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.89/1.56 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_136_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linord219039673up_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.89/1.56 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_137_less__add__same__cancel2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.89/1.56 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_138_less__add__same__cancel1,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.89/1.56 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_139_add__less__same__cancel2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.89/1.56 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_140_add__less__same__cancel1,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere516151231imp_le @ A )
% 2.89/1.56 => ! [B: A,A2: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.89/1.56 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_141_length__0__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( zero_zero @ nat ) )
% 2.89/1.56 = ( Xs
% 2.89/1.56 = ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_142_length__greater__0__conv,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A] :
% 2.89/1.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.89/1.56 = ( Xs
% 2.89/1.56 != ( nil @ A ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_143_nth__append__length,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
% 2.89/1.56 ( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.89/1.56 = X ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_144_less__eq__nat_Osimps_I1_J,axiom,
% 2.89/1.56 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_145_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ab_semigroup_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_146_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( I = J )
% 2.89/1.56 & ( K = L ) )
% 2.89/1.56 => ( ( plus_plus @ A @ I @ K )
% 2.89/1.56 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_147_le__0__eq,axiom,
% 2.89/1.56 ! [N: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.89/1.56 = ( N
% 2.89/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_148_impossible__Cons,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 => ( Xs
% 2.89/1.56 != ( cons @ A @ X @ Ys ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_149_Ex__list__of__length,axiom,
% 2.89/1.56 ! [A: $tType,N: nat] :
% 2.89/1.56 ? [Xs3: list @ A] :
% 2.89/1.56 ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.89/1.56 = N ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_150_neq__if__length__neq,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 != ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 => ( Xs != Ys ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_151_group__cancel_Oadd1,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( comm_monoid_add @ A )
% 2.89/1.56 => ! [A4: A,K: A,A2: A,B: A] :
% 2.89/1.56 ( ( A4
% 2.89/1.56 = ( plus_plus @ A @ K @ A2 ) )
% 2.89/1.56 => ( ( plus_plus @ A @ A4 @ B )
% 2.89/1.56 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_152_group__cancel_Oadd2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( comm_monoid_add @ A )
% 2.89/1.56 => ! [B3: A,K: A,B: A,A2: A] :
% 2.89/1.56 ( ( B3
% 2.89/1.56 = ( plus_plus @ A @ K @ B ) )
% 2.89/1.56 => ( ( plus_plus @ A @ A2 @ B3 )
% 2.89/1.56 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_153_bot__nat__0_Oextremum__unique,axiom,
% 2.89/1.56 ! [A2: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.89/1.56 = ( A2
% 2.89/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_154_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.89/1.56 ! [A2: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.89/1.56 => ( A2
% 2.89/1.56 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_155_add_Oassoc,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( semigroup_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_156_add_Oleft__cancel,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( group_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ A2 @ B )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ C ) )
% 2.89/1.56 = ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_157_add_Oright__cancel,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( group_add @ A )
% 2.89/1.56 => ! [B: A,A2: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ B @ A2 )
% 2.89/1.56 = ( plus_plus @ A @ C @ A2 ) )
% 2.89/1.56 = ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_158_add_Ocommute,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ab_semigroup_add @ A )
% 2.89/1.56 => ( ( plus_plus @ A )
% 2.89/1.56 = ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_159_add_Oleft__commute,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ab_semigroup_add @ A )
% 2.89/1.56 => ! [B: A,A2: A,C: A] :
% 2.89/1.56 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_160_add__left__imp__eq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel_semigroup_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ A2 @ B )
% 2.89/1.56 = ( plus_plus @ A @ A2 @ C ) )
% 2.89/1.56 => ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_161_add__right__imp__eq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( cancel_semigroup_add @ A )
% 2.89/1.56 => ! [B: A,A2: A,C: A] :
% 2.89/1.56 ( ( ( plus_plus @ A @ B @ A2 )
% 2.89/1.56 = ( plus_plus @ A @ C @ A2 ) )
% 2.89/1.56 => ( B = C ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_162_size__neq__size__imp__neq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( size @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ( size_size @ A @ X )
% 2.89/1.56 != ( size_size @ A @ Y ) )
% 2.89/1.56 => ( X != Y ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_163_less__mono__imp__le__mono,axiom,
% 2.89/1.56 ! [F: nat > nat,I: nat,J: nat] :
% 2.89/1.56 ( ! [I2: nat,J2: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I2 @ J2 )
% 2.89/1.56 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.89/1.56 => ( ( ord_less_eq @ nat @ I @ J )
% 2.89/1.56 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_164_le__neq__implies__less,axiom,
% 2.89/1.56 ! [M: nat,N: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ M @ N )
% 2.89/1.56 => ( ( M != N )
% 2.89/1.56 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_165_less__or__eq__imp__le,axiom,
% 2.89/1.56 ! [M: nat,N: nat] :
% 2.89/1.56 ( ( ( ord_less @ nat @ M @ N )
% 2.89/1.56 | ( M = N ) )
% 2.89/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_166_le__eq__less__or__eq,axiom,
% 2.89/1.56 ( ( ord_less_eq @ nat )
% 2.89/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ M2 @ N2 )
% 2.89/1.56 | ( M2 = N2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_167_less__imp__le__nat,axiom,
% 2.89/1.56 ! [M: nat,N: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ M @ N )
% 2.89/1.56 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_168_ex__least__nat__le,axiom,
% 2.89/1.56 ! [P: nat > $o,N: nat] :
% 2.89/1.56 ( ( P @ N )
% 2.89/1.56 => ( ~ ( P @ ( zero_zero @ nat ) )
% 2.89/1.56 => ? [K2: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.89/1.56 & ! [I3: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I3 @ K2 )
% 2.89/1.56 => ~ ( P @ I3 ) )
% 2.89/1.56 & ( P @ K2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_169_nat__less__le,axiom,
% 2.89/1.56 ( ( ord_less @ nat )
% 2.89/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.89/1.56 & ( M2 != N2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_170_comm__monoid__add__class_Oadd__0,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( comm_monoid_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 = A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_171_add_Ocomm__neutral,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( comm_monoid_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 = A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_172_add_Ogroup__left__neutral,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( group_add @ A )
% 2.89/1.56 => ! [A2: A] :
% 2.89/1.56 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 = A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_173_add__le__imp__le__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [A2: A,C: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_174_add__le__imp__le__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [C: A,A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_175_le__iff__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ( ( ord_less_eq @ A )
% 2.89/1.56 = ( ^ [A5: A,B4: A] :
% 2.89/1.56 ? [C2: A] :
% 2.89/1.56 ( B4
% 2.89/1.56 = ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_176_add__right__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_177_less__eqE,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ~ ! [C3: A] :
% 2.89/1.56 ( B
% 2.89/1.56 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_178_add__left__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_179_add__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A,D: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less_eq @ A @ C @ D )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_180_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.89/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_181_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( I = J )
% 2.89/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_182_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere779506340up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.89/1.56 & ( K = L ) )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_183_add__less__imp__less__right,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [A2: A,C: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.89/1.56 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_184_add__less__imp__less__left,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere236663937imp_le @ A )
% 2.89/1.56 => ! [C: A,A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.89/1.56 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_185_add__strict__right__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ B )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_186_add__strict__left__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ B )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_187_add__strict__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( strict2144017051up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A,D: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less @ A @ C @ D )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_188_add__mono__thms__linordered__field_I1_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.89/1.56 & ( K = L ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_189_add__mono__thms__linordered__field_I2_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( I = J )
% 2.89/1.56 & ( ord_less @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_190_add__mono__thms__linordered__field_I5_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.89/1.56 & ( ord_less @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_191_list_Osize_I3_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 2.89/1.56 = ( zero_zero @ nat ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_192_length__induct,axiom,
% 2.89/1.56 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.89/1.56 ( ! [Xs3: list @ A] :
% 2.89/1.56 ( ! [Ys6: list @ A] :
% 2.89/1.56 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 2.89/1.56 => ( P @ Ys6 ) )
% 2.89/1.56 => ( P @ Xs3 ) )
% 2.89/1.56 => ( P @ Xs ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_193_nth__take__lemma,axiom,
% 2.89/1.56 ! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.89/1.56 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 => ( ! [I2: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I2 @ K )
% 2.89/1.56 => ( ( nth @ A @ Xs @ I2 )
% 2.89/1.56 = ( nth @ A @ Ys @ I2 ) ) )
% 2.89/1.56 => ( ( take @ A @ K @ Xs )
% 2.89/1.56 = ( take @ A @ K @ Ys ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_194_add__nonpos__eq__0__iff,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ( plus_plus @ A @ X @ Y )
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 = ( ( X
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 & ( Y
