%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP134^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.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 08:55:49 AM UTC 2026
% Result : Theorem 6.95s 2.29s
% Output : Refutation 6.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP134^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n005.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 08:27:29 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.47 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP134^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer Number_Partition problem prob_94__5324562_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : Number_Partition/prob_94__5324562_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.00 v9.0.0, 0.33 v8.1.0, 0.25 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 334 ( 74 unt; 44 typ; 0 def)
% 0.20/0.47 % Number of atoms : 929 ( 217 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.20/0.47 % Number of connectives : 3092 ( 55 ~; 15 |; 57 &;2532 @)
% 0.20/0.47 % ( 0 <=>; 433 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 19 ( 8 avg)
% 0.20/0.47 % Number of types : 2 ( 1 usr)
% 0.20/0.47 % Number of type conns : 115 ( 115 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 45 ( 43 usr; 2 con; 0-3 aty)
% 0.20/0.47 % Number of variables : 943 ( 87 ^; 802 !; 14 ?; 943 :)
% 0.20/0.47 % ( 40 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.47 % SPC : TH1_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:30.035
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (2)
% 0.20/0.47 thf(ty_t_Set_Oset,type,
% 0.20/0.47 set: $tType > $tType ).
% 0.20/0.47
% 0.20/0.47 thf(ty_t_Nat_Onat,type,
% 0.20/0.47 nat: $tType ).
% 0.20/0.47
% 0.20/0.47 % Explicit typings (42)
% 0.20/0.47 thf(sy_cl_HOL_Otype,type,
% 0.20/0.47 type:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oone,type,
% 0.20/0.47 one:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.47 zero:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.47 ord:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Num_Oneg__numeral,type,
% 0.20/0.47 neg_numeral:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.47 order:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.47 group_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Omonoid__add,type,
% 0.20/0.47 monoid_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.47 linorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.47 preorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.20/0.47 zero_neq_one:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.20/0.47 wellorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.20/0.47 semigroup_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.20/0.47 linordered_idom:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.20/0.47 comm_monoid_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.20/0.47 ab_semigroup_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.20/0.47 dense_linorder:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.20/0.47 cancel_semigroup_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.20/0.47 cancel1352612707id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.20/0.47 linord219039673up_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.20/0.47 ordere216010020id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.20/0.47 ordere779506340up_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.47 linord1659791738miring:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.47 canoni770627133id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 0.20/0.47 strict797366125id_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.20/0.47 ordere236663937imp_le:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.20/0.47 ordere223160158up_add:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.20/0.47 ordere516151231imp_le:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 0.20/0.47 semiri456707255roduct:
% 0.20/0.47 !>[A: $tType] : $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_OFpow,type,
% 0.20/0.47 finite_Fpow:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.20/0.47 finite_finite2:
% 0.20/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.20/0.47 one_one:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.20/0.47 plus_plus:
% 0.20/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.47 zero_zero:
% 0.20/0.47 !>[A: $tType] : A ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 0.20/0.47 neg_numeral_dbl_inc:
% 0.20/0.47 !>[A: $tType] : ( A > A ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 0.20/0.47 number2016821345itions: ( nat > nat ) > nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.47 ord_less:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.47 ord_less_eq:
% 0.20/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_Set_OCollect,type,
% 0.20/0.47 collect:
% 0.20/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member,type,
% 0.20/0.47 member:
% 0.20/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_n,type,
% 0.20/0.47 n: nat ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (253)
% 0.20/0.47 thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 0.20/0.47 ( finite_finite2 @ ( nat > nat )
% 0.20/0.47 @ ( collect @ ( nat > nat )
% 0.20/0.47 @ ^ [F: nat > nat] :
% 0.20/0.47 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
% 0.20/0.47 & ! [I: nat] :
% 0.20/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
% 0.20/0.47 => ( ( F @ I )
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
% 0.20/0.47 thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 0.20/0.47 ( ord_less_eq @ ( set @ ( nat > nat ) )
% 0.20/0.47 @ ( collect @ ( nat > nat )
% 0.20/0.47 @ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) )
% 0.20/0.47 @ ( collect @ ( nat > nat )
% 0.20/0.47 @ ^ [F: nat > nat] :
% 0.20/0.47 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
% 0.20/0.47 & ! [I: nat] :
% 0.20/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
% 0.20/0.47 => ( ( F @ I )
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>{p. p partitions n} \<subseteq> {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
% 0.20/0.47 thf(fact_2_finite__Collect__conjI,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.20/0.47 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.20/0.47 => ( finite_finite2 @ A
% 0.20/0.47 @ ( collect @ A
% 0.20/0.47 @ ^ [X: A] :
% 0.20/0.47 ( ( P2 @ X )
% 0.20/0.47 & ( Q @ X ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_Collect_conjI
% 0.20/0.47 thf(fact_3_finite__Collect__disjI,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ( finite_finite2 @ A
% 0.20/0.47 @ ( collect @ A
% 0.20/0.47 @ ^ [X: A] :
% 0.20/0.47 ( ( P2 @ X )
% 0.20/0.47 | ( Q @ X ) ) ) )
% 0.20/0.47 = ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.20/0.47 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_Collect_disjI
% 0.20/0.47 thf(fact_4_finite__code,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ( ( finite_finite2 @ A )
% 0.20/0.47 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_code
% 0.20/0.47 thf(fact_5_partitions__zero,axiom,
% 0.20/0.47 ! [P3: nat > nat] :
% 0.20/0.47 ( ( number2016821345itions @ P3 @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( P3
% 0.20/0.47 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % partitions_zero
% 0.20/0.47 thf(fact_6_partitions__bounds,axiom,
% 0.20/0.47 ! [P3: nat > nat,N: nat,I2: nat] :
% 0.20/0.47 ( ( number2016821345itions @ P3 @ N )
% 0.20/0.47 => ( ord_less_eq @ nat @ ( P3 @ I2 ) @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % partitions_bounds
% 0.20/0.47 thf(fact_7_finite__set__of__finite__funs,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,D: B] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ( finite_finite2 @ B @ B2 )
% 0.20/0.47 => ( finite_finite2 @ ( A > B )
% 0.20/0.47 @ ( collect @ ( A > B )
% 0.20/0.47 @ ^ [F: A > B] :
% 0.20/0.47 ! [X: A] :
% 0.20/0.47 ( ( ( member @ A @ X @ A3 )
% 0.20/0.47 => ( member @ B @ ( F @ X ) @ B2 ) )
% 0.20/0.47 & ( ~ ( member @ A @ X @ A3 )
% 0.20/0.47 => ( ( F @ X )
% 0.20/0.47 = D ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_set_of_finite_funs
% 0.20/0.47 thf(fact_8_not__finite__existsD,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o] :
% 0.20/0.47 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 0.20/0.47 => ? [X_1: A] : ( P2 @ X_1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_finite_existsD
% 0.20/0.47 thf(fact_9_pigeonhole__infinite__rel,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
% 0.20/0.47 ( ~ ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ( finite_finite2 @ B @ B2 )
% 0.20/0.47 => ( ! [X2: A] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ? [Xa: B] :
% 0.20/0.47 ( ( member @ B @ Xa @ B2 )
% 0.20/0.47 & ( R @ X2 @ Xa ) ) )
% 0.20/0.47 => ? [X2: B] :
% 0.20/0.47 ( ( member @ B @ X2 @ B2 )
% 0.20/0.47 & ~ ( finite_finite2 @ A
% 0.20/0.47 @ ( collect @ A
% 0.20/0.47 @ ^ [A4: A] :
% 0.20/0.47 ( ( member @ A @ A4 @ A3 )
% 0.20/0.47 & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % pigeonhole_infinite_rel
% 0.20/0.47 thf(fact_10_finite__Collect__subsets,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( finite_finite2 @ ( set @ A )
% 0.20/0.47 @ ( collect @ ( set @ A )
% 0.20/0.47 @ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_Collect_subsets
% 0.20/0.47 thf(fact_11_finite__set__choice,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ! [X2: A] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ? [X_12: B] : ( P2 @ X2 @ X_12 ) )
% 0.20/0.47 => ? [F2: A > B] :
% 0.20/0.47 ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A3 )
% 0.20/0.47 => ( P2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_set_choice
% 0.20/0.47 thf(fact_12_finite,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( finite_finite @ A )
% 0.20/0.47 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite
% 0.20/0.47 thf(fact_13_bound__domain__and__range__impl__finitely__many__functions,axiom,
% 0.20/0.47 ! [N: nat,M: nat] :
% 0.20/0.47 ( finite_finite2 @ ( nat > nat )
% 0.20/0.47 @ ( collect @ ( nat > nat )
% 0.20/0.47 @ ^ [F: nat > nat] :
% 0.20/0.47 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ N )
% 0.20/0.47 & ! [I: nat] :
% 0.20/0.47 ( ( ord_less_eq @ nat @ M @ I )
% 0.20/0.47 => ( ( F @ I )
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bound_domain_and_range_impl_finitely_many_functions
% 0.20/0.47 thf(fact_14_finite__Collect__le__nat,axiom,
% 0.20/0.47 ! [K: nat] :
% 0.20/0.47 ( finite_finite2 @ nat
% 0.20/0.47 @ ( collect @ nat
% 0.20/0.47 @ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_Collect_le_nat
% 0.20/0.47 thf(fact_15_finite__has__minimal2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A3: set @ A,A5: A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ( member @ A @ A5 @ A3 )
% 0.20/0.47 => ? [X2: A] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 & ( ord_less_eq @ A @ X2 @ A5 )
% 0.20/0.47 & ! [Xa: A] :
% 0.20/0.47 ( ( member @ A @ Xa @ A3 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 0.20/0.47 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_has_minimal2
% 0.20/0.47 thf(fact_16_finite__has__maximal2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A3: set @ A,A5: A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.20/0.47 => ( ( member @ A @ A5 @ A3 )
% 0.20/0.47 => ? [X2: A] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 & ( ord_less_eq @ A @ A5 @ X2 )
% 0.20/0.47 & ! [Xa: A] :
% 0.20/0.47 ( ( member @ A @ Xa @ A3 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ X2 @ Xa )
% 0.20/0.47 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_has_maximal2
% 0.20/0.47 thf(fact_17_rev__finite__subset,axiom,
% 0.20/0.47 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 0.20/0.47 ( ( finite_finite2 @ A @ B2 )
% 0.20/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.20/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % rev_finite_subset
% 0.20/0.47 thf(fact_18_infinite__super,axiom,
% 0.20/0.47 ! [A: $tType,S: set @ A,T: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ S @ T )
% 0.20/0.47 => ( ~ ( finite_finite2 @ A @ S )
% 0.20/0.47 => ~ ( finite_finite2 @ A @ T ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_super
% 0.20/0.47 thf(fact_19_finite__subset,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.20/0.47 => ( ( finite_finite2 @ A @ B2 )
% 0.20/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % finite_subset
% 0.20/0.47 thf(fact_20_add__le__same__cancel1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere516151231imp_le @ A )
% 0.20/0.47 => ! [B4: A,A5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
% 0.20/0.47 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_le_same_cancel1
% 0.20/0.47 thf(fact_21_add__le__same__cancel2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere516151231imp_le @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
% 0.20/0.47 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_le_same_cancel2
% 0.20/0.47 thf(fact_22_le__add__same__cancel1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere516151231imp_le @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_add_same_cancel1
% 0.20/0.47 thf(fact_23_le__add__same__cancel2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere516151231imp_le @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_add_same_cancel2
% 0.20/0.47 thf(fact_24_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % double_add_le_zero_iff_single_add_le_zero
% 0.20/0.47 thf(fact_25_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_le_double_add_iff_zero_le_single_add
% 0.20/0.47 thf(fact_26_nat__add__left__cancel__le,axiom,
% 0.20/0.47 ! [K: nat,M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.20/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_add_left_cancel_le
% 0.20/0.47 thf(fact_27_add__is__0,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( plus_plus @ nat @ M @ N )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( ( M
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 & ( N
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_is_0
% 0.20/0.47 thf(fact_28_Nat_Oadd__0__right,axiom,
% 0.20/0.47 ! [M: nat] :
% 0.20/0.47 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.20/0.47 = M ) ).
% 0.20/0.47
% 0.20/0.47 % Nat.add_0_right
% 0.20/0.47 thf(fact_29_le0,axiom,
% 0.20/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.20/0.47
% 0.20/0.47 % le0
% 0.20/0.47 thf(fact_30_bot__nat__0_Oextremum,axiom,
% 0.20/0.47 ! [A5: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A5 ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum
% 0.20/0.47 thf(fact_31_add__right__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [B4: A,A5: A,C: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B4 @ A5 )
% 0.20/0.47 = ( plus_plus @ A @ C @ A5 ) )
% 0.20/0.47 = ( B4 = C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_right_cancel
% 0.20/0.47 thf(fact_32_add__left__cancel,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [A5: A,B4: A,C: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A5 @ B4 )
% 0.20/0.47 = ( plus_plus @ A @ A5 @ C ) )
% 0.20/0.47 = ( B4 = C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_left_cancel
% 0.20/0.47 thf(fact_33_le__zero__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_zero_eq
% 0.20/0.47 thf(fact_34_add__le__cancel__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere236663937imp_le @ A )
% 0.20/0.47 => ! [A5: A,C: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_le_cancel_right
% 0.20/0.47 thf(fact_35_add__le__cancel__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordere236663937imp_le @ A )
% 0.20/0.47 => ! [C: A,A5: A,B4: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_le_cancel_left
% 0.20/0.47 thf(fact_36_zero__eq__add__iff__both__eq__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [X4: A,Y: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( plus_plus @ A @ X4 @ Y ) )
% 0.20/0.47 = ( ( X4
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 & ( Y
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_eq_add_iff_both_eq_0
% 0.20/0.47 thf(fact_37_add__eq__0__iff__both__eq__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [X4: A,Y: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ X4 @ Y )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( ( X4
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 & ( Y
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_eq_0_iff_both_eq_0
% 0.20/0.47 thf(fact_38_add__cancel__right__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel1352612707id_add @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( plus_plus @ A @ A5 @ B4 ) )
% 0.20/0.47 = ( B4
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_cancel_right_right
% 0.20/0.47 thf(fact_39_add__cancel__right__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel1352612707id_add @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( A5
% 0.20/0.47 = ( plus_plus @ A @ B4 @ A5 ) )
% 0.20/0.47 = ( B4
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_cancel_right_left
% 0.20/0.47 thf(fact_40_add__cancel__left__right,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel1352612707id_add @ A )
% 0.20/0.47 => ! [A5: A,B4: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A5 @ B4 )
% 0.20/0.47 = A5 )
% 0.20/0.47 = ( B4
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_cancel_left_right
% 0.20/0.47 thf(fact_41_add__cancel__left__left,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel1352612707id_add @ A )
% 0.20/0.47 => ! [B4: A,A5: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B4 @ A5 )
% 0.20/0.47 = A5 )
% 0.20/0.47 = ( B4
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_cancel_left_left
% 0.20/0.47 thf(fact_42_double__zero__sym,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( plus_plus @ A @ A5 @ A5 ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % double_zero_sym
% 0.20/0.47 thf(fact_43_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,A5: A,P2: A > $o] :
% 0.20/0.47 ( ( member @ A @ A5 @ ( collect @ A @ P2 ) )
% 0.20/0.47 = ( P2 @ A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_44_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.20/0.47 = A3 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_45_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X2: A] :
% 0.20/0.47 ( ( P2 @ X2 )
% 0.20/0.47 = ( Q @ X2 ) )
% 0.20/0.47 => ( ( collect @ A @ P2 )
% 0.20/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_46_double__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A5 @ A5 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( A5
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % double_zero
% 0.20/0.47 thf(fact_47_add_Oright__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( monoid_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 0.20/0.47 = A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.right_neutral
% 0.20/0.47 thf(fact_48_add_Oleft__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( monoid_add @ A )
% 0.20/0.47 => ! [A5: A] :
% 0.20/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 0.20/0.47 = A5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.left_neutral
% 0.20/0.47 thf(fact_49_zero__reorient,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( zero @ A )
% 0.20/0.47 => ! [X4: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = X4 )
% 0.20/0.47 = ( X4
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_50_add__right__imp__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [B4: A,A5: A,C: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ B4 @ A5 )
% 0.20/0.47 = ( plus_plus @ A @ C @ A5 ) )
% 0.20/0.47 => ( B4 = C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_right_imp_eq
% 0.20/0.47 thf(fact_51_add__left__imp__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( cancel_semigroup_add @ A )
% 0.20/0.47 => ! [A5: A,B4: A,C: A] :
% 0.20/0.47 ( ( ( plus_plus @ A @ A5 @ B4 )
% 0.20/0.47 = ( plus_plus @ A @ A5 @ C ) )
% 0.20/0.47 => ( B4 = C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add_left_imp_eq
% 0.20/0.47 thf(fact_52_add_Oleft__commute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A )
% 0.20/0.47 => ! [B4: A,A5: A,C: A] :
% 0.20/0.47 ( ( plus_plus @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) )
% 0.20/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.left_commute
% 0.20/0.47 thf(fact_53_add_Ocommute,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ( ( plus_plus @ A )
% 0.31/0.47 = ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.commute
% 0.31/0.47 thf(fact_54_add_Oright__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [B4: A,A5: A,C: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ B4 @ A5 )
% 0.31/0.47 = ( plus_plus @ A @ C @ A5 ) )
% 0.31/0.47 = ( B4 = C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.right_cancel
% 0.31/0.47 thf(fact_55_add_Oleft__cancel,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ( plus_plus @ A @ A5 @ B4 )
% 0.31/0.47 = ( plus_plus @ A @ A5 @ C ) )
% 0.31/0.47 = ( B4 = C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.left_cancel
% 0.31/0.47 thf(fact_56_add_Oassoc,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semigroup_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 0.31/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.assoc
% 0.31/0.47 thf(fact_57_group__cancel_Oadd2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [B2: A,K: A,B4: A,A5: A] :
% 0.31/0.47 ( ( B2
% 0.31/0.47 = ( plus_plus @ A @ K @ B4 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A5 @ B2 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add2
% 0.31/0.47 thf(fact_58_group__cancel_Oadd1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [A3: A,K: A,A5: A,B4: A] :
% 0.31/0.47 ( ( A3
% 0.31/0.47 = ( plus_plus @ A @ K @ A5 ) )
% 0.31/0.47 => ( ( plus_plus @ A @ A3 @ B4 )
% 0.31/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % group_cancel.add1
% 0.31/0.47 thf(fact_59_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( I2 = J )
% 0.31/0.47 & ( K = L ) )
% 0.31/0.47 => ( ( plus_plus @ A @ I2 @ K )
% 0.31/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(4)
% 0.31/0.47 thf(fact_60_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ab_semigroup_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 0.31/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.31/0.47 thf(fact_61_one__reorient,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( one @ A )
% 0.31/0.47 => ! [X4: A] :
% 0.31/0.47 ( ( ( one_one @ A )
% 0.31/0.47 = X4 )
% 0.31/0.47 = ( X4
% 0.31/0.47 = ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % one_reorient
% 0.31/0.47 thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
% 0.31/0.47 ! [P2: nat > $o,K: nat,B4: nat] :
% 0.31/0.47 ( ( P2 @ K )
% 0.31/0.47 => ( ! [Y2: nat] :
% 0.31/0.47 ( ( P2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y2 @ B4 ) )
% 0.31/0.47 => ? [X2: nat] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 & ! [Y3: nat] :
% 0.31/0.47 ( ( P2 @ Y3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Nat.ex_has_greatest_nat
% 0.31/0.47 thf(fact_63_nat__le__linear,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_le_linear
% 0.31/0.47 thf(fact_64_le__antisym,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.31/0.47 => ( M = N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_antisym
% 0.31/0.47 thf(fact_65_eq__imp__le,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( M = N )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_imp_le
% 0.31/0.47 thf(fact_66_le__trans,axiom,
% 0.31/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.31/0.47 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_trans
% 0.31/0.47 thf(fact_67_le__refl,axiom,
% 0.31/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.31/0.47
% 0.31/0.47 % le_refl
% 0.31/0.47 thf(fact_68_zero__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le
% 0.31/0.47 thf(fact_69_add__le__imp__le__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [A5: A,C: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_imp_le_right
% 0.31/0.47 thf(fact_70_add__le__imp__le__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [C: A,A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_imp_le_left
% 0.31/0.47 thf(fact_71_le__iff__add,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [A4: A,B5: A] :
% 0.31/0.47 ? [C2: A] :
% 0.31/0.47 ( B5
% 0.31/0.47 = ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_iff_add
% 0.31/0.47 thf(fact_72_add__right__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_right_mono
% 0.31/0.47 thf(fact_73_less__eqE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ~ ! [C3: A] :
% 0.31/0.47 ( B4
% 0.31/0.47 != ( plus_plus @ A @ A5 @ C3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eqE
% 0.31/0.47 thf(fact_74_add__left__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_left_mono
% 0.31/0.47 thf(fact_75_add__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A,D: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono
% 0.31/0.47 thf(fact_76_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.31/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(1)
% 0.31/0.47 thf(fact_77_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( I2 = J )
% 0.31/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(2)
% 0.31/0.47 thf(fact_78_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere779506340up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.31/0.47 & ( K = L ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_semiring(3)
% 0.31/0.47 thf(fact_79_add_Ogroup__left__neutral,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( group_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 = A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.group_left_neutral
% 0.31/0.47 thf(fact_80_add_Ocomm__neutral,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 0.31/0.47 = A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % add.comm_neutral
% 0.31/0.47 thf(fact_81_comm__monoid__add__class_Oadd__0,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 = A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % comm_monoid_add_class.add_0
% 0.31/0.47 thf(fact_82_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.31/0.47 ! [A5: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
% 0.31/0.47 => ( A5
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum_uniqueI
% 0.31/0.47 thf(fact_83_bot__nat__0_Oextremum__unique,axiom,
% 0.31/0.47 ! [A5: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
% 0.31/0.47 = ( A5
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum_unique
% 0.31/0.47 thf(fact_84_le__0__eq,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_0_eq
% 0.31/0.47 thf(fact_85_less__eq__nat_Osimps_I1_J,axiom,
% 0.31/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_nat.simps(1)
% 0.31/0.47 thf(fact_86_add__eq__self__zero,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ( plus_plus @ nat @ M @ N )
% 0.31/0.47 = M )
% 0.31/0.47 => ( N
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_eq_self_zero
% 0.31/0.47 thf(fact_87_plus__nat_Oadd__0,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.31/0.47 = N ) ).
