%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP151^2 : 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 : n001.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 08:55:58 AM UTC 2026
% Result : Theorem 2.85s 1.61s
% Output : Refutation 2.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP151^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n001.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:50 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.21/0.47 ---- Original DTF file ---
% 0.21/0.47 thf(spec,logic,$$dhol).
% 0.21/0.47 %------------------------------------------------------------------------------
% 0.21/0.47 % File : ITP151^2 : TPTP v9.2.1. Released v7.5.0.
% 0.21/0.47 % Domain : Interactive Theorem Proving
% 0.21/0.47 % Problem : Sledgehammer Polynomial_Expression problem prob_414__8350168_1
% 0.21/0.47 % Version : Especial.
% 0.21/0.47 % English :
% 0.21/0.47
% 0.21/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.21/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.21/0.47 % Source : [Des21]
% 0.21/0.47 % Names : Polynomial_Expression/prob_414__8350168_1 [Des21]
% 0.21/0.47
% 0.21/0.47 % Status : Theorem
% 0.21/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.21/0.47 % Syntax : Number of formulae : 350 ( 102 unt; 46 typ; 0 def)
% 0.21/0.47 % Number of atoms : 847 ( 224 equ; 0 cnn)
% 0.21/0.47 % Maximal formula atoms : 13 ( 2 avg)
% 0.21/0.47 % Number of connectives : 2903 ( 101 ~; 15 |; 38 &;2308 @)
% 0.21/0.47 % ( 0 <=>; 441 =>; 0 <=; 0 <~>)
% 0.21/0.47 % Maximal formula depth : 21 ( 7 avg)
% 0.21/0.47 % Number of types : 4 ( 3 usr)
% 0.21/0.47 % Number of type conns : 82 ( 82 >; 0 *; 0 +; 0 <<)
% 0.21/0.47 % Number of symbols : 44 ( 43 usr; 2 con; 0-3 aty)
% 0.21/0.47 % Number of variables : 838 ( 27 ^; 740 !; 32 ?; 838 :)
% 0.21/0.47 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.21/0.47 % SPC : TH1_THM_EQU_NAR
% 0.21/0.47
% 0.21/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:19.210
% 0.21/0.47 %------------------------------------------------------------------------------
% 0.21/0.47 % Could-be-implicit typings (4)
% 0.21/0.47 thf(ty_t_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly,type,
% 0.21/0.47 polyno1783536151e_poly: $tType > $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_t_Nat_Onat,type,
% 0.21/0.47 nat: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_t_Int_Oint,type,
% 0.21/0.47 int: $tType ).
% 0.21/0.47
% 0.21/0.47 thf(ty_tf_a,type,
% 0.21/0.47 a: $tType ).
% 0.21/0.47
% 0.21/0.47 % Explicit typings (42)
% 0.21/0.47 thf(sy_cl_HOL_Otype,type,
% 0.21/0.47 type:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.21/0.47 zero:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.21/0.47 ord:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.21/0.47 order:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.21/0.47 group_add:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.21/0.47 no_bot:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.21/0.47 no_top:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Osemiring__1,type,
% 0.21/0.47 semiring_1:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.21/0.47 monoid_mult:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.21/0.47 linorder:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.21/0.47 preorder:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.21/0.47 semiring_char_0:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.21/0.47 wellorder:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.21/0.47 dense_order:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.21/0.47 linordered_idom:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Fields_Olinordered__field,type,
% 0.21/0.47 linordered_field:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.21/0.47 comm_monoid_diff:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 0.21/0.47 semiring_1_cancel:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.21/0.47 dense_linorder:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.21/0.47 linordered_semidom:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.21/0.47 ordered_ab_group_add:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.21/0.47 cancel1352612707id_add:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.21/0.47 cancel146912293up_add:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.21/0.47 linord1659791738miring:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.21/0.47 semiri134348788visors:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.21/0.47 archim1804426504_field:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.21/0.47 canoni770627133id_add:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.21/0.47 condit1656338222tinuum:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.21/0.47 condit1037483654norder:
% 0.21/0.47 !>[A: $tType] : $o ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.21/0.47 minus_minus:
% 0.21/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.21/0.47 zero_zero:
% 0.21/0.47 !>[A: $tType] : A ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Nat_OSuc,type,
% 0.21/0.47 suc: nat > nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.21/0.47 semiring_1_of_nat:
% 0.21/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.21/0.47 ord_less:
% 0.21/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.21/0.47 ord_less_eq:
% 0.21/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegree,type,
% 0.21/0.47 polyno498386536degree:
% 0.21/0.47 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegreen,type,
% 0.21/0.47 polyno367318022egreen:
% 0.21/0.47 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > nat ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpoly,type,
% 0.21/0.47 polyno2122670676snpoly:
% 0.21/0.47 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > $o ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpolyh,type,
% 0.21/0.47 polyno86455060npolyh:
% 0.21/0.47 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > $o ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_c_Power_Opower__class_Opower,type,
% 0.21/0.47 power_power:
% 0.21/0.47 !>[A: $tType] : ( A > nat > A ) ).
% 0.21/0.47
% 0.21/0.47 thf(sy_v_n,type,
% 0.21/0.47 n: nat ).
% 0.21/0.47
% 0.21/0.47 thf(sy_v_p,type,
% 0.21/0.47 p: polyno1783536151e_poly @ a ).
% 0.21/0.47
% 0.21/0.47 % Relevant facts (253)
% 0.21/0.47 thf(fact_0_neq0__conv,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( ( N
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % neq0_conv
% 0.21/0.47 thf(fact_1_less__nat__zero__code,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_nat_zero_code
% 0.21/0.47 thf(fact_2_bot__nat__0_Onot__eq__extremum,axiom,
% 0.21/0.47 ! [A2: nat] :
% 0.21/0.47 ( ( A2
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.21/0.47
% 0.21/0.47 % bot_nat_0.not_eq_extremum
% 0.21/0.47 thf(fact_3_not__gr__zero,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( canoni770627133id_add @ A )
% 0.21/0.47 => ! [N: A] :
% 0.21/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.21/0.47 = ( N
% 0.21/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_gr_zero
% 0.21/0.47 thf(fact_4_gr0I,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( ( N
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % gr0I
% 0.21/0.47 thf(fact_5_not__gr0,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.21/0.47 = ( N
% 0.21/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_gr0
% 0.21/0.47 thf(fact_6_not__less0,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_less0
% 0.21/0.47 thf(fact_7_less__zeroE,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_zeroE
% 0.21/0.47 thf(fact_8_gr__implies__not0,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( N
% 0.21/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % gr_implies_not0
% 0.21/0.47 thf(fact_9_infinite__descent0,axiom,
% 0.21/0.47 ! [P: nat > $o,N: nat] :
% 0.21/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.21/0.47 => ( ! [N2: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.21/0.47 => ( ~ ( P @ N2 )
% 0.21/0.47 => ? [M2: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.21/0.47 & ~ ( P @ M2 ) ) ) )
% 0.21/0.47 => ( P @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % infinite_descent0
% 0.21/0.47 thf(fact_10_bot__nat__0_Oextremum__strict,axiom,
% 0.21/0.47 ! [A2: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % bot_nat_0.extremum_strict
% 0.21/0.47 thf(fact_11_zero__reorient,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( zero @ A )
% 0.21/0.47 => ! [X: A] :
% 0.21/0.47 ( ( ( zero_zero @ A )
% 0.21/0.47 = X )
% 0.21/0.47 = ( X
% 0.21/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % zero_reorient
% 0.21/0.47 thf(fact_12_measure__induct__rule,axiom,
% 0.21/0.47 ! [B: $tType,A: $tType] :
% 0.21/0.47 ( ( wellorder @ B )
% 0.21/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.21/0.47 ( ! [X2: A] :
% 0.21/0.47 ( ! [Y: A] :
% 0.21/0.47 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.21/0.47 => ( P @ Y ) )
% 0.21/0.47 => ( P @ X2 ) )
% 0.21/0.47 => ( P @ A2 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % measure_induct_rule
% 0.21/0.47 thf(fact_13_measure__induct,axiom,
% 0.21/0.47 ! [B: $tType,A: $tType] :
% 0.21/0.47 ( ( wellorder @ B )
% 0.21/0.47 => ! [F: A > B,P: A > $o,A2: A] :
% 0.21/0.47 ( ! [X2: A] :
% 0.21/0.47 ( ! [Y: A] :
% 0.21/0.47 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.21/0.47 => ( P @ Y ) )
% 0.21/0.47 => ( P @ X2 ) )
% 0.21/0.47 => ( P @ A2 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % measure_induct
% 0.21/0.47 thf(fact_14_infinite__descent__measure,axiom,
% 0.21/0.47 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.21/0.47 ( ! [X2: A] :
% 0.21/0.47 ( ~ ( P @ X2 )
% 0.21/0.47 => ? [Y: A] :
% 0.21/0.47 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.21/0.47 & ~ ( P @ Y ) ) )
% 0.21/0.47 => ( P @ X ) ) ).
% 0.21/0.47
% 0.21/0.47 % infinite_descent_measure
% 0.21/0.47 thf(fact_15_linorder__neqE__nat,axiom,
% 0.21/0.47 ! [X: nat,Y2: nat] :
% 0.21/0.47 ( ( X != Y2 )
% 0.21/0.47 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 0.21/0.47 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % linorder_neqE_nat
% 0.21/0.47 thf(fact_16_infinite__descent,axiom,
% 0.21/0.47 ! [P: nat > $o,N: nat] :
% 0.21/0.47 ( ! [N2: nat] :
% 0.21/0.47 ( ~ ( P @ N2 )
% 0.21/0.47 => ? [M2: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.21/0.47 & ~ ( P @ M2 ) ) )
% 0.21/0.47 => ( P @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % infinite_descent
% 0.21/0.47 thf(fact_17_nat__less__induct,axiom,
% 0.21/0.47 ! [P: nat > $o,N: nat] :
% 0.21/0.47 ( ! [N2: nat] :
% 0.21/0.47 ( ! [M2: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.21/0.47 => ( P @ M2 ) )
% 0.21/0.47 => ( P @ N2 ) )
% 0.21/0.47 => ( P @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat_less_induct
% 0.21/0.47 thf(fact_18_less__irrefl__nat,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.21/0.47
% 0.21/0.47 % less_irrefl_nat
% 0.21/0.47 thf(fact_19_less__not__refl3,axiom,
% 0.21/0.47 ! [S: nat,T: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ S @ T )
% 0.21/0.47 => ( S != T ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_not_refl3
% 0.21/0.47 thf(fact_20_less__not__refl2,axiom,
% 0.21/0.47 ! [N: nat,M: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ N @ M )
% 0.21/0.47 => ( M != N ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_not_refl2
% 0.21/0.47 thf(fact_21_less__not__refl,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.21/0.47
% 0.21/0.47 % less_not_refl
% 0.21/0.47 thf(fact_22_nat__neq__iff,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( M != N )
% 0.21/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat_neq_iff
% 0.21/0.47 thf(fact_23_zero__less__iff__neq__zero,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( canoni770627133id_add @ A )
% 0.21/0.47 => ! [N: A] :
% 0.21/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.21/0.47 = ( N
% 0.21/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % zero_less_iff_neq_zero
% 0.21/0.47 thf(fact_24_gr__implies__not__zero,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( canoni770627133id_add @ A )
% 0.21/0.47 => ! [M: A,N: A] :
% 0.21/0.47 ( ( ord_less @ A @ M @ N )
% 0.21/0.47 => ( N
% 0.21/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % gr_implies_not_zero
% 0.21/0.47 thf(fact_25_not__less__zero,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( canoni770627133id_add @ A )
% 0.21/0.47 => ! [N: A] :
% 0.21/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % not_less_zero
% 0.21/0.47 thf(fact_26_gr__zeroI,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( canoni770627133id_add @ A )
% 0.21/0.47 => ! [N: A] :
% 0.21/0.47 ( ( N
% 0.21/0.47 != ( zero_zero @ A ) )
% 0.21/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % gr_zeroI
% 0.21/0.47 thf(fact_27_infinite__descent0__measure,axiom,
% 0.21/0.47 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.21/0.47 ( ! [X2: A] :
% 0.21/0.47 ( ( ( V @ X2 )
% 0.21/0.47 = ( zero_zero @ nat ) )
% 0.21/0.47 => ( P @ X2 ) )
% 0.21/0.47 => ( ! [X2: A] :
% 0.21/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.21/0.47 => ( ~ ( P @ X2 )
% 0.21/0.47 => ? [Y: A] :
% 0.21/0.47 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.21/0.47 & ~ ( P @ Y ) ) ) )
% 0.21/0.47 => ( P @ X ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % infinite_descent0_measure
% 0.21/0.47 thf(fact_28_field__lbound__gt__zero,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linordered_field @ A )
% 0.21/0.47 => ! [D1: A,D2: A] :
% 0.21/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.21/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.21/0.47 => ? [E: A] :
% 0.21/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.21/0.47 & ( ord_less @ A @ E @ D1 )
% 0.21/0.47 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % field_lbound_gt_zero
% 0.21/0.47 thf(fact_29_less__numeral__extra_I3_J,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linord1659791738miring @ A )
% 0.21/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_numeral_extra(3)
% 0.21/0.47 thf(fact_30_isnpoly__def,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( zero @ A )
% 0.21/0.47 => ( ( polyno2122670676snpoly @ A )
% 0.21/0.47 = ( ^ [P2: polyno1783536151e_poly @ A] : ( polyno86455060npolyh @ A @ P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % isnpoly_def
% 0.21/0.47 thf(fact_31_degreen__0,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( zero @ A )
% 0.21/0.47 => ! [P3: polyno1783536151e_poly @ A,N: nat,M: nat] :
% 0.21/0.47 ( ( polyno86455060npolyh @ A @ P3 @ N )
% 0.21/0.47 => ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( ( polyno367318022egreen @ A @ P3 @ M )
% 0.21/0.47 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % degreen_0
% 0.21/0.47 thf(fact_32_of__nat__0__less__iff,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linord1659791738miring @ A )
% 0.21/0.47 => ! [N: nat] :
% 0.21/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.21/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_0_less_iff
% 0.21/0.47 thf(fact_33_degree__isnpolyh__Suc,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( zero @ A )
% 0.21/0.47 => ! [P3: polyno1783536151e_poly @ A,N: nat] :
% 0.21/0.47 ( ( polyno86455060npolyh @ A @ P3 @ ( suc @ N ) )
% 0.21/0.47 => ( ( polyno498386536degree @ A @ P3 )
% 0.21/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % degree_isnpolyh_Suc
% 0.21/0.47 thf(fact_34_zero__natural_Orsp,axiom,
% 0.21/0.47 ( ( zero_zero @ nat )
% 0.21/0.47 = ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % zero_natural.rsp
% 0.21/0.47 thf(fact_35_ex__gt__or__lt,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( condit1656338222tinuum @ A )
% 0.21/0.47 => ! [A2: A] :
% 0.21/0.47 ? [B2: A] :
% 0.21/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.21/0.47 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % ex_gt_or_lt
% 0.21/0.47 thf(fact_36_linorder__neqE__linordered__idom,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linordered_idom @ A )
% 0.21/0.47 => ! [X: A,Y2: A] :
% 0.21/0.47 ( ( X != Y2 )
% 0.21/0.47 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.21/0.47 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % linorder_neqE_linordered_idom
% 0.21/0.47 thf(fact_37_dual__order_Ostrict__implies__not__eq,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( order @ A )
% 0.21/0.47 => ! [B3: A,A2: A] :
% 0.21/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.21/0.47 => ( A2 != B3 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % dual_order.strict_implies_not_eq
% 0.21/0.47 thf(fact_38_linordered__field__no__ub,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linordered_field @ A )
% 0.21/0.47 => ! [X3: A] :
% 0.21/0.47 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 0.21/0.47
% 0.21/0.47 % linordered_field_no_ub
% 0.21/0.47 thf(fact_39_nat_Oinject,axiom,
% 0.21/0.47 ! [X22: nat,Y22: nat] :
% 0.21/0.47 ( ( ( suc @ X22 )
% 0.21/0.47 = ( suc @ Y22 ) )
% 0.21/0.47 = ( X22 = Y22 ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat.inject
% 0.21/0.47 thf(fact_40_old_Onat_Oinject,axiom,
% 0.21/0.47 ! [Nat: nat,Nat2: nat] :
% 0.21/0.47 ( ( ( suc @ Nat )
% 0.21/0.47 = ( suc @ Nat2 ) )
% 0.21/0.47 = ( Nat = Nat2 ) ) ).
% 0.21/0.47
% 0.21/0.47 % old.nat.inject
% 0.21/0.47 thf(fact_41_of__nat__eq__iff,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( semiring_char_0 @ A )
% 0.21/0.47 => ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.21/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.21/0.47 = ( M = N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_eq_iff
% 0.21/0.47 thf(fact_42_lessI,axiom,
% 0.21/0.47 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % lessI
% 0.21/0.47 thf(fact_43_Suc__mono,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_mono
% 0.21/0.47 thf(fact_44_Suc__less__eq,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.21/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_less_eq
% 0.21/0.47 thf(fact_45_of__nat__0,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( semiring_1 @ A )
% 0.21/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.21/0.47 = ( zero_zero @ A ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_0
% 0.21/0.47 thf(fact_46_of__nat__0__eq__iff,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( semiring_char_0 @ A )
% 0.21/0.47 => ! [N: nat] :
% 0.21/0.47 ( ( ( zero_zero @ A )
% 0.21/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.21/0.47 = ( ( zero_zero @ nat )
% 0.21/0.47 = N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_0_eq_iff
% 0.21/0.47 thf(fact_47_of__nat__eq__0__iff,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( semiring_char_0 @ A )
% 0.21/0.47 => ! [M: nat] :
% 0.21/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.21/0.47 = ( zero_zero @ A ) )
% 0.21/0.47 = ( M
% 0.21/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_eq_0_iff
% 0.21/0.47 thf(fact_48_of__nat__less__iff,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( linord1659791738miring @ A )
% 0.21/0.47 => ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.21/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_less_iff
% 0.21/0.47 thf(fact_49_less__Suc0,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.21/0.47 = ( N
% 0.21/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_Suc0
% 0.21/0.47 thf(fact_50_zero__less__Suc,axiom,
% 0.21/0.47 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % zero_less_Suc
% 0.21/0.47 thf(fact_51_Suc__inject,axiom,
% 0.21/0.47 ! [X: nat,Y2: nat] :
% 0.21/0.47 ( ( ( suc @ X )
% 0.21/0.47 = ( suc @ Y2 ) )
% 0.21/0.47 => ( X = Y2 ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_inject
% 0.21/0.47 thf(fact_52_n__not__Suc__n,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( N
% 0.21/0.47 != ( suc @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % n_not_Suc_n
% 0.21/0.47 thf(fact_53_of__nat__neq__0,axiom,
% 0.21/0.47 ! [A: $tType] :
% 0.21/0.47 ( ( semiring_char_0 @ A )
% 0.21/0.47 => ! [N: nat] :
% 0.21/0.47 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 0.21/0.47 != ( zero_zero @ A ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % of_nat_neq_0
% 0.21/0.47 thf(fact_54_polypow_Ocases,axiom,
% 0.21/0.47 ! [X: nat] :
% 0.21/0.47 ( ( X
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 => ~ ! [V2: nat] :
% 0.21/0.47 ( X
% 0.21/0.47 != ( suc @ V2 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % polypow.cases
% 0.21/0.47 thf(fact_55_nat_Odistinct_I1_J,axiom,
% 0.21/0.47 ! [X22: nat] :
% 0.21/0.47 ( ( zero_zero @ nat )
% 0.21/0.47 != ( suc @ X22 ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat.distinct(1)
% 0.21/0.47 thf(fact_56_old_Onat_Odistinct_I2_J,axiom,
% 0.21/0.47 ! [Nat3: nat] :
% 0.21/0.47 ( ( suc @ Nat3 )
% 0.21/0.47 != ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % old.nat.distinct(2)
% 0.21/0.47 thf(fact_57_old_Onat_Odistinct_I1_J,axiom,
% 0.21/0.47 ! [Nat2: nat] :
% 0.21/0.47 ( ( zero_zero @ nat )
% 0.21/0.47 != ( suc @ Nat2 ) ) ).