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_195_add__nonneg__eq__0__iff,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.89/1.56 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.89/1.56 => ( ( ( plus_plus @ A @ X @ Y )
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 = ( ( X
% 2.89/1.56 = ( zero_zero @ A ) )
% 2.89/1.56 & ( Y
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_196_add__nonpos__nonpos,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_197_add__nonneg__nonneg,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_198_add__increasing2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [C: A,B: A,A2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.89/1.56 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_199_add__decreasing2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [C: A,A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_200_add__increasing,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ C )
% 2.89/1.56 => ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_201_add__decreasing,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,C: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less_eq @ A @ C @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_202_pos__add__strict,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( strict797366125id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less @ A @ B @ C )
% 2.89/1.56 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_203_canonically__ordered__monoid__add__class_OlessE,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( canoni770627133id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ B )
% 2.89/1.56 => ~ ! [C3: A] :
% 2.89/1.56 ( ( B
% 2.89/1.56 = ( plus_plus @ A @ A2 @ C3 ) )
% 2.89/1.56 => ( C3
% 2.89/1.56 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_204_add__pos__pos,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.89/1.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_205_add__neg__neg,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_206_add__less__le__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A,D: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less_eq @ A @ C @ D )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_207_add__le__less__mono,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A,D: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less @ A @ C @ D )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_208_add__mono__thms__linordered__field_I3_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less @ A @ I @ J )
% 2.89/1.56 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_209_add__mono__thms__linordered__field_I4_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere223160158up_add @ A )
% 2.89/1.56 => ! [I: A,J: A,K: A,L: A] :
% 2.89/1.56 ( ( ( ord_less_eq @ A @ I @ J )
% 2.89/1.56 & ( ord_less @ A @ K @ L ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_210_list__induct4,axiom,
% 2.89/1.56 ! [C4: $tType,A: $tType,B2: $tType,D2: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,Ws: list @ D2,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > ( list @ D2 ) > $o] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ B2 ) @ Ys )
% 2.89/1.56 = ( size_size @ ( list @ C4 ) @ Zs ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ C4 ) @ Zs )
% 2.89/1.56 = ( size_size @ ( list @ D2 ) @ Ws ) )
% 2.89/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) @ ( nil @ D2 ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4,W: D2,Ws2: list @ D2] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
% 2.89/1.56 = ( size_size @ ( list @ C4 ) @ Zs2 ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ C4 ) @ Zs2 )
% 2.89/1.56 = ( size_size @ ( list @ D2 ) @ Ws2 ) )
% 2.89/1.56 => ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) @ ( cons @ D2 @ W @ Ws2 ) ) ) ) ) )
% 2.89/1.56 => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_211_list__induct3,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType,C4: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > $o] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ B2 ) @ Ys )
% 2.89/1.56 = ( size_size @ ( list @ C4 ) @ Zs ) )
% 2.89/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 2.89/1.56 => ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
% 2.89/1.56 = ( size_size @ ( list @ C4 ) @ Zs2 ) )
% 2.89/1.56 => ( ( P @ Xs3 @ Ys3 @ Zs2 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) ) ) ) )
% 2.89/1.56 => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_212_list__induct2,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType,Xs: list @ A,Ys: list @ B2,P: ( list @ A ) > ( list @ B2 ) > $o] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys ) )
% 2.89/1.56 => ( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
% 2.89/1.56 => ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 2.89/1.56 = ( size_size @ ( list @ B2 ) @ Ys3 ) )
% 2.89/1.56 => ( ( P @ Xs3 @ Ys3 )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) ) )
% 2.89/1.56 => ( P @ Xs @ Ys ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_213_psums_Oinduct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 2.89/1.56 => ( P @ A0 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_214_list__eq__iff__nth__eq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ^ [Y4: list @ A,Z2: list @ A] : ( Y4 = Z2 ) )
% 2.89/1.56 = ( ^ [Xs2: list @ A,Ys4: list @ A] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Ys4 ) )
% 2.89/1.56 & ! [I4: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.89/1.56 => ( ( nth @ A @ Xs2 @ I4 )
% 2.89/1.56 = ( nth @ A @ Ys4 @ I4 ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_215_Skolem__list__nth,axiom,
% 2.89/1.56 ! [A: $tType,K: nat,P: nat > A > $o] :
% 2.89/1.56 ( ( ! [I4: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I4 @ K )
% 2.89/1.56 => ? [X4: A] : ( P @ I4 @ X4 ) ) )
% 2.89/1.56 = ( ? [Xs2: list @ A] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.89/1.56 = K )
% 2.89/1.56 & ! [I4: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I4 @ K )
% 2.89/1.56 => ( P @ I4 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_216_nth__equalityI,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 => ( ! [I2: nat] :
% 2.89/1.56 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.89/1.56 => ( ( nth @ A @ Xs @ I2 )
% 2.89/1.56 = ( nth @ A @ Ys @ I2 ) ) )
% 2.89/1.56 => ( Xs = Ys ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_217_psums_Osimps_I3_J,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( plus @ A )
% 2.89/1.56 => ! [X: A,Y: A,Xs: list @ A] :
% 2.89/1.56 ( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
% 2.89/1.56 = ( cons @ A @ X @ ( descar1668888542_psums @ A @ ( cons @ A @ ( plus_plus @ A @ X @ Y ) @ Xs ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_218_add__strict__increasing2,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere1818651114id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less @ A @ B @ C )
% 2.89/1.56 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_219_add__strict__increasing,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere1818651114id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ C )
% 2.89/1.56 => ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_220_add__pos__nonneg,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.89/1.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_221_add__nonpos__neg,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_222_add__nonneg__pos,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.89/1.56 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.89/1.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_223_add__neg__nonpos,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ordere216010020id_add @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
% 2.89/1.56 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_224_sign__changes__induct,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ( plus @ A )
% 2.89/1.56 & ( zero @ A ) )
% 2.89/1.56 => ! [P: ( list @ A ) > $o,Xs: list @ A] :
% 2.89/1.56 ( ( P @ ( nil @ A ) )
% 2.89/1.56 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.89/1.56 => ( ! [Xs3: list @ A] :
% 2.89/1.56 ( ( P @ Xs3 )
% 2.89/1.56 => ( P @ ( cons @ A @ ( zero_zero @ A ) @ Xs3 ) ) )
% 2.89/1.56 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.89/1.56 ( ( X2
% 2.89/1.56 != ( zero_zero @ A ) )
% 2.89/1.56 => ( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
% 2.89/1.56 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 2.89/1.56 => ( P @ Xs ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_225_same__length__different,axiom,
% 2.89/1.56 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.89/1.56 ( ( Xs != Ys )
% 2.89/1.56 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.89/1.56 = ( size_size @ ( list @ A ) @ Ys ) )
% 2.89/1.56 => ? [Pre: list @ A,X2: A,Xs4: list @ A,Y2: A,Ys2: list @ A] :
% 2.89/1.56 ( ( X2 != Y2 )
% 2.89/1.56 & ( Xs
% 2.89/1.56 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Xs4 ) ) )
% 2.89/1.56 & ( Ys
% 2.89/1.56 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys2 ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_226_dual__order_Oantisym,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [B: A,A2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ B @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( A2 = B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_227_dual__order_Oeq__iff,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.89/1.56 = ( ^ [A5: A,B4: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.56 & ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_228_dual__order_Otrans,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [B: A,A2: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ B @ A2 )
% 2.89/1.56 => ( ( ord_less_eq @ A @ C @ B )
% 2.89/1.56 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_229_linorder__wlog,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [P: A > A > $o,A2: A,B: A] :
% 2.89/1.56 ( ! [A3: A,B5: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A3 @ B5 )
% 2.89/1.56 => ( P @ A3 @ B5 ) )
% 2.89/1.56 => ( ! [A3: A,B5: A] :
% 2.89/1.56 ( ( P @ B5 @ A3 )
% 2.89/1.56 => ( P @ A3 @ B5 ) )
% 2.89/1.56 => ( P @ A2 @ B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_230_dual__order_Orefl,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_231_order__trans,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( preorder @ A )
% 2.89/1.56 => ! [X: A,Y: A,Z3: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.89/1.56 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_232_order__class_Oorder_Oantisym,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [A2: A,B: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.89/1.56 => ( A2 = B ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_233_ord__le__eq__trans,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ord @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( B = C )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_234_ord__eq__le__trans,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( ord @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( A2 = B )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ C )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_235_order__class_Oorder_Oeq__iff,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.89/1.56 = ( ^ [A5: A,B4: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.56 & ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_236_antisym__conv,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [Y: A,X: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ Y @ X )
% 2.89/1.56 => ( ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 = ( X = Y ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_237_le__cases3,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [X: A,Y: A,Z3: A] :
% 2.89/1.56 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 2.89/1.56 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 2.89/1.56 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 2.89/1.56 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.89/1.56 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 2.89/1.56 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 2.89/1.56 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_238_order_Otrans,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [A2: A,B: A,C: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less_eq @ A @ B @ C )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_239_le__cases,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_240_eq__refl,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( preorder @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( X = Y )
