%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP132^2 : 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 : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:48 AM UTC 2026
% Result : Theorem 3.63s 1.77s
% Output : Refutation 3.63s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP132^2 : 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
% 0.15/0.34 % Computer : n021.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:27:38 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP132^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Number_Partition problem prob_191__5326608_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Number_Partition/prob_191__5326608_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.00 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 362 ( 96 unt; 62 typ; 0 def)
% 0.30/0.46 % Number of atoms : 893 ( 283 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.30/0.46 % Number of connectives : 3364 ( 66 ~; 26 |; 46 &;2788 @)
% 0.30/0.46 % ( 0 <=>; 438 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 22 ( 7 avg)
% 0.30/0.46 % Number of types : 2 ( 1 usr)
% 0.30/0.46 % Number of type conns : 130 ( 130 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 64 ( 61 usr; 4 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 981 ( 64 ^; 850 !; 9 ?; 981 :)
% 0.30/0.46 % ( 58 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:28:18.421
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (2)
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Nat_Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (60)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Odvd,type,
% 0.30/0.46 dvd:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oone,type,
% 0.30/0.46 one:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oring,type,
% 0.30/0.46 ring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.30/0.46 zero:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.30/0.46 minus:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.46 ord:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.30/0.46 neg_numeral:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.46 order:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Ocomm__ring,type,
% 0.30/0.46 comm_ring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Omult__zero,type,
% 0.30/0.46 mult_zero:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.46 group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.46 no_bot:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.46 no_top:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Osemiring__0,type,
% 0.30/0.46 semiring_0:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.30/0.46 monoid_mult:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.46 preorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oordered__ring,type,
% 0.30/0.46 ordered_ring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.30/0.46 zero_neq_one:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oab__group__add,type,
% 0.30/0.46 ab_group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.46 wellorder:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.30/0.46 semigroup_mult:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.46 dense_order:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.30/0.46 linordered_idom:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.30/0.46 linordered_ring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.30/0.46 comm_monoid_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oordered__semiring,type,
% 0.30/0.46 ordered_semiring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.30/0.46 comm_monoid_diff:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.30/0.46 comm_monoid_mult:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.30/0.46 ab_semigroup_mult:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 0.30/0.46 ordered_semiring_0:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.46 ordered_ab_group_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oordered__comm__semiring,type,
% 0.30/0.46 ordere1490568538miring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 0.30/0.46 comm_s1003936772cancel:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 0.30/0.46 linord581940658strict:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.30/0.46 cancel1352612707id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 0.30/0.46 ring_11004092258visors:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.30/0.46 cancel146912293up_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 0.30/0.46 ordere216010020id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.30/0.46 semiri1193490041visors:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.30/0.46 linord1659791738miring:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.30/0.46 canoni770627133id_add:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.30/0.46 semiri1923998003cancel:
% 0.30/0.46 !>[A: $tType] : $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ofun__upd,type,
% 0.30/0.46 fun_upd:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.30/0.46 minus_minus:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.30/0.46 one_one:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.30/0.46 times_times:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.30/0.46 zero_zero:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 0.30/0.46 groups1340683514dd_sum:
% 0.30/0.46 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If,type,
% 0.30/0.46 if:
% 0.30/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 0.30/0.46 number2016821345itions: ( nat > nat ) > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.46 ord_less:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 0.30/0.46 set_ord_atMost:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_k,type,
% 0.30/0.46 k: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_n,type,
% 0.30/0.46 n: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_p,type,
% 0.30/0.46 p: nat > nat ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (253)
% 0.30/0.46 thf(fact_0_partitions,axiom,
% 0.30/0.46 number2016821345itions @ p @ n ).
% 0.30/0.46
% 0.30/0.46 % partitions
% 0.30/0.46 thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 0.30/0.46 ord_less_eq @ nat @ k @ n ).
% 0.30/0.46
% 0.30/0.46 % \<open>k \<le> n\<close>
% 0.30/0.46 thf(fact_2_atMost__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( order @ A )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ( set_ord_atMost @ A @ X )
% 0.30/0.46 = ( set_ord_atMost @ A @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_eq_iff
% 0.30/0.46 thf(fact_3_minus__apply,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_apply
% 0.30/0.46 thf(fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062,axiom,
% 0.30/0.46 ( ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
% 0.30/0.46 = ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ n ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>(\<Sum>i\<le>n - k. (p(k := p k - 1)) i * i) = (\<Sum>i\<le>n. (p(k := p k - 1)) i * i)\<close>
% 0.30/0.46 thf(fact_5_calculation,axiom,
% 0.30/0.46 ( ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
% 0.30/0.46 = ( minus_minus @ nat
% 0.30/0.46 @ ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ n ) )
% 0.30/0.46 @ k ) ) ).
% 0.30/0.46
% 0.30/0.46 % calculation
% 0.30/0.46 thf(fact_6_sum__subtractf,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ab_group_add @ A )
% 0.30/0.46 => ! [F: B > A,G: B > A,A3: set @ B] :
% 0.30/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [X2: B] : ( minus_minus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.46 @ A3 )
% 0.30/0.46 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_subtractf
% 0.30/0.46 thf(fact_7_sum__product,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( semiring_0 @ B )
% 0.30/0.46 => ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
% 0.30/0.46 ( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
% 0.30/0.46 = ( groups1340683514dd_sum @ A @ B
% 0.30/0.46 @ ^ [I: A] :
% 0.30/0.46 ( groups1340683514dd_sum @ C @ B
% 0.30/0.46 @ ^ [J: C] : ( times_times @ B @ ( F @ I ) @ ( G @ J ) )
% 0.30/0.46 @ B3 )
% 0.30/0.46 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_product
% 0.30/0.46 thf(fact_8_sum__distrib__left,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( semiring_0 @ A )
% 0.30/0.46 => ! [R: A,F: B > A,A3: set @ B] :
% 0.30/0.46 ( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
% 0.30/0.46 = ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
% 0.30/0.46 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_distrib_left
% 0.30/0.46 thf(fact_9_sum__distrib__right,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( semiring_0 @ A )
% 0.30/0.46 => ! [F: B > A,A3: set @ B,R: A] :
% 0.30/0.46 ( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 0.30/0.46 = ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
% 0.30/0.46 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_distrib_right
% 0.30/0.46 thf(fact_10_diff__mult__distrib,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.46 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
% 0.30/0.46 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mult_distrib
% 0.30/0.46 thf(fact_11_diff__mult__distrib2,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
% 0.30/0.46 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mult_distrib2
% 0.30/0.46 thf(fact_12_inf__period_I2_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ( comm_ring @ A )
% 0.30/0.46 & ( dvd @ A ) )
% 0.30/0.46 => ! [P: A > $o,D: A,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A,K2: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 0.30/0.46 => ( ! [X3: A,K2: A] :
% 0.30/0.46 ( ( Q @ X3 )
% 0.30/0.46 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 0.30/0.46 => ! [X4: A,K3: A] :
% 0.30/0.46 ( ( ( P @ X4 )
% 0.30/0.46 | ( Q @ X4 ) )
% 0.30/0.46 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
% 0.30/0.46 | ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inf_period(2)
% 0.30/0.46 thf(fact_13_atMost__subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X: A,Y: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
% 0.30/0.46 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_subset_iff
% 0.30/0.46 thf(fact_14_atMost__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ! [I2: A,K: A] :