% 0.21/0.47
% 0.21/0.47 % old.nat.distinct(1)
% 0.21/0.47 thf(fact_58_nat_OdiscI,axiom,
% 0.21/0.47 ! [Nat: nat,X22: nat] :
% 0.21/0.47 ( ( Nat
% 0.21/0.47 = ( suc @ X22 ) )
% 0.21/0.47 => ( Nat
% 0.21/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat.discI
% 0.21/0.47 thf(fact_59_nat__induct,axiom,
% 0.21/0.47 ! [P: nat > $o,N: nat] :
% 0.21/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.21/0.47 => ( ! [N2: nat] :
% 0.21/0.47 ( ( P @ N2 )
% 0.21/0.47 => ( P @ ( suc @ N2 ) ) )
% 0.21/0.47 => ( P @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % nat_induct
% 0.21/0.47 thf(fact_60_diff__induct,axiom,
% 0.21/0.47 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.21/0.47 ( ! [X2: nat] : ( P @ X2 @ ( zero_zero @ nat ) )
% 0.21/0.47 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 0.21/0.47 => ( ! [X2: nat,Y3: nat] :
% 0.21/0.47 ( ( P @ X2 @ Y3 )
% 0.21/0.47 => ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
% 0.21/0.47 => ( P @ M @ N ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % diff_induct
% 0.21/0.47 thf(fact_61_zero__induct,axiom,
% 0.21/0.47 ! [P: nat > $o,K: nat] :
% 0.21/0.47 ( ( P @ K )
% 0.21/0.47 => ( ! [N2: nat] :
% 0.21/0.47 ( ( P @ ( suc @ N2 ) )
% 0.21/0.47 => ( P @ N2 ) )
% 0.21/0.47 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % zero_induct
% 0.21/0.47 thf(fact_62_Suc__neq__Zero,axiom,
% 0.21/0.47 ! [M: nat] :
% 0.21/0.47 ( ( suc @ M )
% 0.21/0.47 != ( zero_zero @ nat ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_neq_Zero
% 0.21/0.47 thf(fact_63_Zero__neq__Suc,axiom,
% 0.21/0.47 ! [M: nat] :
% 0.21/0.47 ( ( zero_zero @ nat )
% 0.21/0.47 != ( suc @ M ) ) ).
% 0.21/0.47
% 0.21/0.47 % Zero_neq_Suc
% 0.21/0.47 thf(fact_64_Zero__not__Suc,axiom,
% 0.21/0.47 ! [M: nat] :
% 0.21/0.47 ( ( zero_zero @ nat )
% 0.21/0.47 != ( suc @ M ) ) ).
% 0.21/0.47
% 0.21/0.47 % Zero_not_Suc
% 0.21/0.47 thf(fact_65_old_Onat_Oexhaust,axiom,
% 0.21/0.47 ! [Y2: nat] :
% 0.21/0.47 ( ( Y2
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 => ~ ! [Nat4: nat] :
% 0.21/0.47 ( Y2
% 0.21/0.47 != ( suc @ Nat4 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % old.nat.exhaust
% 0.21/0.47 thf(fact_66_old_Onat_Oinducts,axiom,
% 0.21/0.47 ! [P: nat > $o,Nat: nat] :
% 0.21/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.21/0.47 => ( ! [Nat4: nat] :
% 0.21/0.47 ( ( P @ Nat4 )
% 0.21/0.47 => ( P @ ( suc @ Nat4 ) ) )
% 0.21/0.47 => ( P @ Nat ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % old.nat.inducts
% 0.21/0.47 thf(fact_67_not0__implies__Suc,axiom,
% 0.21/0.47 ! [N: nat] :
% 0.21/0.47 ( ( N
% 0.21/0.47 != ( zero_zero @ nat ) )
% 0.21/0.47 => ? [M3: nat] :
% 0.21/0.47 ( N
% 0.21/0.47 = ( suc @ M3 ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % not0_implies_Suc
% 0.21/0.47 thf(fact_68_Nat_OlessE,axiom,
% 0.21/0.47 ! [I: nat,K: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ I @ K )
% 0.21/0.47 => ( ( K
% 0.21/0.47 != ( suc @ I ) )
% 0.21/0.47 => ~ ! [J: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ I @ J )
% 0.21/0.47 => ( K
% 0.21/0.47 != ( suc @ J ) ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % Nat.lessE
% 0.21/0.47 thf(fact_69_Suc__lessD,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.21/0.47 => ( ord_less @ nat @ M @ N ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_lessD
% 0.21/0.47 thf(fact_70_Suc__lessE,axiom,
% 0.21/0.47 ! [I: nat,K: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ ( suc @ I ) @ K )
% 0.21/0.47 => ~ ! [J: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ I @ J )
% 0.21/0.47 => ( K
% 0.21/0.47 != ( suc @ J ) ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_lessE
% 0.21/0.47 thf(fact_71_Suc__lessI,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( ( ( suc @ M )
% 0.21/0.47 != N )
% 0.21/0.47 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % Suc_lessI
% 0.21/0.47 thf(fact_72_less__SucE,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.21/0.47 => ( ~ ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( M = N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_SucE
% 0.21/0.47 thf(fact_73_less__SucI,axiom,
% 0.21/0.47 ! [M: nat,N: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ M @ N )
% 0.21/0.47 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.21/0.47
% 0.21/0.47 % less_SucI
% 0.21/0.47 thf(fact_74_Ex__less__Suc,axiom,
% 0.21/0.47 ! [N: nat,P: nat > $o] :
% 0.21/0.47 ( ( ? [I2: nat] :
% 0.21/0.47 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.21/0.47 & ( P @ I2 ) ) )
% 0.21/0.47 = ( ( P @ N )
% 0.32/0.47 | ? [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ N )
% 0.32/0.47 & ( P @ I2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ex_less_Suc
% 0.32/0.47 thf(fact_75_less__Suc__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 | ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_eq
% 0.32/0.47 thf(fact_76_not__less__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.32/0.47 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_eq
% 0.32/0.47 thf(fact_77_All__less__Suc,axiom,
% 0.32/0.47 ! [N: nat,P: nat > $o] :
% 0.32/0.47 ( ( ! [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.32/0.47 => ( P @ I2 ) ) )
% 0.32/0.47 = ( ( P @ N )
% 0.32/0.47 & ! [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ N )
% 0.32/0.47 => ( P @ I2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % All_less_Suc
% 0.32/0.47 thf(fact_78_Suc__less__eq2,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.32/0.47 = ( ? [M4: nat] :
% 0.32/0.47 ( ( M
% 0.32/0.47 = ( suc @ M4 ) )
% 0.32/0.47 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_less_eq2
% 0.32/0.47 thf(fact_79_less__antisym,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_antisym
% 0.32/0.47 thf(fact_80_Suc__less__SucD,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.47 => ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_less_SucD
% 0.32/0.47 thf(fact_81_less__trans__Suc,axiom,
% 0.32/0.47 ! [I: nat,J2: nat,K: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I @ J2 )
% 0.32/0.47 => ( ( ord_less @ nat @ J2 @ K )
% 0.32/0.47 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_trans_Suc
% 0.32/0.47 thf(fact_82_less__Suc__induct,axiom,
% 0.32/0.47 ! [I: nat,J2: nat,P: nat > nat > $o] :
% 0.32/0.47 ( ( ord_less @ nat @ I @ J2 )
% 0.32/0.47 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.32/0.47 => ( ! [I3: nat,J: nat,K2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J )
% 0.32/0.47 => ( ( ord_less @ nat @ J @ K2 )
% 0.32/0.47 => ( ( P @ I3 @ J )
% 0.32/0.47 => ( ( P @ J @ K2 )
% 0.32/0.47 => ( P @ I3 @ K2 ) ) ) ) )
% 0.32/0.47 => ( P @ I @ J2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_induct
% 0.32/0.47 thf(fact_83_strict__inc__induct,axiom,
% 0.32/0.47 ! [I: nat,J2: nat,P: nat > $o] :
% 0.32/0.47 ( ( ord_less @ nat @ I @ J2 )
% 0.32/0.47 => ( ! [I3: nat] :
% 0.32/0.47 ( ( J2
% 0.32/0.47 = ( suc @ I3 ) )
% 0.32/0.47 => ( P @ I3 ) )
% 0.32/0.47 => ( ! [I3: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I3 @ J2 )
% 0.32/0.47 => ( ( P @ ( suc @ I3 ) )
% 0.32/0.47 => ( P @ I3 ) ) )
% 0.32/0.47 => ( P @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_inc_induct
% 0.32/0.47 thf(fact_84_not__less__less__Suc__eq,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ~ ( ord_less @ nat @ N @ M )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.32/0.47 = ( N = M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_less_Suc_eq
% 0.32/0.47 thf(fact_85_of__nat__less__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat] :
% 0.32/0.47 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_less_0_iff
% 0.32/0.47 thf(fact_86_less__imp__of__nat__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ N )
% 0.32/0.47 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_of_nat_less
% 0.32/0.47 thf(fact_87_of__nat__less__imp__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.32/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_less_imp_less
% 0.32/0.47 thf(fact_88_lift__Suc__mono__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,N3: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( ( ord_less @ nat @ N @ N3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_mono_less
% 0.32/0.47 thf(fact_89_lift__Suc__mono__less__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,M: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.47 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_mono_less_iff
% 0.32/0.47 thf(fact_90_Ex__less__Suc2,axiom,
% 0.32/0.47 ! [N: nat,P: nat > $o] :
% 0.32/0.47 ( ( ? [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.32/0.47 & ( P @ I2 ) ) )
% 0.32/0.47 = ( ( P @ ( zero_zero @ nat ) )
% 0.32/0.47 | ? [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ N )
% 0.32/0.47 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Ex_less_Suc2
% 0.32/0.47 thf(fact_91_gr0__conv__Suc,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 = ( ? [M5: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( suc @ M5 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr0_conv_Suc
% 0.32/0.47 thf(fact_92_All__less__Suc2,axiom,
% 0.32/0.47 ! [N: nat,P: nat > $o] :
% 0.32/0.47 ( ( ! [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 0.32/0.47 => ( P @ I2 ) ) )
% 0.32/0.47 = ( ( P @ ( zero_zero @ nat ) )
% 0.32/0.47 & ! [I2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ I2 @ N )
% 0.32/0.47 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % All_less_Suc2
% 0.32/0.47 thf(fact_93_gr0__implies__Suc,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ? [M3: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( suc @ M3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % gr0_implies_Suc
% 0.32/0.47 thf(fact_94_less__Suc__eq__0__disj,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.32/0.47 = ( ( M
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 | ? [J3: nat] :
% 0.32/0.47 ( ( M
% 0.32/0.47 = ( suc @ J3 ) )
% 0.32/0.47 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_Suc_eq_0_disj
% 0.32/0.47 thf(fact_95_ord__eq__less__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_subst
% 0.32/0.47 thf(fact_96_ord__less__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A2: A,B3: A,F: A > B,C: B] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_subst
% 0.32/0.47 thf(fact_97_order__less__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst1
% 0.32/0.47 thf(fact_98_order__less__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_less_subst2
% 0.32/0.47 thf(fact_99_lt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_bot @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % lt_ex
% 0.32/0.47 thf(fact_100_gt__ex,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( no_top @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % gt_ex
% 0.32/0.47 thf(fact_101_neqE,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( X != Y2 )
% 0.32/0.47 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neqE
% 0.32/0.47 thf(fact_102_neq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( X != Y2 )
% 0.32/0.47 = ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq_iff
% 0.32/0.47 thf(fact_103_order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.asym
% 0.32/0.47 thf(fact_104_dense,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( dense_order @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ? [Z: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Z )
% 0.32/0.47 & ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dense
% 0.32/0.47 thf(fact_105_less__imp__neq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( X != Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_neq
% 0.32/0.47 thf(fact_106_less__asym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym
% 0.32/0.47 thf(fact_107_less__asym_H,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_asym'
% 0.32/0.47 thf(fact_108_less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_trans
% 0.32/0.47 thf(fact_109_less__linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 | ( X = Y2 )
% 0.32/0.47 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_linear
% 0.32/0.47 thf(fact_110_less__irrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_irrefl
% 0.32/0.47 thf(fact_111_ord__eq__less__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( A2 = B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_less_trans
% 0.32/0.47 thf(fact_112_ord__less__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_less_eq_trans
% 0.32/0.47 thf(fact_113_dual__order_Oasym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A2: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.32/0.47 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.asym
% 0.32/0.47 thf(fact_114_less__imp__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( X != Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq
% 0.32/0.47 thf(fact_115_less__not__sym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_not_sym
% 0.32/0.47 thf(fact_116_less__induct,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ! [P: A > $o,A2: A] :
% 0.32/0.47 ( ! [X2: A] :
% 0.32/0.47 ( ! [Y: A] :
% 0.32/0.47 ( ( ord_less @ A @ Y @ X2 )
% 0.32/0.47 => ( P @ Y ) )
% 0.32/0.47 => ( P @ X2 ) )
% 0.32/0.47 => ( P @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_induct
% 0.32/0.47 thf(fact_117_antisym__conv3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [Y2: A,X: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ Y2 @ X )
% 0.32/0.47 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.32/0.47 = ( X = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv3
% 0.32/0.47 thf(fact_118_less__imp__not__eq2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( Y2 != X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_eq2
% 0.32/0.47 thf(fact_119_less__imp__triv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A,P: $o] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( ( ord_less @ A @ Y2 @ X )
% 0.32/0.47 => P ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_triv
% 0.32/0.47 thf(fact_120_linorder__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ( ( X != Y2 )
% 0.32/0.47 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_cases
% 0.32/0.47 thf(fact_121_dual__order_Oirrefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.irrefl
% 0.32/0.47 thf(fact_122_order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_trans
% 0.32/0.47 thf(fact_123_less__imp__not__less,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.47 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_imp_not_less
% 0.32/0.47 thf(fact_124_exists__least__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( wellorder @ A )
% 0.32/0.47 => ( ( ^ [P4: A > $o] :
% 0.32/0.47 ? [X4: A] : ( P4 @ X4 ) )
% 0.32/0.47 = ( ^ [P5: A > $o] :
% 0.32/0.47 ? [N4: A] :
% 0.32/0.47 ( ( P5 @ N4 )
% 0.32/0.47 & ! [M5: A] :
% 0.32/0.47 ( ( ord_less @ A @ M5 @ N4 )
% 0.32/0.47 => ~ ( P5 @ M5 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_least_iff
% 0.32/0.47 thf(fact_125_linorder__less__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.32/0.47 ( ! [A3: A,B2: A] :
% 0.32/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.32/0.47 => ( P @ A3 @ B2 ) )
% 0.32/0.47 => ( ! [A3: A] : ( P @ A3 @ A3 )
% 0.32/0.47 => ( ! [A3: A,B2: A] :
% 0.32/0.47 ( ( P @ B2 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B2 ) )
% 0.32/0.47 => ( P @ A2 @ B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_less_wlog
% 0.32/0.47 thf(fact_126_dual__order_Ostrict__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A2: A,C: A] :
% 0.32/0.47 ( ( ord_less @ A @ B3 @ A2 )
% 0.32/0.47 => ( ( ord_less @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.strict_trans
% 0.32/0.47 thf(fact_127_not__less__iff__gr__or__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.32/0.47 = ( ( ord_less @ A @ Y2 @ X )
% 0.32/0.47 | ( X = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_less_iff_gr_or_eq
% 0.32/0.47 thf(fact_128_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.47 => ( A2 != B3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.strict_implies_not_eq
% 0.32/0.47 thf(fact_129_linordered__field__no__lb,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_field @ A )
% 0.32/0.47 => ! [X3: A] :
% 0.32/0.47 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linordered_field_no_lb
% 0.32/0.47 thf(fact_130_pos__int__cases,axiom,
% 0.32/0.47 ! [K: int] :
% 0.32/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.32/0.47 => ~ ! [N2: nat] :
% 0.32/0.47 ( ( K
% 0.32/0.47 = ( semiring_1_of_nat @ int @ N2 ) )
% 0.32/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % pos_int_cases
% 0.32/0.47 thf(fact_131_zero__less__imp__eq__int,axiom,
% 0.32/0.47 ! [K: int] :
% 0.32/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.32/0.47 => ? [N2: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.32/0.47 & ( K
% 0.32/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_less_imp_eq_int
% 0.32/0.47 thf(fact_132_dependent__nat__choice,axiom,
% 0.32/0.47 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 0.32/0.47 ( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
% 0.32/0.47 => ( ! [X2: A,N2: nat] :
% 0.32/0.47 ( ( P @ N2 @ X2 )
% 0.32/0.47 => ? [Y: A] :
% 0.32/0.47 ( ( P @ ( suc @ N2 ) @ Y )
% 0.32/0.47 & ( Q @ N2 @ X2 @ Y ) ) )
% 0.32/0.47 => ? [F2: nat > A] :
% 0.32/0.47 ! [N5: nat] :
% 0.32/0.47 ( ( P @ N5 @ ( F2 @ N5 ) )
% 0.32/0.47 & ( Q @ N5 @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dependent_nat_choice
% 0.32/0.47 thf(fact_133_exists__least__lemma,axiom,
% 0.32/0.47 ! [P: nat > $o] :
% 0.32/0.47 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.32/0.47 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.32/0.47 => ? [N2: nat] :
% 0.32/0.47 ( ~ ( P @ N2 )
% 0.32/0.47 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % exists_least_lemma
% 0.32/0.47 thf(fact_134_reals__Archimedean2,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( archim1804426504_field @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % reals_Archimedean2
% 0.32/0.47 thf(fact_135_of__nat__zero__less__power__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [X: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 0.32/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.32/0.47 | ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_zero_less_power_iff
% 0.32/0.47 thf(fact_136_of__nat__le__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_le_0_iff
% 0.32/0.47 thf(fact_137_Suc__pred,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.32/0.47 = N ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_pred
% 0.32/0.47 thf(fact_138_order__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_refl
% 0.32/0.47 thf(fact_139_bot__nat__0_Oextremum,axiom,
% 0.32/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum
% 0.32/0.47 thf(fact_140_le0,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le0
% 0.32/0.47 thf(fact_141_Suc__le__mono,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_le_mono
% 0.32/0.47 thf(fact_142_diff__diff__cancel,axiom,
% 0.32/0.47 ! [I: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.32/0.47 = I ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_diff_cancel
% 0.32/0.47 thf(fact_143_le__zero__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( canoni770627133id_add @ A )
% 0.32/0.47 => ! [N: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.32/0.47 = ( N
% 0.32/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_144_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel1352612707id_add @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( minus_minus @ A @ A2 @ A2 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.47 thf(fact_145_diff__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel1352612707id_add @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_zero
% 0.32/0.47 thf(fact_146_zero__diff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_diff @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_diff
% 0.32/0.47 thf(fact_147_diff__0__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_0_right
% 0.32/0.47 thf(fact_148_diff__self,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( minus_minus @ A @ A2 @ A2 )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_self
% 0.32/0.47 thf(fact_149_of__nat__le__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_le_iff
% 0.32/0.47 thf(fact_150_nat__power__eq__Suc__0__iff,axiom,
% 0.32/0.47 ! [X: nat,M: nat] :
% 0.32/0.47 ( ( ( power_power @ nat @ X @ M )
% 0.32/0.47 = ( suc @ ( zero_zero @ nat ) ) )
% 0.32/0.47 = ( ( M
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 | ( X
% 0.32/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_power_eq_Suc_0_iff
% 0.32/0.47 thf(fact_151_power__Suc__0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.32/0.47 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_Suc_0
% 0.32/0.47 thf(fact_152_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_char_0 @ A )
% 0.32/0.47 => ! [X: nat,B3: nat,W: nat] :
% 0.32/0.47 ( ( ( semiring_1_of_nat @ A @ X )
% 0.32/0.47 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 0.32/0.47 = ( X
% 0.32/0.47 = ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.47 thf(fact_153_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_char_0 @ A )
% 0.32/0.47 => ! [B3: nat,W: nat,X: nat] :
% 0.32/0.47 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W )
% 0.32/0.47 = ( semiring_1_of_nat @ A @ X ) )
% 0.32/0.47 = ( ( power_power @ nat @ B3 @ W )
% 0.32/0.47 = X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.47 thf(fact_154_of__nat__power,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_1 @ A )
% 0.32/0.47 => ! [M: nat,N: nat] :
% 0.32/0.47 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 0.32/0.47 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_power
% 0.32/0.47 thf(fact_155_nat__zero__less__power__iff,axiom,
% 0.32/0.47 ! [X: nat,N: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.32/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.32/0.47 | ( N
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_zero_less_power_iff
% 0.32/0.47 thf(fact_156_diff__is__0__eq_H,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ( minus_minus @ nat @ M @ N )
% 0.32/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_is_0_eq'
% 0.32/0.47 thf(fact_157_diff__is__0__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( minus_minus @ nat @ M @ N )
% 0.32/0.47 = ( zero_zero @ nat ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_is_0_eq
% 0.32/0.47 thf(fact_158_diff__self__eq__0,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ M @ M )
% 0.32/0.47 = ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_self_eq_0
% 0.32/0.47 thf(fact_159_diff__0__eq__0,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 = ( zero_zero @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_0_eq_0
% 0.32/0.47 thf(fact_160_Suc__diff__diff,axiom,
% 0.32/0.47 ! [M: nat,N: nat,K: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.32/0.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_diff_diff
% 0.32/0.47 thf(fact_161_diff__Suc__Suc,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.47 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_Suc_Suc
% 0.32/0.47 thf(fact_162_diff__ge__0__iff__ge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B3 ) )
% 0.32/0.47 = ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_ge_0_iff_ge
% 0.32/0.47 thf(fact_163_diff__gt__0__iff__gt,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B3 ) )
% 0.32/0.47 = ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_gt_0_iff_gt
% 0.32/0.47 thf(fact_164_power__0__Suc,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_1 @ A )
% 0.32/0.47 => ! [N: nat] :
% 0.32/0.47 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 0.32/0.47 = ( zero_zero @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_0_Suc
% 0.32/0.47 thf(fact_165_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [X: nat,B3: nat,W: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.47 thf(fact_166_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [B3: nat,W: nat,X: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.47 thf(fact_167_power__Suc0__right,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( monoid_mult @ A )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_Suc0_right
% 0.32/0.47 thf(fact_168_zero__less__diff,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 0.32/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_less_diff
% 0.32/0.47 thf(fact_169_power__eq__0__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiri134348788visors @ A )
% 0.32/0.47 => ! [A2: A,N: nat] :
% 0.32/0.47 ( ( ( power_power @ A @ A2 @ N )
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 = ( ( A2
% 0.32/0.47 = ( zero_zero @ A ) )
% 0.32/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_eq_0_iff