% 0.31/0.47
% 0.31/0.47 % plus_nat.add_0
% 0.31/0.47 thf(fact_88_nat__le__iff__add,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.31/0.47 ? [K2: nat] :
% 0.31/0.47 ( N2
% 0.31/0.47 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_le_iff_add
% 0.31/0.47 thf(fact_89_trans__le__add2,axiom,
% 0.31/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trans_le_add2
% 0.31/0.47 thf(fact_90_trans__le__add1,axiom,
% 0.31/0.47 ! [I2: nat,J: nat,M: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % trans_le_add1
% 0.31/0.47 thf(fact_91_add__le__mono1,axiom,
% 0.31/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_mono1
% 0.31/0.47 thf(fact_92_add__le__mono,axiom,
% 0.31/0.47 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ K @ L )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_mono
% 0.31/0.47 thf(fact_93_le__Suc__ex,axiom,
% 0.31/0.47 ! [K: nat,L: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ K @ L )
% 0.31/0.47 => ? [N3: nat] :
% 0.31/0.47 ( L
% 0.31/0.47 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_Suc_ex
% 0.31/0.47 thf(fact_94_add__leD2,axiom,
% 0.31/0.47 ! [M: nat,K: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.31/0.47 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leD2
% 0.31/0.47 thf(fact_95_add__leD1,axiom,
% 0.31/0.47 ! [M: nat,K: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leD1
% 0.31/0.47 thf(fact_96_le__add2,axiom,
% 0.31/0.47 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_add2
% 0.31/0.47 thf(fact_97_le__add1,axiom,
% 0.31/0.47 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_add1
% 0.31/0.47 thf(fact_98_add__leE,axiom,
% 0.31/0.47 ! [M: nat,K: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.31/0.47 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_leE
% 0.31/0.47 thf(fact_99_add__nonpos__eq__0__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ( plus_plus @ A @ X4 @ Y )
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 = ( ( X4
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 & ( Y
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_nonpos_eq_0_iff
% 0.31/0.47 thf(fact_100_add__nonneg__eq__0__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.31/0.47 => ( ( ( plus_plus @ A @ X4 @ Y )
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 = ( ( X4
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 & ( Y
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_nonneg_eq_0_iff
% 0.31/0.47 thf(fact_101_add__nonpos__nonpos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_nonpos_nonpos
% 0.31/0.47 thf(fact_102_add__nonneg__nonneg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_nonneg_nonneg
% 0.31/0.47 thf(fact_103_add__increasing2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [C: A,B4: A,A5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.31/0.47 => ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_increasing2
% 0.31/0.47 thf(fact_104_add__decreasing2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [C: A,A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_decreasing2
% 0.31/0.47 thf(fact_105_add__increasing,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_increasing
% 0.31/0.47 thf(fact_106_add__decreasing,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,C: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_decreasing
% 0.31/0.47 thf(fact_107_sum_Ofinite__Collect__op,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( comm_monoid_add @ A )
% 0.31/0.47 => ! [I3: set @ B,X4: B > A,Y: B > A] :
% 0.31/0.47 ( ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [I: B] :
% 0.31/0.47 ( ( member @ B @ I @ I3 )
% 0.31/0.47 & ( ( X4 @ I )
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.31/0.47 => ( ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [I: B] :
% 0.31/0.47 ( ( member @ B @ I @ I3 )
% 0.31/0.47 & ( ( Y @ I )
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.31/0.47 => ( finite_finite2 @ B
% 0.31/0.47 @ ( collect @ B
% 0.31/0.47 @ ^ [I: B] :
% 0.31/0.47 ( ( member @ B @ I @ I3 )
% 0.31/0.47 & ( ( plus_plus @ A @ ( X4 @ I ) @ ( Y @ I ) )
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sum.finite_Collect_op
% 0.31/0.47 thf(fact_108_not__one__le__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_one_le_zero
% 0.31/0.47 thf(fact_109_zero__le__one,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le_one
% 0.31/0.47 thf(fact_110_finite__less__ub,axiom,
% 0.31/0.47 ! [F3: nat > nat,U: nat] :
% 0.31/0.47 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
% 0.31/0.47 => ( finite_finite2 @ nat
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_less_ub
% 0.31/0.47 thf(fact_111_subset__antisym,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.31/0.47 => ( A3 = B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_antisym
% 0.31/0.47 thf(fact_112_subsetI,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A3 )
% 0.31/0.47 => ( member @ A @ X2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetI
% 0.31/0.47 thf(fact_113_order__refl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_refl
% 0.31/0.47 thf(fact_114_finite__nat__set__iff__bounded__le,axiom,
% 0.31/0.47 ( ( finite_finite2 @ nat )
% 0.31/0.47 = ( ^ [N4: set @ nat] :
% 0.31/0.47 ? [M2: nat] :
% 0.31/0.47 ! [X: nat] :
% 0.31/0.47 ( ( member @ nat @ X @ N4 )
% 0.31/0.47 => ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_nat_set_iff_bounded_le
% 0.31/0.47 thf(fact_115_Euclid__induct,axiom,
% 0.31/0.47 ! [P2: nat > nat > $o,A5: nat,B4: nat] :
% 0.31/0.47 ( ! [A6: nat,B6: nat] :
% 0.31/0.47 ( ( P2 @ A6 @ B6 )
% 0.31/0.47 = ( P2 @ B6 @ A6 ) )
% 0.31/0.47 => ( ! [A6: nat] : ( P2 @ A6 @ ( zero_zero @ nat ) )
% 0.31/0.47 => ( ! [A6: nat,B6: nat] :
% 0.31/0.47 ( ( P2 @ A6 @ B6 )
% 0.31/0.47 => ( P2 @ A6 @ ( plus_plus @ nat @ A6 @ B6 ) ) )
% 0.31/0.47 => ( P2 @ A5 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Euclid_induct
% 0.31/0.47 thf(fact_116_le__funD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B )
% 0.31/0.47 => ! [F3: A > B,G: A > B,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funD
% 0.31/0.47 thf(fact_117_le__funE,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B )
% 0.31/0.47 => ! [F3: A > B,G: A > B,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funE
% 0.31/0.47 thf(fact_118_le__funI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B )
% 0.31/0.47 => ! [F3: A > B,G: A > B] :
% 0.31/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funI
% 0.31/0.47 thf(fact_119_le__fun__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B )
% 0.31/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.31/0.47 = ( ^ [F: A > B,G2: A > B] :
% 0.31/0.47 ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_fun_def
% 0.31/0.47 thf(fact_120_order__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.31/0.47 => ( ! [X2: B,Y2: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_121_order__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C4: $tType] :
% 0.31/0.47 ( ( ( order @ C4 )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_122_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B )
% 0.31/0.47 & ( ord @ A ) )
% 0.31/0.47 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 0.31/0.47 ( ( A5
% 0.31/0.47 = ( F3 @ B4 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.31/0.47 => ( ! [X2: B,Y2: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_123_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B )
% 0.31/0.47 & ( ord @ A ) )
% 0.31/0.47 => ! [A5: A,B4: A,F3: A > B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ( F3 @ B4 )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X2: A,Y2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_124_eq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [X: A,Y5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.31/0.47 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_iff
% 0.31/0.47 thf(fact_125_antisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y @ X4 )
% 0.31/0.47 => ( X4 = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym
% 0.31/0.47 thf(fact_126_linear,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 | ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linear
% 0.31/0.47 thf(fact_127_eq__refl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( X4 = Y )
% 0.31/0.47 => ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_refl
% 0.31/0.47 thf(fact_128_le__cases,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ~ ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases
% 0.31/0.47 thf(fact_129_order_Otrans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.trans
% 0.31/0.47 thf(fact_130_le__cases3,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Y @ X4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Y @ X4 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 0.31/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases3
% 0.31/0.47 thf(fact_131_antisym__conv,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [Y: A,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ Y @ X4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 = ( X4 = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv
% 0.31/0.47 thf(fact_132_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A4: A,B5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A4 @ B5 )
% 0.31/0.47 & ( ord_less_eq @ A @ B5 @ A4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.eq_iff
% 0.31/0.47 thf(fact_133_ord__eq__le__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( A5 = B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_trans
% 0.31/0.47 thf(fact_134_ord__le__eq__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( B4 = C )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_trans
% 0.31/0.47 thf(fact_135_order__class_Oorder_Oantisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.31/0.47 => ( A5 = B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.antisym
% 0.31/0.47 thf(fact_136_order__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.31/0.47 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_trans
% 0.31/0.47 thf(fact_137_dual__order_Orefl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.refl
% 0.31/0.47 thf(fact_138_linorder__wlog,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [P2: A > A > $o,A5: A,B4: A] :
% 0.31/0.47 ( ! [A6: A,B6: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.31/0.47 => ( P2 @ A6 @ B6 ) )
% 0.31/0.47 => ( ! [A6: A,B6: A] :
% 0.31/0.47 ( ( P2 @ B6 @ A6 )
% 0.31/0.47 => ( P2 @ A6 @ B6 ) )
% 0.31/0.47 => ( P2 @ A5 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_wlog
% 0.31/0.47 thf(fact_139_dual__order_Otrans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [B4: A,A5: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ C @ A5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.trans
% 0.31/0.47 thf(fact_140_dual__order_Oeq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A4: A,B5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B5 @ A4 )
% 0.31/0.47 & ( ord_less_eq @ A @ A4 @ B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.eq_iff
% 0.31/0.47 thf(fact_141_dual__order_Oantisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [B4: A,A5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( A5 = B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.antisym
% 0.31/0.47 thf(fact_142_in__mono,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.31/0.47 => ( ( member @ A @ X4 @ A3 )
% 0.31/0.47 => ( member @ A @ X4 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_mono
% 0.31/0.47 thf(fact_143_subsetD,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.31/0.47 => ( ( member @ A @ C @ A3 )
% 0.31/0.47 => ( member @ A @ C @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetD
% 0.31/0.47 thf(fact_144_equalityE,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.31/0.47 ( ( A3 = B2 )
% 0.31/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.31/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityE
% 0.31/0.47 thf(fact_145_subset__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.47 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.31/0.47 ! [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ A2 )
% 0.31/0.47 => ( member @ A @ X @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_eq
% 0.31/0.47 thf(fact_146_equalityD1,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.31/0.47 ( ( A3 = B2 )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityD1
% 0.31/0.47 thf(fact_147_equalityD2,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.31/0.47 ( ( A3 = B2 )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % equalityD2
% 0.31/0.47 thf(fact_148_subset__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.47 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.31/0.47 ! [T2: A] :
% 0.31/0.47 ( ( member @ A @ T2 @ A2 )
% 0.31/0.47 => ( member @ A @ T2 @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_iff
% 0.31/0.47 thf(fact_149_subset__refl,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.31/0.47
% 0.31/0.47 % subset_refl
% 0.31/0.47 thf(fact_150_Collect__mono,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 => ( Q @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono
% 0.31/0.47 thf(fact_151_subset__trans,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B2: set @ A,C5: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_trans
% 0.31/0.47 thf(fact_152_set__eq__subset,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.31/0.47 & ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % set_eq_subset
% 0.31/0.47 thf(fact_153_Collect__mono__iff,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.31/0.47 = ( ! [X: A] :
% 0.31/0.47 ( ( P2 @ X )
% 0.31/0.47 => ( Q @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono_iff
% 0.31/0.47 thf(fact_154_bounded__Max__nat,axiom,
% 0.31/0.47 ! [P2: nat > $o,X4: nat,M3: nat] :
% 0.31/0.47 ( ( P2 @ X4 )
% 0.31/0.47 => ( ! [X2: nat] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 => ( ord_less_eq @ nat @ X2 @ M3 ) )
% 0.31/0.47 => ~ ! [M4: nat] :
% 0.31/0.47 ( ( P2 @ M4 )
% 0.31/0.47 => ~ ! [X3: nat] :
% 0.31/0.47 ( ( P2 @ X3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ X3 @ M4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bounded_Max_nat
% 0.31/0.47 thf(fact_155_Collect__subset,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,P2: A > $o] :
% 0.31/0.47 ( ord_less_eq @ ( set @ A )
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ A3 )
% 0.31/0.47 & ( P2 @ X ) ) )
% 0.31/0.47 @ A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_subset
% 0.31/0.47 thf(fact_156_less__eq__set__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.47 = ( ^ [A2: set @ A,B3: set @ A] :
% 0.31/0.47 ( ord_less_eq @ ( A > $o )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ A2 )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_set_def
% 0.31/0.47 thf(fact_157_zero__neq__one,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A )
% 0.31/0.47 => ( ( zero_zero @ A )
% 0.31/0.47 != ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_neq_one
% 0.31/0.47 thf(fact_158_le__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(4)
% 0.31/0.47 thf(fact_159_verit__sum__simplify,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( cancel1352612707id_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 0.31/0.47 = A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % verit_sum_simplify
% 0.31/0.47 thf(fact_160_add__0__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semiri456707255roduct @ A )
% 0.31/0.47 => ! [B4: A,A5: A] :
% 0.31/0.47 ( ( B4
% 0.31/0.47 = ( plus_plus @ A @ B4 @ A5 ) )
% 0.31/0.47 = ( A5
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_0_iff
% 0.31/0.47 thf(fact_161_le__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(3)
% 0.31/0.47 thf(fact_162_predicate1I,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 => ( Q @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 0.31/0.47
% 0.31/0.47 % predicate1I
% 0.31/0.47 thf(fact_163_predicate1D,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.31/0.47 => ( ( P2 @ X4 )
% 0.31/0.47 => ( Q @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % predicate1D
% 0.31/0.47 thf(fact_164_rev__predicate1D,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,X4: A,Q: A > $o] :
% 0.31/0.47 ( ( P2 @ X4 )
% 0.31/0.47 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 0.31/0.47 => ( Q @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % rev_predicate1D
% 0.31/0.47 thf(fact_165_verit__la__disequality,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( A5 = B4 )
% 0.31/0.47 | ~ ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 | ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % verit_la_disequality
% 0.31/0.47 thf(fact_166_is__num__normalize_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( neg_numeral @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 0.31/0.47 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % is_num_normalize(1)
% 0.31/0.47 thf(fact_167_pred__subset__eq,axiom,
% 0.31/0.47 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > $o )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ R )
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ S ) )
% 0.31/0.47 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 0.31/0.47
% 0.31/0.47 % pred_subset_eq
% 0.31/0.47 thf(fact_168_dbl__inc__simps_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( neg_numeral @ A )
% 0.31/0.47 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 0.31/0.47 = ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dbl_inc_simps(2)
% 0.31/0.47 thf(fact_169_conj__subset__def,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( P2 @ X )
% 0.31/0.47 & ( Q @ X ) ) ) )
% 0.31/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P2 ) )
% 0.31/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % conj_subset_def
% 0.31/0.47 thf(fact_170_dbl__inc__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( neg_numeral @ A )
% 0.31/0.47 => ( ( neg_numeral_dbl_inc @ A )
% 0.31/0.47 = ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dbl_inc_def
% 0.31/0.47 thf(fact_171_subset__Collect__iff,axiom,
% 0.31/0.47 ! [A: $tType,B2: set @ A,A3: set @ A,P2: A > $o] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ A3 )
% 0.31/0.47 & ( P2 @ X ) ) ) )
% 0.31/0.47 = ( ! [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ B2 )
% 0.31/0.47 => ( P2 @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_Collect_iff
% 0.31/0.47 thf(fact_172_subset__CollectI,axiom,
% 0.31/0.47 ! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P2: A > $o] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ B2 )
% 0.31/0.47 => ( ( Q @ X2 )
% 0.31/0.47 => ( P2 @ X2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A )
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ B2 )
% 0.31/0.47 & ( Q @ X ) ) )
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ A3 )
% 0.31/0.47 & ( P2 @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_CollectI
% 0.31/0.47 thf(fact_173_Collect__restrict,axiom,
% 0.31/0.47 ! [A: $tType,X5: set @ A,P2: A > $o] :
% 0.31/0.47 ( ord_less_eq @ ( set @ A )
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ X5 )
% 0.31/0.47 & ( P2 @ X ) ) )
% 0.31/0.47 @ X5 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_restrict
% 0.31/0.47 thf(fact_174_prop__restrict,axiom,
% 0.31/0.47 ! [A: $tType,X4: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
% 0.31/0.47 ( ( member @ A @ X4 @ Z3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 0.31/0.47 @ ( collect @ A
% 0.31/0.47 @ ^ [X: A] :
% 0.31/0.47 ( ( member @ A @ X @ X5 )
% 0.31/0.47 & ( P2 @ X ) ) ) )
% 0.31/0.47 => ( P2 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % prop_restrict
% 0.31/0.47 thf(fact_175_fun__cong__unused__0,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType,C4: $tType] :
% 0.31/0.47 ( ( zero @ B )
% 0.31/0.47 => ! [F3: ( A > B ) > C4,G: C4] :
% 0.31/0.47 ( ( F3
% 0.31/0.47 = ( ^ [X: A > B] : G ) )
% 0.31/0.47 => ( ( F3
% 0.31/0.47 @ ^ [X: A] : ( zero_zero @ B ) )
% 0.31/0.47 = G ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % fun_cong_unused_0
% 0.31/0.47 thf(fact_176_Fpow__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_Fpow @ A )
% 0.31/0.47 = ( ^ [A2: set @ A] :
% 0.31/0.47 ( collect @ ( set @ A )
% 0.31/0.47 @ ^ [X6: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
% 0.31/0.47 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Fpow_def
% 0.31/0.47 thf(fact_177_partitions__imp__finite__elements,axiom,
% 0.31/0.47 ! [P3: nat > nat,N: nat] :
% 0.31/0.47 ( ( number2016821345itions @ P3 @ N )
% 0.31/0.47 => ( finite_finite2 @ nat
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [I: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P3 @ I ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % partitions_imp_finite_elements
% 0.31/0.47 thf(fact_178_not__gr__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_gr_zero
% 0.31/0.47 thf(fact_179_add__less__cancel__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [A5: A,C: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 0.31/0.47 = ( ord_less @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_cancel_right
% 0.31/0.47 thf(fact_180_add__less__cancel__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere236663937imp_le @ A )
% 0.31/0.47 => ! [C: A,A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 0.31/0.47 = ( ord_less @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_cancel_left
% 0.31/0.47 thf(fact_181_bot__nat__0_Onot__eq__extremum,axiom,
% 0.31/0.47 ! [A5: nat] :
% 0.31/0.47 ( ( A5
% 0.31/0.47 != ( zero_zero @ nat ) )
% 0.31/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.not_eq_extremum
% 0.31/0.47 thf(fact_182_less__nat__zero__code,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_nat_zero_code
% 0.31/0.47 thf(fact_183_neq0__conv,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ nat ) )
% 0.31/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq0_conv
% 0.31/0.47 thf(fact_184_nat__add__left__cancel__less,axiom,
% 0.31/0.47 ! [K: nat,M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.31/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_add_left_cancel_less
% 0.31/0.47 thf(fact_185_finite__Collect__less__nat,axiom,
% 0.31/0.47 ! [K: nat] :
% 0.31/0.47 ( finite_finite2 @ nat
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_Collect_less_nat
% 0.31/0.47 thf(fact_186_add__less__same__cancel1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere516151231imp_le @ A )
% 0.31/0.47 => ! [B4: A,A5: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
% 0.31/0.47 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_same_cancel1
% 0.31/0.47 thf(fact_187_add__less__same__cancel2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere516151231imp_le @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
% 0.31/0.47 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_same_cancel2
% 0.31/0.47 thf(fact_188_less__add__same__cancel1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere516151231imp_le @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_add_same_cancel1
% 0.31/0.47 thf(fact_189_less__add__same__cancel2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere516151231imp_le @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_add_same_cancel2
% 0.31/0.47 thf(fact_190_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % double_add_less_zero_iff_single_add_less_zero
% 0.31/0.47 thf(fact_191_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_double_add_iff_zero_less_single_add
% 0.31/0.47 thf(fact_192_add__gr__0,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 0.31/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.31/0.47 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_gr_0
% 0.31/0.47 thf(fact_193_less__one,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_one
% 0.31/0.47 thf(fact_194_add__less__le__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere223160158up_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A,D: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.31/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_le_mono
% 0.31/0.47 thf(fact_195_add__le__less__mono,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere223160158up_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A,D: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less @ A @ C @ D )
% 0.31/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_le_less_mono
% 0.31/0.47 thf(fact_196_add__mono__thms__linordered__field_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere223160158up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less @ A @ I2 @ J )
% 0.31/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_field(3)
% 0.31/0.47 thf(fact_197_add__mono__thms__linordered__field_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere223160158up_add @ A )
% 0.31/0.47 => ! [I2: A,J: A,K: A,L: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.31/0.47 & ( ord_less @ A @ K @ L ) )
% 0.31/0.47 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_mono_thms_linordered_field(4)
% 0.31/0.47 thf(fact_198_pos__add__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( strict797366125id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 => ( ( ord_less @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pos_add_strict
% 0.31/0.47 thf(fact_199_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.31/0.47 => ~ ! [C3: A] :
% 0.31/0.47 ( ( B4
% 0.31/0.47 = ( plus_plus @ A @ A5 @ C3 ) )
% 0.31/0.47 => ( C3
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % canonically_ordered_monoid_add_class.lessE
% 0.31/0.47 thf(fact_200_add__pos__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 0.31/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_pos_pos
% 0.31/0.47 thf(fact_201_add__neg__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordere216010020id_add @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_neg_neg
% 0.31/0.47 thf(fact_202_add__less__zeroD,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ X4 @ Y ) @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less @ A @ X4 @ ( zero_zero @ A ) )
% 0.31/0.47 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % add_less_zeroD
% 0.31/0.47 thf(fact_203_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( A5 != B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.not_eq_order_implies_strict
% 0.31/0.47 thf(fact_204_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [B4: A,A5: A] :
% 0.31/0.47 ( ( ord_less @ A @ B4 @ A5 )
% 0.31/0.47 => ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_implies_order
% 0.31/0.47 thf(fact_205_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [B5: A,A4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B5 @ A4 )
% 0.31/0.47 & ( A4 != B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_iff_order
% 0.31/0.47 thf(fact_206_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [B5: A,A4: A] :
% 0.31/0.47 ( ( ord_less @ A @ B5 @ A4 )
% 0.31/0.47 | ( A4 = B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.order_iff_strict
% 0.31/0.47 thf(fact_207_order_Ostrict__implies__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_implies_order
% 0.31/0.47 thf(fact_208_dense__le__bounded,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 => ( ! [W: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ W )
% 0.31/0.47 => ( ( ord_less @ A @ W @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_le_bounded
% 0.31/0.47 thf(fact_209_dense__ge__bounded,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A )
% 0.31/0.47 => ! [Z2: A,X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ X4 )
% 0.31/0.47 => ( ! [W: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ W )
% 0.31/0.47 => ( ( ord_less @ A @ W @ X4 )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_ge_bounded
% 0.31/0.47 thf(fact_210_dual__order_Ostrict__trans2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [B4: A,A5: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ B4 @ A5 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ B4 )
% 0.31/0.47 => ( ord_less @ A @ C @ A5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans2
% 0.31/0.47 thf(fact_211_dual__order_Ostrict__trans1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [B4: A,A5: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.31/0.47 => ( ( ord_less @ A @ C @ B4 )
% 0.31/0.47 => ( ord_less @ A @ C @ A5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans1
% 0.31/0.47 thf(fact_212_order_Ostrict__iff__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [A4: A,B5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A4 @ B5 )
% 0.31/0.47 & ( A4 != B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_iff_order
% 0.31/0.47 thf(fact_213_order_Oorder__iff__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [A4: A,B5: A] :
% 0.31/0.47 ( ( ord_less @ A @ A4 @ B5 )
% 0.31/0.47 | ( A4 = B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.order_iff_strict
% 0.31/0.47 thf(fact_214_order_Ostrict__trans2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less @ A @ A5 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans2
% 0.31/0.47 thf(fact_215_order_Ostrict__trans1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less @ A @ B4 @ C )
% 0.31/0.47 => ( ord_less @ A @ A5 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans1
% 0.31/0.47 thf(fact_216_not__le__imp__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [Y: A,X4: A] :
% 0.31/0.47 ( ~ ( ord_less_eq @ A @ Y @ X4 )
% 0.31/0.47 => ( ord_less @ A @ X4 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_le_imp_less
% 0.31/0.47 thf(fact_217_less__le__not__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [X: A,Y5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.31/0.47 & ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le_not_le
% 0.31/0.47 thf(fact_218_le__imp__less__or__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 | ( X4 = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_imp_less_or_eq
% 0.31/0.47 thf(fact_219_le__less__linear,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 | ( ord_less @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less_linear
% 0.31/0.47 thf(fact_220_dense__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A )
% 0.31/0.47 => ! [Y: A,Z2: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_le
% 0.31/0.47 thf(fact_221_dense__ge,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A )
% 0.31/0.47 => ! [Z2: A,Y: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ X2 )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_ge
% 0.31/0.47 thf(fact_222_less__le__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.31/0.47 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le_trans
% 0.31/0.47 thf(fact_223_le__less__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A,Z2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less @ A @ Y @ Z2 )
% 0.31/0.47 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less_trans
% 0.31/0.47 thf(fact_224_less__imp__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_le
% 0.31/0.47 thf(fact_225_antisym__conv2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ~ ( ord_less @ A @ X4 @ Y ) )
% 0.31/0.47 = ( X4 = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv2
% 0.31/0.47 thf(fact_226_antisym__conv1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X4 @ Y )
% 0.31/0.47 = ( X4 = Y ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv1
% 0.31/0.47 thf(fact_227_le__neq__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A,B4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( A5 != B4 )
% 0.31/0.47 => ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_neq_trans
% 0.31/0.47 thf(fact_228_not__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ X4 @ Y ) )
% 0.31/0.47 = ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less
% 0.31/0.47 thf(fact_229_not__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ( ~ ( ord_less_eq @ A @ X4 @ Y ) )
% 0.31/0.47 = ( ord_less @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_le
% 0.31/0.47 thf(fact_230_order__less__le__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C4: $tType] :
% 0.31/0.47 ( ( ( order @ C4 )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst2
% 0.31/0.47 thf(fact_231_order__less__le__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ ( F3 @ B4 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B4 @ C )
% 0.31/0.47 => ( ! [X2: B,Y2: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst1
% 0.31/0.47 thf(fact_232_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C4: $tType] :
% 0.31/0.47 ( ( ( order @ C4 )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.31/0.47 => ( ( ord_less @ C4 @ ( F3 @ B4 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_233_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B )
% 0.31/0.47 & ( order @ A ) )
% 0.31/0.47 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
% 0.31/0.47 => ( ( ord_less @ B @ B4 @ C )
% 0.31/0.47 => ( ! [X2: B,Y2: B] :
% 0.31/0.47 ( ( ord_less @ B @ X2 @ Y2 )
% 0.31/0.47 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.31/0.47 => ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_234_less__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [X: A,Y5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.31/0.47 & ( X != Y5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le
% 0.31/0.47 thf(fact_235_le__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [X: A,Y5: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y5 )
% 0.31/0.47 | ( X = Y5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less
% 0.31/0.47 thf(fact_236_leI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A )
% 0.31/0.47 => ! [X4: A,Y: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ X4 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % leI
% 0.31/0.47 thf(fact_237_leD,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [Y: A,X4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ Y @ X4 )
% 0.31/0.47 => ~ ( ord_less @ A @ X4 @ Y ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % leD
% 0.31/0.47 thf(fact_238_less__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(4)
% 0.31/0.47 thf(fact_239_nat__less__le,axiom,
% 0.31/0.47 ( ( ord_less @ nat )
% 0.31/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.47 & ( M2 != N2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_less_le
% 0.31/0.47 thf(fact_240_less__imp__le__nat,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_le_nat
% 0.31/0.47 thf(fact_241_le__eq__less__or__eq,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.31/0.47 | ( M2 = N2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_eq_less_or_eq
% 0.31/0.47 thf(fact_242_less__or__eq__imp__le,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 | ( M = N ) )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_or_eq_imp_le
% 0.31/0.47 thf(fact_243_le__neq__implies__less,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 => ( ( M != N )
% 0.31/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_neq_implies_less
% 0.31/0.47 thf(fact_244_less__mono__imp__le__mono,axiom,
% 0.31/0.47 ! [F3: nat > nat,I2: nat,J: nat] :
% 0.31/0.47 ( ! [I4: nat,J2: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ I4 @ J2 )
% 0.31/0.47 => ( ord_less @ nat @ ( F3 @ I4 ) @ ( F3 @ J2 ) ) )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ I2 @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_mono_imp_le_mono
% 0.31/0.47 thf(fact_245_verit__comp__simplify1_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A5: A] :
% 0.31/0.47 ~ ( ord_less @ A @ A5 @ A5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % verit_comp_simplify1(1)
% 0.31/0.47 thf(fact_246_less__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A )
% 0.31/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(3)
% 0.31/0.47 thf(fact_247_verit__comp__simplify1_I3_J,axiom,
% 0.31/0.47 ! [B: $tType] :
% 0.31/0.47 ( ( linorder @ B )
% 0.31/0.47 => ! [B7: B,A7: B] :
% 0.31/0.47 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
% 0.31/0.47 = ( ord_less @ B @ A7 @ B7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % verit_comp_simplify1(3)
% 0.31/0.47 thf(fact_248_bounded__nat__set__is__finite,axiom,
% 0.31/0.47 ! [N5: set @ nat,N: nat] :
% 0.31/0.47 ( ! [X2: nat] :
% 0.31/0.47 ( ( member @ nat @ X2 @ N5 )
% 0.31/0.47 => ( ord_less @ nat @ X2 @ N ) )
% 0.31/0.47 => ( finite_finite2 @ nat @ N5 ) ) ).