% 2.89/1.56 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_241_linear,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linorder @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_242_antisym,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [X: A,Y: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X @ Y )
% 2.89/1.56 => ( ( ord_less_eq @ A @ Y @ X )
% 2.89/1.56 => ( X = Y ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_243_eq__iff,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.89/1.56 = ( ^ [X5: A,Y3: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X5 @ Y3 )
% 2.89/1.56 & ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_244_ord__le__eq__subst,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType] :
% 2.89/1.56 ( ( ( ord @ B2 )
% 2.89/1.56 & ( ord @ A ) )
% 2.89/1.56 => ! [A2: A,B: A,F: A > B2,C: B2] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ( F @ B )
% 2.89/1.56 = C )
% 2.89/1.56 => ( ! [X2: A,Y2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.56 => ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.56 => ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_245_ord__eq__le__subst,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType] :
% 2.89/1.56 ( ( ( ord @ B2 )
% 2.89/1.56 & ( ord @ A ) )
% 2.89/1.56 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 2.89/1.56 ( ( A2
% 2.89/1.56 = ( F @ B ) )
% 2.89/1.56 => ( ( ord_less_eq @ B2 @ B @ C )
% 2.89/1.56 => ( ! [X2: B2,Y2: B2] :
% 2.89/1.56 ( ( ord_less_eq @ B2 @ X2 @ Y2 )
% 2.89/1.56 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_246_order__subst2,axiom,
% 2.89/1.56 ! [A: $tType,C4: $tType] :
% 2.89/1.56 ( ( ( order @ C4 )
% 2.89/1.56 & ( order @ A ) )
% 2.89/1.56 => ! [A2: A,B: A,F: A > C4,C: C4] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ B )
% 2.89/1.56 => ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
% 2.89/1.56 => ( ! [X2: A,Y2: A] :
% 2.89/1.56 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.56 => ( ord_less_eq @ C4 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.56 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_247_order__subst1,axiom,
% 2.89/1.56 ! [A: $tType,B2: $tType] :
% 2.89/1.56 ( ( ( order @ B2 )
% 2.89/1.56 & ( order @ A ) )
% 2.89/1.56 => ! [A2: A,F: B2 > A,B: B2,C: B2] :
% 2.89/1.56 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 2.89/1.56 => ( ( ord_less_eq @ B2 @ B @ C )
% 2.89/1.56 => ( ! [X2: B2,Y2: B2] :
% 2.89/1.56 ( ( ord_less_eq @ B2 @ X2 @ Y2 )
% 2.89/1.56 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.89/1.56 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_248_le__fun__def,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType] :
% 2.89/1.56 ( ( ord @ B2 )
% 2.89/1.56 => ( ( ord_less_eq @ ( A > B2 ) )
% 2.89/1.56 = ( ^ [F3: A > B2,G2: A > B2] :
% 2.89/1.56 ! [X5: A] : ( ord_less_eq @ B2 @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_249_le__funI,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType] :
% 2.89/1.56 ( ( ord @ B2 )
% 2.89/1.56 => ! [F: A > B2,G: A > B2] :
% 2.89/1.56 ( ! [X2: A] : ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.89/1.56 => ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_250_le__funE,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType] :
% 2.89/1.56 ( ( ord @ B2 )
% 2.89/1.56 => ! [F: A > B2,G: A > B2,X: A] :
% 2.89/1.56 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 2.89/1.56 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_251_le__funD,axiom,
% 2.89/1.56 ! [B2: $tType,A: $tType] :
% 2.89/1.56 ( ( ord @ B2 )
% 2.89/1.56 => ! [F: A > B2,G: A > B2,X: A] :
% 2.89/1.56 ( ( ord_less_eq @ ( A > B2 ) @ F @ G )
% 2.89/1.56 => ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(fact_252_dual__order_Ostrict__implies__not__eq,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( order @ A )
% 2.89/1.56 => ! [B: A,A2: A] :
% 2.89/1.56 ( ( ord_less @ A @ B @ A2 )
% 2.89/1.56 => ( A2 != B ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( type @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oplus,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( plus @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( zero @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ord @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( order @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( group_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( monoid_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( linorder @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( preorder @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( semigroup_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( comm_monoid_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ab_semigroup_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( cancel_semigroup_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__comm__monoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( cancel1352612707id_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Olinordered__ab__group__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( linord219039673up_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__comm__monoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere216010020id_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere779506340up_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere1818651114id_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( strict797366125id_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere236663937imp_le @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere223160158up_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( strict2144017051up_add @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.89/1.56 ! [A: $tType] :
% 2.89/1.56 ( ( linordered_idom @ A )
% 2.89/1.56 => ( ordere516151231imp_le @ A ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.89/1.56 ! [A6: $tType,A7: $tType] :
% 2.89/1.56 ( ( preorder @ A7 )
% 2.89/1.56 => ( preorder @ ( A6 > A7 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.89/1.56 ! [A6: $tType,A7: $tType] :
% 2.89/1.56 ( ( order @ A7 )
% 2.89/1.56 => ( order @ ( A6 > A7 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_fun___Orderings_Oord,axiom,
% 2.89/1.56 ! [A6: $tType,A7: $tType] :
% 2.89/1.56 ( ( ord @ A7 )
% 2.89/1.56 => ( ord @ ( A6 > A7 ) ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.89/1.56 ordere516151231imp_le @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.89/1.56 strict2144017051up_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.89/1.56 ordere223160158up_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.89/1.56 ordere236663937imp_le @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 2.89/1.56 strict797366125id_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 2.89/1.56 ordere1818651114id_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 2.89/1.56 ordere779506340up_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
% 2.89/1.56 ordere216010020id_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 2.89/1.56 linord219039673up_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.89/1.56 cancel1352612707id_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 2.89/1.56 cancel_semigroup_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 2.89/1.56 ab_semigroup_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 2.89/1.56 comm_monoid_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 2.89/1.56 semigroup_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.89/1.56 preorder @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.89/1.56 linorder @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 2.89/1.56 monoid_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.89/1.56 group_add @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.89/1.56 order @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.89/1.56 ord @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.89/1.56 zero @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Int_Oint___Groups_Oplus,axiom,
% 2.89/1.56 plus @ int ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
% 2.89/1.56 ordere516151231imp_le @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_5,axiom,
% 2.89/1.56 strict2144017051up_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_6,axiom,
% 2.89/1.56 ordere223160158up_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_7,axiom,
% 2.89/1.56 ordere236663937imp_le @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_8,axiom,
% 2.89/1.56 strict797366125id_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_9,axiom,
% 2.89/1.56 ordere1818651114id_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.89/1.56 canoni770627133id_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_10,axiom,
% 2.89/1.56 ordere779506340up_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_11,axiom,
% 2.89/1.56 ordere216010020id_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_12,axiom,
% 2.89/1.56 cancel1352612707id_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_13,axiom,
% 2.89/1.56 cancel_semigroup_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_14,axiom,
% 2.89/1.56 ab_semigroup_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_15,axiom,
% 2.89/1.56 comm_monoid_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Osemigroup__add_16,axiom,
% 2.89/1.56 semigroup_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
% 2.89/1.56 preorder @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Orderings_Olinorder_18,axiom,
% 2.89/1.56 linorder @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Omonoid__add_19,axiom,
% 2.89/1.56 monoid_add @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Orderings_Oorder_20,axiom,
% 2.89/1.56 order @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Orderings_Oord_21,axiom,
% 2.89/1.56 ord @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Ozero_22,axiom,
% 2.89/1.56 zero @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Groups_Oplus_23,axiom,
% 2.89/1.56 plus @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.89/1.56 size @ nat ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_HOL_Obool___Orderings_Opreorder_24,axiom,
% 2.89/1.56 preorder @ $o ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_HOL_Obool___Orderings_Olinorder_25,axiom,
% 2.89/1.56 linorder @ $o ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_HOL_Obool___Orderings_Oorder_26,axiom,
% 2.89/1.56 order @ $o ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
% 2.89/1.56 ord @ $o ).
% 2.89/1.56
% 2.89/1.56 thf(tcon_List_Olist___Nat_Osize_28,axiom,
% 2.89/1.56 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 2.89/1.56
% 2.89/1.56 thf(tfree_0,hypothesis,
% 2.89/1.56 linordered_idom @ a ).
% 2.89/1.56
% 2.89/1.56 thf(conj_0,conjecture,
% 2.89/1.56 ( ( descar1668888542_psums @ a @ xsa )
% 2.89/1.56 = ( append @ a @ ( replicate @ a @ p @ ( zero_zero @ a ) ) @ ( descar1668888542_psums @ a @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
% 2.89/1.56 ------------------------
% 2.89/1.59 % (10796)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_10522 for (2999ds/27Mi)
% 2.89/1.59 % (10797)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/2Mi)
% 2.89/1.59 % (10794)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/183Mi)
% 2.89/1.59 % (10800)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/18Mi)
% 2.89/1.59 % (10797)Instruction limit reached!
% 2.89/1.59 % (10797)------------------------------
% 2.89/1.59 % (10797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.59 % (10797)Termination reason: Unknown
% 2.89/1.59 % (10797)Termination phase: shuffling
% 2.89/1.59
% 2.89/1.59 % (10797)Memory used [KB]: 1407
% 2.89/1.59 % (10797)Time elapsed: 0.003 s
% 2.89/1.59 % (10797)Instructions burned: 3 (million)
% 2.89/1.59 % (10797)------------------------------
% 2.89/1.59 % (10797)------------------------------
% 2.89/1.60 % (10795)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_10522 for (2999ds/4Mi)
% 2.95/1.60 % (10800)Instruction limit reached!