% 0.32/0.47 thf(fact_170_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [B3: nat,W: nat,X: nat] :
% 0.32/0.47 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.32/0.47 = ( ord_less @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.47 thf(fact_171_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [X: nat,B3: nat,W: nat] :
% 0.32/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 0.32/0.47 = ( ord_less @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.47 thf(fact_172_power__mono__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,B3: A,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) )
% 0.32/0.47 = ( ord_less_eq @ A @ A2 @ B3 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_mono_iff
% 0.32/0.47 thf(fact_173_Suc__diff__le,axiom,
% 0.32/0.47 ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 0.32/0.47 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Suc_diff_le
% 0.32/0.47 thf(fact_174_nonneg__int__cases,axiom,
% 0.32/0.47 ! [K: int] :
% 0.32/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.32/0.47 => ~ ! [N2: nat] :
% 0.32/0.47 ( K
% 0.32/0.47 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nonneg_int_cases
% 0.32/0.47 thf(fact_175_zero__le__imp__eq__int,axiom,
% 0.32/0.47 ! [K: int] :
% 0.32/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.32/0.47 => ? [N2: nat] :
% 0.32/0.47 ( K
% 0.32/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_imp_eq_int
% 0.32/0.47 thf(fact_176_zero__integer_Orsp,axiom,
% 0.32/0.47 ( ( zero_zero @ int )
% 0.32/0.47 = ( zero_zero @ int ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_integer.rsp
% 0.32/0.47 thf(fact_177_lift__Suc__mono__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,N3: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_mono_le
% 0.32/0.47 thf(fact_178_lift__Suc__antimono__le,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [F: nat > A,N: nat,N3: nat] :
% 0.32/0.47 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lift_Suc_antimono_le
% 0.32/0.47 thf(fact_179_of__nat__diff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiring_1_cancel @ A )
% 0.32/0.47 => ! [N: nat,M: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ N @ M )
% 0.32/0.47 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 0.32/0.47 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_diff
% 0.32/0.47 thf(fact_180_power__inject__base,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,N: nat,B3: A] :
% 0.32/0.47 ( ( ( power_power @ A @ A2 @ ( suc @ N ) )
% 0.32/0.47 = ( power_power @ A @ B3 @ ( suc @ N ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.47 => ( A2 = B3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_inject_base
% 0.32/0.47 thf(fact_181_of__nat__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linord1659791738miring @ A )
% 0.32/0.47 => ! [I: nat,J2: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ I @ J2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % of_nat_mono
% 0.32/0.47 thf(fact_182_power__le__imp__le__base,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,N: nat,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B3 @ ( suc @ N ) ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_le_imp_le_base
% 0.32/0.47 thf(fact_183_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( A2 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_184_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.47 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_185_dual__order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [B3: A,A2: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_186_linorder__wlog,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.32/0.47 ( ! [A3: A,B2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.32/0.47 => ( P @ A3 @ B2 ) )
% 0.32/0.47 => ( ! [A3: A,B2: A] :
% 0.32/0.47 ( ( P @ B2 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B2 ) )
% 0.32/0.47 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_187_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_188_order__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_189_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A,B3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.32/0.47 => ( A2 = B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_190_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( B3 = C )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_191_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( A2 = B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_192_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.32/0.47 = ( ^ [A4: A,B4: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.32/0.47 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_193_antisym__conv,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [Y2: A,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.32/0.47 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 = ( X = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_194_le__cases3,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A,Z2: A] :
% 0.32/0.47 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 0.32/0.47 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.32/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.32/0.47 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_195_real__arch__simple,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( archim1804426504_field @ A )
% 0.32/0.47 => ! [X: A] :
% 0.32/0.47 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_arch_simple
% 0.32/0.47 thf(fact_196_order_Otrans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_197_le__cases,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_198_eq__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( preorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( X = Y2 )
% 0.32/0.47 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_199_linear,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_200_antisym,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ! [X: A,Y2: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ Y2 @ X )
% 0.32/0.47 => ( X = Y2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_201_eq__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( order @ A )
% 0.32/0.47 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.32/0.47 = ( ^ [X5: A,Y5: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 0.32/0.47 & ( ord_less_eq @ A @ Y5 @ X5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_202_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A2: A,B3: A,F: A > B,C: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ( F @ B3 )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_203_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( ord @ B )
% 0.32/0.47 & ( ord @ A ) )
% 0.32/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 = ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_204_order__subst2,axiom,
% 0.32/0.47 ! [A: $tType,C2: $tType] :
% 0.32/0.47 ( ( ( order @ C2 )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 0.32/0.47 => ( ! [X2: A,Y3: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst2
% 0.32/0.47 thf(fact_205_order__subst1,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType] :
% 0.32/0.47 ( ( ( order @ B )
% 0.32/0.47 & ( order @ A ) )
% 0.32/0.47 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 0.32/0.47 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.32/0.47 => ( ! [X2: B,Y3: B] :
% 0.32/0.47 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.32/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_subst1
% 0.32/0.47 thf(fact_206_le__fun__def,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/0.47 = ( ^ [F3: A > B,G: A > B] :
% 0.32/0.47 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_fun_def
% 0.32/0.47 thf(fact_207_le__funI,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G2: A > B] :
% 0.32/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funI
% 0.32/0.47 thf(fact_208_le__funE,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G2: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funE
% 0.32/0.47 thf(fact_209_le__funD,axiom,
% 0.32/0.47 ! [B: $tType,A: $tType] :
% 0.32/0.47 ( ( ord @ B )
% 0.32/0.47 => ! [F: A > B,G2: A > B,X: A] :
% 0.32/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.32/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_funD
% 0.32/0.47 thf(fact_210_int__int__eq,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ( semiring_1_of_nat @ int @ M )
% 0.32/0.47 = ( semiring_1_of_nat @ int @ N ) )
% 0.32/0.47 = ( M = N ) ) ).
% 0.32/0.47
% 0.32/0.47 % int_int_eq
% 0.32/0.47 thf(fact_211_zle__int,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zle_int
% 0.32/0.47 thf(fact_212_le__iff__diff__le__0,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ( ( ord_less_eq @ A )
% 0.32/0.47 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_iff_diff_le_0
% 0.32/0.47 thf(fact_213_diff__less__mono,axiom,
% 0.32/0.47 ! [A2: nat,B3: nat,C: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ C @ A2 )
% 0.32/0.47 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_less_mono
% 0.32/0.47 thf(fact_214_less__diff__iff,axiom,
% 0.32/0.47 ! [K: nat,M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.47 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.47 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_diff_iff
% 0.32/0.47 thf(fact_215_power__not__zero,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( semiri134348788visors @ A )
% 0.32/0.47 => ! [A2: A,N: nat] :
% 0.32/0.47 ( ( A2
% 0.32/0.47 != ( zero_zero @ A ) )
% 0.32/0.47 => ( ( power_power @ A @ A2 @ N )
% 0.32/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_not_zero
% 0.32/0.47 thf(fact_216_power__less__imp__less__base,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,N: nat,B3: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.47 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_less_imp_less_base
% 0.32/0.47 thf(fact_217_diff__right__commute,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( cancel146912293up_add @ A )
% 0.32/0.47 => ! [A2: A,C: A,B3: A] :
% 0.32/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_right_commute
% 0.32/0.47 thf(fact_218_diff__eq__diff__less__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A2 @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_less_eq
% 0.32/0.47 thf(fact_219_diff__right__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_right_mono
% 0.32/0.47 thf(fact_220_diff__left__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [B3: A,A2: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_left_mono
% 0.32/0.47 thf(fact_221_zero__le__power,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_power
% 0.32/0.47 thf(fact_222_diff__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ordered_ab_group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A,D: A,C: A] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ D @ C )
% 0.32/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_mono
% 0.32/0.47 thf(fact_223_power__mono,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A )
% 0.32/0.47 => ! [A2: A,B3: A,N: nat] :
% 0.32/0.47 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.47 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % power_mono
% 0.32/0.47 thf(fact_224_diff__eq__diff__eq,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( group_add @ A )
% 0.32/0.47 => ! [A2: A,B3: A,C: A,D: A] :
% 0.32/0.47 ( ( ( minus_minus @ A @ A2 @ B3 )
% 0.32/0.47 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.47 => ( ( A2 = B3 )
% 0.32/0.47 = ( C = D ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_eq_diff_eq
% 0.32/0.47 thf(fact_225_diff__le__mono2,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_mono2
% 0.32/0.47 thf(fact_226_le__diff__iff_H,axiom,
% 0.32/0.47 ! [A2: nat,C: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ B3 @ C )
% 0.32/0.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 0.32/0.47 = ( ord_less_eq @ nat @ B3 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_diff_iff'
% 0.32/0.47 thf(fact_227_diff__le__self,axiom,
% 0.32/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.32/0.47
% 0.32/0.47 % diff_le_self
% 0.32/0.47 thf(fact_228_diff__le__mono,axiom,
% 0.32/0.47 ! [M: nat,N: nat,L: nat] :
% 0.32/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.32/0.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_le_mono
% 0.32/0.48 thf(fact_229_Nat_Odiff__diff__eq,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.48 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.48 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % Nat.diff_diff_eq
% 0.32/0.48 thf(fact_230_diff__commute,axiom,
% 0.32/0.48 ! [I: nat,J2: nat,K: nat] :
% 0.32/0.48 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K )
% 0.32/0.48 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_commute
% 0.32/0.48 thf(fact_231_le__diff__iff,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.32/0.48 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_diff_iff
% 0.32/0.48 thf(fact_232_eq__diff__iff,axiom,
% 0.32/0.48 ! [K: nat,M: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ K @ M )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ K @ N )
% 0.32/0.48 => ( ( ( minus_minus @ nat @ M @ K )
% 0.32/0.48 = ( minus_minus @ nat @ N @ K ) )
% 0.32/0.48 = ( M = N ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_diff_iff
% 0.32/0.48 thf(fact_233_less__eq__int__code_I1_J,axiom,
% 0.32/0.48 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.32/0.48
% 0.32/0.48 % less_eq_int_code(1)
% 0.32/0.48 thf(fact_234_less__int__code_I1_J,axiom,
% 0.32/0.48 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_int_code(1)
% 0.32/0.48 thf(fact_235_nat__one__le__power,axiom,
% 0.32/0.48 ! [I: nat,N: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_one_le_power
% 0.32/0.48 thf(fact_236_power__eq__imp__eq__base,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linordered_semidom @ A )
% 0.32/0.48 => ! [A2: A,N: nat,B3: A] :
% 0.32/0.48 ( ( ( power_power @ A @ A2 @ N )
% 0.32/0.48 = ( power_power @ A @ B3 @ N ) )
% 0.32/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.48 => ( A2 = B3 ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % power_eq_imp_eq_base
% 0.32/0.48 thf(fact_237_power__eq__iff__eq__base,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linordered_semidom @ A )
% 0.32/0.48 => ! [N: nat,A2: A,B3: A] :
% 0.32/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.32/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 0.32/0.48 => ( ( ( power_power @ A @ A2 @ N )
% 0.32/0.48 = ( power_power @ A @ B3 @ N ) )
% 0.32/0.48 = ( A2 = B3 ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % power_eq_iff_eq_base
% 0.32/0.48 thf(fact_238_eq__iff__diff__eq__0,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( group_add @ A )
% 0.32/0.48 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.32/0.48 = ( ^ [A4: A,B4: A] :
% 0.32/0.48 ( ( minus_minus @ A @ A4 @ B4 )
% 0.32/0.48 = ( zero_zero @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_iff_diff_eq_0
% 0.32/0.48 thf(fact_239_diff__strict__right__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordered_ab_group_add @ A )
% 0.32/0.48 => ! [A2: A,B3: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.48 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_strict_right_mono
% 0.32/0.48 thf(fact_240_diff__strict__left__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordered_ab_group_add @ A )
% 0.32/0.48 => ! [B3: A,A2: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ B3 @ A2 )
% 0.32/0.48 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_strict_left_mono
% 0.32/0.48 thf(fact_241_diff__eq__diff__less,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordered_ab_group_add @ A )
% 0.32/0.48 => ! [A2: A,B3: A,C: A,D: A] :
% 0.32/0.48 ( ( ( minus_minus @ A @ A2 @ B3 )
% 0.32/0.48 = ( minus_minus @ A @ C @ D ) )
% 0.32/0.48 => ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.48 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_eq_diff_less
% 0.32/0.48 thf(fact_242_diff__strict__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ordered_ab_group_add @ A )
% 0.32/0.48 => ! [A2: A,B3: A,D: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.48 => ( ( ord_less @ A @ D @ C )
% 0.32/0.48 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % diff_strict_mono
% 0.32/0.48 thf(fact_243_zero__le,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( canoni770627133id_add @ A )
% 0.32/0.48 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.32/0.48
% 0.32/0.48 % zero_le
% 0.32/0.48 thf(fact_244_le__numeral__extra_I3_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linord1659791738miring @ A )
% 0.32/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_numeral_extra(3)
% 0.32/0.48 thf(fact_245_complete__interval,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( condit1037483654norder @ A )
% 0.32/0.48 => ! [A2: A,B3: A,P: A > $o] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.48 => ( ( P @ A2 )
% 0.32/0.48 => ( ~ ( P @ B3 )
% 0.32/0.48 => ? [C3: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.32/0.48 & ( ord_less_eq @ A @ C3 @ B3 )
% 0.32/0.48 & ! [X3: A] :
% 0.32/0.48 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.32/0.48 & ( ord_less @ A @ X3 @ C3 ) )
% 0.32/0.48 => ( P @ X3 ) )
% 0.32/0.48 & ! [D3: A] :
% 0.32/0.48 ( ! [X2: A] :
% 0.32/0.48 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.32/0.48 & ( ord_less @ A @ X2 @ D3 ) )
% 0.32/0.48 => ( P @ X2 ) )
% 0.32/0.48 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % complete_interval
% 0.32/0.48 thf(fact_246_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A2: A,B3: A] :
% 0.32/0.48 ( ( A2 != B3 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.32/0.48 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.not_eq_order_implies_strict
% 0.32/0.48 thf(fact_247_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [B3: A,A2: A] :
% 0.32/0.48 ( ( ord_less @ A @ B3 @ A2 )
% 0.32/0.48 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_implies_order
% 0.32/0.48 thf(fact_248_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ( ( ord_less @ A )
% 0.32/0.48 = ( ^ [B4: A,A4: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.32/0.48 & ( A4 != B4 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_iff_order
% 0.32/0.48 thf(fact_249_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ( ( ord_less_eq @ A )
% 0.32/0.48 = ( ^ [B4: A,A4: A] :
% 0.32/0.48 ( ( ord_less @ A @ B4 @ A4 )
% 0.32/0.48 | ( A4 = B4 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.order_iff_strict
% 0.32/0.48 thf(fact_250_order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A )
% 0.32/0.48 => ! [A2: A,B3: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B3 )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_implies_order
% 0.32/0.48 thf(fact_251_dense__le__bounded,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dense_linorder @ A )
% 0.32/0.48 => ! [X: A,Y2: A,Z2: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y2 )
% 0.32/0.48 => ( ! [W2: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ W2 )
% 0.32/0.48 => ( ( ord_less @ A @ W2 @ Y2 )
% 0.32/0.48 => ( ord_less_eq @ A @ W2 @ Z2 ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dense_le_bounded
% 0.32/0.48 thf(fact_252_dense__ge__bounded,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dense_linorder @ A )
% 0.32/0.48 => ! [Z2: A,X: A,Y2: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z2 @ X )
% 0.32/0.48 => ( ! [W2: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z2 @ W2 )
% 0.32/0.48 => ( ( ord_less @ A @ W2 @ X )
% 0.32/0.48 => ( ord_less_eq @ A @ Y2 @ W2 ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dense_ge_bounded
% 0.32/0.48
% 0.32/0.48 % Subclasses (1)
% 0.32/0.48 thf(subcl_Groups_Ozero___HOL_Otype,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( zero @ A )
% 0.32/0.48 => ( type @ A ) ) ).
% 0.32/0.48
% 0.32/0.48 % Type constructors (46)
% 0.32/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.48 ! [A5: $tType,A6: $tType] :
% 0.32/0.48 ( ( preorder @ A6 )
% 0.32/0.48 => ( preorder @ ( A5 > A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.48 ! [A5: $tType,A6: $tType] :
% 0.32/0.48 ( ( order @ A6 )
% 0.32/0.48 => ( order @ ( A5 > A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.48 ! [A5: $tType,A6: $tType] :
% 0.32/0.48 ( ( ord @ A6 )
% 0.32/0.48 => ( ord @ ( A5 > A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.32/0.48 condit1037483654norder @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.32/0.48 semiri134348788visors @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.32/0.48 linord1659791738miring @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.32/0.48 cancel146912293up_add @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.32/0.48 cancel1352612707id_add @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.32/0.48 ordered_ab_group_add @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.32/0.48 linordered_semidom @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
% 0.32/0.48 semiring_1_cancel @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.32/0.48 linordered_idom @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.32/0.48 semiring_char_0 @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.32/0.48 preorder @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.32/0.48 linorder @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 0.32/0.48 monoid_mult @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.32/0.48 semiring_1 @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.32/0.48 no_top @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.32/0.48 no_bot @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.32/0.48 group_add @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.32/0.48 order @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.32/0.48 ord @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.32/0.48 zero @ int ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,axiom,
% 0.32/0.48 condit1037483654norder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.32/0.48 canoni770627133id_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 0.32/0.48 semiri134348788visors @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6,axiom,
% 0.32/0.48 linord1659791738miring @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 0.32/0.48 cancel146912293up_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 0.32/0.48 cancel1352612707id_add @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 0.32/0.48 linordered_semidom @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_10,axiom,
% 0.32/0.48 semiring_1_cancel @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.32/0.48 comm_monoid_diff @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/0.48 wellorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 0.32/0.48 semiring_char_0 @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_12,axiom,
% 0.32/0.48 preorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder_13,axiom,
% 0.32/0.48 linorder @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Omonoid__mult_14,axiom,
% 0.32/0.48 monoid_mult @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Osemiring__1_15,axiom,
% 0.32/0.48 semiring_1 @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Ono__top_16,axiom,
% 0.32/0.48 no_top @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_17,axiom,
% 0.32/0.48 order @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oord_18,axiom,
% 0.32/0.48 ord @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ozero_19,axiom,
% 0.32/0.48 zero @ nat ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_20,axiom,
% 0.32/0.48 preorder @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_21,axiom,
% 0.32/0.48 linorder @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_22,axiom,
% 0.32/0.48 order @ $o ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
% 0.32/0.48 ord @ $o ).
% 0.32/0.48
% 0.32/0.48 % Free types (1)
% 0.32/0.48 thf(tfree_0,hypothesis,
% 0.32/0.48 zero @ a ).
% 0.32/0.48
% 0.32/0.48 % Conjectures (3)
% 0.32/0.48 thf(conj_0,hypothesis,
% 0.32/0.48 ord_less @ nat @ ( zero_zero @ nat ) @ n ).
% 0.32/0.48
% 0.32/0.48 thf(conj_1,hypothesis,
% 0.32/0.48 polyno86455060npolyh @ a @ p @ n ).
% 0.32/0.48
% 0.32/0.48 thf(conj_2,conjecture,
% 0.32/0.48 ( ( polyno498386536degree @ a @ p )
% 0.32/0.48 = ( zero_zero @ nat ) ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 2.49/1.49 ---- Embedded in THF ---
% 2.49/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.49/1.49 %%% Generated on Tue Feb 24 08:28:51 EST 2026
% 2.49/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.49/1.49 %%% Logic specification used:
% 2.49/1.49 %%% thf(spec, logic, $$dhol).