% 0.31/0.47
% 0.31/0.47 % bounded_nat_set_is_finite
% 0.31/0.47 thf(fact_249_finite__nat__set__iff__bounded,axiom,
% 0.31/0.47 ( ( finite_finite2 @ nat )
% 0.31/0.47 = ( ^ [N4: set @ nat] :
% 0.31/0.47 ? [M2: nat] :
% 0.31/0.47 ! [X: nat] :
% 0.31/0.47 ( ( member @ nat @ X @ N4 )
% 0.31/0.47 => ( ord_less @ nat @ X @ M2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_nat_set_iff_bounded
% 0.31/0.47 thf(fact_250_finite__M__bounded__by__nat,axiom,
% 0.31/0.47 ! [P2: nat > $o,I2: nat] :
% 0.31/0.47 ( finite_finite2 @ nat
% 0.31/0.47 @ ( collect @ nat
% 0.31/0.47 @ ^ [K2: nat] :
% 0.31/0.47 ( ( P2 @ K2 )
% 0.31/0.47 & ( ord_less @ nat @ K2 @ I2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_M_bounded_by_nat
% 0.31/0.47 thf(fact_251_infinite__descent__measure,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,V: A > nat,X4: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ~ ( P2 @ X2 )
% 0.31/0.47 => ? [Y3: A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.31/0.47 & ~ ( P2 @ Y3 ) ) )
% 0.31/0.47 => ( P2 @ X4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_descent_measure
% 0.31/0.47 thf(fact_252_measure__induct__rule,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( wellorder @ B )
% 0.31/0.47 => ! [F3: A > B,P2: A > $o,A5: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ! [Y3: A] :
% 0.31/0.47 ( ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X2 ) )
% 0.31/0.47 => ( P2 @ Y3 ) )
% 0.31/0.47 => ( P2 @ X2 ) )
% 0.31/0.47 => ( P2 @ A5 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % measure_induct_rule
% 0.31/0.47
% 0.31/0.47 % Type constructors (36)
% 0.31/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( preorder @ A9 )
% 0.31/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A8 )
% 0.31/0.47 & ( finite_finite @ A9 ) )
% 0.31/0.47 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( order @ A9 )
% 0.31/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.47 ! [A8: $tType,A9: $tType] :
% 0.31/0.47 ( ( ord @ A9 )
% 0.31/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 0.31/0.47 semiri456707255roduct @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.31/0.47 ordere516151231imp_le @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.31/0.47 ordere223160158up_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.31/0.47 ordere236663937imp_le @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.31/0.47 strict797366125id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/0.47 canoni770627133id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.31/0.47 linord1659791738miring @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.31/0.47 ordere779506340up_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 0.31/0.47 ordere216010020id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.31/0.47 cancel1352612707id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.31/0.47 cancel_semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.31/0.47 ab_semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.31/0.47 comm_monoid_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.31/0.47 semigroup_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.31/0.47 wellorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.31/0.47 zero_neq_one @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.31/0.48 preorder @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.31/0.48 linorder @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.31/0.48 monoid_add @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.31/0.48 order @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.31/0.48 ord @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.31/0.48 zero @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.31/0.48 one @ nat ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.31/0.48 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.31/0.48 ! [A8: $tType] :
% 0.31/0.48 ( ( finite_finite @ A8 )
% 0.31/0.48 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.31/0.48 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.31/0.48 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.31/0.48 preorder @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 0.31/0.48 linorder @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 0.31/0.48 finite_finite @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.31/0.48 order @ $o ).
% 0.31/0.48
% 0.31/0.48 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.31/0.48 ord @ $o ).
% 0.31/0.48
% 0.31/0.48 % Conjectures (1)
% 0.31/0.48 thf(conj_0,conjecture,
% 0.31/0.48 ( finite_finite2 @ ( nat > nat )
% 0.31/0.48 @ ( collect @ ( nat > nat )
% 0.31/0.48 @ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 2.57/1.48 ---- Embedded in THF ---
% 2.57/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.57/1.48 %%% Generated on Tue Feb 24 08:27:30 EST 2026
% 2.57/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.57/1.48 %%% Logic specification used:
% 2.57/1.48 %%% thf(spec, logic, $$dhol).
% 2.57/1.48
% 2.57/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2DTF13930.p
% 2.57/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.57/1.48 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.57/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict797366125id_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri456707255roduct: (!> [A:$tType]: ($o))).
% 2.57/1.48 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.57/1.48 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.57/1.48 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.57/1.48 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.57/1.48 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.57/1.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc, type, neg_numeral_dbl_inc: (!> [A:$tType]: ((A > A)))).
% 2.57/1.48 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions, type, number2016821345itions: ((nat > nat) > (nat > $o))).
% 2.57/1.48 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.57/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.57/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.57/1.48 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.57/1.48 thf(sy_v_n, type, n: nat).
% 2.57/1.48 thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062, axiom, ((finite_finite2 @ (nat > nat)) @ ((collect @ (nat > nat)) @ (^ [F:(nat > nat)]: (((! [I:nat]: ((((ord_less_eq @ nat) @ (F @ I)) @ n))) & (! [I:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ n) @ (one_one @ nat))) @ I) => ((F @ I) = (zero_zero @ nat))))))))))).
% 2.57/1.48 thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062, axiom, (((ord_less_eq @ (set @ (nat > nat))) @ ((collect @ (nat > nat)) @ (^ [P:(nat > nat)]: (((number2016821345itions @ P) @ n))))) @ ((collect @ (nat > nat)) @ (^ [F:(nat > nat)]: (((! [I:nat]: ((((ord_less_eq @ nat) @ (F @ I)) @ n))) & (! [I:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ n) @ (one_one @ nat))) @ I) => ((F @ I) = (zero_zero @ nat))))))))))).
% 2.57/1.48 thf(fact_2_finite__Collect__conjI, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P2)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) & (Q @ X)))))))))).
% 2.57/1.48 thf(fact_3_finite__Collect__disjI, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) | (Q @ X)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P2)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 2.57/1.48 thf(fact_4_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A2:(set @ A)]: ($true))))))).
% 2.57/1.48 thf(fact_5_partitions__zero, axiom, (! [P3:(nat > nat)]: ((((number2016821345itions @ P3) @ (zero_zero @ nat)) = (P3 = (^ [I:nat]: ((zero_zero @ nat)))))))).
% 2.57/1.48 thf(fact_6_partitions__bounds, axiom, (! [P3:(nat > nat),N:nat,I2:nat]: ((((number2016821345itions @ P3) @ N) => (((ord_less_eq @ nat) @ (P3 @ I2)) @ N))))).
% 2.57/1.48 thf(fact_7_finite__set__of__finite__funs, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B),D:B]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B2) => ((finite_finite2 @ (A > B)) @ ((collect @ (A > B)) @ (^ [F:(A > B)]: ((! [X:A]: ((((((member @ A) @ X) @ A3) => (((member @ B) @ (F @ X)) @ B2)) & ((~ ((((member @ A) @ X) @ A3))) => ((F @ X) = D)))))))))))))).
% 2.57/1.48 thf(fact_8_not__finite__existsD, axiom, (! [A:$tType,P2:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P2)))) => (? [X_1:A]: ((P2 @ X_1))))))).
% 2.57/1.48 thf(fact_9_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B2:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B2) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B2) & ((R @ X2) @ Xa))))))) => (? [X2:B]: (((((member @ B) @ X2) @ B2) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A3) & ((R @ A4) @ X2))))))))))))))))).
% 2.57/1.48 thf(fact_10_finite__Collect__subsets, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B3) @ A3))))))))).
% 2.57/1.48 thf(fact_11_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P2:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [X_12:B]: (((P2 @ X2) @ X_12)))))) => (? [F2:(A > B)]: ((! [X3:A]: (((((member @ A) @ X3) @ A3) => ((P2 @ X3) @ (F2 @ X3)))))))))))).
% 2.57/1.48 thf(fact_12_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.57/1.48 thf(fact_13_bound__domain__and__range__impl__finitely__many__functions, axiom, (! [N:nat,M:nat]: (((finite_finite2 @ (nat > nat)) @ ((collect @ (nat > nat)) @ (^ [F:(nat > nat)]: (((! [I:nat]: ((((ord_less_eq @ nat) @ (F @ I)) @ N))) & (! [I:nat]: (((((ord_less_eq @ nat) @ M) @ I) => ((F @ I) = (zero_zero @ nat))))))))))))).
% 2.57/1.48 thf(fact_14_finite__Collect__le__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ K)))))))).
% 2.57/1.48 thf(fact_15_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A5:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A5) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & ((((ord_less_eq @ A) @ X2) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X2) => (X2 = Xa))))))))))))))))).
% 2.57/1.48 thf(fact_16_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A5:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A5) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & ((((ord_less_eq @ A) @ A5) @ X2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X2) @ Xa) => (X2 = Xa))))))))))))))))).
% 2.57/1.48 thf(fact_17_rev__finite__subset, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B2) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((finite_finite2 @ A) @ A3)))))).
% 2.57/1.48 thf(fact_18_infinite__super, axiom, (! [A:$tType,S:(set @ A),T:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T)))))))).
% 2.57/1.48 thf(fact_19_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((finite_finite2 @ A) @ B2) => ((finite_finite2 @ A) @ A3)))))).
% 2.57/1.48 thf(fact_20_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B4:A,A5:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B4) @ A5)) @ B4) = (((ord_less_eq @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_21_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ B4) = (((ord_less_eq @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_22_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ (((plus_plus @ A) @ A5) @ B4)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)))))))).
% 2.57/1.48 thf(fact_23_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ (((plus_plus @ A) @ B4) @ A5)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)))))))).
% 2.57/1.48 thf(fact_24_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ A5)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_25_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A5) @ A5)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A5)))))))).
% 2.57/1.48 thf(fact_26_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_27_add__is__0, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 2.57/1.48 thf(fact_28_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.57/1.48 thf(fact_29_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.57/1.48 thf(fact_30_bot__nat__0_Oextremum, axiom, (! [A5:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A5)))).
% 2.57/1.48 thf(fact_31_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B4:A,A5:A,C:A]: ((((((plus_plus @ A) @ B4) @ A5) = (((plus_plus @ A) @ C) @ A5)) = (B4 = C)))))))).
% 2.57/1.48 thf(fact_32_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A,B4:A,C:A]: ((((((plus_plus @ A) @ A5) @ B4) = (((plus_plus @ A) @ A5) @ C)) = (B4 = C)))))))).
% 2.57/1.48 thf(fact_33_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_34_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ C)) = (((ord_less_eq @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_35_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A5:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A5)) @ (((plus_plus @ A) @ C) @ B4)) = (((ord_less_eq @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_36_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X4:A,Y:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X4) @ Y)) = ((X4 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.57/1.48 thf(fact_37_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X4:A,Y:A]: ((((((plus_plus @ A) @ X4) @ Y) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 2.57/1.48 thf(fact_38_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A5:A,B4:A]: (((A5 = (((plus_plus @ A) @ A5) @ B4)) = (B4 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_39_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A5:A,B4:A]: (((A5 = (((plus_plus @ A) @ B4) @ A5)) = (B4 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_40_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A5:A,B4:A]: ((((((plus_plus @ A) @ A5) @ B4) = A5) = (B4 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_41_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [B4:A,A5:A]: ((((((plus_plus @ A) @ B4) @ A5) = A5) = (B4 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_42_double__zero__sym, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A5) @ A5)) = (A5 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A5:A,P2:(A > $o)]: (((((member @ A) @ A5) @ ((collect @ A) @ P2)) = (P2 @ A5))))).
% 2.57/1.48 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 2.57/1.48 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.57/1.48 thf(fact_46_double__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: ((((((plus_plus @ A) @ A5) @ A5) = (zero_zero @ A)) = (A5 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_47_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ A5) @ (zero_zero @ A)) = A5))))))).
% 2.57/1.48 thf(fact_48_add_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A5) = A5))))))).
% 2.57/1.48 thf(fact_49_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X4:A]: ((((zero_zero @ A) = X4) = (X4 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_50_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B4:A,A5:A,C:A]: ((((((plus_plus @ A) @ B4) @ A5) = (((plus_plus @ A) @ C) @ A5)) => (B4 = C)))))))).
% 2.57/1.48 thf(fact_51_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A5:A,B4:A,C:A]: ((((((plus_plus @ A) @ A5) @ B4) = (((plus_plus @ A) @ A5) @ C)) => (B4 = C)))))))).
% 2.57/1.48 thf(fact_52_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B4:A,A5:A,C:A]: (((((plus_plus @ A) @ B4) @ (((plus_plus @ A) @ A5) @ C)) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B4) @ C))))))))).
% 2.57/1.48 thf(fact_53_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A4:A,B5:A]: ((((plus_plus @ A) @ B5) @ A4)))))))).
% 2.57/1.48 thf(fact_54_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B4:A,A5:A,C:A]: ((((((plus_plus @ A) @ B4) @ A5) = (((plus_plus @ A) @ C) @ A5)) = (B4 = C)))))))).
% 2.57/1.48 thf(fact_55_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A5:A,B4:A,C:A]: ((((((plus_plus @ A) @ A5) @ B4) = (((plus_plus @ A) @ A5) @ C)) = (B4 = C)))))))).
% 2.57/1.48 thf(fact_56_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A5:A,B4:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ C) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B4) @ C))))))))).
% 2.57/1.48 thf(fact_57_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B2:A,K:A,B4:A,A5:A]: (((B2 = (((plus_plus @ A) @ K) @ B4)) => ((((plus_plus @ A) @ A5) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A5) @ B4)))))))))).
% 2.57/1.48 thf(fact_58_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A,K:A,A5:A,B4:A]: (((A3 = (((plus_plus @ A) @ K) @ A5)) => ((((plus_plus @ A) @ A3) @ B4) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A5) @ B4)))))))))).
% 2.57/1.48 thf(fact_59_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((I2 = J) & (K = L)) => ((((plus_plus @ A) @ I2) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_60_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A5:A,B4:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ C) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B4) @ C))))))))).
% 2.57/1.48 thf(fact_61_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X4:A]: ((((one_one @ A) = X4) = (X4 = (one_one @ A))))))))).
% 2.57/1.48 thf(fact_62_Nat_Oex__has__greatest__nat, axiom, (! [P2:(nat > $o),K:nat,B4:nat]: (((P2 @ K) => ((! [Y2:nat]: (((P2 @ Y2) => (((ord_less_eq @ nat) @ Y2) @ B4)))) => (? [X2:nat]: (((P2 @ X2) & (! [Y3:nat]: (((P2 @ Y3) => (((ord_less_eq @ nat) @ Y3) @ X2)))))))))))).
% 2.57/1.48 thf(fact_63_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.57/1.48 thf(fact_64_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.57/1.48 thf(fact_65_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_66_le__trans, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I2) @ K)))))).
% 2.57/1.48 thf(fact_67_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.57/1.48 thf(fact_68_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X4))))))).
% 2.57/1.48 thf(fact_69_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ C)) => (((ord_less_eq @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_70_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A5:A,B4:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A5)) @ (((plus_plus @ A) @ C) @ B4)) => (((ord_less_eq @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_71_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B5:A]: ((? [C2:A]: ((B5 = (((plus_plus @ A) @ A4) @ C2))))))))))).
% 2.57/1.48 thf(fact_72_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A5) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ C))))))))).
% 2.57/1.48 thf(fact_73_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => (~ ((! [C3:A]: ((B4 != (((plus_plus @ A) @ A5) @ C3))))))))))))).
% 2.57/1.48 thf(fact_74_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A5) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A5)) @ (((plus_plus @ A) @ C) @ B4))))))))).
% 2.57/1.48 thf(fact_75_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A5:A,B4:A,C:A,D:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ D)))))))))).
% 2.57/1.48 thf(fact_76_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I2) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_77_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((I2 = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_78_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I2) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_79_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A5) = A5))))))).
% 2.57/1.48 thf(fact_80_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ A5) @ (zero_zero @ A)) = A5))))))).
% 2.57/1.48 thf(fact_81_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A5) = A5))))))).
% 2.57/1.48 thf(fact_82_bot__nat__0_Oextremum__uniqueI, axiom, (! [A5:nat]: (((((ord_less_eq @ nat) @ A5) @ (zero_zero @ nat)) => (A5 = (zero_zero @ nat)))))).
% 2.57/1.48 thf(fact_83_bot__nat__0_Oextremum__unique, axiom, (! [A5:nat]: (((((ord_less_eq @ nat) @ A5) @ (zero_zero @ nat)) = (A5 = (zero_zero @ nat)))))).
% 2.57/1.48 thf(fact_84_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.57/1.48 thf(fact_85_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.57/1.48 thf(fact_86_add__eq__self__zero, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = M) => (N = (zero_zero @ nat)))))).
% 2.57/1.48 thf(fact_87_plus__nat_Oadd__0, axiom, (! [N:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 2.57/1.48 thf(fact_88_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: ((? [K2:nat]: ((N2 = (((plus_plus @ nat) @ M2) @ K2)))))))).
% 2.57/1.48 thf(fact_89_trans__le__add2, axiom, (! [I2:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ nat) @ I2) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.57/1.48 thf(fact_90_trans__le__add1, axiom, (! [I2:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ nat) @ I2) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.57/1.48 thf(fact_91_add__le__mono1, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I2) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.57/1.48 thf(fact_92_add__le__mono, axiom, (! [I2:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I2) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.57/1.48 thf(fact_93_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K) @ N3)))))))).
% 2.57/1.48 thf(fact_94_add__leD2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 2.57/1.48 thf(fact_95_add__leD1, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_96_le__add2, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_97_le__add1, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M))))).
% 2.57/1.48 thf(fact_98_add__leE, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 2.57/1.48 thf(fact_99_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X4) @ Y) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.57/1.48 thf(fact_100_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X4) @ Y) = (zero_zero @ A)) = ((X4 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 2.57/1.48 thf(fact_101_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ (zero_zero @ A)))))))))).
% 2.57/1.48 thf(fact_102_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B4) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A5) @ B4)))))))))).
% 2.57/1.48 thf(fact_103_add__increasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,B4:A,A5:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => ((((ord_less_eq @ A) @ B4) @ A5) => (((ord_less_eq @ A) @ B4) @ (((plus_plus @ A) @ A5) @ C)))))))))).
% 2.57/1.48 thf(fact_104_add__decreasing2, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [C:A,A5:A,B4:A]: (((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A5) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ B4))))))))).
% 2.57/1.48 thf(fact_105_add__increasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ B4) @ (((plus_plus @ A) @ A5) @ C)))))))))).
% 2.57/1.48 thf(fact_106_add__decreasing, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,C:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A5) @ C)) @ B4))))))))).
% 2.57/1.48 thf(fact_107_sum_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I3:(set @ B),X4:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I:B]: (((((member @ B) @ I) @ I3) & ((X4 @ I) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I:B]: (((((member @ B) @ I) @ I3) & ((Y @ I) != (zero_zero @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I:B]: (((((member @ B) @ I) @ I3) & ((((plus_plus @ A) @ (X4 @ I)) @ (Y @ I)) != (zero_zero @ A))))))))))))))).
% 2.57/1.48 thf(fact_108_not__one__le__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.48 thf(fact_109_zero__le__one, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.57/1.48 thf(fact_110_finite__less__ub, axiom, (! [F3:(nat > nat),U:nat]: (((! [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ (F3 @ N3)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N2:nat]: ((((ord_less_eq @ nat) @ (F3 @ N2)) @ U))))))))).
% 2.57/1.48 thf(fact_111_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A3) => (A3 = B2)))))).
% 2.57/1.48 thf(fact_112_subsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.57/1.48 thf(fact_113_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 2.57/1.48 thf(fact_114_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N4:(set @ nat)]: ((? [M2:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N4) => (((ord_less_eq @ nat) @ X) @ M2)))))))))).
% 2.57/1.48 thf(fact_115_Euclid__induct, axiom, (! [P2:(nat > (nat > $o)),A5:nat,B4:nat]: (((! [A6:nat,B6:nat]: ((((P2 @ A6) @ B6) = ((P2 @ B6) @ A6)))) => ((! [A6:nat]: (((P2 @ A6) @ (zero_zero @ nat)))) => ((! [A6:nat,B6:nat]: ((((P2 @ A6) @ B6) => ((P2 @ A6) @ (((plus_plus @ nat) @ A6) @ B6))))) => ((P2 @ A5) @ B4))))))).
% 2.57/1.48 thf(fact_116_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (G @ X4))))))))).
% 2.57/1.48 thf(fact_117_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (G @ X4))))))))).
% 2.57/1.48 thf(fact_118_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F3 @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F3) @ G)))))))).
% 2.57/1.48 thf(fact_119_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))))).
% 2.57/1.48 thf(fact_120_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F3:(B > A),B4:B,C:B]: (((((ord_less_eq @ A) @ A5) @ (F3 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F3 @ C))))))))))).
% 2.57/1.48 thf(fact_121_order__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A5:A,B4:A,F3:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less_eq @ C4) @ (F3 @ B4)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C4) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ C4) @ (F3 @ A5)) @ C)))))))))).
% 2.57/1.48 thf(fact_122_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,F3:(B > A),B4:B,C:B]: (((A5 = (F3 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F3 @ C))))))))))).
% 2.57/1.48 thf(fact_123_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,B4:A,F3:(A > B),C:B]: (((((ord_less_eq @ A) @ A5) @ B4) => (((F3 @ B4) = C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ B) @ (F3 @ A5)) @ C)))))))))).
% 2.57/1.48 thf(fact_124_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X))))))))).
% 2.57/1.48 thf(fact_125_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ Y) => ((((ord_less_eq @ A) @ Y) @ X4) => (X4 = Y))))))))).
% 2.57/1.48 thf(fact_126_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ Y) | (((ord_less_eq @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_127_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y:A]: (((X4 = Y) => (((ord_less_eq @ A) @ X4) @ Y)))))))).
% 2.57/1.48 thf(fact_128_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y))) => (((ord_less_eq @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_129_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A5) @ C))))))))).
% 2.57/1.48 thf(fact_130_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X4) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z2)))) => (((((ord_less_eq @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X4)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y))))))))))))))))).
% 2.57/1.48 thf(fact_131_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X4:A]: (((((ord_less_eq @ A) @ Y) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y) = (X4 = Y))))))))).
% 2.57/1.48 thf(fact_132_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A4:A,B5:A]: (((((ord_less_eq @ A) @ A4) @ B5) & (((ord_less_eq @ A) @ B5) @ A4))))))))).
% 2.57/1.48 thf(fact_133_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B4:A,C:A]: (((A5 = B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less_eq @ A) @ A5) @ C))))))))).
% 2.57/1.48 thf(fact_134_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((B4 = C) => (((ord_less_eq @ A) @ A5) @ C))))))))).
% 2.57/1.48 thf(fact_135_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less_eq @ A) @ B4) @ A5) => (A5 = B4))))))))).
% 2.57/1.48 thf(fact_136_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X4) @ Z2))))))))).
% 2.57/1.48 thf(fact_137_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((((ord_less_eq @ A) @ A5) @ A5))))))).
% 2.57/1.48 thf(fact_138_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A5:A,B4:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P2 @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P2 @ B6) @ A6) => ((P2 @ A6) @ B6)))) => ((P2 @ A5) @ B4))))))))).
% 2.57/1.48 thf(fact_139_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A5:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A5) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less_eq @ A) @ C) @ A5))))))))).
% 2.57/1.48 thf(fact_140_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z:A]: ((Y4 = Z))) = (^ [A4:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A4) & (((ord_less_eq @ A) @ A4) @ B5))))))))).
% 2.57/1.48 thf(fact_141_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A5:A]: (((((ord_less_eq @ A) @ B4) @ A5) => ((((ord_less_eq @ A) @ A5) @ B4) => (A5 = B4))))))))).
% 2.57/1.48 thf(fact_142_in__mono, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),X4:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ X4) @ A3) => (((member @ A) @ X4) @ B2)))))).
% 2.57/1.48 thf(fact_143_subsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B2)))))).
% 2.57/1.48 thf(fact_144_equalityE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))))).
% 2.57/1.48 thf(fact_145_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B3)))))))))).
% 2.57/1.48 thf(fact_146_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.57/1.48 thf(fact_147_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3))))).
% 2.57/1.48 thf(fact_148_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A2) => (((member @ A) @ T2) @ B3)))))))))).
% 2.57/1.48 thf(fact_149_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.57/1.48 thf(fact_150_Collect__mono, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)))))).
% 2.57/1.48 thf(fact_151_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C5) => (((ord_less_eq @ (set @ A)) @ A3) @ C5)))))).
% 2.57/1.48 thf(fact_152_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) & (((ord_less_eq @ (set @ A)) @ B3) @ A2)))))))).
% 2.57/1.48 thf(fact_153_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P2 @ X) => (Q @ X)))))))).
% 2.57/1.48 thf(fact_154_bounded__Max__nat, axiom, (! [P2:(nat > $o),X4:nat,M3:nat]: (((P2 @ X4) => ((! [X2:nat]: (((P2 @ X2) => (((ord_less_eq @ nat) @ X2) @ M3)))) => (~ ((! [M4:nat]: (((P2 @ M4) => (~ ((! [X3:nat]: (((P2 @ X3) => (((ord_less_eq @ nat) @ X3) @ M4)))))))))))))))).
% 2.57/1.48 thf(fact_155_Collect__subset, axiom, (! [A:$tType,A3:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P2 @ X)))))) @ A3)))).
% 2.57/1.48 thf(fact_156_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A2:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) @ (^ [X:A]: ((((member @ A) @ X) @ B3)))))))))).
% 2.57/1.48 thf(fact_157_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 2.57/1.48 thf(fact_158_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.57/1.48 thf(fact_159_verit__sum__simplify, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A5:A]: (((((plus_plus @ A) @ A5) @ (zero_zero @ A)) = A5))))))).