% 2.95/1.60 % (10800)------------------------------
% 2.95/1.60 % (10800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (10800)Termination reason: Unknown
% 2.95/1.60 % (10800)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (10800)Memory used [KB]: 1791
% 2.95/1.60 % (10800)Time elapsed: 0.008 s
% 2.95/1.60 % (10800)Instructions burned: 19 (million)
% 2.95/1.60 % (10800)------------------------------
% 2.95/1.60 % (10800)------------------------------
% 2.95/1.60 % (10798)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_10522 for (2999ds/2Mi)
% 2.95/1.60 % (10799)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_10522 for (2999ds/275Mi)
% 2.95/1.60 % (10796)Instruction limit reached!
% 2.95/1.60 % (10796)------------------------------
% 2.95/1.60 % (10796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (10796)Termination reason: Unknown
% 2.95/1.60 % (10796)Termination phase: Property scanning
% 2.95/1.60
% 2.95/1.60 % (10796)Memory used [KB]: 1791
% 2.95/1.60 % (10796)Time elapsed: 0.010 s
% 2.95/1.60 % (10796)Instructions burned: 28 (million)
% 2.95/1.60 % (10796)------------------------------
% 2.95/1.60 % (10796)------------------------------
% 2.95/1.60 % (10798)Instruction limit reached!
% 2.95/1.60 % (10798)------------------------------
% 2.95/1.60 % (10798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (10798)Termination reason: Unknown
% 2.95/1.60 % (10798)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (10798)Memory used [KB]: 1407
% 2.95/1.60 % (10798)Time elapsed: 0.004 s
% 2.95/1.60 % (10798)Instructions burned: 3 (million)
% 2.95/1.60 % (10798)------------------------------
% 2.95/1.60 % (10798)------------------------------
% 2.95/1.60 % (10795)Instruction limit reached!
% 2.95/1.60 % (10795)------------------------------
% 2.95/1.60 % (10795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (10795)Termination reason: Unknown
% 2.95/1.60 % (10795)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (10795)Memory used [KB]: 1407
% 2.95/1.60 % (10795)Time elapsed: 0.005 s
% 2.95/1.60 % (10795)Instructions burned: 5 (million)
% 2.95/1.60 % (10795)------------------------------
% 2.95/1.60 % (10795)------------------------------
% 2.95/1.60 % (10801)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/3Mi)
% 2.95/1.60 % (10801)Instruction limit reached!
% 2.95/1.60 % (10801)------------------------------
% 2.95/1.60 % (10801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (10801)Termination reason: Unknown
% 2.95/1.60 % (10801)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (10801)Memory used [KB]: 1407
% 2.95/1.60 % (10801)Time elapsed: 0.002 s
% 2.95/1.60 % (10801)Instructions burned: 3 (million)
% 2.95/1.60 % (10801)------------------------------
% 2.95/1.60 % (10801)------------------------------
% 2.95/1.61 % (10803)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_10522 for (2999ds/15Mi)
% 2.95/1.61 % (10802)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_10522 for (2999ds/37Mi)
% 2.95/1.61 % (10806)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/7Mi)
% 2.95/1.61 % (10803)Instruction limit reached!
% 2.95/1.61 % (10803)------------------------------
% 2.95/1.61 % (10803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (10803)Termination reason: Unknown
% 2.95/1.61 % (10803)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (10803)Memory used [KB]: 1663
% 2.95/1.61 % (10803)Time elapsed: 0.006 s
% 2.95/1.61 % (10803)Instructions burned: 15 (million)
% 2.95/1.61 % (10803)------------------------------
% 2.95/1.61 % (10803)------------------------------
% 2.95/1.61 % (10806)Instruction limit reached!
% 2.95/1.61 % (10806)------------------------------
% 2.95/1.61 % (10806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (10806)Termination reason: Unknown
% 2.95/1.61 % (10806)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (10806)Memory used [KB]: 1535
% 2.95/1.61 % (10806)Time elapsed: 0.004 s
% 2.95/1.61 % (10806)Instructions burned: 8 (million)
% 2.95/1.61 % (10806)------------------------------
% 2.95/1.61 % (10806)------------------------------
% 2.95/1.62 % (10804)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_10522 for (2999ds/3Mi)
% 2.95/1.62 % (10805)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_10522 for (2999ds/1041Mi)
% 2.95/1.62 % (10804)Instruction limit reached!
% 2.95/1.62 % (10804)------------------------------
% 2.95/1.62 % (10804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.62 % (10804)Termination reason: Unknown
% 2.95/1.62 % (10804)Termination phase: shuffling
% 2.95/1.62
% 2.95/1.62 % (10804)Memory used [KB]: 1407
% 2.95/1.62 % (10804)Time elapsed: 0.004 s
% 2.95/1.62 % (10804)Instructions burned: 4 (million)
% 2.95/1.62 % (10804)------------------------------
% 2.95/1.62 % (10804)------------------------------
% 2.95/1.62 % (10802)Instruction limit reached!
% 2.95/1.62 % (10802)------------------------------
% 2.95/1.62 % (10802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.62 % (10802)Termination reason: Unknown
% 2.95/1.62 % (10802)Termination phase: shuffling
% 2.95/1.62
% 2.95/1.62 % (10802)Memory used [KB]: 2174
% 2.95/1.62 % (10802)Time elapsed: 0.013 s
% 2.95/1.62 % (10802)Instructions burned: 38 (million)
% 2.95/1.62 % (10802)------------------------------
% 2.95/1.62 % (10802)------------------------------
% 2.95/1.62 % (10807)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/16Mi)
% 2.95/1.62 % (10808)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_10522 for (2999ds/3Mi)
% 2.95/1.62 % (10808)Instruction limit reached!
% 2.95/1.62 % (10808)------------------------------
% 2.95/1.62 % (10808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.62 % (10808)Termination reason: Unknown
% 2.95/1.62 % (10808)Termination phase: shuffling
% 2.95/1.62
% 2.95/1.62 % (10808)Memory used [KB]: 1407
% 2.95/1.62 % (10808)Time elapsed: 0.002 s
% 2.95/1.62 % (10808)Instructions burned: 3 (million)
% 2.95/1.62 % (10808)------------------------------
% 2.95/1.62 % (10808)------------------------------
% 2.95/1.63 % (10807)Instruction limit reached!
% 2.95/1.63 % (10807)------------------------------
% 2.95/1.63 % (10807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (10807)Termination reason: Unknown
% 2.95/1.63 % (10807)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (10807)Memory used [KB]: 1663
% 2.95/1.63 % (10807)Time elapsed: 0.007 s
% 2.95/1.63 % (10807)Instructions burned: 18 (million)
% 2.95/1.63 % (10807)------------------------------
% 2.95/1.63 % (10807)------------------------------
% 2.95/1.63 % (10810)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/7Mi)
% 2.95/1.63 % (10810)Instruction limit reached!
% 2.95/1.63 % (10810)------------------------------
% 2.95/1.63 % (10810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (10810)Termination reason: Unknown
% 2.95/1.63 % (10810)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (10810)Memory used [KB]: 1535
% 2.95/1.63 % (10810)Time elapsed: 0.004 s
% 2.95/1.63 % (10810)Instructions burned: 7 (million)
% 2.95/1.63 % (10810)------------------------------
% 2.95/1.63 % (10810)------------------------------
% 2.95/1.63 % (10811)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/3Mi)
% 2.95/1.63 % (10811)Instruction limit reached!
% 2.95/1.63 % (10811)------------------------------
% 2.95/1.63 % (10811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (10811)Termination reason: Unknown
% 2.95/1.63 % (10811)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (10811)Memory used [KB]: 1407
% 2.95/1.63 % (10811)Time elapsed: 0.002 s
% 2.95/1.63 % (10811)Instructions burned: 4 (million)
% 2.95/1.63 % (10811)------------------------------
% 2.95/1.63 % (10811)------------------------------
% 2.95/1.64 % (10809)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_10522 for (2999ds/3Mi)
% 2.95/1.64 % (10812)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/4Mi)
% 2.95/1.64 % (10809)Instruction limit reached!
% 2.95/1.64 % (10809)------------------------------
% 2.95/1.64 % (10809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (10809)Termination reason: Unknown
% 2.95/1.64 % (10809)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (10809)Memory used [KB]: 1407
% 2.95/1.64 % (10809)Time elapsed: 0.004 s
% 2.95/1.64 % (10809)Instructions burned: 3 (million)
% 2.95/1.64 % (10809)------------------------------
% 2.95/1.64 % (10809)------------------------------
% 2.95/1.64 % (10812)Instruction limit reached!