% 2.49/1.49
% 2.49/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2DTF21321.p
% 2.49/1.49 thf(ty_t_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly, type, polyno1783536151e_poly: ($tType > $tType)).
% 2.49/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.49/1.49 thf(ty_t_Int_Oint, type, int: $tType).
% 2.49/1.49 thf(ty_tf_a, type, a: $tType).
% 2.49/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Osemiring__1__cancel, type, semiring_1_cancel: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: ($o))).
% 2.49/1.49 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.49/1.49 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.49/1.49 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.49/1.49 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.49/1.49 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.49/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.49/1.49 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegree, type, polyno498386536degree: (!> [A:$tType]: (((polyno1783536151e_poly @ A) > nat)))).
% 2.49/1.49 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegreen, type, polyno367318022egreen: (!> [A:$tType]: (((polyno1783536151e_poly @ A) > (nat > nat))))).
% 2.49/1.49 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpoly, type, polyno2122670676snpoly: (!> [A:$tType]: (((polyno1783536151e_poly @ A) > $o)))).
% 2.49/1.49 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpolyh, type, polyno86455060npolyh: (!> [A:$tType]: (((polyno1783536151e_poly @ A) > (nat > $o))))).
% 2.49/1.49 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.49/1.49 thf(sy_v_n, type, n: nat).
% 2.49/1.49 thf(sy_v_p, type, p: (polyno1783536151e_poly @ a)).
% 2.49/1.49 thf(fact_0_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.49/1.49 thf(fact_1_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_2_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.49/1.49 thf(fact_3_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_4_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.49/1.49 thf(fact_5_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_6_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_7_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_8_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_9_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 2.49/1.49 thf(fact_10_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_11_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_12_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.49/1.49 thf(fact_13_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.49/1.49 thf(fact_14_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y))))))))) => (P @ X))))).
% 2.49/1.49 thf(fact_15_linorder__neqE__nat, axiom, (! [X:nat,Y2:nat]: (((X != Y2) => ((~ ((((ord_less @ nat) @ X) @ Y2))) => (((ord_less @ nat) @ Y2) @ X)))))).
% 2.49/1.49 thf(fact_16_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.49/1.49 thf(fact_17_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.49/1.49 thf(fact_18_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.49/1.49 thf(fact_19_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.49/1.49 thf(fact_20_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.49/1.49 thf(fact_21_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.49/1.49 thf(fact_22_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.49/1.49 thf(fact_23_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_24_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_25_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.49/1.49 thf(fact_26_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.49/1.49 thf(fact_27_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X:A]: (((! [X2:A]: ((((V @ X2) = (zero_zero @ nat)) => (P @ X2)))) => ((! [X2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X2)) => ((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y)))))))))) => (P @ X)))))).
% 2.49/1.49 thf(fact_28_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D2))))))))))))).
% 2.49/1.49 thf(fact_29_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_30_isnpoly__def, axiom, (! [A:$tType]: (((zero @ A) => ((polyno2122670676snpoly @ A) = (^ [P2:(polyno1783536151e_poly @ A)]: ((((polyno86455060npolyh @ A) @ P2) @ (zero_zero @ nat))))))))).
% 2.49/1.49 thf(fact_31_degreen__0, axiom, (! [A:$tType]: (((zero @ A) => (! [P3:(polyno1783536151e_poly @ A),N:nat,M:nat]: (((((polyno86455060npolyh @ A) @ P3) @ N) => ((((ord_less @ nat) @ M) @ N) => ((((polyno367318022egreen @ A) @ P3) @ M) = (zero_zero @ nat)))))))))).
% 2.49/1.49 thf(fact_32_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 2.49/1.49 thf(fact_33_degree__isnpolyh__Suc, axiom, (! [A:$tType]: (((zero @ A) => (! [P3:(polyno1783536151e_poly @ A),N:nat]: (((((polyno86455060npolyh @ A) @ P3) @ (suc @ N)) => (((polyno498386536degree @ A) @ P3) = (zero_zero @ nat))))))))).
% 2.49/1.49 thf(fact_34_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.49/1.49 thf(fact_35_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit1656338222tinuum @ A) => (! [A2:A]: ((? [B2:A]: (((((ord_less @ A) @ A2) @ B2) | (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.49/1.49 thf(fact_36_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.49/1.49 thf(fact_37_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (A2 != B3)))))))).
% 2.49/1.49 thf(fact_38_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: ((? [X_1:A]: ((((ord_less @ A) @ X3) @ X_1))))))))).
% 2.49/1.49 thf(fact_39_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))).
% 2.49/1.49 thf(fact_40_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.49/1.49 thf(fact_41_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.49/1.49 thf(fact_42_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 2.49/1.49 thf(fact_43_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 2.49/1.49 thf(fact_44_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_45_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.49/1.49 thf(fact_46_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.49/1.49 thf(fact_47_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.49/1.49 thf(fact_48_of__nat__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 2.49/1.49 thf(fact_49_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_50_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 2.49/1.49 thf(fact_51_Suc__inject, axiom, (! [X:nat,Y2:nat]: ((((suc @ X) = (suc @ Y2)) => (X = Y2))))).
% 2.49/1.49 thf(fact_52_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.49/1.49 thf(fact_53_of__nat__neq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N)) != (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_54_polypow_Ocases, axiom, (! [X:nat]: (((X != (zero_zero @ nat)) => (~ ((! [V2:nat]: ((X != (suc @ V2)))))))))).
% 2.49/1.49 thf(fact_55_nat_Odistinct_I1_J, axiom, (! [X22:nat]: (((zero_zero @ nat) != (suc @ X22))))).
% 2.49/1.49 thf(fact_56_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))).
% 2.49/1.49 thf(fact_57_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.49/1.49 thf(fact_58_nat_OdiscI, axiom, (! [Nat:nat,X22:nat]: (((Nat = (suc @ X22)) => (Nat != (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_59_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((P @ N2) => (P @ (suc @ N2))))) => (P @ N)))))).
% 2.49/1.49 thf(fact_60_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X2:nat]: (((P @ X2) @ (zero_zero @ nat)))) => ((! [Y3:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y3)))) => ((! [X2:nat,Y3:nat]: ((((P @ X2) @ Y3) => ((P @ (suc @ X2)) @ (suc @ Y3))))) => ((P @ M) @ N))))))).
% 2.49/1.49 thf(fact_61_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_62_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 2.49/1.49 thf(fact_63_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.49/1.49 thf(fact_64_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.49/1.49 thf(fact_65_old_Onat_Oexhaust, axiom, (! [Y2:nat]: (((Y2 != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y2 != (suc @ Nat4)))))))))).
% 2.49/1.49 thf(fact_66_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat4:nat]: (((P @ Nat4) => (P @ (suc @ Nat4))))) => (P @ Nat)))))).
% 2.49/1.49 thf(fact_67_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M3:nat]: ((N = (suc @ M3)))))))).
% 2.49/1.49 thf(fact_68_Nat_OlessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ I) @ K) => ((K != (suc @ I)) => (~ ((! [J:nat]: (((((ord_less @ nat) @ I) @ J) => (K != (suc @ J)))))))))))).
% 2.49/1.49 thf(fact_69_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_70_Suc__lessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I)) @ K) => (~ ((! [J:nat]: (((((ord_less @ nat) @ I) @ J) => (K != (suc @ J))))))))))).
% 2.49/1.49 thf(fact_71_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 2.49/1.49 thf(fact_72_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 2.49/1.49 thf(fact_73_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 2.49/1.49 thf(fact_74_Ex__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((? [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) & (P @ I2)))) = ((P @ N) | (? [I2:nat]: (((((ord_less @ nat) @ I2) @ N) & (P @ I2))))))))).
% 2.49/1.49 thf(fact_75_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 2.49/1.49 thf(fact_76_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 2.49/1.49 thf(fact_77_All__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) => (P @ I2)))) = ((P @ N) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ N) => (P @ I2))))))))).
% 2.49/1.49 thf(fact_78_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N) @ M4)))))))).
% 2.49/1.49 thf(fact_79_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 2.49/1.49 thf(fact_80_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_81_less__trans__Suc, axiom, (! [I:nat,J2:nat,K:nat]: (((((ord_less @ nat) @ I) @ J2) => ((((ord_less @ nat) @ J2) @ K) => (((ord_less @ nat) @ (suc @ I)) @ K)))))).
% 2.49/1.49 thf(fact_82_less__Suc__induct, axiom, (! [I:nat,J2:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I) @ J2) => ((! [I3:nat]: (((P @ I3) @ (suc @ I3)))) => ((! [I3:nat,J:nat,K2:nat]: (((((ord_less @ nat) @ I3) @ J) => ((((ord_less @ nat) @ J) @ K2) => (((P @ I3) @ J) => (((P @ J) @ K2) => ((P @ I3) @ K2))))))) => ((P @ I) @ J2))))))).
% 2.49/1.49 thf(fact_83_strict__inc__induct, axiom, (! [I:nat,J2:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I) @ J2) => ((! [I3:nat]: (((J2 = (suc @ I3)) => (P @ I3)))) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ J2) => ((P @ (suc @ I3)) => (P @ I3))))) => (P @ I))))))).
% 2.49/1.49 thf(fact_84_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 2.49/1.49 thf(fact_85_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 2.49/1.49 thf(fact_86_less__imp__of__nat__less, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.49/1.49 thf(fact_87_of__nat__less__imp__less, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) => (((ord_less @ nat) @ M) @ N)))))))).
% 2.49/1.49 thf(fact_88_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ nat) @ N) @ N3) => (((ord_less @ A) @ (F @ N)) @ (F @ N3)))))))))).
% 2.49/1.49 thf(fact_89_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 2.49/1.49 thf(fact_90_Ex__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((? [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) & (P @ I2)))) = ((P @ (zero_zero @ nat)) | (? [I2:nat]: (((((ord_less @ nat) @ I2) @ N) & (P @ (suc @ I2)))))))))).
% 2.49/1.49 thf(fact_91_gr0__conv__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) = (? [M5:nat]: ((N = (suc @ M5)))))))).
% 2.49/1.49 thf(fact_92_All__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ (suc @ N)) => (P @ I2)))) = ((P @ (zero_zero @ nat)) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ N) => (P @ (suc @ I2)))))))))).
% 2.49/1.49 thf(fact_93_gr0__implies__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (? [M3:nat]: ((N = (suc @ M3)))))))).
% 2.49/1.49 thf(fact_94_less__Suc__eq__0__disj, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((M = (zero_zero @ nat)) | (? [J3:nat]: (((M = (suc @ J3)) & (((ord_less @ nat) @ J3) @ N))))))))).
% 2.49/1.49 thf(fact_95_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B3:B,C:B]: (((A2 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less @ B) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.49/1.49 thf(fact_96_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B3) => (((F @ B3) = C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.49/1.49 thf(fact_97_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less @ B) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.49/1.49 thf(fact_98_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.49/1.49 thf(fact_99_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X))))))))).
% 2.49/1.49 thf(fact_100_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 2.49/1.49 thf(fact_101_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.49/1.49 thf(fact_102_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((X != Y2) = ((((ord_less @ A) @ X) @ Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.49/1.49 thf(fact_103_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.49/1.49 thf(fact_104_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (? [Z:A]: (((((ord_less @ A) @ X) @ Z) & (((ord_less @ A) @ Z) @ Y2))))))))))).
% 2.49/1.49 thf(fact_105_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.49/1.49 thf(fact_106_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.49/1.49 thf(fact_107_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.49/1.49 thf(fact_108_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.49/1.49 thf(fact_109_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) | ((X = Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.49/1.49 thf(fact_110_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.49/1.49 thf(fact_111_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C:A]: (((A2 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_112_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((B3 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_113_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ A2) @ B3)))))))))).
% 2.49/1.49 thf(fact_114_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.49/1.49 thf(fact_115_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.49/1.49 thf(fact_116_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X2) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.49/1.49 thf(fact_117_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y2:A,X:A]: (((~ ((((ord_less @ A) @ Y2) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y2))) = (X = Y2))))))))).
% 2.49/1.49 thf(fact_118_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (Y2 != X)))))))).
% 2.49/1.49 thf(fact_119_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A,P:$o]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ X) => P)))))))).
% 2.49/1.49 thf(fact_120_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => ((X != Y2) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.49/1.49 thf(fact_121_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.49/1.49 thf(fact_122_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_123_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.49/1.49 thf(fact_124_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P4:(A > $o)]: ((? [X4:A]: ((P4 @ X4))))) = (^ [P5:(A > $o)]: ((? [N4:A]: (((P5 @ N4) & (! [M5:A]: (((((ord_less @ A) @ M5) @ N4) => (~ ((P5 @ M5)))))))))))))))).
% 2.49/1.49 thf(fact_125_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((P @ A3) @ B2)))) => ((! [A3:A]: (((P @ A3) @ A3))) => ((! [A3:A,B2:A]: ((((P @ B2) @ A3) => ((P @ A3) @ B2)))) => ((P @ A2) @ B3)))))))))).
% 2.49/1.49 thf(fact_126_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.49/1.49 thf(fact_127_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) = ((((ord_less @ A) @ Y2) @ X) | (X = Y2))))))))).
% 2.49/1.49 thf(fact_128_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (A2 != B3)))))))).
% 2.49/1.49 thf(fact_129_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X3))))))))).
% 2.49/1.49 thf(fact_130_pos__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: (((K = ((semiring_1_of_nat @ int) @ N2)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))))))))).
% 2.49/1.49 thf(fact_131_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (K = ((semiring_1_of_nat @ int) @ N2))))))))).
% 2.49/1.49 thf(fact_132_dependent__nat__choice, axiom, (! [A:$tType,P:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X_12:A]: (((P @ (zero_zero @ nat)) @ X_12))) => ((! [X2:A,N2:nat]: ((((P @ N2) @ X2) => (? [Y:A]: ((((P @ (suc @ N2)) @ Y) & (((Q @ N2) @ X2) @ Y))))))) => (? [F2:(nat > A)]: ((! [N5:nat]: ((((P @ N5) @ (F2 @ N5)) & (((Q @ N5) @ (F2 @ N5)) @ (F2 @ (suc @ N5))))))))))))).
% 2.49/1.49 thf(fact_133_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X_12:nat]: ((P @ X_12))) => (? [N2:nat]: (((~ ((P @ N2))) & (P @ (suc @ N2)))))))))).
% 2.49/1.49 thf(fact_134_reals__Archimedean2, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [N2:nat]: ((((ord_less @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.49/1.49 thf(fact_135_of__nat__zero__less__power__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ X)) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat)))))))))).
% 2.49/1.49 thf(fact_136_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.49/1.49 thf(fact_137_Suc__pred, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) = N))))).
% 2.49/1.49 thf(fact_138_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.49/1.49 thf(fact_139_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.49/1.49 thf(fact_140_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.49/1.49 thf(fact_141_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.49/1.49 thf(fact_142_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.49/1.49 thf(fact_143_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_144_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_145_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.49/1.49 thf(fact_146_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_147_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.49/1.49 thf(fact_148_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_149_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.49/1.49 thf(fact_150_nat__power__eq__Suc__0__iff, axiom, (! [X:nat,M:nat]: ((((((power_power @ nat) @ X) @ M) = (suc @ (zero_zero @ nat))) = ((M = (zero_zero @ nat)) | (X = (suc @ (zero_zero @ nat)))))))).
% 2.49/1.49 thf(fact_151_power__Suc__0, axiom, (! [N:nat]: (((((power_power @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (suc @ (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_152_of__nat__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:nat,B3:nat,W:nat]: (((((semiring_1_of_nat @ A) @ X) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W)) = (X = (((power_power @ nat) @ B3) @ W))))))))).
% 2.49/1.49 thf(fact_153_of__nat__eq__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [B3:nat,W:nat,X:nat]: ((((((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W) = ((semiring_1_of_nat @ A) @ X)) = ((((power_power @ nat) @ B3) @ W) = X)))))))).
% 2.49/1.49 thf(fact_154_of__nat__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((power_power @ nat) @ M) @ N)) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N)))))))).
% 2.49/1.49 thf(fact_155_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 2.49/1.49 thf(fact_156_diff__is__0__eq_H, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.49/1.49 thf(fact_157_diff__is__0__eq, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_158_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 2.49/1.49 thf(fact_159_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.49/1.49 thf(fact_160_Suc__diff__diff, axiom, (! [M:nat,N:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ K))))).
% 2.49/1.49 thf(fact_161_diff__Suc__Suc, axiom, (! [M:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M)) @ (suc @ N)) = (((minus_minus @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_162_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B3)) = (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.49/1.49 thf(fact_163_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B3)) = (((ord_less @ A) @ B3) @ A2)))))))).
% 2.49/1.49 thf(fact_164_power__0__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ (zero_zero @ A)) @ (suc @ N)) = (zero_zero @ A)))))))).
% 2.49/1.49 thf(fact_165_of__nat__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B3:nat,W:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ B3) @ W))))))))).
% 2.49/1.49 thf(fact_166_of__nat__le__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B3:nat,W:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ B3) @ W)) @ X)))))))).
% 2.49/1.49 thf(fact_167_power__Suc0__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))).
% 2.49/1.49 thf(fact_168_zero__less__diff, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)) = (((ord_less @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_169_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: ((((((power_power @ A) @ A2) @ N) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 2.49/1.49 thf(fact_170_of__nat__less__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B3:nat,W:nat,X:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less @ nat) @ (((power_power @ nat) @ B3) @ W)) @ X)))))))).
% 2.49/1.49 thf(fact_171_of__nat__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B3:nat,W:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B3)) @ W)) = (((ord_less @ nat) @ X) @ (((power_power @ nat) @ B3) @ W))))))))).
% 2.49/1.49 thf(fact_172_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B3) @ N)) = (((ord_less_eq @ A) @ A2) @ B3))))))))))).
% 2.49/1.49 thf(fact_173_Suc__diff__le, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => ((((minus_minus @ nat) @ (suc @ M)) @ N) = (suc @ (((minus_minus @ nat) @ M) @ N))))))).
% 2.49/1.49 thf(fact_174_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: ((K != ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.49/1.49 thf(fact_175_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: ((K = ((semiring_1_of_nat @ int) @ N2)))))))).
% 2.49/1.49 thf(fact_176_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 2.49/1.49 thf(fact_177_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N3)))))))))).
% 2.49/1.49 thf(fact_178_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N2))) @ (F @ N2)))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N3)) @ (F @ N)))))))))).
% 2.49/1.49 thf(fact_179_of__nat__diff, axiom, (! [A:$tType]: (((semiring_1_cancel @ A) => (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ N) @ M) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M) @ N)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 2.49/1.49 thf(fact_180_power__inject__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B3:A]: ((((((power_power @ A) @ A2) @ (suc @ N)) = (((power_power @ A) @ B3) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => (A2 = B3)))))))))).
% 2.49/1.49 thf(fact_181_of__nat__mono, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [I:nat,J2:nat]: (((((ord_less_eq @ nat) @ I) @ J2) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I)) @ ((semiring_1_of_nat @ A) @ J2))))))))).
% 2.49/1.49 thf(fact_182_power__le__imp__le__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B3:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ (suc @ N))) @ (((power_power @ A) @ B3) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => (((ord_less_eq @ A) @ A2) @ B3))))))))).
% 2.49/1.49 thf(fact_183_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ A2) @ B3) => (A2 = B3))))))))).
% 2.49/1.49 thf(fact_184_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.49/1.49 thf(fact_185_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.49/1.49 thf(fact_186_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((P @ A3) @ B2)))) => ((! [A3:A,B2:A]: ((((P @ B2) @ A3) => ((P @ A3) @ B2)))) => ((P @ A2) @ B3))))))))).
% 2.49/1.49 thf(fact_187_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.49/1.49 thf(fact_188_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.49/1.49 thf(fact_189_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ A2) => (A2 = B3))))))))).
% 2.49/1.49 thf(fact_190_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_191_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B3:A,C:A]: (((A2 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_192_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.49/1.49 thf(fact_193_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.49/1.49 thf(fact_194_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y2))))))))))))))))).
% 2.49/1.49 thf(fact_195_real__arch__simple, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [N2:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.49/1.49 thf(fact_196_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.49/1.49 thf(fact_197_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.49/1.49 thf(fact_198_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y2:A]: (((X = Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.49/1.49 thf(fact_199_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.49/1.49 thf(fact_200_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X) => (X = Y2))))))))).
% 2.49/1.49 thf(fact_201_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X5:A,Y5:A]: (((((ord_less_eq @ A) @ X5) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X5))))))))).
% 2.49/1.49 thf(fact_202_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B3) => (((F @ B3) = C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.49/1.49 thf(fact_203_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B3:B,C:B]: (((A2 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less_eq @ B) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.49/1.49 thf(fact_204_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ C2) @ (F @ B3)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.49/1.49 thf(fact_205_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X2:B,Y3:B]: (((((ord_less_eq @ B) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.49/1.49 thf(fact_206_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G:(A > B)]: ((! [X5:A]: ((((ord_less_eq @ B) @ (F3 @ X5)) @ (G @ X5))))))))))).