% 2.57/1.48 thf(fact_160_add__0__iff, axiom, (! [A:$tType]: (((semiri456707255roduct @ A) => (! [B4:A,A5:A]: (((B4 = (((plus_plus @ A) @ B4) @ A5)) = (A5 = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_161_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.57/1.48 thf(fact_162_predicate1I, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) => (Q @ X2)))) => (((ord_less_eq @ (A > $o)) @ P2) @ Q))))).
% 2.57/1.48 thf(fact_163_predicate1D, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o),X4:A]: (((((ord_less_eq @ (A > $o)) @ P2) @ Q) => ((P2 @ X4) => (Q @ X4)))))).
% 2.57/1.48 thf(fact_164_rev__predicate1D, axiom, (! [A:$tType,P2:(A > $o),X4:A,Q:(A > $o)]: (((P2 @ X4) => ((((ord_less_eq @ (A > $o)) @ P2) @ Q) => (Q @ X4)))))).
% 2.57/1.48 thf(fact_165_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A5:A,B4:A]: (((A5 = B4) | ((~ ((((ord_less_eq @ A) @ A5) @ B4))) | (~ ((((ord_less_eq @ A) @ B4) @ A5))))))))))).
% 2.57/1.48 thf(fact_166_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A5:A,B4:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ C) = (((plus_plus @ A) @ A5) @ (((plus_plus @ A) @ B4) @ C))))))))).
% 2.57/1.48 thf(fact_167_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ R)))) @ (^ [X:A]: ((((member @ A) @ X) @ S)))) = (((ord_less_eq @ (set @ A)) @ R) @ S))))).
% 2.57/1.48 thf(fact_168_dbl__inc__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ (zero_zero @ A)) = (one_one @ A)))))).
% 2.57/1.48 thf(fact_169_conj__subset__def, axiom, (! [A:$tType,A3:(set @ A),P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ (^ [X:A]: (((P2 @ X) & (Q @ X)))))) = ((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ P2)) & (((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ Q))))))).
% 2.57/1.48 thf(fact_170_dbl__inc__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_inc @ A) = (^ [X:A]: ((((plus_plus @ A) @ (((plus_plus @ A) @ X) @ X)) @ (one_one @ A))))))))).
% 2.57/1.48 thf(fact_171_subset__Collect__iff, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A3) => ((((ord_less_eq @ (set @ A)) @ B2) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P2 @ X)))))) = (! [X:A]: (((((member @ A) @ X) @ B2) => (P2 @ X))))))))).
% 2.57/1.48 thf(fact_172_subset__CollectI, axiom, (! [A:$tType,B2:(set @ A),A3:(set @ A),Q:(A > $o),P2:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ B2) => ((Q @ X2) => (P2 @ X2))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ B2) & (Q @ X)))))) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P2 @ X))))))))))).
% 2.57/1.48 thf(fact_173_Collect__restrict, axiom, (! [A:$tType,X5:(set @ A),P2:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X5) & (P2 @ X)))))) @ X5)))).
% 2.57/1.48 thf(fact_174_prop__restrict, axiom, (! [A:$tType,X4:A,Z3:(set @ A),X5:(set @ A),P2:(A > $o)]: (((((member @ A) @ X4) @ Z3) => ((((ord_less_eq @ (set @ A)) @ Z3) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X5) & (P2 @ X)))))) => (P2 @ X4)))))).
% 2.57/1.48 thf(fact_175_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C4:$tType]: (((zero @ B) => (! [F3:((A > B) > C4),G:C4]: (((F3 = (^ [X:(A > B)]: (G))) => ((F3 @ (^ [X:A]: ((zero_zero @ B)))) = G)))))))).
% 2.57/1.48 thf(fact_176_Fpow__def, axiom, (! [A:$tType]: (((finite_Fpow @ A) = (^ [A2:(set @ A)]: (((collect @ (set @ A)) @ (^ [X6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ X6) @ A2) & ((finite_finite2 @ A) @ X6))))))))))).
% 2.57/1.48 thf(fact_177_partitions__imp__finite__elements, axiom, (! [P3:(nat > nat),N:nat]: ((((number2016821345itions @ P3) @ N) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [I:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (P3 @ I)))))))))).
% 2.57/1.48 thf(fact_178_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_179_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A5:A,C:A,B4:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ C)) = (((ord_less @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_180_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A5:A,B4:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A5)) @ (((plus_plus @ A) @ C) @ B4)) = (((ord_less @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_181_bot__nat__0_Onot__eq__extremum, axiom, (! [A5:nat]: (((A5 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A5))))).
% 2.57/1.48 thf(fact_182_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.57/1.48 thf(fact_183_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.57/1.48 thf(fact_184_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_185_finite__Collect__less__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N2:nat]: ((((ord_less @ nat) @ N2) @ K)))))))).
% 2.57/1.48 thf(fact_186_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [B4:A,A5:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B4) @ A5)) @ B4) = (((ord_less @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_187_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ B4) = (((ord_less @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_188_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ (((plus_plus @ A) @ A5) @ B4)) = (((ord_less @ A) @ (zero_zero @ A)) @ B4)))))))).
% 2.57/1.48 thf(fact_189_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere516151231imp_le @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ (((plus_plus @ A) @ B4) @ A5)) = (((ord_less @ A) @ (zero_zero @ A)) @ B4)))))))).
% 2.57/1.48 thf(fact_190_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A5) @ A5)) @ (zero_zero @ A)) = (((ord_less @ A) @ A5) @ (zero_zero @ A))))))))).
% 2.57/1.48 thf(fact_191_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A5) @ A5)) = (((ord_less @ A) @ (zero_zero @ A)) @ A5)))))))).
% 2.57/1.48 thf(fact_192_add__gr__0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 2.57/1.48 thf(fact_193_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 2.57/1.48 thf(fact_194_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B4:A,C:A,D:A]: (((((ord_less @ A) @ A5) @ B4) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ D)))))))))).
% 2.57/1.48 thf(fact_195_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A5:A,B4:A,C:A,D:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ C)) @ (((plus_plus @ A) @ B4) @ D)))))))))).
% 2.57/1.48 thf(fact_196_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I2) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_197_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I2:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I2) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I2) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.57/1.48 thf(fact_198_pos__add__strict, axiom, (! [A:$tType]: (((strict797366125id_add @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less @ A) @ B4) @ C) => (((ord_less @ A) @ B4) @ (((plus_plus @ A) @ A5) @ C)))))))))).
% 2.57/1.48 thf(fact_199_canonically__ordered__monoid__add__class_OlessE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) => (~ ((! [C3:A]: (((B4 = (((plus_plus @ A) @ A5) @ C3)) => (C3 = (zero_zero @ A)))))))))))))).
% 2.57/1.48 thf(fact_200_add__pos__pos, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A5) => ((((ord_less @ A) @ (zero_zero @ A)) @ B4) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A5) @ B4)))))))))).
% 2.57/1.48 thf(fact_201_add__neg__neg, axiom, (! [A:$tType]: (((ordere216010020id_add @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ (zero_zero @ A)) => ((((ord_less @ A) @ B4) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A5) @ B4)) @ (zero_zero @ A)))))))))).
% 2.57/1.48 thf(fact_202_add__less__zeroD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X4:A,Y:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ X4) @ Y)) @ (zero_zero @ A)) => ((((ord_less @ A) @ X4) @ (zero_zero @ A)) | (((ord_less @ A) @ Y) @ (zero_zero @ A)))))))))).
% 2.57/1.48 thf(fact_203_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A]: (((A5 != B4) => ((((ord_less_eq @ A) @ A5) @ B4) => (((ord_less @ A) @ A5) @ B4))))))))).
% 2.57/1.48 thf(fact_204_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A5:A]: (((((ord_less @ A) @ B4) @ A5) => (((ord_less_eq @ A) @ B4) @ A5)))))))).
% 2.57/1.48 thf(fact_205_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B5:A,A4:A]: (((((ord_less_eq @ A) @ B5) @ A4) & (A4 != B5))))))))).
% 2.57/1.48 thf(fact_206_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B5:A,A4:A]: (((((ord_less @ A) @ B5) @ A4) | (A4 = B5))))))))).
% 2.57/1.48 thf(fact_207_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) => (((ord_less_eq @ A) @ A5) @ B4)))))))).
% 2.57/1.48 thf(fact_208_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X4:A,Y:A,Z2:A]: (((((ord_less @ A) @ X4) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X4) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.57/1.48 thf(fact_209_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X4:A,Y:A]: (((((ord_less @ A) @ Z2) @ X4) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X4) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 2.57/1.48 thf(fact_210_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A5:A,C:A]: (((((ord_less @ A) @ B4) @ A5) => ((((ord_less_eq @ A) @ C) @ B4) => (((ord_less @ A) @ C) @ A5))))))))).
% 2.57/1.48 thf(fact_211_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A5:A,C:A]: (((((ord_less_eq @ A) @ B4) @ A5) => ((((ord_less @ A) @ C) @ B4) => (((ord_less @ A) @ C) @ A5))))))))).
% 2.57/1.48 thf(fact_212_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A4:A,B5:A]: (((((ord_less_eq @ A) @ A4) @ B5) & (A4 != B5))))))))).
% 2.57/1.48 thf(fact_213_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A4:A,B5:A]: (((((ord_less @ A) @ A4) @ B5) | (A4 = B5))))))))).
% 2.57/1.48 thf(fact_214_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less @ A) @ A5) @ B4) => ((((ord_less_eq @ A) @ B4) @ C) => (((ord_less @ A) @ A5) @ C))))))))).
% 2.57/1.48 thf(fact_215_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A,C:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less @ A) @ B4) @ C) => (((ord_less @ A) @ A5) @ C))))))))).
% 2.57/1.48 thf(fact_216_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X4:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X4))) => (((ord_less @ A) @ X4) @ Y)))))))).
% 2.57/1.48 thf(fact_217_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (~ ((((ord_less_eq @ A) @ Y5) @ X))))))))))).
% 2.57/1.48 thf(fact_218_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ Y) => ((((ord_less @ A) @ X4) @ Y) | (X4 = Y))))))))).
% 2.57/1.48 thf(fact_219_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ Y) | (((ord_less @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_220_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 2.57/1.48 thf(fact_221_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y:A]: (((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 2.57/1.48 thf(fact_222_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y:A,Z2:A]: (((((ord_less @ A) @ X4) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X4) @ Z2))))))))).
% 2.57/1.48 thf(fact_223_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X4) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X4) @ Z2))))))))).
% 2.57/1.48 thf(fact_224_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y:A]: (((((ord_less @ A) @ X4) @ Y) => (((ord_less_eq @ A) @ X4) @ Y)))))))).
% 2.57/1.48 thf(fact_225_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y:A]: (((((ord_less_eq @ A) @ X4) @ Y) => ((~ ((((ord_less @ A) @ X4) @ Y))) = (X4 = Y))))))))).
% 2.57/1.48 thf(fact_226_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y:A]: (((~ ((((ord_less @ A) @ X4) @ Y))) => ((((ord_less_eq @ A) @ X4) @ Y) = (X4 = Y))))))))).
% 2.57/1.48 thf(fact_227_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) => ((A5 != B4) => (((ord_less @ A) @ A5) @ B4))))))))).
% 2.57/1.48 thf(fact_228_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((~ ((((ord_less @ A) @ X4) @ Y))) = (((ord_less_eq @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_229_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y))) = (((ord_less @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_230_order__less__le__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A5:A,B4:A,F3:(A > C4),C:C4]: (((((ord_less @ A) @ A5) @ B4) => ((((ord_less_eq @ C4) @ (F3 @ B4)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C4) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ C4) @ (F3 @ A5)) @ C)))))))))).
% 2.57/1.48 thf(fact_231_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F3:(B > A),B4:B,C:B]: (((((ord_less @ A) @ A5) @ (F3 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A5) @ (F3 @ C))))))))))).
% 2.57/1.48 thf(fact_232_order__le__less__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A5:A,B4:A,F3:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A5) @ B4) => ((((ord_less @ C4) @ (F3 @ B4)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C4) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ C4) @ (F3 @ A5)) @ C)))))))))).
% 2.57/1.48 thf(fact_233_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F3:(B > A),B4:B,C:B]: (((((ord_less_eq @ A) @ A5) @ (F3 @ B4)) => ((((ord_less @ B) @ B4) @ C) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less @ A) @ A5) @ (F3 @ C))))))))))).
% 2.57/1.48 thf(fact_234_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (X != Y5))))))))).
% 2.57/1.48 thf(fact_235_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X:A,Y5:A]: (((((ord_less @ A) @ X) @ Y5) | (X = Y5))))))))).
% 2.57/1.48 thf(fact_236_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y:A]: (((~ ((((ord_less @ A) @ X4) @ Y))) => (((ord_less_eq @ A) @ Y) @ X4)))))))).
% 2.57/1.48 thf(fact_237_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X4:A]: (((((ord_less_eq @ A) @ Y) @ X4) => (~ ((((ord_less @ A) @ X4) @ Y)))))))))).
% 2.57/1.48 thf(fact_238_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.57/1.48 thf(fact_239_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.57/1.48 thf(fact_240_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_241_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.57/1.48 thf(fact_242_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.57/1.48 thf(fact_243_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.57/1.48 thf(fact_244_less__mono__imp__le__mono, axiom, (! [F3:(nat > nat),I2:nat,J:nat]: (((! [I4:nat,J2:nat]: (((((ord_less @ nat) @ I4) @ J2) => (((ord_less @ nat) @ (F3 @ I4)) @ (F3 @ J2))))) => ((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ nat) @ (F3 @ I2)) @ (F3 @ J))))))).
% 2.57/1.48 thf(fact_245_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((~ ((((ord_less @ A) @ A5) @ A5))))))))).
% 2.57/1.48 thf(fact_246_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.48 thf(fact_247_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B7:B,A7:B]: (((~ ((((ord_less_eq @ B) @ B7) @ A7))) = (((ord_less @ B) @ A7) @ B7)))))))).
% 2.57/1.48 thf(fact_248_bounded__nat__set__is__finite, axiom, (! [N5:(set @ nat),N:nat]: (((! [X2:nat]: (((((member @ nat) @ X2) @ N5) => (((ord_less @ nat) @ X2) @ N)))) => ((finite_finite2 @ nat) @ N5))))).
% 2.57/1.48 thf(fact_249_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N4:(set @ nat)]: ((? [M2:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N4) => (((ord_less @ nat) @ X) @ M2)))))))))).
% 2.57/1.48 thf(fact_250_finite__M__bounded__by__nat, axiom, (! [P2:(nat > $o),I2:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K2:nat]: (((P2 @ K2) & (((ord_less @ nat) @ K2) @ I2))))))))).
% 2.57/1.48 thf(fact_251_infinite__descent__measure, axiom, (! [A:$tType,P2:(A > $o),V:(A > nat),X4:A]: (((! [X2:A]: (((~ ((P2 @ X2))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X2)) & (~ ((P2 @ Y3))))))))) => (P2 @ X4))))).
% 2.57/1.48 thf(fact_252_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F3:(A > B),P2:(A > $o),A5:A]: (((! [X2:A]: (((! [Y3:A]: (((((ord_less @ B) @ (F3 @ Y3)) @ (F3 @ X2)) => (P2 @ Y3)))) => (P2 @ X2)))) => (P2 @ A5)))))))).
% 2.57/1.48 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.57/1.48 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: ((((finite_finite @ A8) & (finite_finite @ A9)) => (finite_finite @ (A8 > A9)))))).
% 2.57/1.48 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.57/1.48 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.57/1.48 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, (semiri456707255roduct @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere516151231imp_le @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add, axiom, (strict797366125id_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, axiom, (ordere216010020id_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Rings_Ozero__neq__one, axiom, (zero_neq_one @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, (monoid_add @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.57/1.48 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 2.57/1.48 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.57/1.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A8:$tType]: (((finite_finite @ A8) => (finite_finite @ (set @ A8)))))).
% 2.57/1.48 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.57/1.48 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.57/1.48 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, (preorder @ $o)).
% 2.57/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder_9, axiom, (linorder @ $o)).
% 2.57/1.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10, axiom, (finite_finite @ $o)).
% 2.57/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, (order @ $o)).
% 2.57/1.48 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, (ord @ $o)).
% 2.57/1.48 thf(conj_0, conjecture, ((finite_finite2 @ (nat > nat)) @ ((collect @ (nat > nat)) @ (^ [P:(nat > nat)]: (((number2016821345itions @ P) @ n)))))).
% 2.57/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2DTF13930.p
% 2.57/1.48 ------------------------
% 2.89/1.52 ---- Cleaned THF ---
% 2.89/1.52 thf(ty_t_Set_Oset,type,
% 2.89/1.52 set: $tType > $tType ).
% 2.89/1.52
% 2.89/1.52 thf(ty_t_Nat_Onat,type,
% 2.89/1.52 nat: $tType ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_HOL_Otype,type,
% 2.89/1.52 type:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oone,type,
% 2.89/1.52 one:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ozero,type,
% 2.89/1.52 zero:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Oord,type,
% 2.89/1.52 ord:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Num_Oneg__numeral,type,
% 2.89/1.52 neg_numeral:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Oorder,type,
% 2.89/1.52 order:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ogroup__add,type,
% 2.89/1.52 group_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.89/1.52 finite_finite:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Omonoid__add,type,
% 2.89/1.52 monoid_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.89/1.52 linorder:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Opreorder,type,
% 2.89/1.52 preorder:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.89/1.52 zero_neq_one:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Owellorder,type,
% 2.89/1.52 wellorder:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.89/1.52 semigroup_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.89/1.52 linordered_idom:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.89/1.52 comm_monoid_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.89/1.52 ab_semigroup_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.89/1.52 dense_linorder:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.89/1.52 cancel_semigroup_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.89/1.52 cancel1352612707id_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.89/1.52 linord219039673up_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 2.89/1.52 ordere216010020id_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.89/1.52 ordere779506340up_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.89/1.52 linord1659791738miring:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.89/1.52 canoni770627133id_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 2.89/1.52 strict797366125id_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.89/1.52 ordere236663937imp_le:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.89/1.52 ordere223160158up_add:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.89/1.52 ordere516151231imp_le:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 2.89/1.52 semiri456707255roduct:
% 2.89/1.52 !>[A: $tType] : $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Finite__Set_OFpow,type,
% 2.89/1.52 finite_Fpow:
% 2.89/1.52 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Finite__Set_Ofinite,type,
% 2.89/1.52 finite_finite2:
% 2.89/1.52 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.89/1.52 one_one:
% 2.89/1.52 !>[A: $tType] : A ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.89/1.52 plus_plus:
% 2.89/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.89/1.52 zero_zero:
% 2.89/1.52 !>[A: $tType] : A ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 2.89/1.52 neg_numeral_dbl_inc:
% 2.89/1.52 !>[A: $tType] : ( A > A ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 2.89/1.52 number2016821345itions: ( nat > nat ) > nat > $o ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.89/1.52 ord_less:
% 2.89/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.89/1.52 ord_less_eq:
% 2.89/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_Set_OCollect,type,
% 2.89/1.52 collect:
% 2.89/1.52 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_c_member,type,
% 2.89/1.52 member:
% 2.89/1.52 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.89/1.52
% 2.89/1.52 thf(sy_v_n,type,
% 2.89/1.52 n: nat ).