% 2.95/1.64 % (10812)------------------------------
% 2.95/1.64 % (10812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (10812)Termination reason: Unknown
% 2.95/1.64 % (10812)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (10812)Memory used [KB]: 1407
% 2.95/1.64 % (10812)Time elapsed: 0.003 s
% 2.95/1.64 % (10812)Instructions burned: 6 (million)
% 2.95/1.64 % (10812)------------------------------
% 2.95/1.64 % (10812)------------------------------
% 2.95/1.64 % (10813)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/18Mi)
% 2.95/1.64 % (10814)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/710Mi)
% 2.95/1.65 % (10813)Instruction limit reached!
% 2.95/1.65 % (10813)------------------------------
% 2.95/1.65 % (10813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (10813)Termination reason: Unknown
% 2.95/1.65 % (10813)Termination phase: Property scanning
% 2.95/1.65
% 2.95/1.65 % (10813)Memory used [KB]: 1791
% 2.95/1.65 % (10813)Time elapsed: 0.008 s
% 2.95/1.65 % (10813)Instructions burned: 20 (million)
% 2.95/1.65 % (10813)------------------------------
% 2.95/1.65 % (10813)------------------------------
% 2.95/1.65 % (10816)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/902Mi)
% 2.95/1.65 % (10799)Refutation not found, incomplete strategy
% 2.95/1.65 % (10799)------------------------------
% 2.95/1.65 % (10799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (10799)Termination reason: Refutation not found, incomplete strategy
% 2.95/1.65
% 2.95/1.65
% 2.95/1.65 % (10799)Memory used [KB]: 6908
% 2.95/1.65 % (10799)Time elapsed: 0.054 s
% 2.95/1.65 % (10799)Instructions burned: 99 (million)
% 2.95/1.65 % (10799)------------------------------
% 2.95/1.65 % (10799)------------------------------
% 2.95/1.65 % (10794)Instruction limit reached!
% 2.95/1.65 % (10794)------------------------------
% 2.95/1.65 % (10794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (10794)Termination reason: Unknown
% 2.95/1.65 % (10794)Termination phase: Saturation
% 2.95/1.65
% 2.95/1.65 % (10794)Memory used [KB]: 7291
% 2.95/1.65 % (10794)Time elapsed: 0.061 s
% 2.95/1.65 % (10794)Instructions burned: 185 (million)
% 2.95/1.65 % (10794)------------------------------
% 2.95/1.65 % (10794)------------------------------
% 2.95/1.65 % (10815)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/6Mi)
% 2.95/1.66 % (10817)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/21Mi)
% 2.95/1.66 % (10819)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/6Mi)
% 2.95/1.66 % (10815)Instruction limit reached!
% 2.95/1.66 % (10815)------------------------------
% 2.95/1.66 % (10815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (10815)Termination reason: Unknown
% 2.95/1.66 % (10815)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (10815)Memory used [KB]: 1407
% 2.95/1.66 % (10815)Time elapsed: 0.006 s
% 2.95/1.66 % (10815)Instructions burned: 6 (million)
% 2.95/1.66 % (10815)------------------------------
% 2.95/1.66 % (10815)------------------------------
% 2.95/1.66 % (10818)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/5Mi)
% 2.95/1.66 % (10819)Instruction limit reached!
% 2.95/1.66 % (10819)------------------------------
% 2.95/1.66 % (10819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (10819)Termination reason: Unknown
% 2.95/1.66 % (10819)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (10819)Memory used [KB]: 1407
% 2.95/1.66 % (10819)Time elapsed: 0.003 s
% 2.95/1.66 % (10819)Instructions burned: 7 (million)
% 2.95/1.66 % (10819)------------------------------
% 2.95/1.66 % (10819)------------------------------
% 2.95/1.66 % (10818)Instruction limit reached!
% 2.95/1.66 % (10818)------------------------------
% 2.95/1.66 % (10818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (10818)Termination reason: Unknown
% 2.95/1.66 % (10818)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (10818)Memory used [KB]: 1407
% 2.95/1.66 % (10818)Time elapsed: 0.004 s
% 2.95/1.66 % (10818)Instructions burned: 7 (million)
% 2.95/1.66 % (10818)------------------------------
% 2.95/1.66 % (10818)------------------------------
% 2.95/1.66 % (10817)Instruction limit reached!
% 2.95/1.66 % (10817)------------------------------
% 2.95/1.66 % (10817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (10817)Termination reason: Unknown
% 2.95/1.66 % (10817)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (10817)Memory used [KB]: 1791
% 2.95/1.66 % (10817)Time elapsed: 0.008 s
% 2.95/1.66 % (10817)Instructions burned: 21 (million)
% 2.95/1.66 % (10817)------------------------------
% 2.95/1.66 % (10817)------------------------------
% 2.95/1.67 % (10821)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/779Mi)
% 2.95/1.67 % (10822)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/19Mi)
% 2.95/1.67 % (10823)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/879Mi)
% 2.95/1.68 % (10820)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_10522 for (2999ds/377Mi)
% 2.95/1.68 % (10822)Instruction limit reached!
% 2.95/1.68 % (10822)------------------------------
% 2.95/1.68 % (10822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.68 % (10822)Termination reason: Unknown
% 2.95/1.68 % (10822)Termination phase: shuffling
% 2.95/1.68
% 2.95/1.68 % (10822)Memory used [KB]: 1663
% 2.95/1.68 % (10822)Time elapsed: 0.008 s
% 2.95/1.68 % (10822)Instructions burned: 21 (million)
% 2.95/1.68 % (10822)------------------------------
% 2.95/1.68 % (10822)------------------------------
% 2.95/1.69 % (10816)Refutation not found, incomplete strategy
% 2.95/1.69 % (10816)------------------------------
% 2.95/1.69 % (10816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.69 % (10816)Termination reason: Refutation not found, incomplete strategy
% 2.95/1.69
% 2.95/1.69
% 2.95/1.69 % (10816)Memory used [KB]: 7419
% 2.95/1.69 % (10816)Time elapsed: 0.039 s
% 2.95/1.69 % (10816)Instructions burned: 129 (million)
% 2.95/1.69 % (10816)------------------------------
% 2.95/1.69 % (10816)------------------------------
% 2.95/1.69 % (10824)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/17Mi)
% 2.95/1.69 % (10824)Instruction limit reached!
% 2.95/1.69 % (10824)------------------------------
% 2.95/1.69 % (10824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.69 % (10824)Termination reason: Unknown
% 2.95/1.69 % (10824)Termination phase: shuffling
% 2.95/1.69
% 2.95/1.69 % (10824)Memory used [KB]: 1663
% 2.95/1.69 % (10824)Time elapsed: 0.008 s
% 2.95/1.69 % (10824)Instructions burned: 18 (million)
% 2.95/1.69 % (10824)------------------------------
% 2.95/1.69 % (10824)------------------------------
% 2.95/1.70 % (10825)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/3Mi)
% 2.95/1.70 % (10825)Instruction limit reached!
% 2.95/1.70 % (10825)------------------------------
% 2.95/1.70 % (10825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (10825)Termination reason: Unknown
% 2.95/1.70 % (10825)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (10825)Memory used [KB]: 1407
% 2.95/1.70 % (10825)Time elapsed: 0.002 s
% 2.95/1.70 % (10825)Instructions burned: 3 (million)
% 2.95/1.70 % (10825)------------------------------
% 2.95/1.70 % (10825)------------------------------
% 2.95/1.70 % (10826)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/30Mi)
% 2.95/1.71 % (10827)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/127Mi)
% 2.95/1.71 % (10826)Instruction limit reached!
% 2.95/1.71 % (10826)------------------------------
% 2.95/1.71 % (10826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.71 % (10826)Termination reason: Unknown
% 2.95/1.71 % (10826)Termination phase: Property scanning
% 2.95/1.71
% 2.95/1.71 % (10826)Memory used [KB]: 1791
% 2.95/1.71 % (10826)Time elapsed: 0.012 s
% 2.95/1.71 % (10826)Instructions burned: 31 (million)
% 2.95/1.71 % (10826)------------------------------
% 2.95/1.71 % (10826)------------------------------
% 2.95/1.72 % (10828)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/100Mi)
% 2.95/1.75 % (10827)Instruction limit reached!
% 2.95/1.75 % (10827)------------------------------
% 2.95/1.75 % (10827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.75 % (10827)Termination reason: Unknown
% 2.95/1.75 % (10827)Termination phase: Saturation
% 2.95/1.75
% 2.95/1.75 % (10827)Memory used [KB]: 7036
% 2.95/1.75 % (10827)Time elapsed: 0.040 s
% 2.95/1.75 % (10827)Instructions burned: 128 (million)
% 2.95/1.75 % (10827)------------------------------
% 2.95/1.75 % (10827)------------------------------
% 2.95/1.75 % (10828)Instruction limit reached!
% 2.95/1.75 % (10828)------------------------------
% 2.95/1.75 % (10828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.75 % (10828)Termination reason: Unknown
% 2.95/1.75 % (10828)Termination phase: Property scanning
% 2.95/1.75
% 2.95/1.75 % (10828)Memory used [KB]: 2302
% 2.95/1.75 % (10828)Time elapsed: 0.031 s
% 2.95/1.75 % (10828)Instructions burned: 101 (million)
% 2.95/1.75 % (10828)------------------------------
% 2.95/1.75 % (10828)------------------------------
% 2.95/1.75 % (10829)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/3Mi)
% 2.95/1.76 % (10829)Instruction limit reached!