% 2.49/1.49 thf(fact_207_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G2 @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G2)))))))).
% 2.49/1.49 thf(fact_208_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 2.49/1.49 thf(fact_209_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 2.49/1.49 thf(fact_210_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.49/1.49 thf(fact_211_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.49/1.49 thf(fact_212_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_213_diff__less__mono, axiom, (! [A2:nat,B3:nat,C:nat]: (((((ord_less @ nat) @ A2) @ B3) => ((((ord_less_eq @ nat) @ C) @ A2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A2) @ C)) @ (((minus_minus @ nat) @ B3) @ C))))))).
% 2.49/1.49 thf(fact_214_less__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less @ nat) @ M) @ N))))))).
% 2.49/1.49 thf(fact_215_power__not__zero, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_216_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B3:A]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B3) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.49/1.49 thf(fact_217_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B3:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B3) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B3)) @ C)))))))).
% 2.49/1.49 thf(fact_218_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less_eq @ A) @ A2) @ B3) = (((ord_less_eq @ A) @ C) @ D))))))))).
% 2.49/1.49 thf(fact_219_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.49/1.49 thf(fact_220_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.49/1.49 thf(fact_221_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.49/1.49 thf(fact_222_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,D:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ D) @ C) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.49/1.49 thf(fact_223_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B3:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B3) @ N)))))))))).
% 2.49/1.49 thf(fact_224_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((A2 = B3) = (C = D))))))))).
% 2.49/1.49 thf(fact_225_diff__le__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 2.49/1.49 thf(fact_226_le__diff__iff_H, axiom, (! [A2:nat,C:nat,B3:nat]: (((((ord_less_eq @ nat) @ A2) @ C) => ((((ord_less_eq @ nat) @ B3) @ C) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C) @ A2)) @ (((minus_minus @ nat) @ C) @ B3)) = (((ord_less_eq @ nat) @ B3) @ A2))))))).
% 2.49/1.49 thf(fact_227_diff__le__self, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))).
% 2.49/1.49 thf(fact_228_diff__le__mono, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N) @ L)))))).
% 2.49/1.49 thf(fact_229_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M) @ N))))))).
% 2.49/1.49 thf(fact_230_diff__commute, axiom, (! [I:nat,J2:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J2)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J2))))).
% 2.49/1.49 thf(fact_231_le__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N) @ K)) = (((ord_less_eq @ nat) @ M) @ N))))))).
% 2.49/1.49 thf(fact_232_eq__diff__iff, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N) @ K)) = (M = N))))))).
% 2.49/1.49 thf(fact_233_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.49/1.49 thf(fact_234_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.49/1.49 thf(fact_235_nat__one__le__power, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ I) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((power_power @ nat) @ I) @ N)))))).
% 2.49/1.49 thf(fact_236_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B3:A]: ((((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B3) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (A2 = B3))))))))))).
% 2.49/1.49 thf(fact_237_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,A2:A,B3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B3) => (((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B3) @ N)) = (A2 = B3))))))))))).
% 2.49/1.49 thf(fact_238_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B4:A]: (((((minus_minus @ A) @ A4) @ B4) = (zero_zero @ A))))))))).
% 2.49/1.49 thf(fact_239_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B3) @ C))))))))).
% 2.49/1.49 thf(fact_240_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B3:A,A2:A,C:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B3))))))))).
% 2.49/1.49 thf(fact_241_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B3) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A2) @ B3) = (((ord_less @ A) @ C) @ D))))))))).
% 2.49/1.49 thf(fact_242_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B3:A,D:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B3) @ D)))))))))).
% 2.49/1.49 thf(fact_243_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.49/1.49 thf(fact_244_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.49/1.49 thf(fact_245_complete__interval, axiom, (! [A:$tType]: (((condit1037483654norder @ A) => (! [A2:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B3) => ((P @ A2) => ((~ ((P @ B3))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B3) & ((! [X3:A]: ((((((ord_less_eq @ A) @ A2) @ X3) & (((ord_less @ A) @ X3) @ C3)) => (P @ X3)))) & (! [D3:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D3)) => (P @ X2)))) => (((ord_less_eq @ A) @ D3) @ C3)))))))))))))))))).
% 2.49/1.49 thf(fact_246_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((A2 != B3) => ((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.49/1.49 thf(fact_247_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.49/1.49 thf(fact_248_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.49/1.49 thf(fact_249_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.49/1.49 thf(fact_250_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ A2) @ B3)))))))).
% 2.49/1.49 thf(fact_251_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((! [W2:A]: (((((ord_less @ A) @ X) @ W2) => ((((ord_less @ A) @ W2) @ Y2) => (((ord_less_eq @ A) @ W2) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.49/1.49 thf(fact_252_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W2:A]: (((((ord_less @ A) @ Z2) @ W2) => ((((ord_less @ A) @ W2) @ X) => (((ord_less_eq @ A) @ Y2) @ W2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.49/1.49 thf(subcl_Groups_Ozero___HOL_Otype, axiom, (! [A:$tType]: (((zero @ A) => (type @ A))))).
% 2.49/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A5:$tType,A6:$tType]: (((preorder @ A6) => (preorder @ (A5 > A6)))))).
% 2.49/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A5:$tType,A6:$tType]: (((order @ A6) => (order @ (A5 > A6)))))).
% 2.49/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A5:$tType,A6:$tType]: (((ord @ A6) => (ord @ (A5 > A6)))))).
% 2.49/1.49 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit1037483654norder @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel, axiom, (semiring_1_cancel @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, (preorder @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, (order @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, (ord @ int)).
% 2.49/1.49 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.49/1.49 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4, axiom, (condit1037483654norder @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5, axiom, (semiri134348788visors @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6, axiom, (linord1659791738miring @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7, axiom, (cancel146912293up_add @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8, axiom, (cancel1352612707id_add @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9, axiom, (linordered_semidom @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_10, axiom, (semiring_1_cancel @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11, axiom, (semiring_char_0 @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Opreorder_12, axiom, (preorder @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Olinorder_13, axiom, (linorder @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Omonoid__mult_14, axiom, (monoid_mult @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__1_15, axiom, (semiring_1 @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Ono__top_16, axiom, (no_top @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Oorder_17, axiom, (order @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Orderings_Oord_18, axiom, (ord @ nat)).
% 2.49/1.49 thf(tcon_Nat_Onat___Groups_Ozero_19, axiom, (zero @ nat)).
% 2.49/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_20, axiom, (preorder @ $o)).
% 2.49/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder_21, axiom, (linorder @ $o)).
% 2.49/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_22, axiom, (order @ $o)).
% 2.49/1.49 thf(tcon_HOL_Obool___Orderings_Oord_23, axiom, (ord @ $o)).
% 2.49/1.49 thf(tfree_0, hypothesis, (zero @ a)).
% 2.49/1.49 thf(conj_0, hypothesis, (((ord_less @ nat) @ (zero_zero @ nat)) @ n)).
% 2.49/1.49 thf(conj_1, hypothesis, (((polyno86455060npolyh @ a) @ p) @ n)).
% 2.49/1.49 thf(conj_2, conjecture, (((polyno498386536degree @ a) @ p) = (zero_zero @ nat))).
% 2.49/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2DTF21321.p
% 2.49/1.49 ------------------------
% 2.82/1.52 ---- Cleaned THF ---
% 2.82/1.52 thf(ty_t_Polynomial__Expression__Mirabelle__gfajfqghjc_Opoly,type,
% 2.82/1.52 polyno1783536151e_poly: $tType > $tType ).
% 2.82/1.52
% 2.82/1.52 thf(ty_t_Nat_Onat,type,
% 2.82/1.52 nat: $tType ).
% 2.82/1.52
% 2.82/1.52 thf(ty_t_Int_Oint,type,
% 2.82/1.52 int: $tType ).
% 2.82/1.52
% 2.82/1.52 thf(ty_tf_a,type,
% 2.82/1.52 a: $tType ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_HOL_Otype,type,
% 2.82/1.52 type:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ozero,type,
% 2.82/1.52 zero:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Oord,type,
% 2.82/1.52 ord:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Oorder,type,
% 2.82/1.52 order:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ogroup__add,type,
% 2.82/1.52 group_add:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Ono__bot,type,
% 2.82/1.52 no_bot:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Ono__top,type,
% 2.82/1.52 no_top:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Osemiring__1,type,
% 2.82/1.52 semiring_1:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.82/1.52 monoid_mult:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.82/1.52 linorder:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Opreorder,type,
% 2.82/1.52 preorder:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.82/1.52 semiring_char_0:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Owellorder,type,
% 2.82/1.52 wellorder:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Odense__order,type,
% 2.82/1.52 dense_order:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.82/1.52 linordered_idom:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Fields_Olinordered__field,type,
% 2.82/1.52 linordered_field:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.82/1.52 comm_monoid_diff:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Osemiring__1__cancel,type,
% 2.82/1.52 semiring_1_cancel:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.82/1.52 dense_linorder:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.82/1.52 linordered_semidom:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.82/1.52 ordered_ab_group_add:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.82/1.52 cancel1352612707id_add:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.82/1.52 cancel146912293up_add:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.82/1.52 linord1659791738miring:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.82/1.52 semiri134348788visors:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.82/1.52 archim1804426504_field:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.82/1.52 canoni770627133id_add:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.82/1.52 condit1656338222tinuum:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.82/1.52 condit1037483654norder:
% 2.82/1.52 !>[A: $tType] : $o ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.82/1.52 minus_minus:
% 2.82/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.82/1.52 zero_zero:
% 2.82/1.52 !>[A: $tType] : A ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Nat_OSuc,type,
% 2.82/1.52 suc: nat > nat ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.82/1.52 semiring_1_of_nat:
% 2.82/1.52 !>[A: $tType] : ( nat > A ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.82/1.52 ord_less:
% 2.82/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.82/1.52 ord_less_eq:
% 2.82/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegree,type,
% 2.82/1.52 polyno498386536degree:
% 2.82/1.52 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Odegreen,type,
% 2.82/1.52 polyno367318022egreen:
% 2.82/1.52 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > nat ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpoly,type,
% 2.82/1.52 polyno2122670676snpoly:
% 2.82/1.52 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > $o ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Polynomial__Expression__Mirabelle__gfajfqghjc_Oisnpolyh,type,
% 2.82/1.52 polyno86455060npolyh:
% 2.82/1.52 !>[A: $tType] : ( ( polyno1783536151e_poly @ A ) > nat > $o ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_c_Power_Opower__class_Opower,type,
% 2.82/1.52 power_power:
% 2.82/1.52 !>[A: $tType] : ( A > nat > A ) ).
% 2.82/1.52
% 2.82/1.52 thf(sy_v_n,type,
% 2.82/1.52 n: nat ).
% 2.82/1.52
% 2.82/1.52 thf(sy_v_p,type,
% 2.82/1.52 p: polyno1783536151e_poly @ a ).
% 2.82/1.52
% 2.82/1.52 thf(fact_0_neq0__conv,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( N
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_1_less__nat__zero__code,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_2_bot__nat__0_Onot__eq__extremum,axiom,
% 2.82/1.52 ! [A2: nat] :
% 2.82/1.52 ( ( A2
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_3_not__gr__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [N: A] :
% 2.82/1.52 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.82/1.52 = ( N
% 2.82/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_4_gr0I,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( N
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_5_not__gr0,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.82/1.52 = ( N
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_6_not__less0,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_7_less__zeroE,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_8_gr__implies__not0,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( N
% 2.82/1.52 != ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_9_infinite__descent0,axiom,
% 2.82/1.52 ! [P: nat > $o,N: nat] :
% 2.82/1.52 ( ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 => ( ! [N2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.82/1.52 => ( ~ ( P @ N2 )
% 2.82/1.52 => ? [M2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M2 @ N2 )
% 2.82/1.52 & ~ ( P @ M2 ) ) ) )
% 2.82/1.52 => ( P @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_10_bot__nat__0_Oextremum__strict,axiom,
% 2.82/1.52 ! [A2: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_11_zero__reorient,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( zero @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ( ( ( zero_zero @ A )
% 2.82/1.52 = X )
% 2.82/1.52 = ( X
% 2.82/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_12_measure__induct__rule,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( wellorder @ B )
% 2.82/1.52 => ! [F: A > B,P: A > $o,A2: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ! [Y: A] :
% 2.82/1.52 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 2.82/1.52 => ( P @ Y ) )
% 2.82/1.52 => ( P @ X2 ) )
% 2.82/1.52 => ( P @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_13_measure__induct,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( wellorder @ B )
% 2.82/1.52 => ! [F: A > B,P: A > $o,A2: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ! [Y: A] :
% 2.82/1.52 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 2.82/1.52 => ( P @ Y ) )
% 2.82/1.52 => ( P @ X2 ) )
% 2.82/1.52 => ( P @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_14_infinite__descent__measure,axiom,
% 2.82/1.52 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ~ ( P @ X2 )
% 2.82/1.52 => ? [Y: A] :
% 2.82/1.52 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 2.82/1.52 & ~ ( P @ Y ) ) )
% 2.82/1.52 => ( P @ X ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_15_linorder__neqE__nat,axiom,
% 2.82/1.52 ! [X: nat,Y2: nat] :
% 2.82/1.52 ( ( X != Y2 )
% 2.82/1.52 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 2.82/1.52 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_16_infinite__descent,axiom,
% 2.82/1.52 ! [P: nat > $o,N: nat] :
% 2.82/1.52 ( ! [N2: nat] :
% 2.82/1.52 ( ~ ( P @ N2 )
% 2.82/1.52 => ? [M2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M2 @ N2 )
% 2.82/1.52 & ~ ( P @ M2 ) ) )
% 2.82/1.52 => ( P @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_17_nat__less__induct,axiom,
% 2.82/1.52 ! [P: nat > $o,N: nat] :
% 2.82/1.52 ( ! [N2: nat] :
% 2.82/1.52 ( ! [M2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M2 @ N2 )
% 2.82/1.52 => ( P @ M2 ) )
% 2.82/1.52 => ( P @ N2 ) )
% 2.82/1.52 => ( P @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_18_less__irrefl__nat,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ N @ N ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_19_less__not__refl3,axiom,
% 2.82/1.52 ! [S: nat,T: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ S @ T )
% 2.82/1.52 => ( S != T ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_20_less__not__refl2,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ N @ M )
% 2.82/1.52 => ( M != N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_21_less__not__refl,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ~ ( ord_less @ nat @ N @ N ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_22_nat__neq__iff,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( M != N )
% 2.82/1.52 = ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_23_zero__less__iff__neq__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [N: A] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.82/1.52 = ( N
% 2.82/1.52 != ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_24_gr__implies__not__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [M: A,N: A] :
% 2.82/1.52 ( ( ord_less @ A @ M @ N )
% 2.82/1.52 => ( N
% 2.82/1.52 != ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_25_not__less__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [N: A] :
% 2.82/1.52 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_26_gr__zeroI,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [N: A] :
% 2.82/1.52 ( ( N
% 2.82/1.52 != ( zero_zero @ A ) )
% 2.82/1.52 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_27_infinite__descent0__measure,axiom,
% 2.82/1.52 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ( ( V @ X2 )
% 2.82/1.52 = ( zero_zero @ nat ) )
% 2.82/1.52 => ( P @ X2 ) )
% 2.82/1.52 => ( ! [X2: A] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 2.82/1.52 => ( ~ ( P @ X2 )
% 2.82/1.52 => ? [Y: A] :
% 2.82/1.52 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 2.82/1.52 & ~ ( P @ Y ) ) ) )
% 2.82/1.52 => ( P @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_28_field__lbound__gt__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_field @ A )
% 2.82/1.52 => ! [D1: A,D2: A] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 2.82/1.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 2.82/1.52 => ? [E: A] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 2.82/1.52 & ( ord_less @ A @ E @ D1 )
% 2.82/1.52 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_29_less__numeral__extra_I3_J,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_30_isnpoly__def,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( zero @ A )
% 2.82/1.52 => ( ( polyno2122670676snpoly @ A )
% 2.82/1.52 = ( ^ [P2: polyno1783536151e_poly @ A] : ( polyno86455060npolyh @ A @ P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_31_degreen__0,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( zero @ A )
% 2.82/1.52 => ! [P3: polyno1783536151e_poly @ A,N: nat,M: nat] :
% 2.82/1.52 ( ( polyno86455060npolyh @ A @ P3 @ N )
% 2.82/1.52 => ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( ( polyno367318022egreen @ A @ P3 @ M )
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_32_of__nat__0__less__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [N: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_33_degree__isnpolyh__Suc,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( zero @ A )
% 2.82/1.52 => ! [P3: polyno1783536151e_poly @ A,N: nat] :
% 2.82/1.52 ( ( polyno86455060npolyh @ A @ P3 @ ( suc @ N ) )
% 2.82/1.52 => ( ( polyno498386536degree @ A @ P3 )
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_34_zero__natural_Orsp,axiom,
% 2.82/1.52 ( ( zero_zero @ nat )
% 2.82/1.52 = ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_35_ex__gt__or__lt,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( condit1656338222tinuum @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ? [B2: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B2 )
% 2.82/1.52 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_36_linorder__neqE__linordered__idom,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_idom @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( X != Y2 )
% 2.82/1.52 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_37_dual__order_Ostrict__implies__not__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A] :
% 2.82/1.52 ( ( ord_less @ A @ B3 @ A2 )
% 2.82/1.52 => ( A2 != B3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_38_linordered__field__no__ub,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_field @ A )
% 2.82/1.52 => ! [X3: A] :
% 2.82/1.52 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_39_nat_Oinject,axiom,
% 2.82/1.52 ! [X22: nat,Y22: nat] :
% 2.82/1.52 ( ( ( suc @ X22 )
% 2.82/1.52 = ( suc @ Y22 ) )
% 2.82/1.52 = ( X22 = Y22 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_40_old_Onat_Oinject,axiom,
% 2.82/1.52 ! [Nat: nat,Nat2: nat] :
% 2.82/1.52 ( ( ( suc @ Nat )
% 2.82/1.52 = ( suc @ Nat2 ) )
% 2.82/1.52 = ( Nat = Nat2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_41_of__nat__eq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ( semiring_1_of_nat @ A @ M )
% 2.82/1.52 = ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 = ( M = N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_42_lessI,axiom,
% 2.82/1.52 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_43_Suc__mono,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_44_Suc__less__eq,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.82/1.52 = ( ord_less @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_45_of__nat__0,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_1 @ A )
% 2.82/1.52 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_46_of__nat__0__eq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [N: nat] :
% 2.82/1.52 ( ( ( zero_zero @ A )
% 2.82/1.52 = ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 = ( ( zero_zero @ nat )
% 2.82/1.52 = N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_47_of__nat__eq__0__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [M: nat] :
% 2.82/1.52 ( ( ( semiring_1_of_nat @ A @ M )
% 2.82/1.52 = ( zero_zero @ A ) )
% 2.82/1.52 = ( M
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_48_of__nat__less__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 = ( ord_less @ nat @ M @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_49_less__Suc0,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 2.82/1.52 = ( N
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_50_zero__less__Suc,axiom,
% 2.82/1.52 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_51_Suc__inject,axiom,
% 2.82/1.52 ! [X: nat,Y2: nat] :
% 2.82/1.52 ( ( ( suc @ X )
% 2.82/1.52 = ( suc @ Y2 ) )
% 2.82/1.52 => ( X = Y2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_52_n__not__Suc__n,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( N
% 2.82/1.52 != ( suc @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_53_of__nat__neq__0,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [N: nat] :
% 2.82/1.52 ( ( semiring_1_of_nat @ A @ ( suc @ N ) )
% 2.82/1.52 != ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_54_polypow_Ocases,axiom,
% 2.82/1.52 ! [X: nat] :
% 2.82/1.52 ( ( X
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 => ~ ! [V2: nat] :
% 2.82/1.52 ( X
% 2.82/1.52 != ( suc @ V2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_55_nat_Odistinct_I1_J,axiom,
% 2.82/1.52 ! [X22: nat] :
% 2.82/1.52 ( ( zero_zero @ nat )
% 2.82/1.52 != ( suc @ X22 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_56_old_Onat_Odistinct_I2_J,axiom,
% 2.82/1.52 ! [Nat3: nat] :
% 2.82/1.52 ( ( suc @ Nat3 )
% 2.82/1.52 != ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_57_old_Onat_Odistinct_I1_J,axiom,
% 2.82/1.52 ! [Nat2: nat] :
% 2.82/1.52 ( ( zero_zero @ nat )
% 2.82/1.52 != ( suc @ Nat2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_58_nat_OdiscI,axiom,
% 2.82/1.52 ! [Nat: nat,X22: nat] :