% 0.30/0.46 ( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K ) )
% 0.30/0.46 = ( ord_less_eq @ A @ I2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_iff
% 0.30/0.46 thf(fact_15_diff__diff__cancel,axiom,
% 0.30/0.46 ! [I2: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ I2 @ N2 )
% 0.30/0.46 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I2 ) )
% 0.30/0.46 = I2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_cancel
% 0.30/0.46 thf(fact_16_nat__mult__eq__1__iff,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times @ nat @ M @ N2 )
% 0.30/0.46 = ( one_one @ nat ) )
% 0.30/0.46 = ( ( M
% 0.30/0.46 = ( one_one @ nat ) )
% 0.30/0.46 & ( N2
% 0.30/0.46 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_eq_1_iff
% 0.30/0.46 thf(fact_17_nat__1__eq__mult__iff,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( one_one @ nat )
% 0.30/0.46 = ( times_times @ nat @ M @ N2 ) )
% 0.30/0.46 = ( ( M
% 0.30/0.46 = ( one_one @ nat ) )
% 0.30/0.46 & ( N2
% 0.30/0.46 = ( one_one @ nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_1_eq_mult_iff
% 0.30/0.46 thf(fact_18_le__refl,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % le_refl
% 0.30/0.46 thf(fact_19_le__trans,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ J2 @ K )
% 0.30/0.46 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_trans
% 0.30/0.46 thf(fact_20_eq__imp__le,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( M = N2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_imp_le
% 0.30/0.46 thf(fact_21_le__antisym,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ N2 @ M )
% 0.30/0.46 => ( M = N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_antisym
% 0.30/0.46 thf(fact_22_nat__le__linear,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.46 | ( ord_less_eq @ nat @ N2 @ M ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_le_linear
% 0.30/0.46 thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.46 ! [P: nat > $o,K: nat,B4: nat] :
% 0.30/0.46 ( ( P @ K )
% 0.30/0.46 => ( ! [Y2: nat] :
% 0.30/0.46 ( ( P @ Y2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ Y2 @ B4 ) )
% 0.30/0.46 => ? [X3: nat] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 & ! [Y3: nat] :
% 0.30/0.46 ( ( P @ Y3 )
% 0.30/0.46 => ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nat.ex_has_greatest_nat
% 0.30/0.46 thf(fact_24_bounded__Max__nat,axiom,
% 0.30/0.46 ! [P: nat > $o,X: nat,M2: nat] :
% 0.30/0.46 ( ( P @ X )
% 0.30/0.46 => ( ! [X3: nat] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( ord_less_eq @ nat @ X3 @ M2 ) )
% 0.30/0.46 => ~ ! [M3: nat] :
% 0.30/0.46 ( ( P @ M3 )
% 0.30/0.46 => ~ ! [X4: nat] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 => ( ord_less_eq @ nat @ X4 @ M3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bounded_Max_nat
% 0.30/0.46 thf(fact_25_partitions__bounds,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat,I2: nat] :
% 0.30/0.46 ( ( number2016821345itions @ P2 @ N2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( P2 @ I2 ) @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_bounds
% 0.30/0.46 thf(fact_26_diff__le__mono2,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_mono2
% 0.30/0.46 thf(fact_27_le__diff__iff_H,axiom,
% 0.30/0.46 ! [A4: nat,C2: nat,B4: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ A4 @ C2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ B4 @ C2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A4 ) @ ( minus_minus @ nat @ C2 @ B4 ) )
% 0.30/0.46 = ( ord_less_eq @ nat @ B4 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_diff_iff'
% 0.30/0.46 thf(fact_28_diff__le__self,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_self
% 0.30/0.46 thf(fact_29_diff__le__mono,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_mono
% 0.30/0.46 thf(fact_30_Nat_Odiff__diff__eq,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.30/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 0.30/0.46 = ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nat.diff_diff_eq
% 0.30/0.46 thf(fact_31_le__diff__iff,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 0.30/0.46 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_diff_iff
% 0.30/0.46 thf(fact_32_eq__diff__iff,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ K @ N2 )
% 0.30/0.46 => ( ( ( minus_minus @ nat @ M @ K )
% 0.30/0.46 = ( minus_minus @ nat @ N2 @ K ) )
% 0.30/0.46 = ( M = N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_diff_iff
% 0.30/0.46 thf(fact_33_mult__le__mono2,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono2
% 0.30/0.46 thf(fact_34_mult__le__mono1,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono1
% 0.30/0.46 thf(fact_35_mult__le__mono,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ K @ L )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono
% 0.30/0.46 thf(fact_36_le__square,axiom,
% 0.30/0.46 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_square
% 0.30/0.46 thf(fact_37_le__cube,axiom,
% 0.30/0.46 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cube
% 0.30/0.46 thf(fact_38_nat__mult__1__right,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
% 0.30/0.46 = N2 ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_1_right
% 0.30/0.46 thf(fact_39_nat__mult__1,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
% 0.30/0.46 = N2 ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_1
% 0.30/0.46 thf(fact_40_sum__mono,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ordere216010020id_add @ A )
% 0.30/0.46 => ! [K4: set @ B,F: B > A,G: B > A] :
% 0.30/0.46 ( ! [I3: B] :
% 0.30/0.46 ( ( member @ B @ I3 @ K4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ K4 ) @ ( groups1340683514dd_sum @ B @ A @ G @ K4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_mono
% 0.30/0.46 thf(fact_41_atMost__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord @ A )
% 0.30/0.46 => ( ( set_ord_atMost @ A )
% 0.30/0.46 = ( ^ [U: A] :
% 0.30/0.46 ( collect @ A
% 0.30/0.46 @ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_def
% 0.30/0.46 thf(fact_42_sum__subtractf__nat,axiom,
% 0.30/0.46 ! [A: $tType,A3: set @ A,G: A > nat,F: A > nat] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A3 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ A @ nat
% 0.30/0.46 @ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.46 @ A3 )
% 0.30/0.46 = ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( groups1340683514dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_subtractf_nat
% 0.30/0.46 thf(fact_43_partitions__parts__bounded,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ( number2016821345itions @ P2 @ N2 )
% 0.30/0.46 => ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ P2 @ ( set_ord_atMost @ nat @ N2 ) ) @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_parts_bounded
% 0.30/0.46 thf(fact_44_fun__diff__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_diff_def
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A3: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_47_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( Q @ X3 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_48_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( F @ X3 )
% 0.30/0.46 = ( G @ X3 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_49_sum_Oreindex__bij__witness,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
% 0.30/0.46 ( ! [A5: B] :
% 0.30/0.46 ( ( member @ B @ A5 @ S )
% 0.30/0.46 => ( ( I2 @ ( J2 @ A5 ) )
% 0.30/0.46 = A5 ) )
% 0.30/0.46 => ( ! [A5: B] :
% 0.30/0.46 ( ( member @ B @ A5 @ S )
% 0.30/0.46 => ( member @ C @ ( J2 @ A5 ) @ T ) )
% 0.30/0.46 => ( ! [B5: C] :
% 0.30/0.46 ( ( member @ C @ B5 @ T )
% 0.30/0.46 => ( ( J2 @ ( I2 @ B5 ) )
% 0.30/0.46 = B5 ) )
% 0.30/0.46 => ( ! [B5: C] :
% 0.30/0.46 ( ( member @ C @ B5 @ T )
% 0.30/0.46 => ( member @ B @ ( I2 @ B5 ) @ S ) )
% 0.30/0.46 => ( ! [A5: B] :
% 0.30/0.46 ( ( member @ B @ A5 @ S )
% 0.30/0.46 => ( ( H @ ( J2 @ A5 ) )
% 0.30/0.46 = ( G @ A5 ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ S )
% 0.30/0.46 = ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.reindex_bij_witness
% 0.30/0.46 thf(fact_50_sum_Oeq__general__inverses,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 0.30/0.46 ( ! [Y2: B] :
% 0.30/0.46 ( ( member @ B @ Y2 @ B3 )
% 0.30/0.46 => ( ( member @ C @ ( K @ Y2 ) @ A3 )
% 0.30/0.46 & ( ( H @ ( K @ Y2 ) )
% 0.30/0.46 = Y2 ) ) )
% 0.30/0.46 => ( ! [X3: C] :
% 0.30/0.46 ( ( member @ C @ X3 @ A3 )
% 0.30/0.46 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 0.30/0.46 & ( ( K @ ( H @ X3 ) )
% 0.30/0.46 = X3 )
% 0.30/0.46 & ( ( Gamma @ ( H @ X3 ) )
% 0.30/0.46 = ( Phi @ X3 ) ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 0.30/0.46 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.eq_general_inverses
% 0.30/0.46 thf(fact_51_sum_Oeq__general,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 0.30/0.46 ( ! [Y2: B] :
% 0.30/0.46 ( ( member @ B @ Y2 @ B3 )
% 0.30/0.46 => ? [X4: C] :
% 0.30/0.46 ( ( member @ C @ X4 @ A3 )
% 0.30/0.46 & ( ( H @ X4 )
% 0.30/0.46 = Y2 )
% 0.30/0.46 & ! [Ya: C] :
% 0.30/0.46 ( ( ( member @ C @ Ya @ A3 )
% 0.30/0.46 & ( ( H @ Ya )
% 0.30/0.46 = Y2 ) )
% 0.30/0.46 => ( Ya = X4 ) ) ) )
% 0.30/0.46 => ( ! [X3: C] :
% 0.30/0.46 ( ( member @ C @ X3 @ A3 )
% 0.30/0.46 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 0.30/0.46 & ( ( Gamma @ ( H @ X3 ) )
% 0.30/0.46 = ( Phi @ X3 ) ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 0.30/0.46 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.eq_general
% 0.30/0.46 thf(fact_52_sum_Ocong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
% 0.30/0.46 ( ( A3 = B3 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B3 )
% 0.30/0.46 => ( ( G @ X3 )
% 0.30/0.46 = ( H @ X3 ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 0.30/0.46 = ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.cong
% 0.30/0.46 thf(fact_53_diff__commute,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ K )
% 0.30/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_commute
% 0.30/0.46 thf(fact_54_sum_Oswap,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [G: B > C > A,B3: set @ C,A3: set @ B] :
% 0.30/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [I: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I ) @ B3 )
% 0.30/0.46 @ A3 )
% 0.30/0.46 = ( groups1340683514dd_sum @ C @ A