% 2.89/1.52
% 2.89/1.52 thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 2.89/1.52 ( finite_finite2 @ ( nat > nat )
% 2.89/1.52 @ ( collect @ ( nat > nat )
% 2.89/1.52 @ ^ [F: nat > nat] :
% 2.89/1.52 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
% 2.89/1.52 & ! [I: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
% 2.89/1.52 => ( ( F @ I )
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
% 2.89/1.52 ( ord_less_eq @ ( set @ ( nat > nat ) )
% 2.89/1.52 @ ( collect @ ( nat > nat )
% 2.89/1.52 @ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) )
% 2.89/1.52 @ ( collect @ ( nat > nat )
% 2.89/1.52 @ ^ [F: nat > nat] :
% 2.89/1.52 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
% 2.89/1.52 & ! [I: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
% 2.89/1.52 => ( ( F @ I )
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_2_finite__Collect__conjI,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 2.89/1.52 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 2.89/1.52 => ( finite_finite2 @ A
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( P2 @ X )
% 2.89/1.52 & ( Q @ X ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_3_finite__Collect__disjI,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ( finite_finite2 @ A
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( P2 @ X )
% 2.89/1.52 | ( Q @ X ) ) ) )
% 2.89/1.52 = ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 2.89/1.52 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_4_finite__code,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( finite_finite @ A )
% 2.89/1.52 => ( ( finite_finite2 @ A )
% 2.89/1.52 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_5_partitions__zero,axiom,
% 2.89/1.52 ! [P3: nat > nat] :
% 2.89/1.52 ( ( number2016821345itions @ P3 @ ( zero_zero @ nat ) )
% 2.89/1.52 = ( P3
% 2.89/1.52 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_6_partitions__bounds,axiom,
% 2.89/1.52 ! [P3: nat > nat,N: nat,I2: nat] :
% 2.89/1.52 ( ( number2016821345itions @ P3 @ N )
% 2.89/1.52 => ( ord_less_eq @ nat @ ( P3 @ I2 ) @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_7_finite__set__of__finite__funs,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,D: B] :
% 2.89/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( ( finite_finite2 @ B @ B2 )
% 2.89/1.52 => ( finite_finite2 @ ( A > B )
% 2.89/1.52 @ ( collect @ ( A > B )
% 2.89/1.52 @ ^ [F: A > B] :
% 2.89/1.52 ! [X: A] :
% 2.89/1.52 ( ( ( member @ A @ X @ A3 )
% 2.89/1.52 => ( member @ B @ ( F @ X ) @ B2 ) )
% 2.89/1.52 & ( ~ ( member @ A @ X @ A3 )
% 2.89/1.52 => ( ( F @ X )
% 2.89/1.52 = D ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_8_not__finite__existsD,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o] :
% 2.89/1.52 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
% 2.89/1.52 => ? [X_1: A] : ( P2 @ X_1 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_9_pigeonhole__infinite__rel,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
% 2.89/1.52 ( ~ ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( ( finite_finite2 @ B @ B2 )
% 2.89/1.52 => ( ! [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ A3 )
% 2.89/1.52 => ? [Xa: B] :
% 2.89/1.52 ( ( member @ B @ Xa @ B2 )
% 2.89/1.52 & ( R @ X2 @ Xa ) ) )
% 2.89/1.52 => ? [X2: B] :
% 2.89/1.52 ( ( member @ B @ X2 @ B2 )
% 2.89/1.52 & ~ ( finite_finite2 @ A
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [A4: A] :
% 2.89/1.52 ( ( member @ A @ A4 @ A3 )
% 2.89/1.52 & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_10_finite__Collect__subsets,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A] :
% 2.89/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( finite_finite2 @ ( set @ A )
% 2.89/1.52 @ ( collect @ ( set @ A )
% 2.89/1.52 @ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_11_finite__set__choice,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
% 2.89/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( ! [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ A3 )
% 2.89/1.52 => ? [X_12: B] : ( P2 @ X2 @ X_12 ) )
% 2.89/1.52 => ? [F2: A > B] :
% 2.89/1.52 ! [X3: A] :
% 2.89/1.52 ( ( member @ A @ X3 @ A3 )
% 2.89/1.52 => ( P2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_12_finite,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( finite_finite @ A )
% 2.89/1.52 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_13_bound__domain__and__range__impl__finitely__many__functions,axiom,
% 2.89/1.52 ! [N: nat,M: nat] :
% 2.89/1.52 ( finite_finite2 @ ( nat > nat )
% 2.89/1.52 @ ( collect @ ( nat > nat )
% 2.89/1.52 @ ^ [F: nat > nat] :
% 2.89/1.52 ( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ N )
% 2.89/1.52 & ! [I: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ M @ I )
% 2.89/1.52 => ( ( F @ I )
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_14_finite__Collect__le__nat,axiom,
% 2.89/1.52 ! [K: nat] :
% 2.89/1.52 ( finite_finite2 @ nat
% 2.89/1.52 @ ( collect @ nat
% 2.89/1.52 @ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_15_finite__has__minimal2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [A3: set @ A,A5: A] :
% 2.89/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( ( member @ A @ A5 @ A3 )
% 2.89/1.52 => ? [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ A3 )
% 2.89/1.52 & ( ord_less_eq @ A @ X2 @ A5 )
% 2.89/1.52 & ! [Xa: A] :
% 2.89/1.52 ( ( member @ A @ Xa @ A3 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 2.89/1.52 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_16_finite__has__maximal2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [A3: set @ A,A5: A] :
% 2.89/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.89/1.52 => ( ( member @ A @ A5 @ A3 )
% 2.89/1.52 => ? [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ A3 )
% 2.89/1.52 & ( ord_less_eq @ A @ A5 @ X2 )
% 2.89/1.52 & ! [Xa: A] :
% 2.89/1.52 ( ( member @ A @ Xa @ A3 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ X2 @ Xa )
% 2.89/1.52 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_17_rev__finite__subset,axiom,
% 2.89/1.52 ! [A: $tType,B2: set @ A,A3: set @ A] :
% 2.89/1.52 ( ( finite_finite2 @ A @ B2 )
% 2.89/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_18_infinite__super,axiom,
% 2.89/1.52 ! [A: $tType,S: set @ A,T: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ S @ T )
% 2.89/1.52 => ( ~ ( finite_finite2 @ A @ S )
% 2.89/1.52 => ~ ( finite_finite2 @ A @ T ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_19_finite__subset,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( ( finite_finite2 @ A @ B2 )
% 2.89/1.52 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_20_add__le__same__cancel1,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere516151231imp_le @ A )
% 2.89/1.52 => ! [B4: A,A5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
% 2.89/1.52 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_21_add__le__same__cancel2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere516151231imp_le @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
% 2.89/1.52 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_22_le__add__same__cancel1,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere516151231imp_le @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
% 2.89/1.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_23_le__add__same__cancel2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere516151231imp_le @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
% 2.89/1.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_24_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord219039673up_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
% 2.89/1.52 = ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_25_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord219039673up_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
% 2.89/1.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_26_nat__add__left__cancel__le,axiom,
% 2.89/1.52 ! [K: nat,M: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.89/1.52 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_27_add__is__0,axiom,
% 2.89/1.52 ! [M: nat,N: nat] :
% 2.89/1.52 ( ( ( plus_plus @ nat @ M @ N )
% 2.89/1.52 = ( zero_zero @ nat ) )
% 2.89/1.52 = ( ( M
% 2.89/1.52 = ( zero_zero @ nat ) )
% 2.89/1.52 & ( N
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_28_Nat_Oadd__0__right,axiom,
% 2.89/1.52 ! [M: nat] :
% 2.89/1.52 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 2.89/1.52 = M ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_29_le0,axiom,
% 2.89/1.52 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_30_bot__nat__0_Oextremum,axiom,
% 2.89/1.52 ! [A5: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A5 ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_31_add__right__cancel,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel_semigroup_add @ A )
% 2.89/1.52 => ! [B4: A,A5: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ B4 @ A5 )
% 2.89/1.52 = ( plus_plus @ A @ C @ A5 ) )
% 2.89/1.52 = ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_32_add__left__cancel,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel_semigroup_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ A5 @ B4 )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ C ) )
% 2.89/1.52 = ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_33_le__zero__eq,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [N: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.89/1.52 = ( N
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_34_add__le__cancel__right,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [A5: A,C: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 2.89/1.52 = ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_35_add__le__cancel__left,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [C: A,A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 2.89/1.52 = ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_36_zero__eq__add__iff__both__eq__0,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ( zero_zero @ A )
% 2.89/1.52 = ( plus_plus @ A @ X4 @ Y ) )
% 2.89/1.52 = ( ( X4
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 & ( Y
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_37_add__eq__0__iff__both__eq__0,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ X4 @ Y )
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 = ( ( X4
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 & ( Y
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_38_add__cancel__right__right,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel1352612707id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( A5
% 2.89/1.52 = ( plus_plus @ A @ A5 @ B4 ) )
% 2.89/1.52 = ( B4
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_39_add__cancel__right__left,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel1352612707id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( A5
% 2.89/1.52 = ( plus_plus @ A @ B4 @ A5 ) )
% 2.89/1.52 = ( B4
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_40_add__cancel__left__right,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel1352612707id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ A5 @ B4 )
% 2.89/1.52 = A5 )
% 2.89/1.52 = ( B4
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_41_add__cancel__left__left,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel1352612707id_add @ A )
% 2.89/1.52 => ! [B4: A,A5: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ B4 @ A5 )
% 2.89/1.52 = A5 )
% 2.89/1.52 = ( B4
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_42_double__zero__sym,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord219039673up_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( ( zero_zero @ A )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ A5 ) )
% 2.89/1.52 = ( A5
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_43_mem__Collect__eq,axiom,
% 2.89/1.52 ! [A: $tType,A5: A,P2: A > $o] :
% 2.89/1.52 ( ( member @ A @ A5 @ ( collect @ A @ P2 ) )
% 2.89/1.52 = ( P2 @ A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_44_Collect__mem__eq,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A] :
% 2.89/1.52 ( ( collect @ A
% 2.89/1.52 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 2.89/1.52 = A3 ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_45_Collect__cong,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ! [X2: A] :
% 2.89/1.52 ( ( P2 @ X2 )
% 2.89/1.52 = ( Q @ X2 ) )
% 2.89/1.52 => ( ( collect @ A @ P2 )
% 2.89/1.52 = ( collect @ A @ Q ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_46_double__zero,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord219039673up_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ A5 @ A5 )
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 = ( A5
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_47_add_Oright__neutral,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( monoid_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_48_add_Oleft__neutral,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( monoid_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_49_zero__reorient,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( zero @ A )
% 2.89/1.52 => ! [X4: A] :
% 2.89/1.52 ( ( ( zero_zero @ A )
% 2.89/1.52 = X4 )
% 2.89/1.52 = ( X4
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_50_add__right__imp__eq,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel_semigroup_add @ A )
% 2.89/1.52 => ! [B4: A,A5: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ B4 @ A5 )
% 2.89/1.52 = ( plus_plus @ A @ C @ A5 ) )
% 2.89/1.52 => ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_51_add__left__imp__eq,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel_semigroup_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ A5 @ B4 )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ C ) )
% 2.89/1.52 => ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_52_add_Oleft__commute,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ab_semigroup_add @ A )
% 2.89/1.52 => ! [B4: A,A5: A,C: A] :
% 2.89/1.52 ( ( plus_plus @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_53_add_Ocommute,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ab_semigroup_add @ A )
% 2.89/1.52 => ( ( plus_plus @ A )
% 2.89/1.52 = ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_54_add_Oright__cancel,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( group_add @ A )
% 2.89/1.52 => ! [B4: A,A5: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ B4 @ A5 )
% 2.89/1.52 = ( plus_plus @ A @ C @ A5 ) )
% 2.89/1.52 = ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_55_add_Oleft__cancel,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( group_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ( plus_plus @ A @ A5 @ B4 )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ C ) )
% 2.89/1.52 = ( B4 = C ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_56_add_Oassoc,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( semigroup_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_57_group__cancel_Oadd2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( comm_monoid_add @ A )
% 2.89/1.52 => ! [B2: A,K: A,B4: A,A5: A] :
% 2.89/1.52 ( ( B2
% 2.89/1.52 = ( plus_plus @ A @ K @ B4 ) )
% 2.89/1.52 => ( ( plus_plus @ A @ A5 @ B2 )
% 2.89/1.52 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_58_group__cancel_Oadd1,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( comm_monoid_add @ A )
% 2.89/1.52 => ! [A3: A,K: A,A5: A,B4: A] :
% 2.89/1.52 ( ( A3
% 2.89/1.52 = ( plus_plus @ A @ K @ A5 ) )
% 2.89/1.52 => ( ( plus_plus @ A @ A3 @ B4 )
% 2.89/1.52 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_59_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.52 ( ( ( I2 = J )
% 2.89/1.52 & ( K = L ) )
% 2.89/1.52 => ( ( plus_plus @ A @ I2 @ K )
% 2.89/1.52 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_60_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ab_semigroup_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_61_one__reorient,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( one @ A )
% 2.89/1.52 => ! [X4: A] :
% 2.89/1.52 ( ( ( one_one @ A )
% 2.89/1.52 = X4 )
% 2.89/1.52 = ( X4
% 2.89/1.52 = ( one_one @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
% 2.89/1.52 ! [P2: nat > $o,K: nat,B4: nat] :
% 2.89/1.52 ( ( P2 @ K )
% 2.89/1.52 => ( ! [Y2: nat] :
% 2.89/1.52 ( ( P2 @ Y2 )
% 2.89/1.52 => ( ord_less_eq @ nat @ Y2 @ B4 ) )
% 2.89/1.52 => ? [X2: nat] :
% 2.89/1.52 ( ( P2 @ X2 )
% 2.89/1.52 & ! [Y3: nat] :
% 2.89/1.52 ( ( P2 @ Y3 )
% 2.89/1.52 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_63_nat__le__linear,axiom,
% 2.89/1.52 ! [M: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ M @ N )
% 2.89/1.52 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_64_le__antisym,axiom,
% 2.89/1.52 ! [M: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ M @ N )
% 2.89/1.52 => ( ( ord_less_eq @ nat @ N @ M )
% 2.89/1.52 => ( M = N ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_65_eq__imp__le,axiom,
% 2.89/1.52 ! [M: nat,N: nat] :
% 2.89/1.52 ( ( M = N )
% 2.89/1.52 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_66_le__trans,axiom,
% 2.89/1.52 ! [I2: nat,J: nat,K: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.52 => ( ( ord_less_eq @ nat @ J @ K )
% 2.89/1.52 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_67_le__refl,axiom,
% 2.89/1.52 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_68_zero__le,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_69_add__le__imp__le__right,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [A5: A,C: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_70_add__le__imp__le__left,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [C: A,A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_71_le__iff__add,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ( ( ord_less_eq @ A )
% 2.89/1.52 = ( ^ [A4: A,B5: A] :
% 2.89/1.52 ? [C2: A] :
% 2.89/1.52 ( B5
% 2.89/1.52 = ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_72_add__right__mono,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_73_less__eqE,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ~ ! [C3: A] :
% 2.89/1.52 ( B4
% 2.89/1.52 != ( plus_plus @ A @ A5 @ C3 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_74_add__left__mono,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_75_add__mono,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A,D: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ C @ D )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_76_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.52 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.89/1.52 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_77_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.52 ( ( ( I2 = J )
% 2.89/1.52 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_78_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere779506340up_add @ A )
% 2.89/1.52 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.52 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.89/1.52 & ( K = L ) )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_79_add_Ogroup__left__neutral,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( group_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_80_add_Ocomm__neutral,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( comm_monoid_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_81_comm__monoid__add__class_Oadd__0,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( comm_monoid_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_82_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.89/1.52 ! [A5: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
% 2.89/1.52 => ( A5
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_83_bot__nat__0_Oextremum__unique,axiom,
% 2.89/1.52 ! [A5: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
% 2.89/1.52 = ( A5
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_84_le__0__eq,axiom,
% 2.89/1.52 ! [N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.89/1.52 = ( N
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_85_less__eq__nat_Osimps_I1_J,axiom,
% 2.89/1.52 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_86_add__eq__self__zero,axiom,
% 2.89/1.52 ! [M: nat,N: nat] :
% 2.89/1.52 ( ( ( plus_plus @ nat @ M @ N )
% 2.89/1.52 = M )
% 2.89/1.52 => ( N
% 2.89/1.52 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_87_plus__nat_Oadd__0,axiom,
% 2.89/1.52 ! [N: nat] :
% 2.89/1.52 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 2.89/1.52 = N ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_88_nat__le__iff__add,axiom,
% 2.89/1.52 ( ( ord_less_eq @ nat )
% 2.89/1.52 = ( ^ [M2: nat,N2: nat] :
% 2.89/1.52 ? [K2: nat] :
% 2.89/1.52 ( N2
% 2.89/1.52 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_89_trans__le__add2,axiom,
% 2.89/1.52 ! [I2: nat,J: nat,M: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.52 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_90_trans__le__add1,axiom,
% 2.89/1.52 ! [I2: nat,J: nat,M: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.52 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_91_add__le__mono1,axiom,
% 2.89/1.52 ! [I2: nat,J: nat,K: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.52 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_92_add__le__mono,axiom,
% 2.89/1.52 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.52 => ( ( ord_less_eq @ nat @ K @ L )
% 2.89/1.52 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_93_le__Suc__ex,axiom,
% 2.89/1.52 ! [K: nat,L: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ K @ L )
% 2.89/1.52 => ? [N3: nat] :
% 2.89/1.52 ( L
% 2.89/1.52 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_94_add__leD2,axiom,
% 2.89/1.52 ! [M: nat,K: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.89/1.52 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_95_add__leD1,axiom,
% 2.89/1.52 ! [M: nat,K: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.89/1.52 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_96_le__add2,axiom,
% 2.89/1.52 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_97_le__add1,axiom,
% 2.89/1.52 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_98_add__leE,axiom,
% 2.89/1.52 ! [M: nat,K: nat,N: nat] :
% 2.89/1.52 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.89/1.52 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 2.89/1.52 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_99_add__nonpos__eq__0__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ( ( plus_plus @ A @ X4 @ Y )
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 = ( ( X4
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 & ( Y
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_100_add__nonneg__eq__0__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.89/1.52 => ( ( ( plus_plus @ A @ X4 @ Y )
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 = ( ( X4
% 2.89/1.52 = ( zero_zero @ A ) )
% 2.89/1.52 & ( Y
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_101_add__nonpos__nonpos,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_102_add__nonneg__nonneg,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_103_add__increasing2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [C: A,B4: A,A5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.52 => ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_104_add__decreasing2,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [C: A,A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_105_add__increasing,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.89/1.52 => ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_106_add__decreasing,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere216010020id_add @ A )
% 2.89/1.52 => ! [A5: A,C: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.52 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_107_sum_Ofinite__Collect__op,axiom,
% 2.89/1.52 ! [A: $tType,B: $tType] :
% 2.89/1.52 ( ( comm_monoid_add @ A )
% 2.89/1.52 => ! [I3: set @ B,X4: B > A,Y: B > A] :
% 2.89/1.52 ( ( finite_finite2 @ B
% 2.89/1.52 @ ( collect @ B
% 2.89/1.52 @ ^ [I: B] :
% 2.89/1.52 ( ( member @ B @ I @ I3 )
% 2.89/1.52 & ( ( X4 @ I )
% 2.89/1.52 != ( zero_zero @ A ) ) ) ) )
% 2.89/1.52 => ( ( finite_finite2 @ B
% 2.89/1.52 @ ( collect @ B
% 2.89/1.52 @ ^ [I: B] :
% 2.89/1.52 ( ( member @ B @ I @ I3 )
% 2.89/1.52 & ( ( Y @ I )
% 2.89/1.52 != ( zero_zero @ A ) ) ) ) )
% 2.89/1.52 => ( finite_finite2 @ B
% 2.89/1.52 @ ( collect @ B
% 2.89/1.52 @ ^ [I: B] :
% 2.89/1.52 ( ( member @ B @ I @ I3 )
% 2.89/1.52 & ( ( plus_plus @ A @ ( X4 @ I ) @ ( Y @ I ) )
% 2.89/1.52 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_108_not__one__le__zero,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord1659791738miring @ A )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_109_zero__le__one,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord1659791738miring @ A )
% 2.89/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_110_finite__less__ub,axiom,
% 2.89/1.52 ! [F3: nat > nat,U: nat] :
% 2.89/1.52 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
% 2.89/1.52 => ( finite_finite2 @ nat
% 2.89/1.52 @ ( collect @ nat
% 2.89/1.52 @ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_111_subset__antisym,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 2.89/1.52 => ( A3 = B2 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_112_subsetI,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ! [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ A3 )
% 2.89/1.52 => ( member @ A @ X2 @ B2 ) )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_113_order__refl,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( preorder @ A )
% 2.89/1.52 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_114_finite__nat__set__iff__bounded__le,axiom,
% 2.89/1.52 ( ( finite_finite2 @ nat )
% 2.89/1.52 = ( ^ [N4: set @ nat] :
% 2.89/1.52 ? [M2: nat] :
% 2.89/1.52 ! [X: nat] :
% 2.89/1.52 ( ( member @ nat @ X @ N4 )
% 2.89/1.52 => ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_115_Euclid__induct,axiom,
% 2.89/1.52 ! [P2: nat > nat > $o,A5: nat,B4: nat] :
% 2.89/1.52 ( ! [A6: nat,B6: nat] :
% 2.89/1.52 ( ( P2 @ A6 @ B6 )
% 2.89/1.52 = ( P2 @ B6 @ A6 ) )
% 2.89/1.52 => ( ! [A6: nat] : ( P2 @ A6 @ ( zero_zero @ nat ) )
% 2.89/1.52 => ( ! [A6: nat,B6: nat] :
% 2.89/1.52 ( ( P2 @ A6 @ B6 )
% 2.89/1.52 => ( P2 @ A6 @ ( plus_plus @ nat @ A6 @ B6 ) ) )
% 2.89/1.52 => ( P2 @ A5 @ B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_116_le__funD,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType] :
% 2.89/1.52 ( ( ord @ B )
% 2.89/1.52 => ! [F3: A > B,G: A > B,X4: A] :
% 2.89/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.89/1.52 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_117_le__funE,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType] :
% 2.89/1.52 ( ( ord @ B )
% 2.89/1.52 => ! [F3: A > B,G: A > B,X4: A] :
% 2.89/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.89/1.52 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_118_le__funI,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType] :
% 2.89/1.52 ( ( ord @ B )
% 2.89/1.52 => ! [F3: A > B,G: A > B] :
% 2.89/1.52 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) )
% 2.89/1.52 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_119_le__fun__def,axiom,
% 2.89/1.52 ! [B: $tType,A: $tType] :
% 2.89/1.52 ( ( ord @ B )
% 2.89/1.52 => ( ( ord_less_eq @ ( A > B ) )
% 2.89/1.52 = ( ^ [F: A > B,G2: A > B] :
% 2.89/1.52 ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_120_order__subst1,axiom,
% 2.89/1.52 ! [A: $tType,B: $tType] :
% 2.89/1.52 ( ( ( order @ B )
% 2.89/1.52 & ( order @ A ) )
% 2.89/1.52 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
% 2.89/1.52 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.89/1.52 => ( ! [X2: B,Y2: B] :
% 2.89/1.52 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_121_order__subst2,axiom,
% 2.89/1.52 ! [A: $tType,C4: $tType] :
% 2.89/1.52 ( ( ( order @ C4 )
% 2.89/1.52 & ( order @ A ) )
% 2.89/1.52 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
% 2.89/1.52 => ( ! [X2: A,Y2: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.52 => ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.52 => ( ord_less_eq @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_122_ord__eq__le__subst,axiom,
% 2.89/1.52 ! [A: $tType,B: $tType] :
% 2.89/1.52 ( ( ( ord @ B )
% 2.89/1.52 & ( ord @ A ) )
% 2.89/1.52 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 2.89/1.52 ( ( A5
% 2.89/1.52 = ( F3 @ B4 ) )
% 2.89/1.52 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.89/1.52 => ( ! [X2: B,Y2: B] :
% 2.89/1.52 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.89/1.52 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_123_ord__le__eq__subst,axiom,
% 2.89/1.52 ! [A: $tType,B: $tType] :
% 2.89/1.52 ( ( ( ord @ B )
% 2.89/1.52 & ( ord @ A ) )
% 2.89/1.52 => ! [A5: A,B4: A,F3: A > B,C: B] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( ( F3 @ B4 )
% 2.89/1.52 = C )
% 2.89/1.52 => ( ! [X2: A,Y2: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.52 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.52 => ( ord_less_eq @ B @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_124_eq__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.89/1.52 = ( ^ [X: A,Y5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.89/1.52 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_125_antisym,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 => ( ( ord_less_eq @ A @ Y @ X4 )
% 2.89/1.52 => ( X4 = Y ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_126_linear,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linorder @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 | ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_127_eq__refl,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( preorder @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ( X4 = Y )
% 2.89/1.52 => ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_128_le__cases,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linorder @ A )
% 2.89/1.52 => ! [X4: A,Y: A] :
% 2.89/1.52 ( ~ ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 => ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_129_order_Otrans,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_130_le__cases3,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linorder @ A )
% 2.89/1.52 => ! [X4: A,Y: A,Z2: A] :
% 2.89/1.52 ( ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 2.89/1.52 => ( ( ( ord_less_eq @ A @ Y @ X4 )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
% 2.89/1.52 => ( ( ( ord_less_eq @ A @ X4 @ Z2 )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 2.89/1.52 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ Y @ X4 ) )
% 2.89/1.52 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
% 2.89/1.52 => ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
% 2.89/1.52 => ~ ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_131_antisym__conv,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [Y: A,X4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ Y @ X4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 = ( X4 = Y ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_132_order__class_Oorder_Oeq__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.89/1.52 = ( ^ [A4: A,B5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A4 @ B5 )
% 2.89/1.52 & ( ord_less_eq @ A @ B5 @ A4 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_133_ord__eq__le__trans,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ord @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( A5 = B4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_134_ord__le__eq__trans,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ord @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( B4 = C )
% 2.89/1.52 => ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_135_order__class_Oorder_Oantisym,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.52 => ( A5 = B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_136_order__trans,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( preorder @ A )
% 2.89/1.52 => ! [X4: A,Y: A,Z2: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.52 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.89/1.52 => ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_137_dual__order_Orefl,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_138_linorder__wlog,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linorder @ A )
% 2.89/1.52 => ! [P2: A > A > $o,A5: A,B4: A] :
% 2.89/1.52 ( ! [A6: A,B6: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.89/1.52 => ( P2 @ A6 @ B6 ) )
% 2.89/1.52 => ( ! [A6: A,B6: A] :
% 2.89/1.52 ( ( P2 @ B6 @ A6 )
% 2.89/1.52 => ( P2 @ A6 @ B6 ) )
% 2.89/1.52 => ( P2 @ A5 @ B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_139_dual__order_Otrans,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [B4: A,A5: A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.89/1.52 => ( ord_less_eq @ A @ C @ A5 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_140_dual__order_Oeq__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ( ( ^ [Y4: A,Z: A] : ( Y4 = Z ) )
% 2.89/1.52 = ( ^ [A4: A,B5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ B5 @ A4 )
% 2.89/1.52 & ( ord_less_eq @ A @ A4 @ B5 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_141_dual__order_Oantisym,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( order @ A )
% 2.89/1.52 => ! [B4: A,A5: A] :
% 2.89/1.52 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.52 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 => ( A5 = B4 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_142_in__mono,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( ( member @ A @ X4 @ A3 )
% 2.89/1.52 => ( member @ A @ X4 @ B2 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_143_subsetD,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( ( member @ A @ C @ A3 )
% 2.89/1.52 => ( member @ A @ C @ B2 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_144_equalityE,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ( A3 = B2 )
% 2.89/1.52 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_145_subset__eq,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) )
% 2.89/1.52 = ( ^ [A2: set @ A,B3: set @ A] :
% 2.89/1.52 ! [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ A2 )