% 2.82/1.52 ( ( Nat
% 2.82/1.52 = ( suc @ X22 ) )
% 2.82/1.52 => ( Nat
% 2.82/1.52 != ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_59_nat__induct,axiom,
% 2.82/1.52 ! [P: nat > $o,N: nat] :
% 2.82/1.52 ( ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 => ( ! [N2: nat] :
% 2.82/1.52 ( ( P @ N2 )
% 2.82/1.52 => ( P @ ( suc @ N2 ) ) )
% 2.82/1.52 => ( P @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_60_diff__induct,axiom,
% 2.82/1.52 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.82/1.52 ( ! [X2: nat] : ( P @ X2 @ ( zero_zero @ nat ) )
% 2.82/1.52 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 2.82/1.52 => ( ! [X2: nat,Y3: nat] :
% 2.82/1.52 ( ( P @ X2 @ Y3 )
% 2.82/1.52 => ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
% 2.82/1.52 => ( P @ M @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_61_zero__induct,axiom,
% 2.82/1.52 ! [P: nat > $o,K: nat] :
% 2.82/1.52 ( ( P @ K )
% 2.82/1.52 => ( ! [N2: nat] :
% 2.82/1.52 ( ( P @ ( suc @ N2 ) )
% 2.82/1.52 => ( P @ N2 ) )
% 2.82/1.52 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_62_Suc__neq__Zero,axiom,
% 2.82/1.52 ! [M: nat] :
% 2.82/1.52 ( ( suc @ M )
% 2.82/1.52 != ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_63_Zero__neq__Suc,axiom,
% 2.82/1.52 ! [M: nat] :
% 2.82/1.52 ( ( zero_zero @ nat )
% 2.82/1.52 != ( suc @ M ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_64_Zero__not__Suc,axiom,
% 2.82/1.52 ! [M: nat] :
% 2.82/1.52 ( ( zero_zero @ nat )
% 2.82/1.52 != ( suc @ M ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_65_old_Onat_Oexhaust,axiom,
% 2.82/1.52 ! [Y2: nat] :
% 2.82/1.52 ( ( Y2
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 => ~ ! [Nat4: nat] :
% 2.82/1.52 ( Y2
% 2.82/1.52 != ( suc @ Nat4 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_66_old_Onat_Oinducts,axiom,
% 2.82/1.52 ! [P: nat > $o,Nat: nat] :
% 2.82/1.52 ( ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 => ( ! [Nat4: nat] :
% 2.82/1.52 ( ( P @ Nat4 )
% 2.82/1.52 => ( P @ ( suc @ Nat4 ) ) )
% 2.82/1.52 => ( P @ Nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_67_not0__implies__Suc,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( N
% 2.82/1.52 != ( zero_zero @ nat ) )
% 2.82/1.52 => ? [M3: nat] :
% 2.82/1.52 ( N
% 2.82/1.52 = ( suc @ M3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_68_Nat_OlessE,axiom,
% 2.82/1.52 ! [I: nat,K: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ K )
% 2.82/1.52 => ( ( K
% 2.82/1.52 != ( suc @ I ) )
% 2.82/1.52 => ~ ! [J: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ J )
% 2.82/1.52 => ( K
% 2.82/1.52 != ( suc @ J ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_69_Suc__lessD,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 2.82/1.52 => ( ord_less @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_70_Suc__lessE,axiom,
% 2.82/1.52 ! [I: nat,K: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( suc @ I ) @ K )
% 2.82/1.52 => ~ ! [J: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ J )
% 2.82/1.52 => ( K
% 2.82/1.52 != ( suc @ J ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_71_Suc__lessI,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( ( ( suc @ M )
% 2.82/1.52 != N )
% 2.82/1.52 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_72_less__SucE,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.82/1.52 => ( ~ ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( M = N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_73_less__SucI,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_74_Ex__less__Suc,axiom,
% 2.82/1.52 ! [N: nat,P: nat > $o] :
% 2.82/1.52 ( ( ? [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.82/1.52 & ( P @ I2 ) ) )
% 2.82/1.52 = ( ( P @ N )
% 2.82/1.52 | ? [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.82/1.52 & ( P @ I2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_75_less__Suc__eq,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.82/1.52 = ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 | ( M = N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_76_not__less__eq,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 2.82/1.52 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_77_All__less__Suc,axiom,
% 2.82/1.52 ! [N: nat,P: nat > $o] :
% 2.82/1.52 ( ( ! [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.82/1.52 => ( P @ I2 ) ) )
% 2.82/1.52 = ( ( P @ N )
% 2.82/1.52 & ! [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.82/1.52 => ( P @ I2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_78_Suc__less__eq2,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 2.82/1.52 = ( ? [M4: nat] :
% 2.82/1.52 ( ( M
% 2.82/1.52 = ( suc @ M4 ) )
% 2.82/1.52 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_79_less__antisym,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ~ ( ord_less @ nat @ N @ M )
% 2.82/1.52 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.82/1.52 => ( M = N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_80_Suc__less__SucD,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.82/1.52 => ( ord_less @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_81_less__trans__Suc,axiom,
% 2.82/1.52 ! [I: nat,J2: nat,K: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ J2 )
% 2.82/1.52 => ( ( ord_less @ nat @ J2 @ K )
% 2.82/1.52 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_82_less__Suc__induct,axiom,
% 2.82/1.52 ! [I: nat,J2: nat,P: nat > nat > $o] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ J2 )
% 2.82/1.52 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 2.82/1.52 => ( ! [I3: nat,J: nat,K2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I3 @ J )
% 2.82/1.52 => ( ( ord_less @ nat @ J @ K2 )
% 2.82/1.52 => ( ( P @ I3 @ J )
% 2.82/1.52 => ( ( P @ J @ K2 )
% 2.82/1.52 => ( P @ I3 @ K2 ) ) ) ) )
% 2.82/1.52 => ( P @ I @ J2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_83_strict__inc__induct,axiom,
% 2.82/1.52 ! [I: nat,J2: nat,P: nat > $o] :
% 2.82/1.52 ( ( ord_less @ nat @ I @ J2 )
% 2.82/1.52 => ( ! [I3: nat] :
% 2.82/1.52 ( ( J2
% 2.82/1.52 = ( suc @ I3 ) )
% 2.82/1.52 => ( P @ I3 ) )
% 2.82/1.52 => ( ! [I3: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I3 @ J2 )
% 2.82/1.52 => ( ( P @ ( suc @ I3 ) )
% 2.82/1.52 => ( P @ I3 ) ) )
% 2.82/1.52 => ( P @ I ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_84_not__less__less__Suc__eq,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ~ ( ord_less @ nat @ N @ M )
% 2.82/1.52 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.82/1.52 = ( N = M ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_85_of__nat__less__0__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat] :
% 2.82/1.52 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_86_less__imp__of__nat__less,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ N )
% 2.82/1.52 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_87_of__nat__less__imp__less,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_88_lift__Suc__mono__less,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [F: nat > A,N: nat,N3: nat] :
% 2.82/1.52 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.82/1.52 => ( ( ord_less @ nat @ N @ N3 )
% 2.82/1.52 => ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_89_lift__Suc__mono__less__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [F: nat > A,N: nat,M: nat] :
% 2.82/1.52 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.82/1.52 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 2.82/1.52 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_90_Ex__less__Suc2,axiom,
% 2.82/1.52 ! [N: nat,P: nat > $o] :
% 2.82/1.52 ( ( ? [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.82/1.52 & ( P @ I2 ) ) )
% 2.82/1.52 = ( ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 | ? [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.82/1.52 & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_91_gr0__conv__Suc,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 = ( ? [M5: nat] :
% 2.82/1.52 ( N
% 2.82/1.52 = ( suc @ M5 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_92_All__less__Suc2,axiom,
% 2.82/1.52 ! [N: nat,P: nat > $o] :
% 2.82/1.52 ( ( ! [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ ( suc @ N ) )
% 2.82/1.52 => ( P @ I2 ) ) )
% 2.82/1.52 = ( ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 & ! [I2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ I2 @ N )
% 2.82/1.52 => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_93_gr0__implies__Suc,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 => ? [M3: nat] :
% 2.82/1.52 ( N
% 2.82/1.52 = ( suc @ M3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_94_less__Suc__eq__0__disj,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.82/1.52 = ( ( M
% 2.82/1.52 = ( zero_zero @ nat ) )
% 2.82/1.52 | ? [J3: nat] :
% 2.82/1.52 ( ( M
% 2.82/1.52 = ( suc @ J3 ) )
% 2.82/1.52 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_95_ord__eq__less__subst,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( ord @ B )
% 2.82/1.52 & ( ord @ A ) )
% 2.82/1.52 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.82/1.52 ( ( A2
% 2.82/1.52 = ( F @ B3 ) )
% 2.82/1.52 => ( ( ord_less @ B @ B3 @ C )
% 2.82/1.52 => ( ! [X2: B,Y3: B] :
% 2.82/1.52 ( ( ord_less @ B @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_96_ord__less__eq__subst,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( ord @ B )
% 2.82/1.52 & ( ord @ A ) )
% 2.82/1.52 => ! [A2: A,B3: A,F: A > B,C: B] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ( F @ B3 )
% 2.82/1.52 = C )
% 2.82/1.52 => ( ! [X2: A,Y3: A] :
% 2.82/1.52 ( ( ord_less @ A @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_97_order__less__subst1,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( order @ B )
% 2.82/1.52 & ( order @ A ) )
% 2.82/1.52 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.82/1.52 => ( ( ord_less @ B @ B3 @ C )
% 2.82/1.52 => ( ! [X2: B,Y3: B] :
% 2.82/1.52 ( ( ord_less @ B @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_98_order__less__subst2,axiom,
% 2.82/1.52 ! [A: $tType,C2: $tType] :
% 2.82/1.52 ( ( ( order @ C2 )
% 2.82/1.52 & ( order @ A ) )
% 2.82/1.52 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.82/1.52 => ( ! [X2: A,Y3: A] :
% 2.82/1.52 ( ( ord_less @ A @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_99_lt__ex,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( no_bot @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_100_gt__ex,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( no_top @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_101_neqE,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( X != Y2 )
% 2.82/1.52 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_102_neq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( X != Y2 )
% 2.82/1.52 = ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_103_order_Oasym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_104_dense,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( dense_order @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ? [Z: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Z )
% 2.82/1.52 & ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_105_less__imp__neq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( X != Y2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_106_less__asym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_107_less__asym_H,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_108_less__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A,Z2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.82/1.52 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_109_less__linear,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 | ( X = Y2 )
% 2.82/1.52 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_110_less__irrefl,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_111_ord__eq__less__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ord @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( A2 = B3 )
% 2.82/1.52 => ( ( ord_less @ A @ B3 @ C )
% 2.82/1.52 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_112_ord__less__eq__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ord @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( B3 = C )
% 2.82/1.52 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_113_dual__order_Oasym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A] :
% 2.82/1.52 ( ( ord_less @ A @ B3 @ A2 )
% 2.82/1.52 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_114_less__imp__not__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( X != Y2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_115_less__not__sym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_116_less__induct,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( wellorder @ A )
% 2.82/1.52 => ! [P: A > $o,A2: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ! [Y: A] :
% 2.82/1.52 ( ( ord_less @ A @ Y @ X2 )
% 2.82/1.52 => ( P @ Y ) )
% 2.82/1.52 => ( P @ X2 ) )
% 2.82/1.52 => ( P @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_117_antisym__conv3,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [Y2: A,X: A] :
% 2.82/1.52 ( ~ ( ord_less @ A @ Y2 @ X )
% 2.82/1.52 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.82/1.52 = ( X = Y2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_118_less__imp__not__eq2,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( Y2 != X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_119_less__imp__triv,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A,P: $o] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ( ord_less @ A @ Y2 @ X )
% 2.82/1.52 => P ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_120_linorder__cases,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ( X != Y2 )
% 2.82/1.52 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_121_dual__order_Oirrefl,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_122_order_Ostrict__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less @ A @ B3 @ C )
% 2.82/1.52 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_123_less__imp__not__less,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_124_exists__least__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( wellorder @ A )
% 2.82/1.52 => ( ( ^ [P4: A > $o] :
% 2.82/1.52 ? [X4: A] : ( P4 @ X4 ) )
% 2.82/1.52 = ( ^ [P5: A > $o] :
% 2.82/1.52 ? [N4: A] :
% 2.82/1.52 ( ( P5 @ N4 )
% 2.82/1.52 & ! [M5: A] :
% 2.82/1.52 ( ( ord_less @ A @ M5 @ N4 )
% 2.82/1.52 => ~ ( P5 @ M5 ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_125_linorder__less__wlog,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.82/1.52 ( ! [A3: A,B2: A] :
% 2.82/1.52 ( ( ord_less @ A @ A3 @ B2 )
% 2.82/1.52 => ( P @ A3 @ B2 ) )
% 2.82/1.52 => ( ! [A3: A] : ( P @ A3 @ A3 )
% 2.82/1.52 => ( ! [A3: A,B2: A] :
% 2.82/1.52 ( ( P @ B2 @ A3 )
% 2.82/1.52 => ( P @ A3 @ B2 ) )
% 2.82/1.52 => ( P @ A2 @ B3 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_126_dual__order_Ostrict__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ B3 @ A2 )
% 2.82/1.52 => ( ( ord_less @ A @ C @ B3 )
% 2.82/1.52 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_127_not__less__iff__gr__or__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.82/1.52 = ( ( ord_less @ A @ Y2 @ X )
% 2.82/1.52 | ( X = Y2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_128_order_Ostrict__implies__not__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( A2 != B3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_129_linordered__field__no__lb,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_field @ A )
% 2.82/1.52 => ! [X3: A] :
% 2.82/1.52 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_130_pos__int__cases,axiom,
% 2.82/1.52 ! [K: int] :
% 2.82/1.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.82/1.52 => ~ ! [N2: nat] :
% 2.82/1.52 ( ( K
% 2.82/1.52 = ( semiring_1_of_nat @ int @ N2 ) )
% 2.82/1.52 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_131_zero__less__imp__eq__int,axiom,
% 2.82/1.52 ! [K: int] :
% 2.82/1.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.82/1.52 => ? [N2: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.82/1.52 & ( K
% 2.82/1.52 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_132_dependent__nat__choice,axiom,
% 2.82/1.52 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
% 2.82/1.52 ( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
% 2.82/1.52 => ( ! [X2: A,N2: nat] :
% 2.82/1.52 ( ( P @ N2 @ X2 )
% 2.82/1.52 => ? [Y: A] :
% 2.82/1.52 ( ( P @ ( suc @ N2 ) @ Y )
% 2.82/1.52 & ( Q @ N2 @ X2 @ Y ) ) )
% 2.82/1.52 => ? [F2: nat > A] :
% 2.82/1.52 ! [N5: nat] :
% 2.82/1.52 ( ( P @ N5 @ ( F2 @ N5 ) )
% 2.82/1.52 & ( Q @ N5 @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_133_exists__least__lemma,axiom,
% 2.82/1.52 ! [P: nat > $o] :
% 2.82/1.52 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.82/1.52 => ( ? [X_12: nat] : ( P @ X_12 )
% 2.82/1.52 => ? [N2: nat] :
% 2.82/1.52 ( ~ ( P @ N2 )
% 2.82/1.52 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_134_reals__Archimedean2,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( archim1804426504_field @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_135_of__nat__zero__less__power__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [X: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 2.82/1.52 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.82/1.52 | ( N
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_136_of__nat__le__0__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.82/1.52 = ( M
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_137_Suc__pred,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 2.82/1.52 = N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_138_order__refl,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_139_bot__nat__0_Oextremum,axiom,
% 2.82/1.52 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_140_le0,axiom,
% 2.82/1.52 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_141_Suc__le__mono,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_142_diff__diff__cancel,axiom,
% 2.82/1.52 ! [I: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ I @ N )
% 2.82/1.52 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 2.82/1.52 = I ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_143_le__zero__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [N: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.82/1.52 = ( N
% 2.82/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_144_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( cancel1352612707id_add @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( minus_minus @ A @ A2 @ A2 )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_145_diff__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( cancel1352612707id_add @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.82/1.52 = A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_146_zero__diff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( comm_monoid_diff @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_147_diff__0__right,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( group_add @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.82/1.52 = A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_148_diff__self,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( group_add @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( minus_minus @ A @ A2 @ A2 )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_149_of__nat__le__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_150_nat__power__eq__Suc__0__iff,axiom,
% 2.82/1.52 ! [X: nat,M: nat] :
% 2.82/1.52 ( ( ( power_power @ nat @ X @ M )
% 2.82/1.52 = ( suc @ ( zero_zero @ nat ) ) )
% 2.82/1.52 = ( ( M
% 2.82/1.52 = ( zero_zero @ nat ) )
% 2.82/1.52 | ( X
% 2.82/1.52 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_151_power__Suc__0,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 2.82/1.52 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_152_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [X: nat,B3: nat,W: nat] :
% 2.82/1.52 ( ( ( semiring_1_of_nat @ A @ X )
% 2.82/1.52 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 2.82/1.52 = ( X
% 2.82/1.52 = ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_153_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_char_0 @ A )
% 2.82/1.52 => ! [B3: nat,W: nat,X: nat] :
% 2.82/1.52 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W )
% 2.82/1.52 = ( semiring_1_of_nat @ A @ X ) )
% 2.82/1.52 = ( ( power_power @ nat @ B3 @ W )
% 2.82/1.52 = X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_154_of__nat__power,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_1 @ A )
% 2.82/1.52 => ! [M: nat,N: nat] :
% 2.82/1.52 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 2.82/1.52 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_155_nat__zero__less__power__iff,axiom,
% 2.82/1.52 ! [X: nat,N: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 2.82/1.52 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.82/1.52 | ( N
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_156_diff__is__0__eq_H,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ M @ N )
% 2.82/1.52 => ( ( minus_minus @ nat @ M @ N )
% 2.82/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_157_diff__is__0__eq,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ( minus_minus @ nat @ M @ N )
% 2.82/1.52 = ( zero_zero @ nat ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_158_diff__self__eq__0,axiom,
% 2.82/1.52 ! [M: nat] :
% 2.82/1.52 ( ( minus_minus @ nat @ M @ M )
% 2.82/1.52 = ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_159_diff__0__eq__0,axiom,
% 2.82/1.52 ! [N: nat] :
% 2.82/1.52 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 = ( zero_zero @ nat ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_160_Suc__diff__diff,axiom,
% 2.82/1.52 ! [M: nat,N: nat,K: nat] :
% 2.82/1.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 2.82/1.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_161_diff__Suc__Suc,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.82/1.52 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_162_diff__ge__0__iff__ge,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B3 ) )
% 2.82/1.52 = ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_163_diff__gt__0__iff__gt,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B3 ) )
% 2.82/1.52 = ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_164_power__0__Suc,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_1 @ A )
% 2.82/1.52 => ! [N: nat] :
% 2.82/1.52 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_165_of__nat__power__le__of__nat__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [X: nat,B3: nat,W: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_166_of__nat__le__of__nat__power__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [B3: nat,W: nat,X: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_167_power__Suc0__right,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( monoid_mult @ A )
% 2.82/1.52 => ! [A2: A] :
% 2.82/1.52 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 2.82/1.52 = A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_168_zero__less__diff,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 2.82/1.52 = ( ord_less @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_169_power__eq__0__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiri134348788visors @ A )
% 2.82/1.52 => ! [A2: A,N: nat] :
% 2.82/1.52 ( ( ( power_power @ A @ A2 @ N )
% 2.82/1.52 = ( zero_zero @ A ) )
% 2.82/1.52 = ( ( A2
% 2.82/1.52 = ( zero_zero @ A ) )
% 2.82/1.52 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_170_of__nat__less__of__nat__power__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [B3: nat,W: nat,X: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.82/1.52 = ( ord_less @ nat @ ( power_power @ nat @ B3 @ W ) @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_171_of__nat__power__less__of__nat__cancel__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [X: nat,B3: nat,W: nat] :
% 2.82/1.52 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W ) )
% 2.82/1.52 = ( ord_less @ nat @ X @ ( power_power @ nat @ B3 @ W ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_172_power__mono__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,B3: A,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) )
% 2.82/1.52 = ( ord_less_eq @ A @ A2 @ B3 ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_173_Suc__diff__le,axiom,
% 2.82/1.52 ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ N @ M )
% 2.82/1.52 => ( ( minus_minus @ nat @ ( suc @ M ) @ N )
% 2.82/1.52 = ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_174_nonneg__int__cases,axiom,
% 2.82/1.52 ! [K: int] :
% 2.82/1.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.82/1.52 => ~ ! [N2: nat] :
% 2.82/1.52 ( K
% 2.82/1.52 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_175_zero__le__imp__eq__int,axiom,
% 2.82/1.52 ! [K: int] :
% 2.82/1.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.82/1.52 => ? [N2: nat] :
% 2.82/1.52 ( K
% 2.82/1.52 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_176_zero__integer_Orsp,axiom,
% 2.82/1.52 ( ( zero_zero @ int )
% 2.82/1.52 = ( zero_zero @ int ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_177_lift__Suc__mono__le,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [F: nat > A,N: nat,N3: nat] :