% 0.30/0.46 @ ^ [J: C] :
% 0.30/0.46 ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [I: B] : ( G @ I @ J )
% 0.30/0.46 @ A3 )
% 0.30/0.46 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.swap
% 0.30/0.46 thf(fact_55_inf__period_I1_J,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ( comm_ring @ A )
% 0.30/0.46 & ( dvd @ A ) )
% 0.30/0.46 => ! [P: A > $o,D: A,Q: A > $o] :
% 0.30/0.46 ( ! [X3: A,K2: A] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 0.30/0.46 => ( ! [X3: A,K2: A] :
% 0.30/0.46 ( ( Q @ X3 )
% 0.30/0.46 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 0.30/0.46 => ! [X4: A,K3: A] :
% 0.30/0.46 ( ( ( P @ X4 )
% 0.30/0.46 & ( Q @ X4 ) )
% 0.30/0.46 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
% 0.30/0.46 & ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inf_period(1)
% 0.30/0.46 thf(fact_56_mult_Oright__neutral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( monoid_mult @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( times_times @ A @ A4 @ ( one_one @ A ) )
% 0.30/0.46 = A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult.right_neutral
% 0.30/0.46 thf(fact_57_mult_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( monoid_mult @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( times_times @ A @ ( one_one @ A ) @ A4 )
% 0.30/0.46 = A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult.left_neutral
% 0.30/0.46 thf(fact_58_fun__upd__upd,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 0.30/0.46 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 0.30/0.46 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_upd
% 0.30/0.46 thf(fact_59_fun__upd__triv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.30/0.46 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_triv
% 0.30/0.46 thf(fact_60_fun__upd__apply,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( fun_upd @ B @ A )
% 0.30/0.46 = ( ^ [F2: B > A,X2: B,Y4: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_apply
% 0.30/0.46 thf(fact_61_order__refl,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( preorder @ A )
% 0.30/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_62_lambda__one,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( monoid_mult @ A )
% 0.30/0.46 => ( ( ^ [X2: A] : X2 )
% 0.30/0.46 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_one
% 0.30/0.46 thf(fact_63_partitions__def,axiom,
% 0.30/0.46 ( number2016821345itions
% 0.30/0.46 = ( ^ [P3: nat > nat,N: nat] :
% 0.30/0.46 ( ! [I: nat] :
% 0.30/0.46 ( ( ( P3 @ I )
% 0.30/0.46 != ( zero_zero @ nat ) )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I )
% 0.30/0.46 & ( ord_less_eq @ nat @ I @ N ) ) )
% 0.30/0.46 & ( ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( P3 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ N ) )
% 0.30/0.46 = N ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_def
% 0.30/0.46 thf(fact_64_partitionsI,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ! [I3: nat] :
% 0.30/0.46 ( ( ( P2 @ I3 )
% 0.30/0.46 != ( zero_zero @ nat ) )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I3 )
% 0.30/0.46 & ( ord_less_eq @ nat @ I3 @ N2 ) ) )
% 0.30/0.46 => ( ( ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ N2 ) )
% 0.30/0.46 = N2 )
% 0.30/0.46 => ( number2016821345itions @ P2 @ N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitionsI
% 0.30/0.46 thf(fact_65_partitionsE,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ( number2016821345itions @ P2 @ N2 )
% 0.30/0.46 => ~ ( ! [I4: nat] :
% 0.30/0.46 ( ( ( P2 @ I4 )
% 0.30/0.46 != ( zero_zero @ nat ) )
% 0.30/0.46 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I4 )
% 0.30/0.46 & ( ord_less_eq @ nat @ I4 @ N2 ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost @ nat @ N2 ) )
% 0.30/0.46 != N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitionsE
% 0.30/0.46 thf(fact_66_le__zero__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A )
% 0.30/0.46 => ! [N2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 0.30/0.46 = ( N2
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_zero_eq
% 0.30/0.46 thf(fact_67_mult__zero__left,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( mult_zero @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_zero_left
% 0.30/0.46 thf(fact_68_mult__zero__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( mult_zero @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_zero_right
% 0.30/0.46 thf(fact_69_mult__eq__0__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1193490041visors @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ A4 @ B4 )
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 = ( ( A4
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( B4
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_eq_0_iff
% 0.30/0.46 thf(fact_70_mult__cancel__left,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1923998003cancel @ A )
% 0.30/0.46 => ! [C2: A,A4: A,B4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ C2 @ A4 )
% 0.30/0.46 = ( times_times @ A @ C2 @ B4 ) )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_left
% 0.30/0.46 thf(fact_71_mult__cancel__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1923998003cancel @ A )
% 0.30/0.46 => ! [A4: A,C2: A,B4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ A4 @ C2 )
% 0.30/0.46 = ( times_times @ A @ B4 @ C2 ) )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( A4 = B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_right
% 0.30/0.46 thf(fact_72_diff__self,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A4 @ A4 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_self
% 0.30/0.46 thf(fact_73_diff__0__right,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 = A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_0_right
% 0.30/0.46 thf(fact_74_zero__diff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( comm_monoid_diff @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_diff
% 0.30/0.46 thf(fact_75_diff__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel1352612707id_add @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 = A4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_zero
% 0.30/0.46 thf(fact_76_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel1352612707id_add @ A )
% 0.30/0.46 => ! [A4: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A4 @ A4 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.46 thf(fact_77_bot__nat__0_Oextremum,axiom,
% 0.30/0.46 ! [A4: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A4 ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum
% 0.30/0.46 thf(fact_78_le0,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % le0
% 0.30/0.46 thf(fact_79_diff__0__eq__0,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_0_eq_0
% 0.30/0.46 thf(fact_80_diff__self__eq__0,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( minus_minus @ nat @ M @ M )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_self_eq_0
% 0.30/0.46 thf(fact_81_mult__is__0,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times @ nat @ M @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) )
% 0.30/0.46 = ( ( M
% 0.30/0.46 = ( zero_zero @ nat ) )
% 0.30/0.46 | ( N2
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_is_0
% 0.30/0.46 thf(fact_82_mult__0__right,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_0_right
% 0.30/0.46 thf(fact_83_mult__cancel1,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times @ nat @ K @ M )
% 0.30/0.46 = ( times_times @ nat @ K @ N2 ) )
% 0.30/0.46 = ( ( M = N2 )
% 0.30/0.46 | ( K
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel1
% 0.30/0.46 thf(fact_84_mult__cancel2,axiom,
% 0.30/0.46 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times @ nat @ M @ K )
% 0.30/0.46 = ( times_times @ nat @ N2 @ K ) )
% 0.30/0.46 = ( ( M = N2 )
% 0.30/0.46 | ( K
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel2
% 0.30/0.46 thf(fact_85_sum_Oneutral__const,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [A3: set @ B] :
% 0.30/0.46 ( ( groups1340683514dd_sum @ B @ A
% 0.30/0.46 @ ^ [Uu: B] : ( zero_zero @ A )
% 0.30/0.46 @ A3 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.neutral_const
% 0.30/0.46 thf(fact_86_diff__ge__0__iff__ge,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
% 0.30/0.46 = ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_ge_0_iff_ge
% 0.30/0.46 thf(fact_87_mult__cancel__right2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ring_11004092258visors @ A )
% 0.30/0.46 => ! [A4: A,C2: A] :
% 0.30/0.46 ( ( ( times_times @ A @ A4 @ C2 )
% 0.30/0.46 = C2 )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( A4
% 0.30/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_right2
% 0.30/0.46 thf(fact_88_mult__cancel__right1,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ring_11004092258visors @ A )
% 0.30/0.46 => ! [C2: A,B4: A] :
% 0.30/0.46 ( ( C2
% 0.30/0.46 = ( times_times @ A @ B4 @ C2 ) )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( B4
% 0.30/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_right1
% 0.30/0.46 thf(fact_89_mult__cancel__left2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ring_11004092258visors @ A )
% 0.30/0.46 => ! [C2: A,A4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ C2 @ A4 )
% 0.30/0.46 = C2 )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( A4
% 0.30/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_left2
% 0.30/0.46 thf(fact_90_mult__cancel__left1,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ring_11004092258visors @ A )
% 0.30/0.46 => ! [C2: A,B4: A] :
% 0.30/0.46 ( ( C2
% 0.30/0.46 = ( times_times @ A @ C2 @ B4 ) )
% 0.30/0.46 = ( ( C2
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( B4
% 0.30/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_left1
% 0.30/0.46 thf(fact_91_diff__is__0__eq_H,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.30/0.46 => ( ( minus_minus @ nat @ M @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_is_0_eq'
% 0.30/0.46 thf(fact_92_diff__is__0__eq,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( minus_minus @ nat @ M @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) )
% 0.30/0.46 = ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_is_0_eq
% 0.30/0.46 thf(fact_93_gr0,axiom,
% 0.30/0.46 ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) ).