% 2.95/1.76 % (10829)------------------------------
% 2.95/1.76 % (10829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.76 % (10829)Termination reason: Unknown
% 2.95/1.76 % (10829)Termination phase: shuffling
% 2.95/1.76
% 2.95/1.76 % (10829)Memory used [KB]: 1407
% 2.95/1.76 % (10829)Time elapsed: 0.002 s
% 2.95/1.76 % (10829)Instructions burned: 3 (million)
% 2.95/1.76 % (10829)------------------------------
% 2.95/1.76 % (10829)------------------------------
% 2.95/1.76 % (10830)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/20Mi)
% 2.95/1.76 % (10831)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/86Mi)
% 2.95/1.77 % (10830)Instruction limit reached!
% 2.95/1.77 % (10830)------------------------------
% 2.95/1.77 % (10830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.77 % (10830)Termination reason: Unknown
% 2.95/1.77 % (10830)Termination phase: Property scanning
% 2.95/1.77
% 2.95/1.77 % (10830)Memory used [KB]: 1791
% 2.95/1.77 % (10830)Time elapsed: 0.009 s
% 2.95/1.77 % (10830)Instructions burned: 20 (million)
% 2.95/1.77 % (10830)------------------------------
% 2.95/1.77 % (10830)------------------------------
% 2.95/1.78 % (10832)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_10522 for (2998ds/107Mi)
% 2.95/1.79 % (10831)Instruction limit reached!
% 2.95/1.79 % (10831)------------------------------
% 2.95/1.79 % (10831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.79 % (10831)Termination reason: Unknown
% 2.95/1.79 % (10831)Termination phase: Saturation
% 2.95/1.79
% 2.95/1.79 % (10831)Memory used [KB]: 6780
% 2.95/1.79 % (10831)Time elapsed: 0.026 s
% 2.95/1.79 % (10831)Instructions burned: 88 (million)
% 2.95/1.79 % (10831)------------------------------
% 2.95/1.79 % (10831)------------------------------
% 2.95/1.80 % (10833)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/39Mi)
% 2.95/1.81 % (10833)Instruction limit reached!
% 2.95/1.81 % (10833)------------------------------
% 2.95/1.81 % (10833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (10833)Termination reason: Unknown
% 2.95/1.81 % (10833)Termination phase: shuffling
% 2.95/1.81
% 2.95/1.81 % (10833)Memory used [KB]: 2174
% 2.95/1.81 % (10833)Time elapsed: 0.013 s
% 2.95/1.81 % (10833)Instructions burned: 41 (million)
% 2.95/1.81 % (10833)------------------------------
% 2.95/1.81 % (10833)------------------------------
% 2.95/1.81 % (10820)Instruction limit reached!
% 2.95/1.81 % (10820)------------------------------
% 2.95/1.81 % (10820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (10820)Termination reason: Unknown
% 2.95/1.81 % (10820)Termination phase: Saturation
% 2.95/1.81
% 2.95/1.81 % (10820)Memory used [KB]: 7931
% 2.95/1.81 % (10820)Time elapsed: 0.139 s
% 2.95/1.81 % (10820)Instructions burned: 377 (million)
% 2.95/1.81 % (10820)------------------------------
% 2.95/1.81 % (10820)------------------------------
% 2.95/1.82 % (10832)Instruction limit reached!
% 2.95/1.82 % (10832)------------------------------
% 2.95/1.82 % (10832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (10832)Termination reason: Unknown
% 2.95/1.82 % (10832)Termination phase: Saturation
% 2.95/1.82
% 2.95/1.82 % (10832)Memory used [KB]: 6908
% 2.95/1.82 % (10832)Time elapsed: 0.036 s
% 2.95/1.82 % (10832)Instructions burned: 108 (million)
% 2.95/1.82 % (10832)------------------------------
% 2.95/1.82 % (10832)------------------------------
% 2.95/1.82 % (10834)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/448Mi)
% 2.95/1.82 % (10835)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/46Mi)
% 2.95/1.83 % (10836)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/98Mi)
% 2.95/1.84 % (10835)Instruction limit reached!
% 2.95/1.84 % (10835)------------------------------
% 2.95/1.84 % (10835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.84 % (10835)Termination reason: Unknown
% 2.95/1.84 % (10835)Termination phase: Preprocessing 3
% 2.95/1.84
% 2.95/1.84 % (10835)Memory used [KB]: 2046
% 2.95/1.84 % (10835)Time elapsed: 0.018 s
% 2.95/1.84 % (10835)Instructions burned: 48 (million)
% 2.95/1.84 % (10835)------------------------------
% 2.95/1.84 % (10835)------------------------------
% 2.95/1.85 % (10837)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/507Mi)
% 2.95/1.85 % (10836)Instruction limit reached!
% 2.95/1.85 % (10836)------------------------------
% 2.95/1.85 % (10836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.85 % (10836)Termination reason: Unknown
% 2.95/1.85 % (10836)Termination phase: Property scanning
% 2.95/1.85
% 2.95/1.85 % (10836)Memory used [KB]: 2302
% 2.95/1.85 % (10836)Time elapsed: 0.031 s
% 2.95/1.85 % (10836)Instructions burned: 101 (million)
% 2.95/1.85 % (10836)------------------------------
% 2.95/1.85 % (10836)------------------------------
% 2.95/1.86 % (10814)Instruction limit reached!
% 2.95/1.86 % (10814)------------------------------
% 2.95/1.86 % (10814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.86 % (10814)Termination reason: Unknown
% 2.95/1.86 % (10814)Termination phase: Saturation
% 2.95/1.86
% 2.95/1.86 % (10814)Memory used [KB]: 7803
% 2.95/1.86 % (10814)Time elapsed: 0.223 s
% 2.95/1.86 % (10814)Instructions burned: 712 (million)
% 2.95/1.86 % (10814)------------------------------
% 2.95/1.86 % (10814)------------------------------
% 2.95/1.86 % (10838)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/149Mi)
% 2.95/1.87 % (10839)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_10522 for (2997ds/39Mi)
% 2.95/1.89 % (10839)Instruction limit reached!
% 2.95/1.89 % (10839)------------------------------
% 2.95/1.89 % (10839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.89 % (10839)Termination reason: Unknown
% 2.95/1.89 % (10839)Termination phase: Preprocessing 3
% 2.95/1.89
% 2.95/1.89 % (10839)Memory used [KB]: 2046
% 2.95/1.89 % (10839)Time elapsed: 0.013 s
% 2.95/1.89 % (10839)Instructions burned: 39 (million)
% 2.95/1.89 % (10839)------------------------------
% 2.95/1.89 % (10839)------------------------------
% 2.95/1.90 % (10840)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_10522 for (2996ds/6Mi)
% 2.95/1.90 % (10840)Instruction limit reached!
% 2.95/1.90 % (10840)------------------------------
% 2.95/1.90 % (10840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.90 % (10840)Termination reason: Unknown
% 2.95/1.90 % (10840)Termination phase: shuffling
% 2.95/1.90
% 2.95/1.90 % (10840)Memory used [KB]: 1407
% 2.95/1.90 % (10840)Time elapsed: 0.003 s
% 2.95/1.90 % (10840)Instructions burned: 7 (million)
% 2.95/1.90 % (10840)------------------------------
% 2.95/1.90 % (10840)------------------------------
% 2.95/1.90 % (10837)First to succeed.
% 2.95/1.90 % (10837)Refutation found. Thanks to Tanya!
% 2.95/1.90 % SZS status Theorem for DTF2THF_10522
% 2.95/1.90 % SZS output start Proof for DTF2THF_10522
% 2.95/1.90 thf(type_def_5, type, list: ($tType) > $tType).
% 2.95/1.90 thf(type_def_7, type, nat: $tType).
% 2.95/1.90 thf(type_def_8, type, a: $tType).
% 2.95/1.90 thf(type_def_9, type, int: $tType).
% 2.95/1.90 thf(func_def_0, type, list: ($tType) > $tType).
% 2.95/1.90 thf(func_def_1, type, nat: $tType).
% 2.95/1.90 thf(func_def_2, type, int: $tType).
% 2.95/1.90 thf(func_def_3, type, a: $tType).
% 2.95/1.90 thf(func_def_4, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_6, type, size: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_7, type, plus: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_9, type, ord: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_11, type, group_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_12, type, monoid_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_15, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_16, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_17, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_18, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_19, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_20, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_21, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_22, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_23, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_24, type, ordere1818651114id_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_25, type, strict797366125id_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_26, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_27, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_28, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_29, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 2.95/1.90 thf(func_def_30, type, descar1668888542_psums: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_31, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.95/1.90 thf(func_def_32, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.95/1.90 thf(func_def_33, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_34, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_35, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_36, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.95/1.90 thf(func_def_37, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 2.95/1.90 thf(func_def_38, type, replicate: !>[X0: $tType]:(nat > X0 > list @ X0)).