% 2.89/1.52 => ( member @ A @ X @ B3 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_146_equalityD1,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ( A3 = B2 )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_147_equalityD2,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.89/1.52 ( ( A3 = B2 )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_148_subset__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) )
% 2.89/1.52 = ( ^ [A2: set @ A,B3: set @ A] :
% 2.89/1.52 ! [T2: A] :
% 2.89/1.52 ( ( member @ A @ T2 @ A2 )
% 2.89/1.52 => ( member @ A @ T2 @ B3 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_149_subset__refl,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_150_Collect__mono,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ! [X2: A] :
% 2.89/1.52 ( ( P2 @ X2 )
% 2.89/1.52 => ( Q @ X2 ) )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_151_subset__trans,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,B2: set @ A,C5: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.89/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_152_set__eq__subset,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 2.89/1.52 = ( ^ [A2: set @ A,B3: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 2.89/1.52 & ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_153_Collect__mono__iff,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 2.89/1.52 = ( ! [X: A] :
% 2.89/1.52 ( ( P2 @ X )
% 2.89/1.52 => ( Q @ X ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_154_bounded__Max__nat,axiom,
% 2.89/1.52 ! [P2: nat > $o,X4: nat,M3: nat] :
% 2.89/1.52 ( ( P2 @ X4 )
% 2.89/1.52 => ( ! [X2: nat] :
% 2.89/1.52 ( ( P2 @ X2 )
% 2.89/1.52 => ( ord_less_eq @ nat @ X2 @ M3 ) )
% 2.89/1.52 => ~ ! [M4: nat] :
% 2.89/1.52 ( ( P2 @ M4 )
% 2.89/1.52 => ~ ! [X3: nat] :
% 2.89/1.52 ( ( P2 @ X3 )
% 2.89/1.52 => ( ord_less_eq @ nat @ X3 @ M4 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_155_Collect__subset,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,P2: A > $o] :
% 2.89/1.52 ( ord_less_eq @ ( set @ A )
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ A3 )
% 2.89/1.52 & ( P2 @ X ) ) )
% 2.89/1.52 @ A3 ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_156_less__eq__set__def,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) )
% 2.89/1.52 = ( ^ [A2: set @ A,B3: set @ A] :
% 2.89/1.52 ( ord_less_eq @ ( A > $o )
% 2.89/1.52 @ ^ [X: A] : ( member @ A @ X @ A2 )
% 2.89/1.52 @ ^ [X: A] : ( member @ A @ X @ B3 ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_157_zero__neq__one,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( zero_neq_one @ A )
% 2.89/1.52 => ( ( zero_zero @ A )
% 2.89/1.52 != ( one_one @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_158_le__numeral__extra_I4_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord1659791738miring @ A )
% 2.89/1.52 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_159_verit__sum__simplify,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( cancel1352612707id_add @ A )
% 2.89/1.52 => ! [A5: A] :
% 2.89/1.52 ( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.52 = A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_160_add__0__iff,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( semiri456707255roduct @ A )
% 2.89/1.52 => ! [B4: A,A5: A] :
% 2.89/1.52 ( ( B4
% 2.89/1.52 = ( plus_plus @ A @ B4 @ A5 ) )
% 2.89/1.52 = ( A5
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_161_le__numeral__extra_I3_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linord1659791738miring @ A )
% 2.89/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_162_predicate1I,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ! [X2: A] :
% 2.89/1.52 ( ( P2 @ X2 )
% 2.89/1.52 => ( Q @ X2 ) )
% 2.89/1.52 => ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_163_predicate1D,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,Q: A > $o,X4: A] :
% 2.89/1.52 ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 2.89/1.52 => ( ( P2 @ X4 )
% 2.89/1.52 => ( Q @ X4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_164_rev__predicate1D,axiom,
% 2.89/1.52 ! [A: $tType,P2: A > $o,X4: A,Q: A > $o] :
% 2.89/1.52 ( ( P2 @ X4 )
% 2.89/1.52 => ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
% 2.89/1.52 => ( Q @ X4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_165_verit__la__disequality,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( linorder @ A )
% 2.89/1.52 => ! [A5: A,B4: A] :
% 2.89/1.52 ( ( A5 = B4 )
% 2.89/1.52 | ~ ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.52 | ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_166_is__num__normalize_I1_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( neg_numeral @ A )
% 2.89/1.52 => ! [A5: A,B4: A,C: A] :
% 2.89/1.52 ( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
% 2.89/1.52 = ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_167_pred__subset__eq,axiom,
% 2.89/1.52 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( A > $o )
% 2.89/1.52 @ ^ [X: A] : ( member @ A @ X @ R )
% 2.89/1.52 @ ^ [X: A] : ( member @ A @ X @ S ) )
% 2.89/1.52 = ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_168_dbl__inc__simps_I2_J,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( neg_numeral @ A )
% 2.89/1.52 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 2.89/1.52 = ( one_one @ A ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_169_conj__subset__def,axiom,
% 2.89/1.52 ! [A: $tType,A3: set @ A,P2: A > $o,Q: A > $o] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( P2 @ X )
% 2.89/1.52 & ( Q @ X ) ) ) )
% 2.89/1.52 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P2 ) )
% 2.89/1.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_170_dbl__inc__def,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( neg_numeral @ A )
% 2.89/1.52 => ( ( neg_numeral_dbl_inc @ A )
% 2.89/1.52 = ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_171_subset__Collect__iff,axiom,
% 2.89/1.52 ! [A: $tType,B2: set @ A,A3: set @ A,P2: A > $o] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 2.89/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B2
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ A3 )
% 2.89/1.52 & ( P2 @ X ) ) ) )
% 2.89/1.52 = ( ! [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ B2 )
% 2.89/1.52 => ( P2 @ X ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_172_subset__CollectI,axiom,
% 2.89/1.52 ! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P2: A > $o] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 2.89/1.52 => ( ! [X2: A] :
% 2.89/1.52 ( ( member @ A @ X2 @ B2 )
% 2.89/1.52 => ( ( Q @ X2 )
% 2.89/1.52 => ( P2 @ X2 ) ) )
% 2.89/1.52 => ( ord_less_eq @ ( set @ A )
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ B2 )
% 2.89/1.52 & ( Q @ X ) ) )
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ A3 )
% 2.89/1.52 & ( P2 @ X ) ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_173_Collect__restrict,axiom,
% 2.89/1.52 ! [A: $tType,X5: set @ A,P2: A > $o] :
% 2.89/1.52 ( ord_less_eq @ ( set @ A )
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ X5 )
% 2.89/1.52 & ( P2 @ X ) ) )
% 2.89/1.52 @ X5 ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_174_prop__restrict,axiom,
% 2.89/1.52 ! [A: $tType,X4: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
% 2.89/1.52 ( ( member @ A @ X4 @ Z3 )
% 2.89/1.52 => ( ( ord_less_eq @ ( set @ A ) @ Z3
% 2.89/1.52 @ ( collect @ A
% 2.89/1.52 @ ^ [X: A] :
% 2.89/1.52 ( ( member @ A @ X @ X5 )
% 2.89/1.52 & ( P2 @ X ) ) ) )
% 2.89/1.52 => ( P2 @ X4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_175_fun__cong__unused__0,axiom,
% 2.89/1.52 ! [A: $tType,B: $tType,C4: $tType] :
% 2.89/1.52 ( ( zero @ B )
% 2.89/1.52 => ! [F3: ( A > B ) > C4,G: C4] :
% 2.89/1.52 ( ( F3
% 2.89/1.52 = ( ^ [X: A > B] : G ) )
% 2.89/1.52 => ( ( F3
% 2.89/1.52 @ ^ [X: A] : ( zero_zero @ B ) )
% 2.89/1.52 = G ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_176_Fpow__def,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( finite_Fpow @ A )
% 2.89/1.52 = ( ^ [A2: set @ A] :
% 2.89/1.52 ( collect @ ( set @ A )
% 2.89/1.52 @ ^ [X6: set @ A] :
% 2.89/1.52 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
% 2.89/1.52 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_177_partitions__imp__finite__elements,axiom,
% 2.89/1.52 ! [P3: nat > nat,N: nat] :
% 2.89/1.52 ( ( number2016821345itions @ P3 @ N )
% 2.89/1.52 => ( finite_finite2 @ nat
% 2.89/1.52 @ ( collect @ nat
% 2.89/1.52 @ ^ [I: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P3 @ I ) ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_178_not__gr__zero,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( canoni770627133id_add @ A )
% 2.89/1.52 => ! [N: A] :
% 2.89/1.52 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.89/1.52 = ( N
% 2.89/1.52 = ( zero_zero @ A ) ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_179_add__less__cancel__right,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [A5: A,C: A,B4: A] :
% 2.89/1.52 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
% 2.89/1.52 = ( ord_less @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_180_add__less__cancel__left,axiom,
% 2.89/1.52 ! [A: $tType] :
% 2.89/1.52 ( ( ordere236663937imp_le @ A )
% 2.89/1.52 => ! [C: A,A5: A,B4: A] :
% 2.89/1.52 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
% 2.89/1.52 = ( ord_less @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_181_bot__nat__0_Onot__eq__extremum,axiom,
% 2.89/1.52 ! [A5: nat] :
% 2.89/1.52 ( ( A5
% 2.89/1.52 != ( zero_zero @ nat ) )
% 2.89/1.52 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A5 ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_182_less__nat__zero__code,axiom,
% 2.89/1.52 ! [N: nat] :
% 2.89/1.52 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.89/1.52
% 2.89/1.52 thf(fact_183_neq0__conv,axiom,
% 2.89/1.52 ! [N: nat] :
% 2.89/1.52 ( ( N
% 2.89/1.52 != ( zero_zero @ nat ) )
% 2.89/1.52 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_184_nat__add__left__cancel__less,axiom,
% 2.89/1.53 ! [K: nat,M: nat,N: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.89/1.53 = ( ord_less @ nat @ M @ N ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_185_finite__Collect__less__nat,axiom,
% 2.89/1.53 ! [K: nat] :
% 2.89/1.53 ( finite_finite2 @ nat
% 2.89/1.53 @ ( collect @ nat
% 2.89/1.53 @ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_186_add__less__same__cancel1,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere516151231imp_le @ A )
% 2.89/1.53 => ! [B4: A,A5: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
% 2.89/1.53 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_187_add__less__same__cancel2,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere516151231imp_le @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
% 2.89/1.53 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_188_less__add__same__cancel1,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere516151231imp_le @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
% 2.89/1.53 = ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_189_less__add__same__cancel2,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere516151231imp_le @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
% 2.89/1.53 = ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_190_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linord219039673up_add @ A )
% 2.89/1.53 => ! [A5: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
% 2.89/1.53 = ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_191_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linord219039673up_add @ A )
% 2.89/1.53 => ! [A5: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
% 2.89/1.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_192_add__gr__0,axiom,
% 2.89/1.53 ! [M: nat,N: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 2.89/1.53 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.89/1.53 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_193_less__one,axiom,
% 2.89/1.53 ! [N: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 2.89/1.53 = ( N
% 2.89/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_194_add__less__le__mono,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere223160158up_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A,C: A,D: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less_eq @ A @ C @ D )
% 2.89/1.53 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_195_add__le__less__mono,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere223160158up_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A,C: A,D: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less @ A @ C @ D )
% 2.89/1.53 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_196_add__mono__thms__linordered__field_I3_J,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere223160158up_add @ A )
% 2.89/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.53 ( ( ( ord_less @ A @ I2 @ J )
% 2.89/1.53 & ( ord_less_eq @ A @ K @ L ) )
% 2.89/1.53 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_197_add__mono__thms__linordered__field_I4_J,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere223160158up_add @ A )
% 2.89/1.53 => ! [I2: A,J: A,K: A,L: A] :
% 2.89/1.53 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.89/1.53 & ( ord_less @ A @ K @ L ) )
% 2.89/1.53 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_198_pos__add__strict,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( strict797366125id_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A,C: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.53 => ( ( ord_less @ A @ B4 @ C )
% 2.89/1.53 => ( ord_less @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_199_canonically__ordered__monoid__add__class_OlessE,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( canoni770627133id_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ B4 )
% 2.89/1.53 => ~ ! [C3: A] :
% 2.89/1.53 ( ( B4
% 2.89/1.53 = ( plus_plus @ A @ A5 @ C3 ) )
% 2.89/1.53 => ( C3
% 2.89/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_200_add__pos__pos,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere216010020id_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
% 2.89/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 2.89/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_201_add__neg__neg,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( ordere216010020id_add @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
% 2.89/1.53 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 2.89/1.53 => ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_202_add__less__zeroD,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linordered_idom @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less @ A @ ( plus_plus @ A @ X4 @ Y ) @ ( zero_zero @ A ) )
% 2.89/1.53 => ( ( ord_less @ A @ X4 @ ( zero_zero @ A ) )
% 2.89/1.53 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_203_order_Onot__eq__order__implies__strict,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( A5 != B4 )
% 2.89/1.53 => ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.53 => ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_204_dual__order_Ostrict__implies__order,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [B4: A,A5: A] :
% 2.89/1.53 ( ( ord_less @ A @ B4 @ A5 )
% 2.89/1.53 => ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_205_dual__order_Ostrict__iff__order,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less @ A )
% 2.89/1.53 = ( ^ [B5: A,A4: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ B5 @ A4 )
% 2.89/1.53 & ( A4 != B5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_206_dual__order_Oorder__iff__strict,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less_eq @ A )
% 2.89/1.53 = ( ^ [B5: A,A4: A] :
% 2.89/1.53 ( ( ord_less @ A @ B5 @ A4 )
% 2.89/1.53 | ( A4 = B5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_207_order_Ostrict__implies__order,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ B4 )
% 2.89/1.53 => ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_208_dense__le__bounded,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( dense_linorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A,Z2: A] :
% 2.89/1.53 ( ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 => ( ! [W: A] :
% 2.89/1.53 ( ( ord_less @ A @ X4 @ W )
% 2.89/1.53 => ( ( ord_less @ A @ W @ Y )
% 2.89/1.53 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_209_dense__ge__bounded,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( dense_linorder @ A )
% 2.89/1.53 => ! [Z2: A,X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less @ A @ Z2 @ X4 )
% 2.89/1.53 => ( ! [W: A] :
% 2.89/1.53 ( ( ord_less @ A @ Z2 @ W )
% 2.89/1.53 => ( ( ord_less @ A @ W @ X4 )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_210_dual__order_Ostrict__trans2,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [B4: A,A5: A,C: A] :
% 2.89/1.53 ( ( ord_less @ A @ B4 @ A5 )
% 2.89/1.53 => ( ( ord_less_eq @ A @ C @ B4 )
% 2.89/1.53 => ( ord_less @ A @ C @ A5 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_211_dual__order_Ostrict__trans1,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [B4: A,A5: A,C: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ B4 @ A5 )
% 2.89/1.53 => ( ( ord_less @ A @ C @ B4 )
% 2.89/1.53 => ( ord_less @ A @ C @ A5 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_212_order_Ostrict__iff__order,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less @ A )
% 2.89/1.53 = ( ^ [A4: A,B5: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A4 @ B5 )
% 2.89/1.53 & ( A4 != B5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_213_order_Oorder__iff__strict,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less_eq @ A )
% 2.89/1.53 = ( ^ [A4: A,B5: A] :
% 2.89/1.53 ( ( ord_less @ A @ A4 @ B5 )
% 2.89/1.53 | ( A4 = B5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_214_order_Ostrict__trans2,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A,B4: A,C: A] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less_eq @ A @ B4 @ C )
% 2.89/1.53 => ( ord_less @ A @ A5 @ C ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_215_order_Ostrict__trans1,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A,B4: A,C: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less @ A @ B4 @ C )
% 2.89/1.53 => ( ord_less @ A @ A5 @ C ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_216_not__le__imp__less,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linorder @ A )
% 2.89/1.53 => ! [Y: A,X4: A] :
% 2.89/1.53 ( ~ ( ord_less_eq @ A @ Y @ X4 )
% 2.89/1.53 => ( ord_less @ A @ X4 @ Y ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_217_less__le__not__le,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( preorder @ A )
% 2.89/1.53 => ( ( ord_less @ A )
% 2.89/1.53 = ( ^ [X: A,Y5: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.89/1.53 & ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_218_le__imp__less__or__eq,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.53 => ( ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 | ( X4 = Y ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_219_le__less__linear,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.53 | ( ord_less @ A @ Y @ X4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_220_dense__le,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( dense_linorder @ A )
% 2.89/1.53 => ! [Y: A,Z2: A] :
% 2.89/1.53 ( ! [X2: A] :
% 2.89/1.53 ( ( ord_less @ A @ X2 @ Y )
% 2.89/1.53 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_221_dense__ge,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( dense_linorder @ A )
% 2.89/1.53 => ! [Z2: A,Y: A] :
% 2.89/1.53 ( ! [X2: A] :
% 2.89/1.53 ( ( ord_less @ A @ Z2 @ X2 )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ X2 ) )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_222_less__le__trans,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( preorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A,Z2: A] :
% 2.89/1.53 ( ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.89/1.53 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_223_le__less__trans,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( preorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A,Z2: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.53 => ( ( ord_less @ A @ Y @ Z2 )
% 2.89/1.53 => ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_224_less__imp__le,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( preorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 => ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_225_antisym__conv2,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.53 => ( ( ~ ( ord_less @ A @ X4 @ Y ) )
% 2.89/1.53 = ( X4 = Y ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_226_antisym__conv1,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ~ ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 => ( ( ord_less_eq @ A @ X4 @ Y )
% 2.89/1.53 = ( X4 = Y ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_227_le__neq__trans,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A,B4: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( A5 != B4 )
% 2.89/1.53 => ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_228_not__less,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ~ ( ord_less @ A @ X4 @ Y ) )
% 2.89/1.53 = ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_229_not__le,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ( ~ ( ord_less_eq @ A @ X4 @ Y ) )
% 2.89/1.53 = ( ord_less @ A @ Y @ X4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_230_order__less__le__subst2,axiom,
% 2.89/1.53 ! [A: $tType,C4: $tType] :
% 2.89/1.53 ( ( ( order @ C4 )
% 2.89/1.53 & ( order @ A ) )
% 2.89/1.53 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
% 2.89/1.53 => ( ! [X2: A,Y2: A] :
% 2.89/1.53 ( ( ord_less @ A @ X2 @ Y2 )
% 2.89/1.53 => ( ord_less @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.53 => ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_231_order__less__le__subst1,axiom,
% 2.89/1.53 ! [A: $tType,B: $tType] :
% 2.89/1.53 ( ( ( order @ B )
% 2.89/1.53 & ( order @ A ) )
% 2.89/1.53 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 2.89/1.53 ( ( ord_less @ A @ A5 @ ( F3 @ B4 ) )
% 2.89/1.53 => ( ( ord_less_eq @ B @ B4 @ C )
% 2.89/1.53 => ( ! [X2: B,Y2: B] :
% 2.89/1.53 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.89/1.53 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.53 => ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_232_order__le__less__subst2,axiom,
% 2.89/1.53 ! [A: $tType,C4: $tType] :
% 2.89/1.53 ( ( ( order @ C4 )
% 2.89/1.53 & ( order @ A ) )
% 2.89/1.53 => ! [A5: A,B4: A,F3: A > C4,C: C4] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A5 @ B4 )
% 2.89/1.53 => ( ( ord_less @ C4 @ ( F3 @ B4 ) @ C )
% 2.89/1.53 => ( ! [X2: A,Y2: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.89/1.53 => ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.53 => ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_233_order__le__less__subst1,axiom,
% 2.89/1.53 ! [A: $tType,B: $tType] :
% 2.89/1.53 ( ( ( order @ B )
% 2.89/1.53 & ( order @ A ) )
% 2.89/1.53 => ! [A5: A,F3: B > A,B4: B,C: B] :
% 2.89/1.53 ( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
% 2.89/1.53 => ( ( ord_less @ B @ B4 @ C )
% 2.89/1.53 => ( ! [X2: B,Y2: B] :
% 2.89/1.53 ( ( ord_less @ B @ X2 @ Y2 )
% 2.89/1.53 => ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.89/1.53 => ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_234_less__le,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less @ A )
% 2.89/1.53 = ( ^ [X: A,Y5: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ X @ Y5 )
% 2.89/1.53 & ( X != Y5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_235_le__less,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ( ( ord_less_eq @ A )
% 2.89/1.53 = ( ^ [X: A,Y5: A] :
% 2.89/1.53 ( ( ord_less @ A @ X @ Y5 )
% 2.89/1.53 | ( X = Y5 ) ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_236_leI,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linorder @ A )
% 2.89/1.53 => ! [X4: A,Y: A] :
% 2.89/1.53 ( ~ ( ord_less @ A @ X4 @ Y )
% 2.89/1.53 => ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_237_leD,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [Y: A,X4: A] :
% 2.89/1.53 ( ( ord_less_eq @ A @ Y @ X4 )
% 2.89/1.53 => ~ ( ord_less @ A @ X4 @ Y ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_238_less__numeral__extra_I4_J,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linord1659791738miring @ A )
% 2.89/1.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_239_nat__less__le,axiom,
% 2.89/1.53 ( ( ord_less @ nat )
% 2.89/1.53 = ( ^ [M2: nat,N2: nat] :
% 2.89/1.53 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.89/1.53 & ( M2 != N2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_240_less__imp__le__nat,axiom,
% 2.89/1.53 ! [M: nat,N: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ M @ N )
% 2.89/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_241_le__eq__less__or__eq,axiom,
% 2.89/1.53 ( ( ord_less_eq @ nat )
% 2.89/1.53 = ( ^ [M2: nat,N2: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ M2 @ N2 )
% 2.89/1.53 | ( M2 = N2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_242_less__or__eq__imp__le,axiom,
% 2.89/1.53 ! [M: nat,N: nat] :
% 2.89/1.53 ( ( ( ord_less @ nat @ M @ N )
% 2.89/1.53 | ( M = N ) )
% 2.89/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_243_le__neq__implies__less,axiom,
% 2.89/1.53 ! [M: nat,N: nat] :
% 2.89/1.53 ( ( ord_less_eq @ nat @ M @ N )
% 2.89/1.53 => ( ( M != N )
% 2.89/1.53 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_244_less__mono__imp__le__mono,axiom,
% 2.89/1.53 ! [F3: nat > nat,I2: nat,J: nat] :
% 2.89/1.53 ( ! [I4: nat,J2: nat] :
% 2.89/1.53 ( ( ord_less @ nat @ I4 @ J2 )
% 2.89/1.53 => ( ord_less @ nat @ ( F3 @ I4 ) @ ( F3 @ J2 ) ) )
% 2.89/1.53 => ( ( ord_less_eq @ nat @ I2 @ J )
% 2.89/1.53 => ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_245_verit__comp__simplify1_I1_J,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( order @ A )
% 2.89/1.53 => ! [A5: A] :
% 2.89/1.53 ~ ( ord_less @ A @ A5 @ A5 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_246_less__numeral__extra_I3_J,axiom,
% 2.89/1.53 ! [A: $tType] :
% 2.89/1.53 ( ( linord1659791738miring @ A )
% 2.89/1.53 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_247_verit__comp__simplify1_I3_J,axiom,
% 2.89/1.53 ! [B: $tType] :
% 2.89/1.53 ( ( linorder @ B )
% 2.89/1.53 => ! [B7: B,A7: B] :
% 2.89/1.53 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
% 2.89/1.53 = ( ord_less @ B @ A7 @ B7 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_248_bounded__nat__set__is__finite,axiom,
% 2.89/1.53 ! [N5: set @ nat,N: nat] :
% 2.89/1.53 ( ! [X2: nat] :
% 2.89/1.53 ( ( member @ nat @ X2 @ N5 )
% 2.89/1.53 => ( ord_less @ nat @ X2 @ N ) )
% 2.89/1.53 => ( finite_finite2 @ nat @ N5 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_249_finite__nat__set__iff__bounded,axiom,
% 2.89/1.53 ( ( finite_finite2 @ nat )
% 2.89/1.53 = ( ^ [N4: set @ nat] :
% 2.89/1.53 ? [M2: nat] :
% 2.89/1.53 ! [X: nat] :
% 2.89/1.53 ( ( member @ nat @ X @ N4 )
% 2.89/1.53 => ( ord_less @ nat @ X @ M2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_250_finite__M__bounded__by__nat,axiom,
% 2.89/1.53 ! [P2: nat > $o,I2: nat] :
% 2.89/1.53 ( finite_finite2 @ nat
% 2.89/1.53 @ ( collect @ nat
% 2.89/1.53 @ ^ [K2: nat] :
% 2.89/1.53 ( ( P2 @ K2 )
% 2.89/1.53 & ( ord_less @ nat @ K2 @ I2 ) ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_251_infinite__descent__measure,axiom,
% 2.89/1.53 ! [A: $tType,P2: A > $o,V: A > nat,X4: A] :
% 2.89/1.53 ( ! [X2: A] :
% 2.89/1.53 ( ~ ( P2 @ X2 )
% 2.89/1.53 => ? [Y3: A] :
% 2.89/1.53 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 2.89/1.53 & ~ ( P2 @ Y3 ) ) )
% 2.89/1.53 => ( P2 @ X4 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(fact_252_measure__induct__rule,axiom,
% 2.89/1.53 ! [B: $tType,A: $tType] :
% 2.89/1.53 ( ( wellorder @ B )
% 2.89/1.53 => ! [F3: A > B,P2: A > $o,A5: A] :
% 2.89/1.53 ( ! [X2: A] :
% 2.89/1.53 ( ! [Y3: A] :
% 2.89/1.53 ( ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X2 ) )
% 2.89/1.53 => ( P2 @ Y3 ) )
% 2.89/1.53 => ( P2 @ X2 ) )
% 2.89/1.53 => ( P2 @ A5 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.89/1.53 ! [A8: $tType,A9: $tType] :
% 2.89/1.53 ( ( preorder @ A9 )
% 2.89/1.53 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.89/1.53 ! [A8: $tType,A9: $tType] :
% 2.89/1.53 ( ( ( finite_finite @ A8 )
% 2.89/1.53 & ( finite_finite @ A9 ) )
% 2.89/1.53 => ( finite_finite @ ( A8 > A9 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.89/1.53 ! [A8: $tType,A9: $tType] :
% 2.89/1.53 ( ( order @ A9 )
% 2.89/1.53 => ( order @ ( A8 > A9 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_fun___Orderings_Oord,axiom,
% 2.89/1.53 ! [A8: $tType,A9: $tType] :
% 2.89/1.53 ( ( ord @ A9 )
% 2.89/1.53 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 2.89/1.53 semiri456707255roduct @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.89/1.53 ordere516151231imp_le @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.89/1.53 ordere223160158up_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.89/1.53 ordere236663937imp_le @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 2.89/1.53 strict797366125id_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.89/1.53 canoni770627133id_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.89/1.53 linord1659791738miring @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.89/1.53 ordere779506340up_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 2.89/1.53 ordere216010020id_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.89/1.53 cancel1352612707id_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.89/1.53 cancel_semigroup_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.89/1.53 ab_semigroup_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.89/1.53 comm_monoid_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.89/1.53 semigroup_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.89/1.53 wellorder @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 2.89/1.53 zero_neq_one @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.89/1.53 preorder @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.89/1.53 linorder @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 2.89/1.53 monoid_add @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.89/1.53 order @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.89/1.53 ord @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.89/1.53 zero @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.89/1.53 one @ nat ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.89/1.53 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.89/1.53 ! [A8: $tType] :
% 2.89/1.53 ( ( finite_finite @ A8 )
% 2.89/1.53 => ( finite_finite @ ( set @ A8 ) ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.89/1.53 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.89/1.53 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.89/1.53 preorder @ $o ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 2.89/1.53 linorder @ $o ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 2.89/1.53 finite_finite @ $o ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 2.89/1.53 order @ $o ).
% 2.89/1.53
% 2.89/1.53 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 2.89/1.53 ord @ $o ).
% 2.89/1.53
% 2.89/1.53 thf(conj_0,conjecture,
% 2.89/1.53 ( finite_finite2 @ ( nat > nat )
% 2.89/1.53 @ ( collect @ ( nat > nat )
% 2.89/1.53 @ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) ) ) ).
% 2.89/1.53 ------------------------
% 2.89/1.54 % (14124)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_13930 for (2999ds/27Mi)
% 2.89/1.54 % (14123)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_13930 for (2999ds/4Mi)
% 2.89/1.54 % (14122)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/183Mi)
% 2.89/1.55 % (14123)Instruction limit reached!
% 2.89/1.55 % (14123)------------------------------
% 2.89/1.55 % (14123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.55 % (14123)Termination reason: Unknown
% 2.89/1.55 % (14123)Termination phase: shuffling
% 2.89/1.55
% 2.89/1.55 % (14123)Memory used [KB]: 1407
% 2.89/1.55 % (14123)Time elapsed: 0.003 s
% 2.89/1.55 % (14123)Instructions burned: 6 (million)
% 2.89/1.55 % (14123)------------------------------
% 2.89/1.55 % (14123)------------------------------
% 2.89/1.55 % (14126)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_13930 for (2999ds/2Mi)
% 2.89/1.55 % (14128)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/18Mi)
% 2.89/1.55 % (14125)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/2Mi)
% 2.89/1.55 % (14127)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_13930 for (2999ds/275Mi)
% 2.89/1.55 % (14125)Instruction limit reached!
% 2.89/1.55 % (14125)------------------------------
% 2.89/1.55 % (14125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.55 % (14125)Termination reason: Unknown
% 2.89/1.55 % (14126)Instruction limit reached!
% 2.89/1.55 % (14126)------------------------------
% 2.89/1.55 % (14126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.55 % (14126)Termination reason: Unknown
% 2.89/1.55 % (14126)Termination phase: shuffling
% 2.89/1.55
% 2.89/1.55 % (14126)Memory used [KB]: 1407
% 2.89/1.55 % (14126)Time elapsed: 0.004 s
% 2.89/1.55 % (14126)Instructions burned: 3 (million)
% 2.89/1.55 % (14126)------------------------------
% 2.89/1.55 % (14126)------------------------------
% 2.89/1.55 % (14125)Termination phase: shuffling
% 2.89/1.55
% 2.89/1.55 % (14125)Memory used [KB]: 1279
% 2.89/1.55 % (14124)Instruction limit reached!
% 2.89/1.55 % (14124)------------------------------
% 2.89/1.55 % (14124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.55 % (14124)Termination reason: Unknown
% 2.89/1.55 % (14124)Termination phase: Property scanning
% 2.89/1.55
% 2.89/1.55 % (14124)Memory used [KB]: 1791
% 2.89/1.55 % (14124)Time elapsed: 0.010 s
% 2.89/1.55 % (14124)Instructions burned: 27 (million)
% 2.89/1.55 % (14124)------------------------------
% 2.89/1.55 % (14124)------------------------------
% 2.89/1.55 % (14125)Time elapsed: 0.003 s
% 2.89/1.55 % (14125)Instructions burned: 2 (million)
% 2.89/1.55 % (14125)------------------------------
% 2.89/1.55 % (14125)------------------------------
% 2.89/1.56 % (14129)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/3Mi)
% 2.89/1.56 % (14129)Instruction limit reached!