% 0.30/0.46
% 0.30/0.46 % gr0
% 0.30/0.46 thf(fact_94_zero__reorient,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( zero @ A )
% 0.30/0.46 => ! [X: A] :
% 0.30/0.46 ( ( ( zero_zero @ A )
% 0.30/0.46 = X )
% 0.30/0.46 = ( X
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_95_zero__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( canoni770627133id_add @ A )
% 0.30/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le
% 0.30/0.46 thf(fact_96_mult__not__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( mult_zero @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ A4 @ B4 )
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ( ( A4
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 & ( B4
% 0.30/0.46 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_not_zero
% 0.30/0.46 thf(fact_97_divisors__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1193490041visors @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ( times_times @ A @ A4 @ B4 )
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 => ( ( A4
% 0.30/0.46 = ( zero_zero @ A ) )
% 0.30/0.46 | ( B4
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divisors_zero
% 0.30/0.46 thf(fact_98_no__zero__divisors,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1193490041visors @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( A4
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ( ( B4
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ( ( times_times @ A @ A4 @ B4 )
% 0.30/0.46 != ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % no_zero_divisors
% 0.30/0.46 thf(fact_99_mult__left__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1923998003cancel @ A )
% 0.30/0.46 => ! [C2: A,A4: A,B4: A] :
% 0.30/0.46 ( ( C2
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ( ( ( times_times @ A @ C2 @ A4 )
% 0.30/0.46 = ( times_times @ A @ C2 @ B4 ) )
% 0.30/0.46 = ( A4 = B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_left_cancel
% 0.30/0.46 thf(fact_100_mult__right__cancel,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semiri1923998003cancel @ A )
% 0.30/0.46 => ! [C2: A,A4: A,B4: A] :
% 0.30/0.46 ( ( C2
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ( ( ( times_times @ A @ A4 @ C2 )
% 0.30/0.46 = ( times_times @ A @ B4 @ C2 ) )
% 0.30/0.46 = ( A4 = B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_right_cancel
% 0.30/0.46 thf(fact_101_eq__iff__diff__eq__0,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A )
% 0.30/0.46 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [A6: A,B6: A] :
% 0.30/0.46 ( ( minus_minus @ A @ A6 @ B6 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff_diff_eq_0
% 0.30/0.46 thf(fact_102_zero__neq__one,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( zero_neq_one @ A )
% 0.30/0.46 => ( ( zero_zero @ A )
% 0.30/0.46 != ( one_one @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_neq_one
% 0.30/0.46 thf(fact_103_lambda__zero,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( mult_zero @ A )
% 0.30/0.46 => ( ( ^ [H2: A] : ( zero_zero @ A ) )
% 0.30/0.46 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_zero
% 0.30/0.46 thf(fact_104_sum_Oneutral,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [A3: set @ B,G: B > A] :
% 0.30/0.46 ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A3 )
% 0.30/0.46 => ( ( G @ X3 )
% 0.30/0.46 = ( zero_zero @ A ) ) )
% 0.30/0.46 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.neutral
% 0.30/0.46 thf(fact_105_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( comm_monoid_add @ A )
% 0.30/0.46 => ! [G: B > A,A3: set @ B] :
% 0.30/0.46 ( ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 0.30/0.46 != ( zero_zero @ A ) )
% 0.30/0.46 => ~ ! [A5: B] :
% 0.30/0.46 ( ( member @ B @ A5 @ A3 )
% 0.30/0.46 => ( ( G @ A5 )
% 0.30/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.not_neutral_contains_not_neutral
% 0.30/0.46 thf(fact_106_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A4: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
% 0.30/0.46 => ( A4
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum_uniqueI
% 0.30/0.46 thf(fact_107_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.46 ! [A4: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
% 0.30/0.46 = ( A4
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum_unique
% 0.30/0.46 thf(fact_108_le__0__eq,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 0.30/0.46 = ( N2
% 0.30/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_0_eq
% 0.30/0.46 thf(fact_109_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nat.simps(1)
% 0.30/0.46 thf(fact_110_diffs0__imp__equal,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( minus_minus @ nat @ M @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) )
% 0.30/0.46 => ( ( ( minus_minus @ nat @ N2 @ M )
% 0.30/0.46 = ( zero_zero @ nat ) )
% 0.30/0.46 => ( M = N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diffs0_imp_equal
% 0.30/0.46 thf(fact_111_minus__nat_Odiff__0,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.30/0.46 = M ) ).
% 0.30/0.46
% 0.30/0.46 % minus_nat.diff_0
% 0.30/0.46 thf(fact_112_mult__0,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.30/0.46 = ( zero_zero @ nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_0
% 0.30/0.46 thf(fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordere1490568538miring @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.46 thf(fact_114_zero__le__mult__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord581940658strict @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) )
% 0.30/0.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 0.30/0.46 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_mult_iff
% 0.30/0.46 thf(fact_115_mult__nonneg__nonpos2,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring_0 @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ B4 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonpos2
% 0.30/0.46 thf(fact_116_mult__nonpos__nonneg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring_0 @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonpos_nonneg
% 0.30/0.46 thf(fact_117_mult__nonneg__nonpos,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring_0 @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonpos
% 0.30/0.46 thf(fact_118_mult__nonneg__nonneg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring_0 @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonneg
% 0.30/0.46 thf(fact_119_split__mult__neg__le,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring_0 @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 0.30/0.46 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_mult_neg_le
% 0.30/0.46 thf(fact_120_mult__le__0__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linord581940658strict @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
% 0.30/0.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.46 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 0.30/0.46 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_0_iff
% 0.30/0.46 thf(fact_121_mult__right__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_right_mono
% 0.30/0.46 thf(fact_122_mult__right__mono__neg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ring @ A )
% 0.30/0.46 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_right_mono_neg
% 0.30/0.46 thf(fact_123_mult__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_semiring @ A )
% 0.30/0.46 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_left_mono
% 0.30/0.46 thf(fact_124_mult__nonpos__nonpos,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ring @ A )
% 0.30/0.46 => ! [A4: A,B4: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 0.30/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonpos_nonpos
% 0.30/0.46 thf(fact_125_mult__left__mono__neg,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ring @ A )
% 0.30/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_left_mono_neg
% 0.30/0.47 thf(fact_126_split__mult__pos__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ring @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.47 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 0.30/0.47 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 0.30/0.47 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % split_mult_pos_le
% 0.30/0.47 thf(fact_127_zero__le__square,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_ring @ A )
% 0.30/0.47 => ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_square
% 0.30/0.47 thf(fact_128_mult__mono_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_semiring @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_mono'
% 0.30/0.47 thf(fact_129_mult__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_semiring @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_mono
% 0.30/0.47 thf(fact_130_le__iff__diff__le__0,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ( ( ord_less_eq @ A )
% 0.30/0.47 = ( ^ [A6: A,B6: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_iff_diff_le_0
% 0.30/0.47 thf(fact_131_not__one__le__zero,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_one_le_zero
% 0.30/0.47 thf(fact_132_zero__le__one,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_le_one
% 0.30/0.47 thf(fact_133_partitions__zero,axiom,
% 0.30/0.47 ! [P2: nat > nat] :
% 0.30/0.47 ( ( number2016821345itions @ P2 @ ( zero_zero @ nat ) )
% 0.30/0.47 = ( P2
% 0.30/0.47 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % partitions_zero
% 0.30/0.47 thf(fact_134_mult__left__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ! [C2: A,A4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_left_le
% 0.30/0.47 thf(fact_135_mult__le__one,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B4 @ ( one_one @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_one
% 0.30/0.47 thf(fact_136_mult__right__le__one__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_right_le_one_le
% 0.30/0.47 thf(fact_137_mult__left__le__one__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_left_le_one_le
% 0.30/0.47 thf(fact_138_sum__nonpos,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ordere216010020id_add @ A )
% 0.30/0.47 => ! [A3: set @ B,F: B > A] :
% 0.30/0.47 ( ! [X3: B] :
% 0.30/0.47 ( ( member @ B @ X3 @ A3 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum_nonpos
% 0.30/0.47 thf(fact_139_sum__nonneg,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ordere216010020id_add @ A )
% 0.30/0.47 => ! [A3: set @ B,F: B > A] :
% 0.30/0.47 ( ! [X3: B] :
% 0.30/0.47 ( ( member @ B @ X3 @ A3 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sum_nonneg
% 0.30/0.47 thf(fact_140_mult__eq__self__implies__10,axiom,
% 0.30/0.47 ! [M: nat,N2: nat] :
% 0.30/0.47 ( ( M
% 0.30/0.47 = ( times_times @ nat @ M @ N2 ) )
% 0.30/0.47 => ( ( N2
% 0.30/0.47 = ( one_one @ nat ) )
% 0.30/0.47 | ( M
% 0.30/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_eq_self_implies_10
% 0.30/0.47 thf(fact_141_dual__order_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B4: A,A4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.antisym
% 0.30/0.47 thf(fact_142_dual__order_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.30/0.47 = ( ^ [A6: A,B6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B6 @ A6 )
% 0.30/0.47 & ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.eq_iff
% 0.30/0.47 thf(fact_143_dual__order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 0.30/0.47 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.trans
% 0.30/0.47 thf(fact_144_linorder__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.30/0.47 ( ! [A5: A,B5: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.30/0.47 => ( P @ A5 @ B5 ) )
% 0.30/0.47 => ( ! [A5: A,B5: A] :
% 0.30/0.47 ( ( P @ B5 @ A5 )
% 0.30/0.47 => ( P @ A5 @ B5 ) )
% 0.30/0.47 => ( P @ A4 @ B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_wlog
% 0.30/0.47 thf(fact_145_dual__order_Orefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.refl
% 0.30/0.47 thf(fact_146_order__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_trans
% 0.30/0.47 thf(fact_147_order__class_Oorder_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.47 => ( A4 = B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.antisym
% 0.30/0.47 thf(fact_148_ord__le__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( B4 = C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_trans
% 0.30/0.47 thf(fact_149_ord__eq__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( A4 = B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_trans