% 2.95/1.90 thf(func_def_39, type, take: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_40, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.95/1.90 thf(func_def_41, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_42, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_43, type, p: nat).
% 2.95/1.90 thf(func_def_44, type, v: list @ a > int).
% 2.95/1.90 thf(func_def_45, type, xa: a).
% 2.95/1.90 thf(func_def_46, type, xs: list @ a).
% 2.95/1.90 thf(func_def_47, type, xs1: list @ a).
% 2.95/1.90 thf(func_def_48, type, xs2: list @ a).
% 2.95/1.90 thf(func_def_49, type, xsa: list @ a).
% 2.95/1.90 thf(func_def_64, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_65, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_66, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_67, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_68, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_69, type, sK5: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_70, type, sK6: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_71, type, sK7: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_72, type, sK8: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_73, type, sK9: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_74, type, sK10: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_75, type, sK11: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_76, type, sK12: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_77, type, sK13: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_78, type, sK14: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_79, type, sK15: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_80, type, sK16: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_81, type, sK17: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0 > X0 > $o)).
% 2.95/1.90 thf(func_def_82, type, sK18: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_83, type, sK19: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_84, type, sK20: !>[X0: $tType]:(((X0 > X0 > $o) > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_85, type, sK21: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_86, type, sK22: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_87, type, sK23: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_88, type, sK24: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_89, type, sK25: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.95/1.90 thf(func_def_90, type, sK26: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.95/1.90 thf(func_def_91, type, sK27: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.95/1.90 thf(func_def_92, type, sK28: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_93, type, sK29: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_94, type, sK30: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_95, type, sK31: !>[X0: $tType]:(list @ X0 > list @ X0 > nat > nat)).
% 2.95/1.90 thf(func_def_96, type, sK32: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.90 thf(func_def_97, type, sK33: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_98, type, sK34: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_99, type, sK35: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_100, type, sK36: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_101, type, sK37: !>[X0: $tType]:(nat > list @ X0)).
% 2.95/1.90 thf(func_def_102, type, sK38: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.95/1.90 thf(func_def_103, type, sK39: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_104, type, sK40: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_105, type, sK41: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_106, type, sK42: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_107, type, sK43: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_108, type, sK44: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_109, type, sK45: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_110, type, sK46: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_111, type, sK47: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_112, type, sK48: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_113, type, sK49: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_114, type, sK50: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_115, type, sK51: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_116, type, sK52: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_117, type, sK53: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_118, type, sK54: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_119, type, sK55: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_120, type, sK56: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_121, type, sK57: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.95/1.90 thf(func_def_122, type, sK58: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_123, type, sK59: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X0)).
% 2.95/1.90 thf(func_def_124, type, sK60: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_125, type, sK61: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_126, type, sK62: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_127, type, sK63: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_128, type, sK64: !>[X0: $tType, X1: $tType]:((list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_129, type, sK65: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.95/1.90 thf(func_def_130, type, sK66: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 2.95/1.90 thf(func_def_131, type, sK67: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_132, type, sK68: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_133, type, sK69: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_134, type, sK70: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_135, type, sK71: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_136, type, sK72: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_137, type, sK73: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_138, type, sK74: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_139, type, sK75: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_140, type, sK76: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_141, type, sK77: !>[X0: $tType]:(list @ X0 > list @ X0 > X0)).
% 2.95/1.90 thf(func_def_142, type, sK78: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_143, type, sK79: !>[X0: $tType]:(list @ X0 > list @ X0 > X0)).
% 2.95/1.90 thf(func_def_144, type, sK80: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_145, type, sK81: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.90 thf(func_def_146, type, sK82: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_147, type, sK83: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_148, type, sK84: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_149, type, sK85: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_150, type, sK86: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_151, type, sK87: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_152, type, sK88: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_153, type, sK89: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1 > X0)).
% 2.95/1.90 thf(func_def_154, type, sK90: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_155, type, sK91: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_156, type, sK92: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_157, type, sK93: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1 > X0)).
% 2.95/1.90 thf(func_def_158, type, sK94: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1 > X0)).
% 2.95/1.90 thf(func_def_159, type, sK95: !>[X0: $tType, X1: $tType]:(((X1 > X0) > list @ X1 > $o) > X1)).
% 2.95/1.90 thf(func_def_160, type, sK96: !>[X0: $tType]:(list @ list @ X0 > X0)).
% 2.95/1.90 thf(func_def_161, type, sK97: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_162, type, sK98: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.95/1.90 thf(func_def_163, type, sK99: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.95/1.90 thf(func_def_164, type, sK100: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_165, type, sK101: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_166, type, sK102: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_167, type, sK103: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_168, type, sK104: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_169, type, sK105: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_170, type, sK106: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_171, type, sK107: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_172, type, sK108: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_173, type, sK109: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_174, type, sK110: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_175, type, sK111: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_176, type, sK112: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_177, type, sK113: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_178, type, sK114: nat > (nat > $o) > nat).
% 2.95/1.90 thf(func_def_179, type, sK115: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_180, type, sK116: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_181, type, sK117: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_182, type, sK118: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_183, type, sK119: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_184, type, sK120: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_185, type, sK121: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_186, type, sK122: !>[X0: $tType]:((list @ X0 > list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_187, type, sK123: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X1)).
% 2.95/1.90 thf(func_def_188, type, sK124: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_189, type, sK125: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X0)).
% 2.95/1.90 thf(func_def_190, type, sK126: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_191, type, sK127: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > X2)).
% 2.95/1.90 thf(func_def_192, type, sK128: !>[X0: $tType, X1: $tType, X2: $tType]:((list @ X1 > list @ X0 > list @ X2 > $o) > list @ X2)).
% 2.95/1.90 thf(func_def_193, type, sK129: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > X1)).
% 2.95/1.90 thf(func_def_194, type, sK130: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_195, type, sK131: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > X2)).
% 2.95/1.90 thf(func_def_196, type, sK132: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > list @ X2)).
% 2.95/1.90 thf(func_def_197, type, sK133: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > X0)).
% 2.95/1.90 thf(func_def_198, type, sK134: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_199, type, sK135: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > X3)).
% 2.95/1.90 thf(func_def_200, type, sK136: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((list @ X1 > list @ X2 > list @ X0 > list @ X3 > $o) > list @ X3)).
% 2.95/1.90 thf(func_def_201, type, sK137: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.95/1.90 thf(func_def_202, type, sK138: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 2.95/1.90 thf(func_def_203, type, sK139: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_204, type, sK140: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_205, type, sK141: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_206, type, sK142: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_207, type, sK143: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_208, type, sK144: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_209, type, sK145: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0 > X1)).
% 2.95/1.90 thf(func_def_210, type, sK146: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_211, type, sK147: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0 > X1)).
% 2.95/1.90 thf(func_def_212, type, sK148: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > X0)).
% 2.95/1.90 thf(func_def_213, type, sK149: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_214, type, sK150: !>[X0: $tType, X1: $tType]:(((X0 > X1) > list @ X0 > list @ X1 > $o) > list @ X1)).
% 2.95/1.90 thf(func_def_215, type, sK151: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_216, type, sK152: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_217, type, sK153: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_218, type, sK154: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_219, type, sK155: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_220, type, sK156: !>[X0: $tType]:(list @ X0 > X0)).
% 2.95/1.90 thf(func_def_221, type, sK157: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.95/1.90 thf(func_def_222, type, sK158: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.95/1.90 thf(func_def_223, type, sK159: (nat > nat) > nat).
% 2.95/1.90 thf(func_def_224, type, sK160: (nat > nat) > nat).
% 2.95/1.90 thf(func_def_225, type, sK161: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.95/1.90 thf(func_def_226, type, sK162: !>[X0: $tType]:((list @ X0 > $o) > X0)).
% 2.95/1.90 thf(func_def_227, type, sK163: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.95/1.90 thf(func_def_229, type, ph165: !>[X0: $tType]:(X0)).
% 2.95/1.90 thf(f1910,plain,(
% 2.95/1.90 $false),
% 2.95/1.90 inference(avatar_sat_refutation,[status(thm)],[f1900,f1904,f1909])).
% 2.95/1.90 thf(f1909,plain,(
% 2.95/1.90 spl164_3),
% 2.95/1.90 inference(avatar_contradiction_clause,[status(thm)],[f1908])).
% 2.95/1.90 thf(f1908,plain,(
% 2.95/1.90 $false | spl164_3),
% 2.95/1.90 inference(trivial_inequality_removal,[status(thm)],[f1907])).
% 2.95/1.90 thf(f1907,plain,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) != (descar1668888542_psums @ a @ xsa)) | spl164_3),
% 2.95/1.90 inference(superposition,[status(thm)],[f1895,f1841])).
% 2.95/1.90 thf(f1841,plain,(
% 2.95/1.90 (xsa = (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))),
% 2.95/1.90 inference(definition_unfolding,[status(thm)],[f1535,f1659])).