% 2.89/1.56 % (14129)------------------------------
% 2.89/1.56 % (14129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.56 % (14129)Termination reason: Unknown
% 2.89/1.56 % (14129)Termination phase: shuffling
% 2.89/1.56
% 2.89/1.56 % (14129)Memory used [KB]: 1407
% 2.89/1.56 % (14129)Time elapsed: 0.003 s
% 2.89/1.56 % (14129)Instructions burned: 5 (million)
% 2.89/1.56 % (14129)------------------------------
% 2.89/1.56 % (14129)------------------------------
% 2.89/1.56 % (14128)Instruction limit reached!
% 2.89/1.56 % (14128)------------------------------
% 2.89/1.56 % (14128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.56 % (14128)Termination reason: Unknown
% 2.89/1.56 % (14128)Termination phase: shuffling
% 2.89/1.56
% 2.89/1.56 % (14128)Memory used [KB]: 1663
% 2.89/1.56 % (14128)Time elapsed: 0.012 s
% 2.89/1.56 % (14128)Instructions burned: 18 (million)
% 2.89/1.56 % (14128)------------------------------
% 2.89/1.56 % (14128)------------------------------
% 2.89/1.56 % (14130)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_13930 for (2999ds/37Mi)
% 2.89/1.57 % (14133)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_13930 for (2999ds/1041Mi)
% 2.89/1.57 % (14131)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_13930 for (2999ds/15Mi)
% 2.89/1.57 % (14132)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_13930 for (2999ds/3Mi)
% 2.89/1.57 % (14132)Instruction limit reached!
% 2.89/1.57 % (14132)------------------------------
% 2.89/1.57 % (14132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.57 % (14132)Termination reason: Unknown
% 2.89/1.57 % (14132)Termination phase: shuffling
% 2.89/1.57
% 2.89/1.57 % (14132)Memory used [KB]: 1279
% 2.89/1.57 % (14132)Time elapsed: 0.003 s
% 2.89/1.57 % (14132)Instructions burned: 3 (million)
% 2.89/1.57 % (14132)------------------------------
% 2.89/1.57 % (14132)------------------------------
% 2.89/1.57 % (14130)Instruction limit reached!
% 2.89/1.57 % (14130)------------------------------
% 2.89/1.57 % (14130)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.57 % (14130)Termination reason: Unknown
% 2.89/1.57 % (14130)Termination phase: Property scanning
% 2.89/1.57
% 2.89/1.57 % (14130)Memory used [KB]: 2046
% 2.89/1.57 % (14130)Time elapsed: 0.012 s
% 2.89/1.57 % (14130)Instructions burned: 40 (million)
% 2.89/1.57 % (14130)------------------------------
% 2.89/1.57 % (14130)------------------------------
% 2.89/1.58 % (14134)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/7Mi)
% 2.89/1.58 % (14131)Instruction limit reached!
% 2.89/1.58 % (14131)------------------------------
% 2.89/1.58 % (14131)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.58 % (14131)Termination reason: Unknown
% 2.89/1.58 % (14131)Termination phase: shuffling
% 2.89/1.58
% 2.89/1.58 % (14131)Memory used [KB]: 1663
% 2.89/1.58 % (14131)Time elapsed: 0.011 s
% 2.89/1.58 % (14131)Instructions burned: 16 (million)
% 2.89/1.58 % (14131)------------------------------
% 2.89/1.58 % (14131)------------------------------
% 2.89/1.58 % (14134)Instruction limit reached!
% 2.89/1.58 % (14134)------------------------------
% 2.89/1.58 % (14134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.58 % (14134)Termination reason: Unknown
% 2.89/1.58 % (14134)Termination phase: shuffling
% 2.89/1.58
% 2.89/1.58 % (14134)Memory used [KB]: 1535
% 2.89/1.58 % (14134)Time elapsed: 0.006 s
% 2.89/1.58 % (14134)Instructions burned: 8 (million)
% 2.89/1.58 % (14134)------------------------------
% 2.89/1.58 % (14134)------------------------------
% 2.89/1.58 % (14136)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_13930 for (2999ds/3Mi)
% 2.89/1.58 % (14136)Instruction limit reached!
% 2.89/1.58 % (14136)------------------------------
% 2.89/1.58 % (14136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.58 % (14136)Termination reason: Unknown
% 2.89/1.58 % (14136)Termination phase: shuffling
% 2.89/1.58
% 2.89/1.58 % (14136)Memory used [KB]: 1407
% 2.89/1.58 % (14136)Time elapsed: 0.003 s
% 2.89/1.58 % (14136)Instructions burned: 6 (million)
% 2.89/1.58 % (14136)------------------------------
% 2.89/1.58 % (14136)------------------------------
% 2.89/1.59 % (14135)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/16Mi)
% 2.89/1.59 % (14138)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/7Mi)
% 2.89/1.59 % (14139)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/3Mi)
% 2.89/1.59 % (14127)Refutation not found, incomplete strategy
% 2.89/1.59 % (14127)------------------------------
% 2.89/1.59 % (14127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.59 % (14127)Termination reason: Refutation not found, incomplete strategy
% 2.89/1.59
% 2.89/1.59
% 2.89/1.59 % (14127)Memory used [KB]: 6524
% 2.89/1.59 % (14127)Time elapsed: 0.045 s
% 2.89/1.59 % (14127)Instructions burned: 80 (million)
% 2.89/1.59 % (14127)------------------------------
% 2.89/1.59 % (14127)------------------------------
% 2.89/1.59 % (14139)Instruction limit reached!
% 2.89/1.59 % (14139)------------------------------
% 2.89/1.59 % (14139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.59 % (14139)Termination reason: Unknown
% 2.89/1.59 % (14139)Termination phase: shuffling
% 2.89/1.59
% 2.89/1.59 % (14139)Memory used [KB]: 1407
% 2.89/1.59 % (14139)Time elapsed: 0.002 s
% 2.89/1.59 % (14139)Instructions burned: 3 (million)
% 2.89/1.59 % (14139)------------------------------
% 2.89/1.59 % (14139)------------------------------
% 2.89/1.60 % (14138)Instruction limit reached!
% 2.89/1.60 % (14138)------------------------------
% 2.89/1.60 % (14138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.60 % (14138)Termination reason: Unknown
% 2.89/1.60 % (14138)Termination phase: shuffling
% 2.89/1.60
% 2.89/1.60 % (14138)Memory used [KB]: 1535
% 2.89/1.60 % (14138)Time elapsed: 0.004 s
% 2.89/1.60 % (14138)Instructions burned: 8 (million)
% 2.89/1.60 % (14138)------------------------------
% 2.89/1.60 % (14138)------------------------------
% 2.89/1.60 % (14137)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_13930 for (2999ds/3Mi)
% 2.93/1.60 % (14135)Instruction limit reached!
% 2.93/1.60 % (14135)------------------------------
% 2.93/1.60 % (14135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.60 % (14135)Termination reason: Unknown
% 2.93/1.60 % (14135)Termination phase: shuffling
% 2.93/1.60
% 2.93/1.60 % (14135)Memory used [KB]: 1663
% 2.93/1.60 % (14135)Time elapsed: 0.011 s
% 2.93/1.60 % (14135)Instructions burned: 17 (million)
% 2.93/1.60 % (14135)------------------------------
% 2.93/1.60 % (14135)------------------------------
% 2.93/1.60 % (14137)Instruction limit reached!
% 2.93/1.60 % (14137)------------------------------
% 2.93/1.60 % (14137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.60 % (14137)Termination reason: Unknown
% 2.93/1.60 % (14137)Termination phase: shuffling
% 2.93/1.60
% 2.93/1.60 % (14137)Memory used [KB]: 1407
% 2.93/1.60 % (14137)Time elapsed: 0.004 s
% 2.93/1.60 % (14137)Instructions burned: 4 (million)
% 2.93/1.60 % (14137)------------------------------
% 2.93/1.60 % (14137)------------------------------
% 2.93/1.60 % (14140)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/4Mi)
% 2.93/1.61 % (14140)Instruction limit reached!
% 2.93/1.61 % (14140)------------------------------
% 2.93/1.61 % (14140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.61 % (14140)Termination reason: Unknown
% 2.93/1.61 % (14140)Termination phase: shuffling
% 2.93/1.61
% 2.93/1.61 % (14140)Memory used [KB]: 1407
% 2.93/1.61 % (14140)Time elapsed: 0.003 s
% 2.93/1.61 % (14140)Instructions burned: 6 (million)
% 2.93/1.61 % (14140)------------------------------
% 2.93/1.61 % (14140)------------------------------
% 2.93/1.61 % (14122)Instruction limit reached!
% 2.93/1.61 % (14122)------------------------------
% 2.93/1.61 % (14122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.61 % (14122)Termination reason: Unknown
% 2.93/1.61 % (14122)Termination phase: Saturation
% 2.93/1.61
% 2.93/1.61 % (14122)Memory used [KB]: 6908
% 2.93/1.61 % (14122)Time elapsed: 0.063 s
% 2.93/1.61 % (14122)Instructions burned: 184 (million)
% 2.93/1.61 % (14122)------------------------------
% 2.93/1.61 % (14122)------------------------------
% 2.93/1.61 % (14141)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/18Mi)
% 2.93/1.61 % (14142)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/710Mi)
% 2.93/1.61 % (14141)Instruction limit reached!
% 2.93/1.61 % (14141)------------------------------
% 2.93/1.61 % (14141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.61 % (14141)Termination reason: Unknown
% 2.93/1.61 % (14141)Termination phase: Property scanning
% 2.93/1.61
% 2.93/1.61 % (14141)Memory used [KB]: 1791
% 2.93/1.61 % (14141)Time elapsed: 0.009 s
% 2.93/1.61 % (14141)Instructions burned: 20 (million)
% 2.93/1.61 % (14141)------------------------------
% 2.93/1.61 % (14141)------------------------------
% 2.93/1.61 % (14143)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/6Mi)
% 2.93/1.61 % (14144)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/902Mi)
% 2.93/1.62 % (14146)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/5Mi)
% 2.93/1.62 % (14143)Instruction limit reached!
% 2.93/1.62 % (14143)------------------------------
% 2.93/1.62 % (14143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.62 % (14143)Termination reason: Unknown
% 2.93/1.62 % (14143)Termination phase: shuffling
% 2.93/1.62
% 2.93/1.62 % (14143)Memory used [KB]: 1407
% 2.93/1.62 % (14143)Time elapsed: 0.003 s
% 2.93/1.62 % (14143)Instructions burned: 6 (million)
% 2.93/1.62 % (14143)------------------------------
% 2.93/1.62 % (14143)------------------------------
% 2.93/1.62 % (14146)Instruction limit reached!
% 2.93/1.62 % (14146)------------------------------
% 2.93/1.62 % (14146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.62 % (14146)Termination reason: Unknown
% 2.93/1.62 % (14146)Termination phase: shuffling
% 2.93/1.62
% 2.93/1.62 % (14146)Memory used [KB]: 1407
% 2.93/1.62 % (14146)Time elapsed: 0.003 s
% 2.93/1.62 % (14146)Instructions burned: 6 (million)
% 2.93/1.62 % (14146)------------------------------
% 2.93/1.62 % (14146)------------------------------
% 2.93/1.62 % (14147)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/6Mi)
% 2.93/1.63 % (14147)Instruction limit reached!
% 2.93/1.63 % (14147)------------------------------
% 2.93/1.63 % (14147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.63 % (14147)Termination reason: Unknown
% 2.93/1.63 % (14147)Termination phase: shuffling
% 2.93/1.63
% 2.93/1.63 % (14147)Memory used [KB]: 1407
% 2.93/1.63 % (14147)Time elapsed: 0.004 s
% 2.93/1.63 % (14147)Instructions burned: 6 (million)
% 2.93/1.63 % (14147)------------------------------
% 2.93/1.63 % (14147)------------------------------
% 2.93/1.63 % (14148)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/377Mi)
% 2.93/1.63 % (14145)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/21Mi)
% 2.93/1.63 % (14149)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/779Mi)
% 2.93/1.63 % (14145)Instruction limit reached!
% 2.93/1.63 % (14145)------------------------------
% 2.93/1.63 % (14145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.63 % (14145)Termination reason: Unknown
% 2.93/1.63 % (14145)Termination phase: Property scanning
% 2.93/1.63
% 2.93/1.63 % (14145)Memory used [KB]: 1791
% 2.93/1.63 % (14145)Time elapsed: 0.009 s
% 2.93/1.63 % (14145)Instructions burned: 23 (million)
% 2.93/1.63 % (14145)------------------------------
% 2.93/1.63 % (14145)------------------------------
% 2.93/1.64 % (14150)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/19Mi)
% 2.93/1.64 % (14150)Instruction limit reached!
% 2.93/1.64 % (14150)------------------------------
% 2.93/1.64 % (14150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.64 % (14150)Termination reason: Unknown
% 2.93/1.64 % (14150)Termination phase: shuffling
% 2.93/1.64
% 2.93/1.64 % (14150)Memory used [KB]: 1663
% 2.93/1.64 % (14150)Time elapsed: 0.007 s
% 2.93/1.64 % (14150)Instructions burned: 20 (million)
% 2.93/1.64 % (14150)------------------------------
% 2.93/1.64 % (14150)------------------------------
% 2.93/1.64 % (14151)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13930 for (2999ds/879Mi)
% 2.93/1.65 % (14152)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13930 for (2998ds/17Mi)
% 2.93/1.66 % (14152)Instruction limit reached!
% 2.93/1.66 % (14152)------------------------------
% 2.93/1.66 % (14152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.66 % (14152)Termination reason: Unknown
% 2.93/1.66 % (14152)Termination phase: shuffling
% 2.93/1.66
% 2.93/1.66 % (14152)Memory used [KB]: 1663
% 2.93/1.66 % (14152)Time elapsed: 0.007 s
% 2.93/1.66 % (14152)Instructions burned: 17 (million)
% 2.93/1.66 % (14152)------------------------------
% 2.93/1.66 % (14152)------------------------------
% 2.93/1.67 % (14153)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13930 for (2998ds/3Mi)
% 2.93/1.67 % (14153)Instruction limit reached!
% 2.93/1.67 % (14153)------------------------------
% 2.93/1.67 % (14153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.67 % (14153)Termination reason: Unknown
% 2.93/1.67 % (14153)Termination phase: shuffling
% 2.93/1.67
% 2.93/1.67 % (14153)Memory used [KB]: 1407
% 2.93/1.67 % (14153)Time elapsed: 0.003 s
% 2.93/1.67 % (14153)Instructions burned: 4 (million)
% 2.93/1.67 % (14153)------------------------------
% 2.93/1.67 % (14153)------------------------------
% 2.93/1.68 % (14155)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13930 for (2998ds/30Mi)
% 2.93/1.69 % (14155)Instruction limit reached!
% 2.93/1.69 % (14155)------------------------------
% 2.93/1.69 % (14155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.69 % (14155)Termination reason: Unknown
% 2.93/1.69 % (14155)Termination phase: Preprocessing 1
% 2.93/1.69
% 2.93/1.69 % (14155)Memory used [KB]: 1791
% 2.93/1.69 % (14155)Time elapsed: 0.011 s
% 2.93/1.69 % (14155)Instructions burned: 31 (million)
% 2.93/1.69 % (14155)------------------------------
% 2.93/1.69 % (14155)------------------------------
% 2.93/1.70 % (14156)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13930 for (2998ds/127Mi)
% 2.93/1.74 % (14156)Instruction limit reached!
% 2.93/1.74 % (14156)------------------------------
% 2.93/1.74 % (14156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.74 % (14156)Termination reason: Unknown
% 2.93/1.74 % (14156)Termination phase: Saturation
% 2.93/1.74
% 2.93/1.74 % (14156)Memory used [KB]: 6908
% 2.93/1.74 % (14156)Time elapsed: 0.062 s
% 2.93/1.74 % (14156)Instructions burned: 127 (million)
% 2.93/1.74 % (14156)------------------------------
% 2.93/1.74 % (14156)------------------------------
% 2.93/1.75 % (14162)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13930 for (2998ds/100Mi)
% 2.93/1.77 % (14162)Instruction limit reached!
% 2.93/1.77 % (14162)------------------------------
% 2.93/1.77 % (14162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.77 % (14162)Termination reason: Unknown
% 2.93/1.77 % (14162)Termination phase: Saturation
% 2.93/1.77
% 2.93/1.77 % (14162)Memory used [KB]: 6652
% 2.93/1.77 % (14162)Time elapsed: 0.028 s
% 2.93/1.77 % (14162)Instructions burned: 103 (million)
% 2.93/1.77 % (14162)------------------------------
% 2.93/1.77 % (14162)------------------------------
% 2.93/1.77 % (14148)Instruction limit reached!
% 2.93/1.77 % (14148)------------------------------
% 2.93/1.77 % (14148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.77 % (14148)Termination reason: Unknown
% 2.93/1.78 % (14148)Termination phase: Saturation
% 2.93/1.78
% 2.93/1.78 % (14148)Memory used [KB]: 7675
% 2.93/1.78 % (14148)Time elapsed: 0.150 s
% 2.93/1.78 % (14148)Instructions burned: 378 (million)
% 2.93/1.78 % (14148)------------------------------
% 2.93/1.78 % (14148)------------------------------
% 2.93/1.79 % (14172)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_13930 for (2997ds/3Mi)
% 2.93/1.79 % (14172)Instruction limit reached!
% 2.93/1.79 % (14172)------------------------------
% 2.93/1.79 % (14172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.79 % (14172)Termination reason: Unknown
% 2.93/1.79 % (14172)Termination phase: shuffling
% 2.93/1.79
% 2.93/1.79 % (14172)Memory used [KB]: 1407
% 2.93/1.79 % (14172)Time elapsed: 0.003 s
% 2.93/1.79 % (14172)Instructions burned: 3 (million)
% 2.93/1.79 % (14172)------------------------------
% 2.93/1.79 % (14172)------------------------------
% 2.93/1.79 % (14173)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_13930 for (2997ds/20Mi)
% 2.93/1.80 % (14173)Instruction limit reached!
% 2.93/1.80 % (14173)------------------------------
% 2.93/1.80 % (14173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.80 % (14173)Termination reason: Unknown
% 2.93/1.80 % (14173)Termination phase: Property scanning
% 2.93/1.80
% 2.93/1.80 % (14173)Memory used [KB]: 1791
% 2.93/1.80 % (14173)Time elapsed: 0.012 s
% 2.93/1.80 % (14173)Instructions burned: 22 (million)
% 2.93/1.80 % (14173)------------------------------
% 2.93/1.80 % (14173)------------------------------
% 2.93/1.81 % (14178)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_13930 for (2997ds/86Mi)
% 2.93/1.82 % (14181)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_13930 for (2997ds/107Mi)
% 2.93/1.85 % (14178)Instruction limit reached!
% 2.93/1.85 % (14178)------------------------------
% 2.93/1.85 % (14178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.85 % (14178)Termination reason: Unknown
% 2.93/1.85 % (14178)Termination phase: Saturation
% 2.93/1.85
% 2.93/1.85 % (14178)Memory used [KB]: 6652
% 2.93/1.85 % (14178)Time elapsed: 0.044 s
% 2.93/1.85 % (14178)Instructions burned: 87 (million)
% 2.93/1.85 % (14178)------------------------------
% 2.93/1.85 % (14178)------------------------------
% 2.93/1.87 % (14193)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_13930 for (2996ds/39Mi)
% 2.93/1.88 % (14181)Instruction limit reached!
% 2.93/1.88 % (14181)------------------------------
% 2.93/1.88 % (14181)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.88 % (14181)Termination reason: Unknown
% 2.93/1.88 % (14181)Termination phase: Saturation
% 2.93/1.88
% 2.93/1.88 % (14181)Memory used [KB]: 6780
% 2.93/1.88 % (14181)Time elapsed: 0.054 s
% 2.93/1.88 % (14181)Instructions burned: 107 (million)
% 2.93/1.88 % (14181)------------------------------
% 2.93/1.88 % (14181)------------------------------
% 2.93/1.88 % (14193)Instruction limit reached!
% 2.93/1.88 % (14193)------------------------------
% 2.93/1.88 % (14193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.88 % (14193)Termination reason: Unknown
% 2.93/1.88 % (14193)Termination phase: Property scanning
% 2.93/1.88
% 2.93/1.88 % (14193)Memory used [KB]: 2046
% 2.93/1.88 % (14193)Time elapsed: 0.013 s
% 2.93/1.88 % (14193)Instructions burned: 39 (million)
% 2.93/1.88 % (14193)------------------------------
% 2.93/1.88 % (14193)------------------------------
% 2.93/1.89 % (14200)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_13930 for (2996ds/46Mi)
% 2.93/1.89 % (14198)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_13930 for (2996ds/448Mi)
% 2.93/1.90 % (14142)Instruction limit reached!
% 2.93/1.90 % (14142)------------------------------
% 2.93/1.90 % (14142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.90 % (14142)Termination reason: Unknown
% 2.93/1.90 % (14142)Termination phase: Saturation
% 2.93/1.90
% 2.93/1.90 % (14142)Memory used [KB]: 8059
% 2.93/1.90 % (14142)Time elapsed: 0.286 s
% 2.93/1.90 % (14142)Instructions burned: 711 (million)
% 2.93/1.90 % (14142)------------------------------
% 2.93/1.90 % (14142)------------------------------
% 2.93/1.90 % (14200)Instruction limit reached!
% 2.93/1.90 % (14200)------------------------------
% 2.93/1.90 % (14200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.90 % (14200)Termination reason: Unknown
% 2.93/1.90 % (14200)Termination phase: Property scanning
% 2.93/1.90
% 2.93/1.90 % (14200)Memory used [KB]: 2046
% 2.93/1.90 % (14200)Time elapsed: 0.016 s
% 2.93/1.90 % (14200)Instructions burned: 49 (million)
% 2.93/1.90 % (14200)------------------------------
% 2.93/1.90 % (14200)------------------------------
% 2.93/1.91 % (14201)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_13930 for (2996ds/98Mi)
% 2.93/1.92 % (14202)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_13930 for (2996ds/507Mi)
% 2.93/1.93 % (14149)Instruction limit reached!
% 2.93/1.93 % (14149)------------------------------
% 2.93/1.93 % (14149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.93 % (14149)Termination reason: Unknown
% 2.93/1.93 % (14149)Termination phase: Saturation
% 2.93/1.93
% 2.93/1.93 % (14149)Memory used [KB]: 7803
% 2.93/1.93 % (14149)Time elapsed: 0.298 s
% 2.93/1.93 % (14149)Instructions burned: 779 (million)
% 2.93/1.93 % (14149)------------------------------
% 2.93/1.93 % (14149)------------------------------
% 2.93/1.94 % (14205)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/149Mi)
% 2.93/1.96 % (14201)Instruction limit reached!
% 2.93/1.96 % (14201)------------------------------
% 2.93/1.96 % (14201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.96 % (14201)Termination reason: Unknown
% 2.93/1.96 % (14201)Termination phase: Saturation
% 2.93/1.96
% 2.93/1.96 % (14201)Memory used [KB]: 2174
% 2.93/1.96 % (14201)Time elapsed: 0.046 s
% 2.93/1.96 % (14201)Instructions burned: 99 (million)
% 2.93/1.96 % (14201)------------------------------
% 2.93/1.96 % (14201)------------------------------
% 2.93/1.97 % (14208)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/39Mi)
% 2.93/1.98 % (14144)Instruction limit reached!
% 2.93/1.98 % (14144)------------------------------
% 2.93/1.98 % (14144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.98 % (14144)Termination reason: Unknown
% 2.93/1.98 % (14144)Termination phase: Saturation
% 2.93/1.98
% 2.93/1.98 % (14144)Memory used [KB]: 8315
% 2.93/1.98 % (14144)Time elapsed: 0.363 s
% 2.93/1.98 % (14144)Instructions burned: 904 (million)
% 2.93/1.98 % (14144)------------------------------
% 2.93/1.98 % (14144)------------------------------
% 2.93/1.98 % (14151)Instruction limit reached!
% 2.93/1.98 % (14151)------------------------------
% 2.93/1.98 % (14151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.98 % (14151)Termination reason: Unknown
% 2.93/1.98 % (14151)Termination phase: Saturation
% 2.93/1.98
% 2.93/1.98 % (14151)Memory used [KB]: 7931
% 2.93/1.98 % (14151)Time elapsed: 0.341 s
% 2.93/1.98 % (14151)Instructions burned: 881 (million)
% 2.93/1.98 % (14151)------------------------------
% 2.93/1.98 % (14151)------------------------------
% 2.93/1.99 % (14210)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/6Mi)
% 2.93/1.99 % (14208)Instruction limit reached!
% 2.93/1.99 % (14208)------------------------------
% 2.93/1.99 % (14208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.99 % (14208)Termination reason: Unknown
% 2.93/1.99 % (14208)Termination phase: Preprocessing 3
% 2.93/1.99
% 2.93/1.99 % (14208)Memory used [KB]: 2046
% 2.93/1.99 % (14208)Time elapsed: 0.022 s
% 2.93/1.99 % (14208)Instructions burned: 40 (million)
% 2.93/1.99 % (14208)------------------------------
% 2.93/1.99 % (14208)------------------------------
% 2.93/2.00 % (14210)Instruction limit reached!
% 2.93/2.00 % (14210)------------------------------
% 2.93/2.00 % (14210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/2.00 % (14210)Termination reason: Unknown
% 2.93/2.00 % (14210)Termination phase: shuffling
% 2.93/2.00
% 2.93/2.00 % (14210)Memory used [KB]: 1407
% 2.93/2.00 % (14210)Time elapsed: 0.005 s
% 2.93/2.00 % (14210)Instructions burned: 7 (million)
% 2.93/2.00 % (14210)------------------------------
% 2.93/2.00 % (14210)------------------------------
% 2.93/2.00 % (14211)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/1087Mi)
% 2.93/2.01 % (14133)Instruction limit reached!