% 0.30/0.47 thf(fact_150_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.30/0.47 = ( ^ [A6: A,B6: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.30/0.47 & ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.eq_iff
% 0.30/0.47 thf(fact_151_antisym__conv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv
% 0.30/0.47 thf(fact_152_le__cases3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.30/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases3
% 0.30/0.47 thf(fact_153_order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.trans
% 0.30/0.47 thf(fact_154_le__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases
% 0.30/0.47 thf(fact_155_eq__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X = Y )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_refl
% 0.30/0.47 thf(fact_156_linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linear
% 0.30/0.47 thf(fact_157_antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym
% 0.30/0.47 thf(fact_158_eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.30/0.47 = ( ^ [X2: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.47 & ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_iff
% 0.30/0.47 thf(fact_159_ord__le__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ( F @ B4 )
% 0.30/0.47 = C2 )
% 0.30/0.47 => ( ! [X3: A,Y2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_subst
% 0.30/0.47 thf(fact_160_ord__eq__le__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( F @ B4 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.30/0.47 => ( ! [X3: B,Y2: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_161_order__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 0.30/0.47 => ( ! [X3: A,Y2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_162_order__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.30/0.47 => ( ! [X3: B,Y2: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_163_le__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.47 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_fun_def
% 0.30/0.47 thf(fact_164_le__funI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funI
% 0.30/0.47 thf(fact_165_le__funE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funE
% 0.30/0.47 thf(fact_166_le__funD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B )
% 0.30/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funD
% 0.30/0.47 thf(fact_167_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_mult @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
% 0.30/0.47 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.47 thf(fact_168_mult_Oassoc,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( semigroup_mult @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
% 0.30/0.47 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.assoc
% 0.30/0.47 thf(fact_169_mult_Ocommute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_mult @ A )
% 0.30/0.47 => ( ( times_times @ A )
% 0.30/0.47 = ( ^ [A6: A,B6: A] : ( times_times @ A @ B6 @ A6 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.commute
% 0.30/0.47 thf(fact_170_mult_Oleft__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ab_semigroup_mult @ A )
% 0.30/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ B4 @ ( times_times @ A @ A4 @ C2 ) )
% 0.30/0.47 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.left_commute
% 0.30/0.47 thf(fact_171_diff__eq__diff__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( group_add @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A4 @ B4 )
% 0.30/0.47 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.30/0.47 => ( ( A4 = B4 )
% 0.30/0.47 = ( C2 = D2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_eq
% 0.30/0.47 thf(fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( cancel146912293up_add @ A )
% 0.30/0.47 => ! [A4: A,C2: A,B4: A] :
% 0.30/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B4 )
% 0.30/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.30/0.47 thf(fact_173_one__reorient,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( one @ A )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ( ( ( one_one @ A )
% 0.30/0.47 = X )
% 0.30/0.47 = ( X
% 0.30/0.47 = ( one_one @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_reorient
% 0.30/0.47 thf(fact_174_fun__upd__idem__iff,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 0.30/0.47 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.47 = F )
% 0.30/0.47 = ( ( F @ X )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem_iff
% 0.30/0.47 thf(fact_175_fun__upd__twist,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,A4: A,C2: A,M: A > B,B4: B,D2: B] :
% 0.30/0.47 ( ( A4 != C2 )
% 0.30/0.47 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A4 @ B4 ) @ C2 @ D2 )
% 0.30/0.47 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D2 ) @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_twist
% 0.30/0.47 thf(fact_176_fun__upd__other,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 0.30/0.47 ( ( Z != X )
% 0.30/0.47 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 0.30/0.47 = ( F @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_other
% 0.30/0.47 thf(fact_177_fun__upd__same,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.47 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_same
% 0.30/0.47 thf(fact_178_fun__upd__idem,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 0.30/0.47 ( ( ( F @ X )
% 0.30/0.47 = Y )
% 0.30/0.47 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 0.30/0.47 = F ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem
% 0.30/0.47 thf(fact_179_fun__upd__eqD,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 0.30/0.47 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.30/0.47 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 0.30/0.47 => ( Y = Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_eqD
% 0.30/0.47 thf(fact_180_fun__upd__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( fun_upd @ A @ B )
% 0.30/0.47 = ( ^ [F2: A > B,A6: A,B6: B,X2: A] : ( if @ B @ ( X2 = A6 ) @ B6 @ ( F2 @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_def
% 0.30/0.47 thf(fact_181_diff__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A4: A,B4: A,D2: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_mono
% 0.30/0.47 thf(fact_182_diff__left__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_left_mono
% 0.30/0.47 thf(fact_183_diff__right__mono,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_right_mono
% 0.30/0.47 thf(fact_184_diff__eq__diff__less__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A,D2: A] :
% 0.30/0.47 ( ( ( minus_minus @ A @ A4 @ B4 )
% 0.30/0.47 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.47 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_eq_diff_less_eq
% 0.30/0.47 thf(fact_185_left__diff__distrib,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 )
% 0.30/0.47 = ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % left_diff_distrib
% 0.30/0.47 thf(fact_186_right__diff__distrib,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ring @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_diff_distrib
% 0.30/0.47 thf(fact_187_left__diff__distrib_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_s1003936772cancel @ A )
% 0.30/0.47 => ! [B4: A,C2: A,A4: A] :
% 0.30/0.47 ( ( times_times @ A @ ( minus_minus @ A @ B4 @ C2 ) @ A4 )
% 0.30/0.47 = ( minus_minus @ A @ ( times_times @ A @ B4 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % left_diff_distrib'
% 0.30/0.47 thf(fact_188_right__diff__distrib_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_s1003936772cancel @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
% 0.30/0.47 = ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_diff_distrib'
% 0.30/0.47 thf(fact_189_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_monoid_mult @ A )
% 0.30/0.47 => ! [A4: A] :
% 0.30/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A4 )
% 0.30/0.47 = A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % comm_monoid_mult_class.mult_1
% 0.30/0.47 thf(fact_190_mult_Ocomm__neutral,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comm_monoid_mult @ A )
% 0.30/0.47 => ! [A4: A] :
% 0.30/0.47 ( ( times_times @ A @ A4 @ ( one_one @ A ) )
% 0.30/0.47 = A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.comm_neutral
% 0.30/0.47 thf(fact_191_diff__numeral__special_I9_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( neg_numeral @ A )
% 0.30/0.47 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.30/0.47 = ( zero_zero @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_numeral_special(9)
% 0.30/0.47 thf(fact_192_subsetI,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ( member @ A @ X3 @ A3 )
% 0.30/0.47 => ( member @ A @ X3 @ B3 ) )
% 0.30/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_193_subset__antisym,axiom,
% 0.30/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.30/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.30/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.30/0.47 => ( A3 = B3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_194_partitions__remove1__bounds,axiom,
% 0.30/0.47 ! [P2: nat > nat,N2: nat,K: nat,I2: nat] :
% 0.30/0.47 ( ( number2016821345itions @ P2 @ N2 )
% 0.30/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P2 @ K ) )
% 0.30/0.47 => ( ( ( fun_upd @ nat @ nat @ P2 @ K @ ( minus_minus @ nat @ ( P2 @ K ) @ ( one_one @ nat ) ) @ I2 )
% 0.30/0.47 != ( zero_zero @ nat ) )
% 0.30/0.47 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I2 )
% 0.30/0.47 & ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % partitions_remove1_bounds
% 0.30/0.47 thf(fact_195_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ( times_times @ nat @ K @ M )
% 0.30/0.47 = ( times_times @ nat @ K @ N2 ) )
% 0.30/0.47 = ( ( K
% 0.30/0.47 = ( zero_zero @ nat ) )
% 0.30/0.47 | ( M = N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_eq_cancel_disj
% 0.30/0.47 thf(fact_196_le__numeral__extra_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(4)
% 0.30/0.47 thf(fact_197_le__numeral__extra_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(3)
% 0.30/0.47 thf(fact_198_not__gr__zero,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( canoni770627133id_add @ A )
% 0.30/0.47 => ! [N2: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 0.30/0.47 = ( N2
% 0.30/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_gr_zero
% 0.30/0.47 thf(fact_199_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.47 ! [A4: nat] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 != ( zero_zero @ nat ) )
% 0.30/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.not_eq_extremum
% 0.30/0.47 thf(fact_200_less__nat__zero__code,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_nat_zero_code
% 0.30/0.47 thf(fact_201_neq0__conv,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ( ( N2
% 0.30/0.47 != ( zero_zero @ nat ) )
% 0.30/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq0_conv
% 0.30/0.47 thf(fact_202_diff__gt__0__iff__gt,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ordered_ab_group_add @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
% 0.30/0.47 = ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % diff_gt_0_iff_gt
% 0.30/0.47 thf(fact_203_zero__less__diff,axiom,
% 0.30/0.47 ! [N2: nat,M: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
% 0.30/0.47 = ( ord_less @ nat @ M @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_diff
% 0.30/0.47 thf(fact_204_nat__mult__less__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 & ( ord_less @ nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_less_cancel_disj
% 0.30/0.47 thf(fact_205_mult__less__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 & ( ord_less @ nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel2
% 0.30/0.47 thf(fact_206_nat__0__less__mult__iff,axiom,
% 0.30/0.47 ! [M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.30/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_0_less_mult_iff
% 0.30/0.47 thf(fact_207_less__one,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
% 0.30/0.47 = ( N2
% 0.30/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_one
% 0.30/0.47 thf(fact_208_nat__mult__le__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_le_cancel_disj
% 0.30/0.47 thf(fact_209_mult__le__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
% 0.30/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.30/0.47 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel2
% 0.30/0.47 thf(fact_210_linorder__neqE__linordered__idom,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linordered_idom @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_neqE_linordered_idom