% 2.82/1.52 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_178_lift__Suc__antimono__le,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [F: nat > A,N: nat,N3: nat] :
% 2.82/1.52 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_179_of__nat__diff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiring_1_cancel @ A )
% 2.82/1.52 => ! [N: nat,M: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ N @ M )
% 2.82/1.52 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
% 2.82/1.52 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_180_power__inject__base,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,N: nat,B3: A] :
% 2.82/1.52 ( ( ( power_power @ A @ A2 @ ( suc @ N ) )
% 2.82/1.52 = ( power_power @ A @ B3 @ ( suc @ N ) ) )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( A2 = B3 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_181_of__nat__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ! [I: nat,J2: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ I @ J2 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_182_power__le__imp__le__base,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,N: nat,B3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B3 @ ( suc @ N ) ) )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_183_dual__order_Oantisym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( A2 = B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_184_dual__order_Oeq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.82/1.52 = ( ^ [A4: A,B4: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.82/1.52 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_185_dual__order_Otrans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.82/1.52 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_186_linorder__wlog,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.82/1.52 ( ! [A3: A,B2: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A3 @ B2 )
% 2.82/1.52 => ( P @ A3 @ B2 ) )
% 2.82/1.52 => ( ! [A3: A,B2: A] :
% 2.82/1.52 ( ( P @ B2 @ A3 )
% 2.82/1.52 => ( P @ A3 @ B2 ) )
% 2.82/1.52 => ( P @ A2 @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_187_dual__order_Orefl,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_188_order__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A,Z2: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.82/1.52 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_189_order__class_Oorder_Oantisym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.82/1.52 => ( A2 = B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_190_ord__le__eq__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ord @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( B3 = C )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_191_ord__eq__le__trans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ord @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( A2 = B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_192_order__class_Oorder_Oeq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.82/1.52 = ( ^ [A4: A,B4: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.82/1.52 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_193_antisym__conv,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [Y2: A,X: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.82/1.52 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 = ( X = Y2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_194_le__cases3,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A,Z2: A] :
% 2.82/1.52 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.82/1.52 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.82/1.52 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.82/1.52 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 2.82/1.52 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.82/1.52 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.82/1.52 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_195_real__arch__simple,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( archim1804426504_field @ A )
% 2.82/1.52 => ! [X: A] :
% 2.82/1.52 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_196_order_Otrans,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_197_le__cases,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_198_eq__refl,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( preorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( X = Y2 )
% 2.82/1.52 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_199_linear,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_200_antisym,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ Y2 @ X )
% 2.82/1.52 => ( X = Y2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_201_eq__iff,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.82/1.52 = ( ^ [X5: A,Y5: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X5 @ Y5 )
% 2.82/1.52 & ( ord_less_eq @ A @ Y5 @ X5 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_202_ord__le__eq__subst,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( ord @ B )
% 2.82/1.52 & ( ord @ A ) )
% 2.82/1.52 => ! [A2: A,B3: A,F: A > B,C: B] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ( F @ B3 )
% 2.82/1.52 = C )
% 2.82/1.52 => ( ! [X2: A,Y3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_203_ord__eq__le__subst,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( ord @ B )
% 2.82/1.52 & ( ord @ A ) )
% 2.82/1.52 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.82/1.52 ( ( A2
% 2.82/1.52 = ( F @ B3 ) )
% 2.82/1.52 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.82/1.52 => ( ! [X2: B,Y3: B] :
% 2.82/1.52 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_204_order__subst2,axiom,
% 2.82/1.52 ! [A: $tType,C2: $tType] :
% 2.82/1.52 ( ( ( order @ C2 )
% 2.82/1.52 & ( order @ A ) )
% 2.82/1.52 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 2.82/1.52 => ( ! [X2: A,Y3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_205_order__subst1,axiom,
% 2.82/1.52 ! [A: $tType,B: $tType] :
% 2.82/1.52 ( ( ( order @ B )
% 2.82/1.52 & ( order @ A ) )
% 2.82/1.52 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 2.82/1.52 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.82/1.52 => ( ! [X2: B,Y3: B] :
% 2.82/1.52 ( ( ord_less_eq @ B @ X2 @ Y3 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_206_le__fun__def,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( ord @ B )
% 2.82/1.52 => ( ( ord_less_eq @ ( A > B ) )
% 2.82/1.52 = ( ^ [F3: A > B,G: A > B] :
% 2.82/1.52 ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_207_le__funI,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( ord @ B )
% 2.82/1.52 => ! [F: A > B,G2: A > B] :
% 2.82/1.52 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
% 2.82/1.52 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_208_le__funE,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( ord @ B )
% 2.82/1.52 => ! [F: A > B,G2: A > B,X: A] :
% 2.82/1.52 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.82/1.52 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_209_le__funD,axiom,
% 2.82/1.52 ! [B: $tType,A: $tType] :
% 2.82/1.52 ( ( ord @ B )
% 2.82/1.52 => ! [F: A > B,G2: A > B,X: A] :
% 2.82/1.52 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 2.82/1.52 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_210_int__int__eq,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ( semiring_1_of_nat @ int @ M )
% 2.82/1.52 = ( semiring_1_of_nat @ int @ N ) )
% 2.82/1.52 = ( M = N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_211_zle__int,axiom,
% 2.82/1.52 ! [M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_212_le__iff__diff__le__0,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ( ( ord_less_eq @ A )
% 2.82/1.52 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_213_diff__less__mono,axiom,
% 2.82/1.52 ! [A2: nat,B3: nat,C: nat] :
% 2.82/1.52 ( ( ord_less @ nat @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ C @ A2 )
% 2.82/1.52 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_214_less__diff__iff,axiom,
% 2.82/1.52 ! [K: nat,M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ K @ M )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.82/1.52 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.82/1.52 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_215_power__not__zero,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( semiri134348788visors @ A )
% 2.82/1.52 => ! [A2: A,N: nat] :
% 2.82/1.52 ( ( A2
% 2.82/1.52 != ( zero_zero @ A ) )
% 2.82/1.52 => ( ( power_power @ A @ A2 @ N )
% 2.82/1.52 != ( zero_zero @ A ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_216_power__less__imp__less__base,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,N: nat,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_217_diff__right__commute,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( cancel146912293up_add @ A )
% 2.82/1.52 => ! [A2: A,C: A,B3: A] :
% 2.82/1.52 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B3 )
% 2.82/1.52 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_218_diff__eq__diff__less__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A,D: A] :
% 2.82/1.52 ( ( ( minus_minus @ A @ A2 @ B3 )
% 2.82/1.52 = ( minus_minus @ A @ C @ D ) )
% 2.82/1.52 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_219_diff__right__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_220_diff__left__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [B3: A,A2: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_221_zero__le__power,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_222_diff__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,D: A,C: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ D @ C )
% 2.82/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_223_power__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,B3: A,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_224_diff__eq__diff__eq,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A,D: A] :
% 2.82/1.52 ( ( ( minus_minus @ A @ A2 @ B3 )
% 2.82/1.52 = ( minus_minus @ A @ C @ D ) )
% 2.82/1.52 => ( ( A2 = B3 )
% 2.82/1.52 = ( C = D ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_225_diff__le__mono2,axiom,
% 2.82/1.52 ! [M: nat,N: nat,L: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ M @ N )
% 2.82/1.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_226_le__diff__iff_H,axiom,
% 2.82/1.52 ! [A2: nat,C: nat,B3: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ B3 @ C )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B3 ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ B3 @ A2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_227_diff__le__self,axiom,
% 2.82/1.52 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_228_diff__le__mono,axiom,
% 2.82/1.52 ! [M: nat,N: nat,L: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ M @ N )
% 2.82/1.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_229_Nat_Odiff__diff__eq,axiom,
% 2.82/1.52 ! [K: nat,M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ K @ M )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.82/1.52 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.82/1.52 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_230_diff__commute,axiom,
% 2.82/1.52 ! [I: nat,J2: nat,K: nat] :
% 2.82/1.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K )
% 2.82/1.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J2 ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_231_le__diff__iff,axiom,
% 2.82/1.52 ! [K: nat,M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ K @ M )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.82/1.52 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_232_eq__diff__iff,axiom,
% 2.82/1.52 ! [K: nat,M: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ K @ M )
% 2.82/1.52 => ( ( ord_less_eq @ nat @ K @ N )
% 2.82/1.52 => ( ( ( minus_minus @ nat @ M @ K )
% 2.82/1.52 = ( minus_minus @ nat @ N @ K ) )
% 2.82/1.52 = ( M = N ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_233_less__eq__int__code_I1_J,axiom,
% 2.82/1.52 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_234_less__int__code_I1_J,axiom,
% 2.82/1.52 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_235_nat__one__le__power,axiom,
% 2.82/1.52 ! [I: nat,N: nat] :
% 2.82/1.52 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 2.82/1.52 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_236_power__eq__imp__eq__base,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [A2: A,N: nat,B3: A] :
% 2.82/1.52 ( ( ( power_power @ A @ A2 @ N )
% 2.82/1.52 = ( power_power @ A @ B3 @ N ) )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 => ( A2 = B3 ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_237_power__eq__iff__eq__base,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linordered_semidom @ A )
% 2.82/1.52 => ! [N: nat,A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
% 2.82/1.52 => ( ( ( power_power @ A @ A2 @ N )
% 2.82/1.52 = ( power_power @ A @ B3 @ N ) )
% 2.82/1.52 = ( A2 = B3 ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_238_eq__iff__diff__eq__0,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( group_add @ A )
% 2.82/1.52 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.82/1.52 = ( ^ [A4: A,B4: A] :
% 2.82/1.52 ( ( minus_minus @ A @ A4 @ B4 )
% 2.82/1.52 = ( zero_zero @ A ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_239_diff__strict__right__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_240_diff__strict__left__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [B3: A,A2: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ B3 @ A2 )
% 2.82/1.52 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_241_diff__eq__diff__less,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,C: A,D: A] :
% 2.82/1.52 ( ( ( minus_minus @ A @ A2 @ B3 )
% 2.82/1.52 = ( minus_minus @ A @ C @ D ) )
% 2.82/1.52 => ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_242_diff__strict__mono,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( ordered_ab_group_add @ A )
% 2.82/1.52 => ! [A2: A,B3: A,D: A,C: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( ord_less @ A @ D @ C )
% 2.82/1.52 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_243_zero__le,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( canoni770627133id_add @ A )
% 2.82/1.52 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_244_le__numeral__extra_I3_J,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( linord1659791738miring @ A )
% 2.82/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_245_complete__interval,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( condit1037483654norder @ A )
% 2.82/1.52 => ! [A2: A,B3: A,P: A > $o] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ( P @ A2 )
% 2.82/1.52 => ( ~ ( P @ B3 )
% 2.82/1.52 => ? [C3: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.82/1.52 & ( ord_less_eq @ A @ C3 @ B3 )
% 2.82/1.52 & ! [X3: A] :
% 2.82/1.52 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 2.82/1.52 & ( ord_less @ A @ X3 @ C3 ) )
% 2.82/1.52 => ( P @ X3 ) )
% 2.82/1.52 & ! [D3: A] :
% 2.82/1.52 ( ! [X2: A] :
% 2.82/1.52 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.82/1.52 & ( ord_less @ A @ X2 @ D3 ) )
% 2.82/1.52 => ( P @ X2 ) )
% 2.82/1.52 => ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_246_order_Onot__eq__order__implies__strict,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( A2 != B3 )
% 2.82/1.52 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.82/1.52 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_247_dual__order_Ostrict__implies__order,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [B3: A,A2: A] :
% 2.82/1.52 ( ( ord_less @ A @ B3 @ A2 )
% 2.82/1.52 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_248_dual__order_Ostrict__iff__order,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ( ( ord_less @ A )
% 2.82/1.52 = ( ^ [B4: A,A4: A] :
% 2.82/1.52 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.82/1.52 & ( A4 != B4 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_249_dual__order_Oorder__iff__strict,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ( ( ord_less_eq @ A )
% 2.82/1.52 = ( ^ [B4: A,A4: A] :
% 2.82/1.52 ( ( ord_less @ A @ B4 @ A4 )
% 2.82/1.52 | ( A4 = B4 ) ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_250_order_Ostrict__implies__order,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( order @ A )
% 2.82/1.52 => ! [A2: A,B3: A] :
% 2.82/1.52 ( ( ord_less @ A @ A2 @ B3 )
% 2.82/1.52 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_251_dense__le__bounded,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( dense_linorder @ A )
% 2.82/1.52 => ! [X: A,Y2: A,Z2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ Y2 )
% 2.82/1.52 => ( ! [W2: A] :
% 2.82/1.52 ( ( ord_less @ A @ X @ W2 )
% 2.82/1.52 => ( ( ord_less @ A @ W2 @ Y2 )
% 2.82/1.52 => ( ord_less_eq @ A @ W2 @ Z2 ) ) )
% 2.82/1.52 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(fact_252_dense__ge__bounded,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( dense_linorder @ A )
% 2.82/1.52 => ! [Z2: A,X: A,Y2: A] :
% 2.82/1.52 ( ( ord_less @ A @ Z2 @ X )
% 2.82/1.52 => ( ! [W2: A] :
% 2.82/1.52 ( ( ord_less @ A @ Z2 @ W2 )
% 2.82/1.52 => ( ( ord_less @ A @ W2 @ X )
% 2.82/1.52 => ( ord_less_eq @ A @ Y2 @ W2 ) ) )
% 2.82/1.52 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(subcl_Groups_Ozero___HOL_Otype,axiom,
% 2.82/1.52 ! [A: $tType] :
% 2.82/1.52 ( ( zero @ A )
% 2.82/1.52 => ( type @ A ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.82/1.52 ! [A5: $tType,A6: $tType] :
% 2.82/1.52 ( ( preorder @ A6 )
% 2.82/1.52 => ( preorder @ ( A5 > A6 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.82/1.52 ! [A5: $tType,A6: $tType] :
% 2.82/1.52 ( ( order @ A6 )
% 2.82/1.52 => ( order @ ( A5 > A6 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_fun___Orderings_Oord,axiom,
% 2.82/1.52 ! [A5: $tType,A6: $tType] :
% 2.82/1.52 ( ( ord @ A6 )
% 2.82/1.52 => ( ord @ ( A5 > A6 ) ) ) ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.82/1.52 condit1037483654norder @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.82/1.52 semiri134348788visors @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.82/1.52 linord1659791738miring @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.82/1.52 cancel146912293up_add @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.82/1.52 cancel1352612707id_add @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.82/1.52 ordered_ab_group_add @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.82/1.52 linordered_semidom @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
% 2.82/1.52 semiring_1_cancel @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.82/1.52 linordered_idom @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.82/1.52 semiring_char_0 @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.82/1.52 preorder @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.82/1.52 linorder @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 2.82/1.52 monoid_mult @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.82/1.52 semiring_1 @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 2.82/1.52 no_top @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 2.82/1.52 no_bot @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.82/1.52 group_add @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.82/1.52 order @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.82/1.52 ord @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.82/1.52 zero @ int ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,axiom,
% 2.82/1.52 condit1037483654norder @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.82/1.52 canoni770627133id_add @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 2.82/1.52 semiri134348788visors @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6,axiom,
% 2.82/1.52 linord1659791738miring @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 2.82/1.52 cancel146912293up_add @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 2.82/1.52 cancel1352612707id_add @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 2.82/1.52 linordered_semidom @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_10,axiom,
% 2.82/1.52 semiring_1_cancel @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.82/1.52 comm_monoid_diff @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.82/1.52 wellorder @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 2.82/1.52 semiring_char_0 @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_12,axiom,
% 2.82/1.52 preorder @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder_13,axiom,
% 2.82/1.52 linorder @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Omonoid__mult_14,axiom,
% 2.82/1.52 monoid_mult @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Rings_Osemiring__1_15,axiom,
% 2.82/1.52 semiring_1 @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Ono__top_16,axiom,
% 2.82/1.52 no_top @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_17,axiom,
% 2.82/1.52 order @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Orderings_Oord_18,axiom,
% 2.82/1.52 ord @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_Nat_Onat___Groups_Ozero_19,axiom,
% 2.82/1.52 zero @ nat ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_20,axiom,
% 2.82/1.52 preorder @ $o ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_21,axiom,
% 2.82/1.52 linorder @ $o ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_22,axiom,
% 2.82/1.52 order @ $o ).
% 2.82/1.52
% 2.82/1.52 thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
% 2.82/1.52 ord @ $o ).
% 2.82/1.52
% 2.82/1.52 thf(tfree_0,hypothesis,
% 2.82/1.52 zero @ a ).
% 2.82/1.52
% 2.82/1.52 thf(conj_0,hypothesis,
% 2.82/1.52 ord_less @ nat @ ( zero_zero @ nat ) @ n ).
% 2.82/1.52
% 2.82/1.52 thf(conj_1,hypothesis,
% 2.82/1.52 polyno86455060npolyh @ a @ p @ n ).
% 2.82/1.52
% 2.82/1.52 thf(conj_2,conjecture,
% 2.82/1.52 ( ( polyno498386536degree @ a @ p )
% 2.82/1.52 = ( zero_zero @ nat ) ) ).
% 2.82/1.52 ------------------------
% 2.82/1.55 % (21475)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_21321 for (2999ds/2Mi)
% 2.82/1.55 % (21474)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/2Mi)
% 2.82/1.55 % (21473)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_21321 for (2999ds/27Mi)
% 2.82/1.55 % (21472)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_21321 for (2999ds/4Mi)
% 2.82/1.55 % (21474)Instruction limit reached!
% 2.82/1.55 % (21474)------------------------------
% 2.82/1.55 % (21474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.55 % (21474)Termination reason: Unknown
% 2.82/1.55 % (21474)Termination phase: shuffling
% 2.82/1.55
% 2.82/1.55 % (21474)Memory used [KB]: 1279
% 2.82/1.55 % (21474)Time elapsed: 0.002 s
% 2.82/1.55 % (21474)Instructions burned: 2 (million)
% 2.82/1.55 % (21475)Instruction limit reached!
% 2.82/1.55 % (21475)------------------------------
% 2.82/1.55 % (21475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.55 % (21475)Termination reason: Unknown
% 2.82/1.55 % (21475)Termination phase: shuffling
% 2.82/1.55
% 2.82/1.55 % (21475)Memory used [KB]: 1279
% 2.82/1.55 % (21475)Time elapsed: 0.002 s
% 2.82/1.55 % (21475)Instructions burned: 3 (million)
% 2.82/1.55 % (21475)------------------------------
% 2.82/1.55 % (21475)------------------------------
% 2.82/1.55 % (21474)------------------------------
% 2.82/1.55 % (21474)------------------------------
% 2.82/1.55 % (21476)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_21321 for (2999ds/275Mi)
% 2.82/1.55 % (21472)Instruction limit reached!
% 2.82/1.55 % (21472)------------------------------
% 2.82/1.55 % (21472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.55 % (21472)Termination reason: Unknown
% 2.82/1.55 % (21472)Termination phase: shuffling
% 2.82/1.55
% 2.82/1.55 % (21472)Memory used [KB]: 1407
% 2.82/1.55 % (21472)Time elapsed: 0.003 s
% 2.82/1.55 % (21472)Instructions burned: 6 (million)
% 2.82/1.55 % (21472)------------------------------
% 2.82/1.55 % (21472)------------------------------
% 2.82/1.55 % (21471)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/183Mi)
% 2.82/1.55 % (21477)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/18Mi)
% 2.82/1.56 % (21473)Instruction limit reached!
% 2.82/1.56 % (21473)------------------------------
% 2.82/1.56 % (21473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (21473)Termination reason: Unknown
% 2.82/1.56 % (21473)Termination phase: Preprocessing 1
% 2.82/1.56
% 2.82/1.56 % (21473)Memory used [KB]: 1663
% 2.82/1.56 % (21473)Time elapsed: 0.010 s
% 2.82/1.56 % (21473)Instructions burned: 28 (million)
% 2.82/1.56 % (21473)------------------------------
% 2.82/1.56 % (21473)------------------------------
% 2.82/1.56 % (21480)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/3Mi)
% 2.82/1.56 % (21481)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_21321 for (2999ds/37Mi)
% 2.82/1.56 % (21480)Instruction limit reached!
% 2.82/1.56 % (21480)------------------------------
% 2.82/1.56 % (21480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.56 % (21480)Termination reason: Unknown
% 2.82/1.56 % (21482)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_21321 for (2999ds/15Mi)
% 2.82/1.56 % (21480)Termination phase: shuffling
% 2.82/1.56
% 2.82/1.56 % (21480)Memory used [KB]: 1279
% 2.82/1.56 % (21480)Time elapsed: 0.002 s
% 2.82/1.56 % (21480)Instructions burned: 3 (million)
% 2.82/1.56 % (21480)------------------------------
% 2.82/1.56 % (21480)------------------------------
% 2.82/1.57 % (21482)Instruction limit reached!