% 2.93/2.01 % (14133)------------------------------
% 2.93/2.01 % (14133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/2.01 % (14133)Termination reason: Unknown
% 2.93/2.01 % (14133)Termination phase: Saturation
% 2.93/2.01
% 2.93/2.01 % (14133)Memory used [KB]: 9722
% 2.93/2.01 % (14133)Time elapsed: 0.443 s
% 2.93/2.01 % (14212)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/47Mi)
% 2.93/2.01 % (14133)Instructions burned: 1041 (million)
% 2.93/2.01 % (14133)------------------------------
% 2.93/2.01 % (14133)------------------------------
% 2.93/2.01 % (14213)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/44Mi)
% 2.93/2.02 % (14205)Instruction limit reached!
% 2.93/2.02 % (14205)------------------------------
% 2.93/2.02 % (14205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/2.02 % (14205)Termination reason: Unknown
% 2.93/2.02 % (14205)Termination phase: Saturation
% 2.93/2.02
% 2.93/2.02 % (14205)Memory used [KB]: 6908
% 2.93/2.02 % (14205)Time elapsed: 0.077 s
% 2.93/2.02 % (14205)Instructions burned: 150 (million)
% 2.93/2.02 % (14205)------------------------------
% 2.93/2.02 % (14205)------------------------------
% 6.24/2.03 % (14214)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/432Mi)
% 6.24/2.03 % (14212)Instruction limit reached!
% 6.24/2.03 % (14212)------------------------------
% 6.24/2.03 % (14212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.03 % (14212)Termination reason: Unknown
% 6.24/2.03 % (14212)Termination phase: Property scanning
% 6.24/2.03
% 6.24/2.03 % (14212)Memory used [KB]: 2174
% 6.24/2.03 % (14212)Time elapsed: 0.025 s
% 6.24/2.03 % (14212)Instructions burned: 48 (million)
% 6.24/2.03 % (14212)------------------------------
% 6.24/2.03 % (14212)------------------------------
% 6.24/2.03 % (14215)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_13930 for (2995ds/62Mi)
% 6.24/2.03 % (14213)Instruction limit reached!
% 6.24/2.03 % (14213)------------------------------
% 6.24/2.03 % (14213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.03 % (14213)Termination reason: Unknown
% 6.24/2.04 % (14213)Termination phase: Property scanning
% 6.24/2.04
% 6.24/2.04 % (14213)Memory used [KB]: 2046
% 6.24/2.04 % (14213)Time elapsed: 0.022 s
% 6.24/2.04 % (14213)Instructions burned: 45 (million)
% 6.24/2.04 % (14213)------------------------------
% 6.24/2.04 % (14213)------------------------------
% 6.24/2.05 % (14218)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/96Mi)
% 6.24/2.06 % (14217)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/31Mi)
% 6.24/2.06 % (14215)Instruction limit reached!
% 6.24/2.06 % (14215)------------------------------
% 6.24/2.06 % (14215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.06 % (14215)Termination reason: Unknown
% 6.24/2.06 % (14215)Termination phase: Preprocessing 3
% 6.24/2.06
% 6.24/2.06 % (14215)Memory used [KB]: 2174
% 6.24/2.06 % (14215)Time elapsed: 0.029 s
% 6.24/2.06 % (14215)Instructions burned: 62 (million)
% 6.24/2.06 % (14215)------------------------------
% 6.24/2.06 % (14215)------------------------------
% 6.24/2.07 % (14217)Instruction limit reached!
% 6.24/2.07 % (14217)------------------------------
% 6.24/2.07 % (14217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.07 % (14217)Termination reason: Unknown
% 6.24/2.07 % (14217)Termination phase: Preprocessing 2
% 6.24/2.07
% 6.24/2.07 % (14217)Memory used [KB]: 1791
% 6.24/2.07 % (14217)Time elapsed: 0.011 s
% 6.24/2.07 % (14217)Instructions burned: 33 (million)
% 6.24/2.07 % (14217)------------------------------
% 6.24/2.07 % (14217)------------------------------
% 6.24/2.08 % (14221)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/71Mi)
% 6.24/2.08 % (14220)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/453Mi)
% 6.24/2.10 % (14218)Instruction limit reached!
% 6.24/2.10 % (14218)------------------------------
% 6.24/2.10 % (14218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.10 % (14218)Termination reason: Unknown
% 6.24/2.10 % (14218)Termination phase: Saturation
% 6.24/2.10
% 6.24/2.10 % (14218)Memory used [KB]: 6780
% 6.24/2.10 % (14218)Time elapsed: 0.048 s
% 6.24/2.10 % (14218)Instructions burned: 97 (million)
% 6.24/2.10 % (14218)------------------------------
% 6.24/2.10 % (14218)------------------------------
% 6.24/2.10 % (14221)Instruction limit reached!
% 6.24/2.10 % (14221)------------------------------
% 6.24/2.10 % (14221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.10 % (14221)Termination reason: Unknown
% 6.24/2.10 % (14221)Termination phase: Property scanning
% 6.24/2.10
% 6.24/2.10 % (14221)Memory used [KB]: 2174
% 6.24/2.10 % (14221)Time elapsed: 0.029 s
% 6.24/2.10 % (14221)Instructions burned: 71 (million)
% 6.24/2.10 % (14221)------------------------------
% 6.24/2.10 % (14221)------------------------------
% 6.24/2.11 % (14198)Instruction limit reached!
% 6.24/2.11 % (14198)------------------------------
% 6.24/2.11 % (14198)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.11 % (14198)Termination reason: Unknown
% 6.24/2.11 % (14198)Termination phase: Saturation
% 6.24/2.11
% 6.24/2.11 % (14198)Memory used [KB]: 7547
% 6.24/2.11 % (14198)Time elapsed: 0.221 s
% 6.24/2.11 % (14198)Instructions burned: 449 (million)
% 6.24/2.11 % (14198)------------------------------
% 6.24/2.11 % (14198)------------------------------
% 6.24/2.11 % (14202)Instruction limit reached!
% 6.24/2.11 % (14202)------------------------------
% 6.24/2.11 % (14202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.11 % (14202)Termination reason: Unknown
% 6.24/2.11 % (14202)Termination phase: Saturation
% 6.24/2.11
% 6.24/2.11 % (14202)Memory used [KB]: 7675
% 6.24/2.11 % (14202)Time elapsed: 0.196 s
% 6.24/2.11 % (14202)Instructions burned: 508 (million)
% 6.24/2.11 % (14202)------------------------------
% 6.24/2.11 % (14202)------------------------------
% 6.24/2.11 % (14224)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/83Mi)
% 6.24/2.12 % (14226)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/11Mi)
% 6.24/2.13 % (14226)Instruction limit reached!
% 6.24/2.13 % (14226)------------------------------
% 6.24/2.13 % (14226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.24/2.13 % (14226)Termination reason: Unknown
% 6.24/2.13 % (14226)Termination phase: shuffling
% 6.24/2.13
% 6.24/2.13 % (14226)Memory used [KB]: 1535
% 6.24/2.13 % (14226)Time elapsed: 0.007 s
% 6.24/2.13 % (14226)Instructions burned: 12 (million)
% 6.24/2.13 % (14226)------------------------------
% 6.24/2.13 % (14226)------------------------------
% 6.75/2.13 % (14228)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/6Mi)
% 6.75/2.13 % (14227)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/41Mi)
% 6.75/2.13 % (14228)Instruction limit reached!
% 6.75/2.13 % (14228)------------------------------
% 6.75/2.13 % (14228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.13 % (14228)Termination reason: Unknown
% 6.75/2.13 % (14228)Termination phase: shuffling
% 6.75/2.13
% 6.75/2.13 % (14228)Memory used [KB]: 1407
% 6.75/2.13 % (14228)Time elapsed: 0.005 s
% 6.75/2.13 % (14228)Instructions burned: 7 (million)
% 6.75/2.13 % (14228)------------------------------
% 6.75/2.13 % (14228)------------------------------
% 6.75/2.14 % (14229)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/150Mi)
% 6.75/2.15 % (14230)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_13930 for (2994ds/343Mi)
% 6.75/2.15 % (14227)Instruction limit reached!
% 6.75/2.15 % (14227)------------------------------
% 6.75/2.15 % (14227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.15 % (14227)Termination reason: Unknown
% 6.75/2.15 % (14227)Termination phase: Property scanning
% 6.75/2.15
% 6.75/2.15 % (14227)Memory used [KB]: 2046
% 6.75/2.15 % (14227)Time elapsed: 0.021 s
% 6.75/2.15 % (14227)Instructions burned: 42 (million)
% 6.75/2.15 % (14227)------------------------------
% 6.75/2.15 % (14227)------------------------------
% 6.75/2.16 % (14224)Instruction limit reached!
% 6.75/2.16 % (14224)------------------------------
% 6.75/2.16 % (14224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.75/2.16 % (14224)Termination reason: Unknown
% 6.75/2.16 % (14224)Termination phase: Saturation
% 6.75/2.16
% 6.75/2.16 % (14224)Memory used [KB]: 6524
% 6.75/2.16 % (14224)Time elapsed: 0.043 s
% 6.75/2.16 % (14224)Instructions burned: 83 (million)
% 6.75/2.16 % (14224)------------------------------
% 6.75/2.16 % (14224)------------------------------
% 6.75/2.16 % (14232)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_13930 for (2993ds/109Mi)
% 6.75/2.17 % (14233)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_13930 for (2993ds/253Mi)
% 6.95/2.20 % (14232)Instruction limit reached!
% 6.95/2.20 % (14232)------------------------------
% 6.95/2.20 % (14232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.20 % (14232)Termination reason: Unknown
% 6.95/2.20 % (14232)Termination phase: Saturation
% 6.95/2.20
% 6.95/2.20 % (14232)Memory used [KB]: 6908
% 6.95/2.20 % (14232)Time elapsed: 0.040 s
% 6.95/2.20 % (14232)Instructions burned: 110 (million)
% 6.95/2.20 % (14232)------------------------------
% 6.95/2.20 % (14232)------------------------------
% 6.95/2.20 % (14229)Instruction limit reached!
% 6.95/2.20 % (14229)------------------------------
% 6.95/2.20 % (14229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.20 % (14229)Termination reason: Unknown
% 6.95/2.20 % (14229)Termination phase: Saturation
% 6.95/2.20
% 6.95/2.20 % (14229)Memory used [KB]: 6908
% 6.95/2.20 % (14229)Time elapsed: 0.064 s
% 6.95/2.20 % (14229)Instructions burned: 150 (million)
% 6.95/2.20 % (14229)------------------------------
% 6.95/2.20 % (14229)------------------------------
% 6.95/2.22 % (14240)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_13930 for (2993ds/614Mi)
% 6.95/2.22 % (14239)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_13930 for (2993ds/140Mi)
% 6.95/2.24 % (14214)Instruction limit reached!
% 6.95/2.24 % (14214)------------------------------
% 6.95/2.24 % (14214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.24 % (14214)Termination reason: Unknown
% 6.95/2.24 % (14214)Termination phase: Saturation
% 6.95/2.24
% 6.95/2.24 % (14214)Memory used [KB]: 3709
% 6.95/2.24 % (14214)Time elapsed: 0.215 s
% 6.95/2.24 % (14214)Instructions burned: 432 (million)
% 6.95/2.24 % (14214)------------------------------
% 6.95/2.24 % (14214)------------------------------
% 6.95/2.24 % (14220)Instruction limit reached!
% 6.95/2.24 % (14220)------------------------------
% 6.95/2.24 % (14220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.24 % (14220)Termination reason: Unknown
% 6.95/2.24 % (14220)Termination phase: Saturation
% 6.95/2.24
% 6.95/2.24 % (14220)Memory used [KB]: 7547
% 6.95/2.24 % (14220)Time elapsed: 0.164 s
% 6.95/2.24 % (14220)Instructions burned: 453 (million)
% 6.95/2.24 % (14220)------------------------------
% 6.95/2.24 % (14220)------------------------------
% 6.95/2.25 % (14243)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_13930 for (2992ds/339Mi)
% 6.95/2.26 % (14242)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_13930 for (2992ds/202Mi)
% 6.95/2.29 % (14239)Instruction limit reached!
% 6.95/2.29 % (14239)------------------------------
% 6.95/2.29 % (14239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.29 % (14239)Termination reason: Unknown
% 6.95/2.29 % (14239)Termination phase: Saturation
% 6.95/2.29
% 6.95/2.29 % (14239)Memory used [KB]: 6780
% 6.95/2.29 % (14239)Time elapsed: 0.072 s
% 6.95/2.29 % (14239)Instructions burned: 140 (million)
% 6.95/2.29 % (14239)------------------------------
% 6.95/2.29 % (14239)------------------------------
% 6.95/2.29 % (14230)First to succeed.
% 6.95/2.29 % (14230)Refutation found. Thanks to Tanya!
% 6.95/2.29 % SZS status Theorem for DTF2THF_13930
% 6.95/2.29 % SZS output start Proof for DTF2THF_13930
% 6.95/2.29 thf(type_def_5, type, set: ($tType) > $tType).
% 6.95/2.29 thf(type_def_7, type, nat: $tType).
% 6.95/2.29 thf(func_def_0, type, set: ($tType) > $tType).
% 6.95/2.29 thf(func_def_1, type, nat: $tType).
% 6.95/2.29 thf(func_def_2, type, type: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_3, type, one: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_4, type, zero: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_5, type, ord: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_6, type, neg_numeral: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_8, type, group_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_9, type, finite_finite: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_10, type, monoid_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_13, type, zero_neq_one: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_14, type, wellorder: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_15, type, semigroup_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_16, type, linordered_idom: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_17, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_18, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_19, type, dense_linorder: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_20, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_21, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_22, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_23, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_24, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_25, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_27, type, strict797366125id_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_28, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_29, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 6.95/2.29 thf(func_def_30, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 6.95/2.30 thf(func_def_31, type, semiri456707255roduct: !>[X0: $tType]:($o)).
% 6.95/2.30 thf(func_def_32, type, finite_Fpow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 6.95/2.30 thf(func_def_33, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 6.95/2.30 thf(func_def_34, type, one_one: !>[X0: $tType]:(X0)).
% 6.95/2.30 thf(func_def_35, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 6.95/2.30 thf(func_def_36, type, zero_zero: !>[X0: $tType]:(X0)).
% 6.95/2.30 thf(func_def_37, type, neg_numeral_dbl_inc: !>[X0: $tType]:(X0 > X0)).
% 6.95/2.30 thf(func_def_38, type, number2016821345itions: (nat > nat) > nat > $o).
% 6.95/2.30 thf(func_def_39, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 6.95/2.30 thf(func_def_40, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 6.95/2.30 thf(func_def_41, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 6.95/2.30 thf(func_def_42, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 6.95/2.30 thf(func_def_43, type, n: nat).
% 6.95/2.30 thf(func_def_58, type, sP0: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 6.95/2.30 thf(func_def_59, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 6.95/2.30 thf(func_def_60, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 6.95/2.30 thf(func_def_61, type, sK3: (nat > nat) > nat).
% 6.95/2.30 thf(func_def_62, type, sK4: (nat > nat) > nat).
% 6.95/2.30 thf(func_def_63, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_64, type, sK6: !>[X0: $tType]:(X0 > set @ X0 > X0)).
% 6.95/2.30 thf(func_def_65, type, sK7: nat > set @ nat > nat).
% 6.95/2.30 thf(func_def_66, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_67, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_68, type, sK10: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_69, type, sK11: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_70, type, sK12: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_71, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_72, type, sK14: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_73, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_74, type, sK16: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > set @ X0 > X0)).
% 6.95/2.30 thf(func_def_75, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 6.95/2.30 thf(func_def_76, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 6.95/2.30 thf(func_def_77, type, sK19: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_78, type, sK20: (nat > $o) > nat).
% 6.95/2.30 thf(func_def_79, type, sK21: nat > (nat > $o) > nat).
% 6.95/2.30 thf(func_def_80, type, sK22: (nat > $o) > nat).
% 6.95/2.30 thf(func_def_81, type, sK23: nat > (nat > $o) > nat).
% 6.95/2.30 thf(func_def_82, type, sK24: !>[X0: $tType]:(X0 > set @ X0 > X0)).
% 6.95/2.30 thf(func_def_83, type, sK25: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_84, type, sK26: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_85, type, sK27: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_86, type, sK28: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_87, type, sK29: (nat > nat) > nat).
% 6.95/2.30 thf(func_def_88, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_89, type, sK31: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_90, type, sK32: (nat > nat > $o) > nat).
% 6.95/2.30 thf(func_def_91, type, sK33: (nat > nat > $o) > nat).
% 6.95/2.30 thf(func_def_92, type, sK34: (nat > nat > $o) > nat).
% 6.95/2.30 thf(func_def_93, type, sK35: (nat > nat > $o) > nat).
% 6.95/2.30 thf(func_def_94, type, sK36: (nat > nat > $o) > nat).
% 6.95/2.30 thf(func_def_95, type, sK37: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_96, type, sK38: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_97, type, sK39: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_98, type, sK40: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_99, type, sK41: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_100, type, sK42: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_101, type, sK43: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_102, type, sK44: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 6.95/2.30 thf(func_def_103, type, sK45: nat > nat > nat).
% 6.95/2.30 thf(func_def_104, type, sK46: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 6.95/2.30 thf(func_def_105, type, sK47: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 6.95/2.30 thf(func_def_106, type, sK48: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 6.95/2.30 thf(func_def_107, type, sK49: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_108, type, sK50: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 6.95/2.30 thf(func_def_110, type, sK52: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 6.95/2.30 thf(func_def_111, type, sK53: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 6.95/2.30 thf(func_def_112, type, ph54: !>[X0: $tType]:(X0)).
% 6.95/2.30 thf(f1775,plain,(
% 6.95/2.30 $false),
% 6.95/2.30 inference(subsumption_resolution,[status(thm)],[f1774,f1268])).
% 6.95/2.30 thf(f1268,plain,(
% 6.95/2.30 ($true != (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(cnf_transformation,[status(thm)],[f859])).
% 6.95/2.30 thf(f859,plain,(
% 6.95/2.30 ($true != (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(flattening,[status(thm)],[f844])).
% 6.95/2.30 thf(f844,plain,(
% 6.95/2.30 ~($true = (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(fool_elimination,[status(thm)],[f843])).
% 6.95/2.30 thf(f843,plain,(
% 6.95/2.30 ~(finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[X0 : nat > nat] : (number2016821345itions @ X0 @ n))))),
% 6.95/2.30 inference(rectify,[status(thm)],[f291])).
% 6.95/2.30 thf(f291,negated_conjecture,(
% 6.95/2.30 ~(finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[X3 : nat > nat] : (number2016821345itions @ X3 @ n))))),
% 6.95/2.30 inference(negated_conjecture,[status(cth)],[f290])).
% 6.95/2.30 thf(f290,conjecture,(
% 6.95/2.30 (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[X3 : nat > nat] : (number2016821345itions @ X3 @ n))))),
% 6.95/2.30 file('/export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2THF_13930.p',conj_0)).
% 6.95/2.30 thf(f1774,plain,(
% 6.95/2.30 ($true = (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(subsumption_resolution,[status(thm)],[f1773,f1124])).
% 6.95/2.30 thf(f1124,plain,(
% 6.95/2.30 ( ! [X0 : nat,X1 : nat] : (((finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ X1 @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ X0))))))) = $true)) )),
% 6.95/2.30 inference(cnf_transformation,[status(thm)],[f608])).
% 6.95/2.30 thf(f608,plain,(
% 6.95/2.30 ! [X0 : nat,X1 : nat] : ((finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ X1 @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ X0))))))) = $true)),
% 6.95/2.30 inference(fool_elimination,[status(thm)],[f607])).
% 6.95/2.30 thf(f607,plain,(
% 6.95/2.30 ! [X0 : nat,X1 : nat] : (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[X2 : nat > nat] : (! [X3 : nat] : (ord_less_eq @ nat @ (X2 @ X3) @ X0) & ! [X4 : nat] : ((ord_less_eq @ nat @ X1 @ X4) => (zero_zero @ nat = (X2 @ X4)))))))),
% 6.95/2.30 inference(rectify,[status(thm)],[f14])).
% 6.95/2.30 thf(f14,axiom,(
% 6.95/2.30 ! [X9 : nat,X24 : nat] : (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[X1 : nat > nat] : (! [X2 : nat] : (ord_less_eq @ nat @ (X1 @ X2) @ X9) & ! [X2 : nat] : ((ord_less_eq @ nat @ X24 @ X2) => ((X1 @ X2) = zero_zero @ nat))))))),
% 6.95/2.30 file('/export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2THF_13930.p',fact_13_bound__domain__and__range__impl__finitely__many__functions)).
% 6.95/2.30 thf(f1773,plain,(
% 6.95/2.30 ((finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ n))))))) != $true) | ($true = (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(trivial_inequality_removal,[status(thm)],[f1770])).
% 6.95/2.30 thf(f1770,plain,(
% 6.95/2.30 ((finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ n))))))) != $true) | ($true != $true) | ($true = (finite_finite2 @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n)))))),
% 6.95/2.30 inference(superposition,[status(thm)],[f1253,f1131])).
% 6.95/2.30 thf(f1131,plain,(
% 6.95/2.30 ($true = (ord_less_eq @ set @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n))) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ n))))))))),
% 6.95/2.30 inference(cnf_transformation,[status(thm)],[f582])).
% 6.95/2.30 thf(f582,plain,(
% 6.95/2.30 ($true = (ord_less_eq @ set @ (nat > nat) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: (number2016821345itions @ Y0 @ n))) @ (collect @ (nat > nat) @ (^[Y0 : nat > nat]: ((!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ Y1) => (zero_zero @ nat = (Y0 @ Y1))))) & (!! @ nat @ (^[Y1 : nat]: (ord_less_eq @ nat @ (Y0 @ Y1) @ n))))))))),
% 6.95/2.30 inference(fool_elimination,[status(thm)],[f581])).
% 6.95/2.30 thf(f581,plain,(
% 6.95/2.30 (ord_less_eq @ set @ (nat > nat) @ (collect @ (nat > nat) @ (^[X0 : nat > nat] : (number2016821345itions @ X0 @ n))) @ (collect @ (nat > nat) @ (^[X1 : nat > nat] : (! [X2 : nat] : (ord_less_eq @ nat @ (X1 @ X2) @ n) & ! [X3 : nat] : ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ X3) => (zero_zero @ nat = (X1 @ X3)))))))),
% 6.95/2.30 inference(rectify,[status(thm)],[f2])).
% 6.95/2.30 thf(f2,axiom,(
% 6.95/2.30 (ord_less_eq @ set @ (nat > nat) @ (collect @ (nat > nat) @ (^[X3 : nat > nat] : (number2016821345itions @ X3 @ n))) @ (collect @ (nat > nat) @ (^[X1 : nat > nat] : (! [X2 : nat] : (ord_less_eq @ nat @ (X1 @ X2) @ n) & ! [X2 : nat] : ((ord_less_eq @ nat @ (plus_plus @ nat @ n @ one_one @ nat) @ X2) => ((X1 @ X2) = zero_zero @ nat))))))),
% 6.95/2.30 file('/export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2THF_13930.p',fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062)).
% 6.95/2.30 thf(f1253,plain,(
% 6.95/2.30 ( ! [X0 : $tType,X2 : set @ X0,X1 : set @ X0] : (($true != (ord_less_eq @ set @ X0 @ X1 @ X2)) | ($true != (finite_finite2 @ X0 @ X2)) | ($true = (finite_finite2 @ X0 @ X1))) )),
% 6.95/2.30 inference(cnf_transformation,[status(thm)],[f980])).
% 6.95/2.30 thf(f980,plain,(
% 6.95/2.30 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (($true = (finite_finite2 @ X0 @ X1)) | ($true != (finite_finite2 @ X0 @ X2)) | ($true != (ord_less_eq @ set @ X0 @ X1 @ X2)))),
% 6.95/2.30 inference(flattening,[status(thm)],[f979])).
% 6.95/2.30 thf(f979,plain,(
% 6.95/2.30 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((($true = (finite_finite2 @ X0 @ X1)) | ($true != (finite_finite2 @ X0 @ X2))) | ($true != (ord_less_eq @ set @ X0 @ X1 @ X2)))),
% 6.95/2.30 inference(ennf_transformation,[status(thm)],[f468])).
% 6.95/2.30 thf(f468,plain,(
% 6.95/2.30 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : (($true = (ord_less_eq @ set @ X0 @ X1 @ X2)) => (($true = (finite_finite2 @ X0 @ X2)) => ($true = (finite_finite2 @ X0 @ X1))))),
% 6.95/2.30 inference(fool_elimination,[status(thm)],[f467])).
% 6.95/2.30 thf(f467,plain,(
% 6.95/2.30 ! [X0 : $tType,X1 : set @ X0,X2 : set @ X0] : ((ord_less_eq @ set @ X0 @ X1 @ X2) => ((finite_finite2 @ X0 @ X2) => (finite_finite2 @ X0 @ X1)))),
% 6.95/2.30 inference(rectify,[status(thm)],[f20])).
% 6.95/2.30 thf(f20,axiom,(
% 6.95/2.30 ! [X0 : $tType,X12 : set @ X0,X13 : set @ X0] : ((ord_less_eq @ set @ X0 @ X12 @ X13) => ((finite_finite2 @ X0 @ X13) => (finite_finite2 @ X0 @ X12)))),
% 6.95/2.30 file('/export/starexec/sandbox/tmp/tmp.S3svwHMrie/DTF2THF_13930.p',fact_19_finite__subset)).
% 6.95/2.30 % SZS output end Proof for DTF2THF_13930
% 6.95/2.30 % (14230)------------------------------
% 6.95/2.30 % (14230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.30 % (14230)Termination reason: Refutation
% 6.95/2.30
% 6.95/2.30 % (14230)Memory used [KB]: 7419
% 6.95/2.30 % (14230)Time elapsed: 0.147 s
% 6.95/2.30 % (14230)Instructions burned: 284 (million)
% 6.95/2.30 % (14230)------------------------------
% 6.95/2.30 % (14230)------------------------------
% 6.95/2.30 % (14121)Success in time 0.758 s
%------------------------------------------------------------------------------