% 0.30/0.47 thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B4: A,A4: A] :
% 0.30/0.47 ( ( ord_less @ A @ B4 @ A4 )
% 0.30/0.47 => ( A4 != B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_not_eq
% 0.30/0.47 thf(fact_212_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ( A4 != B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_not_eq
% 0.30/0.47 thf(fact_213_not__less__iff__gr__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 | ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_iff_gr_or_eq
% 0.30/0.47 thf(fact_214_dual__order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B4: A,A4: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B4 @ A4 )
% 0.30/0.47 => ( ( ord_less @ A @ C2 @ B4 )
% 0.30/0.47 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans
% 0.30/0.47 thf(fact_215_linorder__less__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [P: A > A > $o,A4: A,B4: A] :
% 0.30/0.47 ( ! [A5: A,B5: A] :
% 0.30/0.47 ( ( ord_less @ A @ A5 @ B5 )
% 0.30/0.47 => ( P @ A5 @ B5 ) )
% 0.30/0.47 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 0.30/0.47 => ( ! [A5: A,B5: A] :
% 0.30/0.47 ( ( P @ B5 @ A5 )
% 0.30/0.47 => ( P @ A5 @ B5 ) )
% 0.30/0.47 => ( P @ A4 @ B4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_less_wlog
% 0.30/0.47 thf(fact_216_exists__least__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( wellorder @ A )
% 0.30/0.47 => ( ( ^ [P4: A > $o] :
% 0.30/0.47 ? [X5: A] : ( P4 @ X5 ) )
% 0.30/0.47 = ( ^ [P5: A > $o] :
% 0.30/0.47 ? [N: A] :
% 0.30/0.47 ( ( P5 @ N )
% 0.30/0.47 & ! [M4: A] :
% 0.30/0.47 ( ( ord_less @ A @ M4 @ N )
% 0.30/0.47 => ~ ( P5 @ M4 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % exists_least_iff
% 0.30/0.47 thf(fact_217_less__imp__not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_less
% 0.30/0.47 thf(fact_218_order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less @ A @ B4 @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans
% 0.30/0.47 thf(fact_219_dual__order_Oirrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.irrefl
% 0.30/0.47 thf(fact_220_linorder__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( X != Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_cases
% 0.30/0.47 thf(fact_221_less__imp__triv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,P: $o] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 => P ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_triv
% 0.30/0.47 thf(fact_222_less__imp__not__eq2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( Y != X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq2
% 0.30/0.47 thf(fact_223_antisym__conv3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ Y @ X )
% 0.30/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv3
% 0.30/0.47 thf(fact_224_less__induct,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( wellorder @ A )
% 0.30/0.47 => ! [P: A > $o,A4: A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ! [Y3: A] :
% 0.30/0.47 ( ( ord_less @ A @ Y3 @ X3 )
% 0.30/0.47 => ( P @ Y3 ) )
% 0.30/0.47 => ( P @ X3 ) )
% 0.30/0.47 => ( P @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_induct
% 0.30/0.47 thf(fact_225_less__not__sym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_not_sym
% 0.30/0.47 thf(fact_226_less__imp__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq
% 0.30/0.47 thf(fact_227_dual__order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [B4: A,A4: A] :
% 0.30/0.47 ( ( ord_less @ A @ B4 @ A4 )
% 0.30/0.47 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.asym
% 0.30/0.47 thf(fact_228_ord__less__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( B4 = C2 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_trans
% 0.30/0.47 thf(fact_229_ord__eq__less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A )
% 0.30/0.47 => ! [A4: A,B4: A,C2: A] :
% 0.30/0.47 ( ( A4 = B4 )
% 0.30/0.47 => ( ( ord_less @ A @ B4 @ C2 )
% 0.30/0.47 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_trans
% 0.30/0.47 thf(fact_230_less__irrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_irrefl
% 0.30/0.47 thf(fact_231_less__linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 | ( X = Y )
% 0.30/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_linear
% 0.30/0.47 thf(fact_232_less__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ Z )
% 0.30/0.47 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_trans
% 0.30/0.47 thf(fact_233_less__asym_H,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym'
% 0.30/0.47 thf(fact_234_less__asym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym
% 0.30/0.47 thf(fact_235_less__imp__neq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_neq
% 0.30/0.47 thf(fact_236_dense,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( dense_order @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ? [Z4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Z4 )
% 0.30/0.47 & ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dense
% 0.30/0.47 thf(fact_237_order_Oasym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A )
% 0.30/0.47 => ! [A4: A,B4: A] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.asym
% 0.30/0.47 thf(fact_238_neq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 = ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq_iff
% 0.30/0.47 thf(fact_239_neqE,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neqE
% 0.30/0.47 thf(fact_240_gt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_top @ A )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.30/0.47
% 0.30/0.47 % gt_ex
% 0.30/0.47 thf(fact_241_lt__ex,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( no_bot @ A )
% 0.30/0.47 => ! [X: A] :
% 0.30/0.47 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % lt_ex
% 0.30/0.47 thf(fact_242_order__less__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C: $tType] :
% 0.30/0.47 ( ( ( order @ C )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 0.30/0.47 => ( ! [X3: A,Y2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst2
% 0.30/0.47 thf(fact_243_order__less__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B )
% 0.30/0.47 & ( order @ A ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B4 @ C2 )
% 0.30/0.47 => ( ! [X3: B,Y2: B] :
% 0.30/0.47 ( ( ord_less @ B @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst1
% 0.30/0.47 thf(fact_244_ord__less__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 0.30/0.47 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.47 => ( ( ( F @ B4 )
% 0.30/0.47 = C2 )
% 0.30/0.47 => ( ! [X3: A,Y2: A] :
% 0.30/0.47 ( ( ord_less @ A @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_subst
% 0.30/0.47 thf(fact_245_ord__eq__less__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B )
% 0.30/0.47 & ( ord @ A ) )
% 0.30/0.47 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 0.30/0.47 ( ( A4
% 0.30/0.47 = ( F @ B4 ) )
% 0.30/0.47 => ( ( ord_less @ B @ B4 @ C2 )
% 0.30/0.47 => ( ! [X3: B,Y2: B] :
% 0.30/0.47 ( ( ord_less @ B @ X3 @ Y2 )
% 0.30/0.47 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_subst
% 0.30/0.47 thf(fact_246_less__numeral__extra_I3_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(3)
% 0.30/0.47 thf(fact_247_less__numeral__extra_I4_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linord1659791738miring @ A )
% 0.30/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(4)
% 0.30/0.47 thf(fact_248_infinite__descent__measure,axiom,
% 0.30/0.47 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ~ ( P @ X3 )
% 0.30/0.47 => ? [Y3: A] :
% 0.30/0.47 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X3 ) )
% 0.30/0.47 & ~ ( P @ Y3 ) ) )
% 0.30/0.47 => ( P @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % infinite_descent_measure
% 0.30/0.47 thf(fact_249_measure__induct__rule,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( wellorder @ B )
% 0.30/0.47 => ! [F: A > B,P: A > $o,A4: A] :
% 0.30/0.47 ( ! [X3: A] :
% 0.30/0.47 ( ! [Y3: A] :
% 0.30/0.47 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 0.30/0.47 => ( P @ Y3 ) )
% 0.30/0.47 => ( P @ X3 ) )
% 0.30/0.47 => ( P @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % measure_induct_rule
% 0.30/0.47 thf(fact_250_linorder__neqE__nat,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.30/0.47 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_neqE_nat
% 0.30/0.47 thf(fact_251_infinite__descent,axiom,
% 0.30/0.47 ! [P: nat > $o,N2: nat] :
% 0.30/0.47 ( ! [N3: nat] :
% 0.30/0.47 ( ~ ( P @ N3 )
% 0.30/0.47 => ? [M5: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ M5 @ N3 )
% 0.30/0.47 & ~ ( P @ M5 ) ) )
% 0.30/0.47 => ( P @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % infinite_descent
% 0.30/0.47 thf(fact_252_nat__less__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,N2: nat] :
% 0.30/0.47 ( ! [N3: nat] :
% 0.30/0.47 ( ! [M5: nat] :
% 0.30/0.47 ( ( ord_less @ nat @ M5 @ N3 )
% 0.30/0.47 => ( P @ M5 ) )
% 0.30/0.47 => ( P @ N3 ) )
% 0.30/0.47 => ( P @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_less_induct
% 0.30/0.47
% 0.30/0.47 % Type constructors (43)
% 0.30/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.47 ! [A7: $tType,A8: $tType] :
% 0.30/0.47 ( ( preorder @ A8 )
% 0.30/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.47 ! [A7: $tType,A8: $tType] :
% 0.30/0.47 ( ( order @ A8 )
% 0.30/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.47 ! [A7: $tType,A8: $tType] :
% 0.30/0.47 ( ( ord @ A8 )
% 0.30/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.30/0.47 ! [A7: $tType,A8: $tType] :
% 0.30/0.47 ( ( minus @ A8 )
% 0.30/0.47 => ( minus @ ( A7 > A8 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.30/0.47 semiri1923998003cancel @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.30/0.47 canoni770627133id_add @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.30/0.47 linord1659791738miring @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 0.30/0.47 semiri1193490041visors @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 0.30/0.47 ordere216010020id_add @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.30/0.47 cancel146912293up_add @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.30/0.47 cancel1352612707id_add @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 0.30/0.47 comm_s1003936772cancel @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
% 0.30/0.47 ordere1490568538miring @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
% 0.30/0.47 ordered_semiring_0 @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.30/0.47 ab_semigroup_mult @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.30/0.47 comm_monoid_mult @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.30/0.47 comm_monoid_diff @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 0.30/0.47 ordered_semiring @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.30/0.47 comm_monoid_add @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.30/0.47 semigroup_mult @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.30/0.47 wellorder @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.30/0.47 zero_neq_one @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.30/0.47 preorder @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.47 linorder @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 0.30/0.47 monoid_mult @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
% 0.30/0.47 semiring_0 @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.30/0.47 no_top @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 0.30/0.47 mult_zero @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.30/0.47 order @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.30/0.47 ord @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 0.30/0.47 minus @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.30/0.47 zero @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.30/0.47 one @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Nat_Onat___Rings_Odvd,axiom,
% 0.30/0.47 dvd @ nat ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.30/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.30/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.30/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
% 0.30/0.47 ! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 0.30/0.47 preorder @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
% 0.30/0.47 linorder @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.30/0.47 order @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.30/0.47 ord @ $o ).