% 2.95/1.90 thf(f1659,plain,(
% 2.95/1.90 (xs1 = (replicate @ a @ p @ zero_zero @ a))),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f653])).
% 2.95/1.90 thf(f653,plain,(
% 2.95/1.90 (xs1 = (replicate @ a @ p @ zero_zero @ a))),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f5])).
% 2.95/1.90 thf(f5,axiom,(
% 2.95/1.90 (xs1 = (replicate @ a @ p @ zero_zero @ a))),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',fact_4_xs1__def)).
% 2.95/1.90 thf(f1535,plain,(
% 2.95/1.90 (xsa = (append @ a @ xs1 @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f626])).
% 2.95/1.90 thf(f626,plain,(
% 2.95/1.90 (xsa = (append @ a @ xs1 @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f6])).
% 2.95/1.90 thf(f6,axiom,(
% 2.95/1.90 (xsa = (append @ a @ xs1 @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',fact_5_xs__decompose)).
% 2.95/1.90 thf(f1895,plain,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) != (descar1668888542_psums @ a @ (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))) | spl164_3),
% 2.95/1.90 inference(avatar_component_clause,[status(thm)],[f1893])).
% 2.95/1.90 thf(f1893,definition,(
% 2.95/1.90 spl164_3 <=> ((descar1668888542_psums @ a @ xsa) = (descar1668888542_psums @ a @ (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 introduced(definition,[new_symbols(naming,[spl164_3])],[avatar_definition])).
% 2.95/1.90 thf(f1904,plain,(
% 2.95/1.90 spl164_4),
% 2.95/1.90 inference(avatar_contradiction_clause,[status(thm)],[f1903])).
% 2.95/1.90 thf(f1903,plain,(
% 2.95/1.90 $false | spl164_4),
% 2.95/1.90 inference(subsumption_resolution,[status(thm)],[f1902,f1723])).
% 2.95/1.90 thf(f1723,plain,(
% 2.95/1.90 (linordered_idom @ a = $true)),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f667])).
% 2.95/1.90 thf(f667,plain,(
% 2.95/1.90 (linordered_idom @ a = $true)),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f666])).
% 2.95/1.90 thf(f666,plain,(
% 2.95/1.90 linordered_idom @ a),
% 2.95/1.90 inference(rectify,[status(thm)],[f329])).
% 2.95/1.90 thf(f329,axiom,(
% 2.95/1.90 linordered_idom @ a),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',tfree_0)).
% 2.95/1.90 thf(f1902,plain,(
% 2.95/1.90 (linordered_idom @ a != $true) | spl164_4),
% 2.95/1.90 inference(trivial_inequality_removal,[status(thm)],[f1901])).
% 2.95/1.90 thf(f1901,plain,(
% 2.95/1.90 ($true != $true) | (linordered_idom @ a != $true) | spl164_4),
% 2.95/1.90 inference(superposition,[status(thm)],[f1899,f1472])).
% 2.95/1.90 thf(f1472,plain,(
% 2.95/1.90 ( ! [X0 : $tType] : ((monoid_add @ X0 = $true) | (linordered_idom @ X0 != $true)) )),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f1184])).
% 2.95/1.90 thf(f1184,plain,(
% 2.95/1.90 ! [X0 : $tType] : ((monoid_add @ X0 = $true) | (linordered_idom @ X0 != $true))),
% 2.95/1.90 inference(ennf_transformation,[status(thm)],[f962])).
% 2.95/1.90 thf(f962,plain,(
% 2.95/1.90 ! [X0 : $tType] : ((linordered_idom @ X0 = $true) => (monoid_add @ X0 = $true))),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f961])).
% 2.95/1.90 thf(f961,plain,(
% 2.95/1.90 ! [X0 : $tType] : (linordered_idom @ X0 => monoid_add @ X0)),
% 2.95/1.90 inference(rectify,[status(thm)],[f260])).
% 2.95/1.90 thf(f260,axiom,(
% 2.95/1.90 ! [X0 : $tType] : (linordered_idom @ X0 => monoid_add @ X0)),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',subcl_Rings_Olinordered__idom___Groups_Omonoid__add)).
% 2.95/1.90 thf(f1899,plain,(
% 2.95/1.90 (monoid_add @ a != $true) | spl164_4),
% 2.95/1.90 inference(avatar_component_clause,[status(thm)],[f1897])).
% 2.95/1.90 thf(f1897,definition,(
% 2.95/1.90 spl164_4 <=> (monoid_add @ a = $true)),
% 2.95/1.90 introduced(definition,[new_symbols(naming,[spl164_4])],[avatar_definition])).
% 2.95/1.90 thf(f1900,plain,(
% 2.95/1.90 ~spl164_3 | ~spl164_4),
% 2.95/1.90 inference(avatar_split_clause,[status(thm)],[f1882,f1897,f1893])).
% 2.95/1.90 thf(f1882,plain,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) != (descar1668888542_psums @ a @ (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2)))) | (monoid_add @ a != $true)),
% 2.95/1.90 inference(superposition,[status(thm)],[f1514,f1651])).
% 2.95/1.90 thf(f1651,plain,(
% 2.95/1.90 ( ! [X0 : $tType,X2 : list @ X0,X1 : nat] : (((descar1668888542_psums @ X0 @ (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ X2)) = (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ (descar1668888542_psums @ X0 @ X2))) | (monoid_add @ X0 != $true)) )),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f1127])).
% 2.95/1.90 thf(f1127,plain,(
% 2.95/1.90 ! [X0 : $tType] : ((monoid_add @ X0 != $true) | ! [X1 : nat,X2 : list @ X0] : ((descar1668888542_psums @ X0 @ (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ X2)) = (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ (descar1668888542_psums @ X0 @ X2))))),
% 2.95/1.90 inference(ennf_transformation,[status(thm)],[f489])).
% 2.95/1.90 thf(f489,plain,(
% 2.95/1.90 ! [X0 : $tType] : ((monoid_add @ X0 = $true) => ! [X1 : nat,X2 : list @ X0] : ((descar1668888542_psums @ X0 @ (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ X2)) = (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ (descar1668888542_psums @ X0 @ X2))))),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f488])).
% 2.95/1.90 thf(f488,plain,(
% 2.95/1.90 ! [X0 : $tType] : (monoid_add @ X0 => ! [X1 : nat,X2 : list @ X0] : ((descar1668888542_psums @ X0 @ (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ X2)) = (append @ X0 @ (replicate @ X0 @ X1 @ zero_zero @ X0) @ (descar1668888542_psums @ X0 @ X2))))),
% 2.95/1.90 inference(rectify,[status(thm)],[f7])).
% 2.95/1.90 thf(f7,axiom,(
% 2.95/1.90 ! [X0 : $tType] : (monoid_add @ X0 => ! [X2 : nat,X1 : list @ X0] : ((descar1668888542_psums @ X0 @ (append @ X0 @ (replicate @ X0 @ X2 @ zero_zero @ X0) @ X1)) = (append @ X0 @ (replicate @ X0 @ X2 @ zero_zero @ X0) @ (descar1668888542_psums @ X0 @ X1))))),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',fact_6_psums__replicate__0__append)).
% 2.95/1.90 thf(f1514,plain,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) != (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (descar1668888542_psums @ a @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 inference(cnf_transformation,[status(thm)],[f998])).
% 2.95/1.90 thf(f998,plain,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) != (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (descar1668888542_psums @ a @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 inference(flattening,[status(thm)],[f416])).
% 2.95/1.90 thf(f416,plain,(
% 2.95/1.90 ~((descar1668888542_psums @ a @ xsa) = (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (descar1668888542_psums @ a @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 inference(fool_elimination,[status(thm)],[f331])).
% 2.95/1.90 thf(f331,negated_conjecture,(
% 2.95/1.90 ~((descar1668888542_psums @ a @ xsa) = (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (descar1668888542_psums @ a @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 inference(negated_conjecture,[status(cth)],[f330])).
% 2.95/1.90 thf(f330,conjecture,(
% 2.95/1.90 ((descar1668888542_psums @ a @ xsa) = (append @ a @ (replicate @ a @ p @ zero_zero @ a) @ (descar1668888542_psums @ a @ (cons @ a @ (nth @ a @ xsa @ p) @ xs2))))),
% 2.95/1.90 file('/export/starexec/sandbox2/tmp/tmp.t9QTaY3Ytb/DTF2THF_10522.p',conj_0)).
% 2.95/1.90 % SZS output end Proof for DTF2THF_10522
% 2.95/1.90 % (10837)------------------------------
% 2.95/1.90 % (10837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.90 % (10837)Termination reason: Refutation
% 2.95/1.90
% 2.95/1.90 % (10837)Memory used [KB]: 7675
% 2.95/1.90 % (10837)Time elapsed: 0.072 s
% 2.95/1.90 % (10837)Instructions burned: 148 (million)
% 2.95/1.90 % (10837)------------------------------
% 2.95/1.90 % (10837)------------------------------
% 2.95/1.90 % (10793)Success in time 0.342 s
%------------------------------------------------------------------------------