% 2.82/1.57 % (21482)------------------------------
% 2.82/1.57 % (21482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.57 % (21482)Termination reason: Unknown
% 2.82/1.57 % (21482)Termination phase: Property scanning
% 2.82/1.57
% 2.82/1.57 % (21482)Memory used [KB]: 1663
% 2.82/1.57 % (21482)Time elapsed: 0.007 s
% 2.82/1.57 % (21482)Instructions burned: 17 (million)
% 2.82/1.57 % (21482)------------------------------
% 2.82/1.57 % (21482)------------------------------
% 2.82/1.57 % (21477)Instruction limit reached!
% 2.82/1.57 % (21477)------------------------------
% 2.82/1.57 % (21477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.57 % (21477)Termination reason: Unknown
% 2.82/1.57 % (21483)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_21321 for (2999ds/3Mi)
% 2.82/1.57 % (21477)Termination phase: Property scanning
% 2.82/1.57
% 2.82/1.57 % (21477)Memory used [KB]: 1663
% 2.82/1.57 % (21477)Time elapsed: 0.013 s
% 2.82/1.57 % (21477)Instructions burned: 19 (million)
% 2.82/1.57 % (21477)------------------------------
% 2.82/1.57 % (21477)------------------------------
% 2.82/1.57 % (21483)Instruction limit reached!
% 2.82/1.57 % (21483)------------------------------
% 2.82/1.57 % (21483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.57 % (21483)Termination reason: Unknown
% 2.82/1.57 % (21483)Termination phase: shuffling
% 2.82/1.57
% 2.82/1.57 % (21483)Memory used [KB]: 1407
% 2.82/1.57 % (21483)Time elapsed: 0.004 s
% 2.82/1.57 % (21483)Instructions burned: 4 (million)
% 2.82/1.57 % (21483)------------------------------
% 2.82/1.57 % (21483)------------------------------
% 2.82/1.57 % (21484)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_21321 for (2999ds/1041Mi)
% 2.82/1.58 % (21481)Instruction limit reached!
% 2.82/1.58 % (21481)------------------------------
% 2.82/1.58 % (21481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (21481)Termination reason: Unknown
% 2.82/1.58 % (21481)Termination phase: Property scanning
% 2.82/1.58
% 2.82/1.58 % (21481)Memory used [KB]: 2046
% 2.82/1.58 % (21481)Time elapsed: 0.018 s
% 2.82/1.58 % (21481)Instructions burned: 37 (million)
% 2.82/1.58 % (21481)------------------------------
% 2.82/1.58 % (21481)------------------------------
% 2.82/1.58 % (21492)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/7Mi)
% 2.82/1.58 % (21492)Instruction limit reached!
% 2.82/1.58 % (21492)------------------------------
% 2.82/1.58 % (21492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.58 % (21492)Termination reason: Unknown
% 2.82/1.58 % (21492)Termination phase: shuffling
% 2.82/1.58
% 2.82/1.58 % (21492)Memory used [KB]: 1535
% 2.82/1.58 % (21492)Time elapsed: 0.005 s
% 2.82/1.58 % (21492)Instructions burned: 9 (million)
% 2.82/1.58 % (21492)------------------------------
% 2.82/1.58 % (21492)------------------------------
% 2.82/1.58 % (21493)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/16Mi)
% 2.82/1.58 % (21494)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_21321 for (2999ds/3Mi)
% 2.82/1.59 % (21494)Instruction limit reached!
% 2.82/1.59 % (21494)------------------------------
% 2.82/1.59 % (21494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.59 % (21494)Termination reason: Unknown
% 2.82/1.59 % (21494)Termination phase: shuffling
% 2.82/1.59
% 2.82/1.59 % (21494)Memory used [KB]: 1407
% 2.82/1.59 % (21494)Time elapsed: 0.003 s
% 2.82/1.59 % (21494)Instructions burned: 3 (million)
% 2.82/1.59 % (21494)------------------------------
% 2.82/1.59 % (21494)------------------------------
% 2.82/1.59 % (21495)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_21321 for (2999ds/3Mi)
% 2.82/1.59 % (21495)Instruction limit reached!
% 2.82/1.59 % (21495)------------------------------
% 2.82/1.59 % (21495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.59 % (21495)Termination reason: Unknown
% 2.82/1.59 % (21495)Termination phase: shuffling
% 2.82/1.59
% 2.82/1.59 % (21495)Memory used [KB]: 1407
% 2.82/1.59 % (21495)Time elapsed: 0.002 s
% 2.82/1.59 % (21495)Instructions burned: 3 (million)
% 2.82/1.59 % (21495)------------------------------
% 2.82/1.59 % (21495)------------------------------
% 2.82/1.59 % (21493)Instruction limit reached!
% 2.82/1.59 % (21493)------------------------------
% 2.82/1.59 % (21493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.82/1.59 % (21493)Termination reason: Unknown
% 2.82/1.59 % (21493)Termination phase: shuffling
% 2.82/1.59
% 2.82/1.59 % (21493)Memory used [KB]: 1663
% 2.82/1.59 % (21493)Time elapsed: 0.011 s
% 2.82/1.59 % (21493)Instructions burned: 16 (million)
% 2.82/1.59 % (21493)------------------------------
% 2.82/1.59 % (21493)------------------------------
% 2.82/1.60 % (21496)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/7Mi)
% 2.85/1.60 % (21496)Instruction limit reached!
% 2.85/1.60 % (21496)------------------------------
% 2.85/1.60 % (21496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (21496)Termination reason: Unknown
% 2.85/1.60 % (21496)Termination phase: shuffling
% 2.85/1.60
% 2.85/1.60 % (21496)Memory used [KB]: 1535
% 2.85/1.60 % (21496)Time elapsed: 0.006 s
% 2.85/1.60 % (21496)Instructions burned: 8 (million)
% 2.85/1.60 % (21496)------------------------------
% 2.85/1.60 % (21496)------------------------------
% 2.85/1.60 % (21476)First to succeed.
% 2.85/1.61 % (21500)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/3Mi)
% 2.85/1.61 % (21502)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21321 for (2999ds/4Mi)
% 2.85/1.61 % (21476)Refutation found. Thanks to Tanya!
% 2.85/1.61 % SZS status Theorem for DTF2THF_21321
% 2.85/1.61 % SZS output start Proof for DTF2THF_21321
% 2.85/1.61 thf(type_def_6, type, nat: $tType).
% 2.85/1.61 thf(type_def_7, type, polyno1783536151e_poly: ($tType) > $tType).
% 2.85/1.61 thf(type_def_8, type, a: $tType).
% 2.85/1.61 thf(type_def_9, type, int: $tType).
% 2.85/1.61 thf(func_def_0, type, polyno1783536151e_poly: ($tType) > $tType).
% 2.85/1.61 thf(func_def_1, type, nat: $tType).
% 2.85/1.61 thf(func_def_2, type, int: $tType).
% 2.85/1.61 thf(func_def_3, type, a: $tType).
% 2.85/1.61 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_5, type, zero: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_8, type, group_add: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_9, type, no_bot: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_10, type, no_top: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_11, type, semiring_1: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_12, type, monoid_mult: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_15, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_16, type, wellorder: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_17, type, dense_order: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_18, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_19, type, linordered_field: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_20, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_21, type, semiring_1_cancel: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_22, type, dense_linorder: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_23, type, linordered_semidom: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_24, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_25, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_26, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_27, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_28, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_29, type, archim1804426504_field: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_30, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_31, type, condit1656338222tinuum: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_32, type, condit1037483654norder: !>[X0: $tType]:($o)).
% 2.85/1.61 thf(func_def_33, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.85/1.61 thf(func_def_34, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.85/1.61 thf(func_def_35, type, suc: nat > nat).
% 2.85/1.61 thf(func_def_36, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.85/1.61 thf(func_def_37, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.85/1.61 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.85/1.61 thf(func_def_39, type, polyno498386536degree: !>[X0: $tType]:(polyno1783536151e_poly @ X0 > nat)).
% 2.85/1.61 thf(func_def_40, type, polyno367318022egreen: !>[X0: $tType]:(polyno1783536151e_poly @ X0 > nat > nat)).
% 2.85/1.61 thf(func_def_41, type, polyno2122670676snpoly: !>[X0: $tType]:(polyno1783536151e_poly @ X0 > $o)).
% 2.85/1.61 thf(func_def_42, type, polyno86455060npolyh: !>[X0: $tType]:(polyno1783536151e_poly @ X0 > nat > $o)).
% 2.85/1.61 thf(func_def_43, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 2.85/1.61 thf(func_def_44, type, n: nat).
% 2.85/1.61 thf(func_def_45, type, p: polyno1783536151e_poly @ a).
% 2.85/1.61 thf(func_def_60, type, sK0: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_61, type, sK1: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > X0)).
% 2.85/1.61 thf(func_def_62, type, sK2: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat)).
% 2.85/1.61 thf(func_def_63, type, sK3: !>[X0: $tType]:((nat > X0 > X0 > $o) > (nat > X0 > $o) > nat > X0)).
% 2.85/1.61 thf(func_def_64, type, sK4: nat > nat > nat).
% 2.85/1.61 thf(func_def_65, type, sK5: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_66, type, sK6: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_67, type, sK7: nat > nat).
% 2.85/1.61 thf(func_def_68, type, sK8: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_69, type, sK9: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_70, type, sK10: nat > nat).
% 2.85/1.61 thf(func_def_71, type, sK11: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_72, type, sK12: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_73, type, sK13: nat > nat > nat).
% 2.85/1.61 thf(func_def_74, type, sK14: nat > nat > nat).
% 2.85/1.61 thf(func_def_75, type, sK15: int > nat).
% 2.85/1.61 thf(func_def_76, type, sK16: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_77, type, sK17: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_78, type, sK18: nat > nat).
% 2.85/1.61 thf(func_def_79, type, sK19: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_80, type, sK20: int > nat).
% 2.85/1.61 thf(func_def_81, type, sK21: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_82, type, sK22: int > nat).
% 2.85/1.61 thf(func_def_83, type, sK23: int > nat).
% 2.85/1.61 thf(func_def_84, type, sK24: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.85/1.61 thf(func_def_85, type, sK25: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.85/1.61 thf(func_def_86, type, sK26: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.85/1.61 thf(func_def_87, type, sK27: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_88, type, sK28: nat > nat > nat).
% 2.85/1.61 thf(func_def_89, type, sK29: nat > nat).
% 2.85/1.61 thf(func_def_90, type, sK30: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_91, type, sK31: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_92, type, sK32: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_93, type, sK33: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_94, type, sK34: (nat > $o) > nat).
% 2.85/1.61 thf(func_def_95, type, sK35: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_96, type, sK36: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_97, type, sK37: nat > nat).
% 2.85/1.61 thf(func_def_98, type, sK38: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_99, type, sK39: (nat > $o) > nat > nat).
% 2.85/1.61 thf(func_def_100, type, sK40: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_101, type, sK41: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_102, type, sK42: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_103, type, sK43: (nat > nat > $o) > nat).
% 2.85/1.61 thf(func_def_105, type, ph45: !>[X0: $tType]:(X0)).
% 2.85/1.61 thf(f1417,plain,(
% 2.85/1.61 $false),
% 2.85/1.61 inference(subsumption_resolution,[status(thm)],[f1416,f1223])).
% 2.85/1.61 thf(f1223,plain,(
% 2.85/1.61 (zero_zero @ nat != (polyno498386536degree @ a @ p))),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f865])).
% 2.85/1.61 thf(f865,plain,(
% 2.85/1.61 (zero_zero @ nat != (polyno498386536degree @ a @ p))),
% 2.85/1.61 inference(flattening,[status(thm)],[f305])).
% 2.85/1.61 thf(f305,negated_conjecture,(
% 2.85/1.61 ~(zero_zero @ nat = (polyno498386536degree @ a @ p))),
% 2.85/1.61 inference(negated_conjecture,[status(cth)],[f304])).
% 2.85/1.61 thf(f304,conjecture,(
% 2.85/1.61 (zero_zero @ nat = (polyno498386536degree @ a @ p))),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',conj_2)).
% 2.85/1.61 thf(f1416,plain,(
% 2.85/1.61 (zero_zero @ nat = (polyno498386536degree @ a @ p))),
% 2.85/1.61 inference(subsumption_resolution,[status(thm)],[f1415,f1101])).
% 2.85/1.61 thf(f1101,plain,(
% 2.85/1.61 (zero @ a = $true)),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f468])).
% 2.85/1.61 thf(f468,plain,(
% 2.85/1.61 (zero @ a = $true)),
% 2.85/1.61 inference(fool_elimination,[status(thm)],[f467])).
% 2.85/1.61 thf(f467,plain,(
% 2.85/1.61 zero @ a),
% 2.85/1.61 inference(rectify,[status(thm)],[f301])).
% 2.85/1.61 thf(f301,axiom,(
% 2.85/1.61 zero @ a),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',tfree_0)).
% 2.85/1.61 thf(f1415,plain,(
% 2.85/1.61 (zero @ a != $true) | (zero_zero @ nat = (polyno498386536degree @ a @ p))),
% 2.85/1.61 inference(trivial_inequality_removal,[status(thm)],[f1414])).
% 2.85/1.61 thf(f1414,plain,(
% 2.85/1.61 ($true != $true) | (zero_zero @ nat = (polyno498386536degree @ a @ p)) | (zero @ a != $true)),
% 2.85/1.61 inference(superposition,[status(thm)],[f1279,f1199])).
% 2.85/1.61 thf(f1199,plain,(
% 2.85/1.61 ((polyno86455060npolyh @ a @ p @ n) = $true)),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f674])).
% 2.85/1.61 thf(f674,plain,(
% 2.85/1.61 ((polyno86455060npolyh @ a @ p @ n) = $true)),
% 2.85/1.61 inference(fool_elimination,[status(thm)],[f673])).
% 2.85/1.61 thf(f673,plain,(
% 2.85/1.61 (polyno86455060npolyh @ a @ p @ n)),
% 2.85/1.61 inference(rectify,[status(thm)],[f303])).
% 2.85/1.61 thf(f303,axiom,(
% 2.85/1.61 (polyno86455060npolyh @ a @ p @ n)),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',conj_1)).
% 2.85/1.61 thf(f1279,plain,(
% 2.85/1.61 ( ! [X0 : $tType,X1 : polyno1783536151e_poly @ X0] : (($true != (polyno86455060npolyh @ X0 @ X1 @ n)) | (zero @ X0 != $true) | (zero_zero @ nat = (polyno498386536degree @ X0 @ X1))) )),
% 2.85/1.61 inference(superposition,[status(thm)],[f1150,f1247])).
% 2.85/1.61 thf(f1247,plain,(
% 2.85/1.61 (n = (suc @ (sK7 @ n)))),
% 2.85/1.61 inference(trivial_inequality_removal,[status(thm)],[f1246])).
% 2.85/1.61 thf(f1246,plain,(
% 2.85/1.61 (n = (suc @ (sK7 @ n))) | ($true != $true)),
% 2.85/1.61 inference(superposition,[status(thm)],[f1084,f1091])).
% 2.85/1.61 thf(f1091,plain,(
% 2.85/1.61 ((ord_less @ nat @ zero_zero @ nat @ n) = $true)),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f708])).
% 2.85/1.61 thf(f708,plain,(
% 2.85/1.61 ((ord_less @ nat @ zero_zero @ nat @ n) = $true)),
% 2.85/1.61 inference(fool_elimination,[status(thm)],[f707])).
% 2.85/1.61 thf(f707,plain,(
% 2.85/1.61 (ord_less @ nat @ zero_zero @ nat @ n)),
% 2.85/1.61 inference(rectify,[status(thm)],[f302])).
% 2.85/1.61 thf(f302,axiom,(
% 2.85/1.61 (ord_less @ nat @ zero_zero @ nat @ n)),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',conj_0)).
% 2.85/1.61 thf(f1084,plain,(
% 2.85/1.61 ( ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X0) != $true) | ((suc @ (sK7 @ X0)) = X0)) )),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f973])).
% 2.85/1.61 thf(f973,plain,(
% 2.85/1.61 ! [X0 : nat] : (((suc @ (sK7 @ X0)) = X0) | ((ord_less @ nat @ zero_zero @ nat @ X0) != $true))),
% 2.85/1.61 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7])],[f943,f972])).
% 2.85/1.61 thf(f972,plain,(
% 2.85/1.61 ! [X0 : nat] : (? [X1 : nat] : ((suc @ X1) = X0) => ((suc @ (sK7 @ X0)) = X0))),
% 2.85/1.61 introduced(definition,[],[choice_axiom])).
% 2.85/1.61 thf(f943,plain,(
% 2.85/1.61 ! [X0 : nat] : (? [X1 : nat] : ((suc @ X1) = X0) | ((ord_less @ nat @ zero_zero @ nat @ X0) != $true))),
% 2.85/1.61 inference(ennf_transformation,[status(thm)],[f394])).
% 2.85/1.61 thf(f394,plain,(
% 2.85/1.61 ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X0) = $true) => ? [X1 : nat] : ((suc @ X1) = X0))),
% 2.85/1.61 inference(fool_elimination,[status(thm)],[f393])).
% 2.85/1.61 thf(f393,plain,(
% 2.85/1.61 ! [X0 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X0) => ? [X1 : nat] : ((suc @ X1) = X0))),
% 2.85/1.61 inference(rectify,[status(thm)],[f94])).
% 2.85/1.61 thf(f94,axiom,(
% 2.85/1.61 ! [X1 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X1) => ? [X34 : nat] : ((suc @ X34) = X1))),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',fact_93_gr0__implies__Suc)).
% 2.85/1.61 thf(f1150,plain,(
% 2.85/1.61 ( ! [X0 : $tType,X2 : nat,X1 : polyno1783536151e_poly @ X0] : (((polyno86455060npolyh @ X0 @ X1 @ (suc @ X2)) != $true) | (zero_zero @ nat = (polyno498386536degree @ X0 @ X1)) | (zero @ X0 != $true)) )),
% 2.85/1.61 inference(cnf_transformation,[status(thm)],[f923])).
% 2.85/1.61 thf(f923,plain,(
% 2.85/1.61 ! [X0 : $tType] : (! [X1 : polyno1783536151e_poly @ X0,X2 : nat] : (((polyno86455060npolyh @ X0 @ X1 @ (suc @ X2)) != $true) | (zero_zero @ nat = (polyno498386536degree @ X0 @ X1))) | (zero @ X0 != $true))),
% 2.85/1.61 inference(ennf_transformation,[status(thm)],[f852])).
% 2.85/1.61 thf(f852,plain,(
% 2.85/1.61 ! [X0 : $tType] : ((zero @ X0 = $true) => ! [X1 : polyno1783536151e_poly @ X0,X2 : nat] : (((polyno86455060npolyh @ X0 @ X1 @ (suc @ X2)) = $true) => (zero_zero @ nat = (polyno498386536degree @ X0 @ X1))))),
% 2.85/1.61 inference(fool_elimination,[status(thm)],[f851])).
% 2.85/1.61 thf(f851,plain,(
% 2.85/1.61 ! [X0 : $tType] : (zero @ X0 => ! [X1 : polyno1783536151e_poly @ X0,X2 : nat] : ((polyno86455060npolyh @ X0 @ X1 @ (suc @ X2)) => (zero_zero @ nat = (polyno498386536degree @ X0 @ X1))))),
% 2.85/1.61 inference(rectify,[status(thm)],[f34])).
% 2.85/1.61 thf(f34,axiom,(
% 2.85/1.61 ! [X0 : $tType] : (zero @ X0 => ! [X20 : polyno1783536151e_poly @ X0,X1 : nat] : ((polyno86455060npolyh @ X0 @ X20 @ (suc @ X1)) => (zero_zero @ nat = (polyno498386536degree @ X0 @ X20))))),
% 2.85/1.61 file('/export/starexec/sandbox2/tmp/tmp.rDW7FOZjHl/DTF2THF_21321.p',fact_33_degree__isnpolyh__Suc)).
% 2.85/1.61 % SZS output end Proof for DTF2THF_21321
% 2.85/1.61 % (21476)------------------------------
% 2.85/1.61 % (21476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.61 % (21476)Termination reason: Refutation
% 2.85/1.61
% 2.85/1.61 % (21476)Memory used [KB]: 6652
% 2.85/1.61 % (21476)Time elapsed: 0.060 s
% 2.85/1.61 % (21476)Instructions burned: 122 (million)
% 2.85/1.61 % (21476)------------------------------
% 2.85/1.61 % (21476)------------------------------
% 2.85/1.61 % (21466)Success in time 0.072 s
%------------------------------------------------------------------------------