% 0.30/0.47
% 0.30/0.47 thf(tcon_HOL_Obool___Groups_Ominus_13,axiom,
% 0.30/0.47 minus @ $o ).
% 0.30/0.47
% 0.30/0.47 % Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_T,axiom,
% 0.30/0.47 ! [P: $o] :
% 0.30/0.47 ( ( P = $true )
% 0.30/0.47 | ( P = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_T,axiom,
% 0.30/0.47 ! [A: $tType,X: A,Y: A] :
% 0.30/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 ( ( minus_minus @ nat
% 0.30/0.47 @ ( groups1340683514dd_sum @ nat @ nat
% 0.30/0.47 @ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
% 0.30/0.47 @ ( set_ord_atMost @ nat @ n ) )
% 0.30/0.47 @ k )
% 0.30/0.47 = ( minus_minus @ nat @ n @ k ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 3.50/1.64 ---- Embedded in THF ---
% 3.50/1.64 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.50/1.64 %%% Generated on Tue Feb 24 08:27:40 EST 2026
% 3.50/1.64 %%% using '$$dhol' embedding, version 1.3.0.
% 3.50/1.64 %%% Logic specification used:
% 3.50/1.64 %%% thf(spec, logic, $$dhol).
% 3.50/1.64
% 3.50/1.64 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.37uXGZ2q2e/DTF2DTF25510.p
% 3.50/1.64 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.50/1.64 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.50/1.64 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Odvd, type, dvd: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Osemiring__0, type, semiring_0: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oordered__ring, type, ordered_ring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oordered__semiring, type, ordered_semiring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oordered__semiring__0, type, ordered_semiring_0: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oordered__comm__semiring, type, ordere1490568538miring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Ocomm__semiring__1__cancel, type, comm_s1003936772cancel: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord581940658strict: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Oring__1__no__zero__divisors, type, ring_11004092258visors: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere216010020id_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri1923998003cancel: (!> [A:$tType]: ($o))).
% 3.50/1.64 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 3.50/1.64 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 3.50/1.64 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 3.50/1.64 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 3.50/1.64 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.50/1.64 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum, type, groups1340683514dd_sum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 3.50/1.64 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.50/1.64 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions, type, number2016821345itions: ((nat > nat) > (nat > $o))).
% 3.50/1.64 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.50/1.64 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.50/1.64 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.50/1.64 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))).
% 3.50/1.64 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.50/1.64 thf(sy_v_k, type, k: nat).
% 3.50/1.64 thf(sy_v_n, type, n: nat).
% 3.50/1.64 thf(sy_v_p, type, p: (nat > nat)).
% 3.50/1.64 thf(fact_0_partitions, axiom, ((number2016821345itions @ p) @ n)).
% 3.50/1.64 thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062, axiom, (((ord_less_eq @ nat) @ k) @ n)).
% 3.50/1.64 thf(fact_2_atMost__eq__iff, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((set_ord_atMost @ A) @ X) = ((set_ord_atMost @ A) @ Y)) = (X = Y)))))))).
% 3.50/1.64 thf(fact_3_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X2:A]: ((((minus_minus @ B) @ (A2 @ X2)) @ (B2 @ X2))))))))).
% 3.50/1.64 thf(fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062, axiom, (((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ ((((((fun_upd @ nat) @ nat) @ p) @ k) @ (((minus_minus @ nat) @ (p @ k)) @ (one_one @ nat))) @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ n) @ k))) = ((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ ((((((fun_upd @ nat) @ nat) @ p) @ k) @ (((minus_minus @ nat) @ (p @ k)) @ (one_one @ nat))) @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ n)))).
% 3.50/1.64 thf(fact_5_calculation, axiom, (((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ ((((((fun_upd @ nat) @ nat) @ p) @ k) @ (((minus_minus @ nat) @ (p @ k)) @ (one_one @ nat))) @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ n) @ k))) = (((minus_minus @ nat) @ ((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ (p @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ n))) @ k))).
% 3.50/1.64 thf(fact_6_sum__subtractf, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [F:(B > A),G:(B > A),A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [X2:B]: ((((minus_minus @ A) @ (F @ X2)) @ (G @ X2))))) @ A3) = (((minus_minus @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ A3))))))))).
% 3.50/1.64 thf(fact_7_sum__product, axiom, (! [B:$tType,C:$tType,A:$tType]: (((semiring_0 @ B) => (! [F:(A > B),A3:(set @ A),G:(C > B),B3:(set @ C)]: (((((times_times @ B) @ ((((groups1340683514dd_sum @ A) @ B) @ F) @ A3)) @ ((((groups1340683514dd_sum @ C) @ B) @ G) @ B3)) = ((((groups1340683514dd_sum @ A) @ B) @ (^ [I:A]: (((((groups1340683514dd_sum @ C) @ B) @ (^ [J:C]: ((((times_times @ B) @ (F @ I)) @ (G @ J))))) @ B3)))) @ A3)))))))).
% 3.50/1.64 thf(fact_8_sum__distrib__left, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [R:A,F:(B > A),A3:(set @ B)]: (((((times_times @ A) @ R) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) = ((((groups1340683514dd_sum @ B) @ A) @ (^ [N:B]: ((((times_times @ A) @ R) @ (F @ N))))) @ A3)))))))).
% 3.50/1.64 thf(fact_9_sum__distrib__right, axiom, (! [A:$tType,B:$tType]: (((semiring_0 @ A) => (! [F:(B > A),A3:(set @ B),R:A]: (((((times_times @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ R) = ((((groups1340683514dd_sum @ B) @ A) @ (^ [N:B]: ((((times_times @ A) @ (F @ N)) @ R)))) @ A3)))))))).
% 3.50/1.64 thf(fact_10_diff__mult__distrib, axiom, (! [M:nat,N2:nat,K:nat]: (((((times_times @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ K) = (((minus_minus @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)))))).
% 3.50/1.64 thf(fact_11_diff__mult__distrib2, axiom, (! [K:nat,M:nat,N2:nat]: (((((times_times @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N2)) = (((minus_minus @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)))))).
% 3.50/1.64 thf(fact_12_inf__period_I2_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [P:(A > $o),D:A,Q:(A > $o)]: (((! [X3:A,K2:A]: (((P @ X3) = (P @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D)))))) => ((! [X3:A,K2:A]: (((Q @ X3) = (Q @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D)))))) => (! [X4:A,K3:A]: ((((P @ X4) | (Q @ X4)) = ((P @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K3) @ D))) | (Q @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K3) @ D))))))))))))))).
% 3.50/1.64 thf(fact_13_atMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X)) @ ((set_ord_atMost @ A) @ Y)) = (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.50/1.64 thf(fact_14_atMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I2:A,K:A]: (((((member @ A) @ I2) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I2) @ K)))))))).
% 3.50/1.64 thf(fact_15_diff__diff__cancel, axiom, (! [I2:nat,N2:nat]: (((((ord_less_eq @ nat) @ I2) @ N2) => ((((minus_minus @ nat) @ N2) @ (((minus_minus @ nat) @ N2) @ I2)) = I2))))).
% 3.50/1.64 thf(fact_16_nat__mult__eq__1__iff, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (one_one @ nat)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))).
% 3.50/1.64 thf(fact_17_nat__1__eq__mult__iff, axiom, (! [M:nat,N2:nat]: ((((one_one @ nat) = (((times_times @ nat) @ M) @ N2)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))).
% 3.50/1.64 thf(fact_18_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 3.50/1.64 thf(fact_19_le__trans, axiom, (! [I2:nat,J2:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J2) => ((((ord_less_eq @ nat) @ J2) @ K) => (((ord_less_eq @ nat) @ I2) @ K)))))).
% 3.50/1.64 thf(fact_20_eq__imp__le, axiom, (! [M:nat,N2:nat]: (((M = N2) => (((ord_less_eq @ nat) @ M) @ N2))))).
% 3.50/1.64 thf(fact_21_le__antisym, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M) => (M = N2)))))).
% 3.50/1.64 thf(fact_22_nat__le__linear, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) | (((ord_less_eq @ nat) @ N2) @ M))))).
% 3.50/1.64 thf(fact_23_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B4:nat]: (((P @ K) => ((! [Y2:nat]: (((P @ Y2) => (((ord_less_eq @ nat) @ Y2) @ B4)))) => (? [X3:nat]: (((P @ X3) & (! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ X3)))))))))))).
% 3.50/1.64 thf(fact_24_bounded__Max__nat, axiom, (! [P:(nat > $o),X:nat,M2:nat]: (((P @ X) => ((! [X3:nat]: (((P @ X3) => (((ord_less_eq @ nat) @ X3) @ M2)))) => (~ ((! [M3:nat]: (((P @ M3) => (~ ((! [X4:nat]: (((P @ X4) => (((ord_less_eq @ nat) @ X4) @ M3)))))))))))))))).
% 3.50/1.64 thf(fact_25_partitions__bounds, axiom, (! [P2:(nat > nat),N2:nat,I2:nat]: ((((number2016821345itions @ P2) @ N2) => (((ord_less_eq @ nat) @ (P2 @ I2)) @ N2))))).
% 3.50/1.64 thf(fact_26_diff__le__mono2, axiom, (! [M:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N2)) @ (((minus_minus @ nat) @ L) @ M)))))).
% 3.50/1.64 thf(fact_27_le__diff__iff_H, axiom, (! [A4:nat,C2:nat,B4:nat]: (((((ord_less_eq @ nat) @ A4) @ C2) => ((((ord_less_eq @ nat) @ B4) @ C2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C2) @ A4)) @ (((minus_minus @ nat) @ C2) @ B4)) = (((ord_less_eq @ nat) @ B4) @ A4))))))).
% 3.50/1.64 thf(fact_28_diff__le__self, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ M)))).
% 3.50/1.64 thf(fact_29_diff__le__mono, axiom, (! [M:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N2) @ L)))))).
% 3.50/1.64 thf(fact_30_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((minus_minus @ nat) @ M) @ N2))))))).
% 3.50/1.64 thf(fact_31_le__diff__iff, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((ord_less_eq @ nat) @ M) @ N2))))))).
% 3.50/1.64 thf(fact_32_eq__diff__iff, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N2) @ K)) = (M = N2))))))).
% 3.50/1.64 thf(fact_33_mult__le__mono2, axiom, (! [I2:nat,J2:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J2) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ I2)) @ (((times_times @ nat) @ K) @ J2)))))).
% 3.50/1.64 thf(fact_34_mult__le__mono1, axiom, (! [I2:nat,J2:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J2) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I2) @ K)) @ (((times_times @ nat) @ J2) @ K)))))).
% 3.50/1.64 thf(fact_35_mult__le__mono, axiom, (! [I2:nat,J2:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I2) @ J2) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I2) @ K)) @ (((times_times @ nat) @ J2) @ L))))))).
% 3.50/1.64 thf(fact_36_le__square, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ M))))).
% 3.50/1.64 thf(fact_37_le__cube, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ (((times_times @ nat) @ M) @ M)))))).
% 3.50/1.64 thf(fact_38_nat__mult__1__right, axiom, (! [N2:nat]: (((((times_times @ nat) @ N2) @ (one_one @ nat)) = N2)))).
% 3.50/1.64 thf(fact_39_nat__mult__1, axiom, (! [N2:nat]: (((((times_times @ nat) @ (one_one @ nat)) @ N2) = N2)))).
% 3.50/1.64 thf(fact_40_sum__mono, axiom, (! [A:$tType,B:$tType]: (((ordere216010020id_add @ A) => (! [K4:(set @ B),F:(B > A),G:(B > A)]: (((! [I3:B]: (((((member @ B) @ I3) @ K4) => (((ord_less_eq @ A) @ (F @ I3)) @ (G @ I3))))) => (((ord_less_eq @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ K4)) @ ((((groups1340683514dd_sum @ B) @ A) @ G) @ K4))))))))).
% 3.50/1.64 thf(fact_41_atMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atMost @ A) = (^ [U:A]: (((collect @ A) @ (^ [X2:A]: ((((ord_less_eq @ A) @ X2) @ U))))))))))).
% 3.50/1.64 thf(fact_42_sum__subtractf__nat, axiom, (! [A:$tType,A3:(set @ A),G:(A > nat),F:(A > nat)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((ord_less_eq @ nat) @ (G @ X3)) @ (F @ X3))))) => (((((groups1340683514dd_sum @ A) @ nat) @ (^ [X2:A]: ((((minus_minus @ nat) @ (F @ X2)) @ (G @ X2))))) @ A3) = (((minus_minus @ nat) @ ((((groups1340683514dd_sum @ A) @ nat) @ F) @ A3)) @ ((((groups1340683514dd_sum @ A) @ nat) @ G) @ A3))))))).
% 3.50/1.64 thf(fact_43_partitions__parts__bounded, axiom, (! [P2:(nat > nat),N2:nat]: ((((number2016821345itions @ P2) @ N2) => (((ord_less_eq @ nat) @ ((((groups1340683514dd_sum @ nat) @ nat) @ P2) @ ((set_ord_atMost @ nat) @ N2))) @ N2))))).
% 3.50/1.64 thf(fact_44_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A2:(A > B),B2:(A > B),X2:A]: ((((minus_minus @ B) @ (A2 @ X2)) @ (B2 @ X2))))))))).
% 3.50/1.64 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 3.50/1.64 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 3.50/1.64 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.50/1.64 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 3.50/1.64 thf(fact_49_sum_Oreindex__bij__witness, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S:(set @ B),I2:(C > B),J2:(B > C),T:(set @ C),H:(C > A),G:(B > A)]: (((! [A5:B]: (((((member @ B) @ A5) @ S) => ((I2 @ (J2 @ A5)) = A5)))) => ((! [A5:B]: (((((member @ B) @ A5) @ S) => (((member @ C) @ (J2 @ A5)) @ T)))) => ((! [B5:C]: (((((member @ C) @ B5) @ T) => ((J2 @ (I2 @ B5)) = B5)))) => ((! [B5:C]: (((((member @ C) @ B5) @ T) => (((member @ B) @ (I2 @ B5)) @ S)))) => ((! [A5:B]: (((((member @ B) @ A5) @ S) => ((H @ (J2 @ A5)) = (G @ A5))))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ S) = ((((groups1340683514dd_sum @ C) @ A) @ H) @ T))))))))))))).
% 3.50/1.64 thf(fact_50_sum_Oeq__general__inverses, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ B),K:(B > C),A3:(set @ C),H:(C > B),Gamma:(B > A),Phi:(C > A)]: (((! [Y2:B]: (((((member @ B) @ Y2) @ B3) => ((((member @ C) @ (K @ Y2)) @ A3) & ((H @ (K @ Y2)) = Y2))))) => ((! [X3:C]: (((((member @ C) @ X3) @ A3) => ((((member @ B) @ (H @ X3)) @ B3) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3))))))) => (((((groups1340683514dd_sum @ C) @ A) @ Phi) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ Gamma) @ B3)))))))))).
% 3.50/1.64 thf(fact_51_sum_Oeq__general, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ B),A3:(set @ C),H:(C > B),Gamma:(B > A),Phi:(C > A)]: (((! [Y2:B]: (((((member @ B) @ Y2) @ B3) => (? [X4:C]: (((((member @ C) @ X4) @ A3) & (((H @ X4) = Y2) & (! [Ya:C]: ((((((member @ C) @ Ya) @ A3) & ((H @ Ya) = Y2)) => (Ya = X4))))))))))) => ((! [X3:C]: (((((member @ C) @ X3) @ A3) => ((((member @ B) @ (H @ X3)) @ B3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => (((((groups1340683514dd_sum @ C) @ A) @ Phi) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ Gamma) @ B3)))))))))).
% 3.50/1.64 thf(fact_52_sum_Ocong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A),H:(B > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((G @ X3) = (H @ X3))))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ A3) = ((((groups1340683514dd_sum @ B) @ A) @ H) @ B3)))))))))).
% 3.50/1.64 thf(fact_53_diff__commute, axiom, (! [I2:nat,J2:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I2) @ J2)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I2) @ K)) @ J2))))).
% 3.50/1.64 thf(fact_54_sum_Oswap, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [G:(B > (C > A)),B3:(set @ C),A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [I:B]: (((((groups1340683514dd_sum @ C) @ A) @ (G @ I)) @ B3)))) @ A3) = ((((groups1340683514dd_sum @ C) @ A) @ (^ [J:C]: (((((groups1340683514dd_sum @ B) @ A) @ (^ [I:B]: (((G @ I) @ J)))) @ A3)))) @ B3)))))))).
% 3.50/1.64 thf(fact_55_inf__period_I1_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [P:(A > $o),D:A,Q:(A > $o)]: (((! [X3:A,K2:A]: (((P @ X3) = (P @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D)))))) => ((! [X3:A,K2:A]: (((Q @ X3) = (Q @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ K2) @ D)))))) => (! [X4:A,K3:A]: ((((P @ X4) & (Q @ X4)) = ((P @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K3) @ D))) & (Q @ (((minus_minus @ A) @ X4) @ (((times_times @ A) @ K3) @ D))))))))))))))).
% 3.50/1.64 thf(fact_56_mult_Oright__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A4:A]: (((((times_times @ A) @ A4) @ (one_one @ A)) = A4))))))).
% 3.50/1.64 thf(fact_57_mult_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A4:A]: (((((times_times @ A) @ (one_one @ A)) @ A4) = A4))))))).
% 3.50/1.64 thf(fact_58_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,Z:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ X) @ Z) = (((((fun_upd @ A) @ B) @ F) @ X) @ Z))))).
% 3.50/1.64 thf(fact_59_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: (((((((fun_upd @ A) @ B) @ F) @ X) @ (F @ X)) = F)))).
% 3.50/1.64 thf(fact_60_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F2:(B > A),X2:B,Y4:A,Z2:B]: (((((if @ A) @ (Z2 = X2)) @ Y4) @ (F2 @ Z2)))))))).
% 3.50/1.64 thf(fact_61_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.50/1.64 thf(fact_62_lambda__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((^ [X2:A]: (X2)) = ((times_times @ A) @ (one_one @ A))))))).
% 3.50/1.64 thf(fact_63_partitions__def, axiom, (number2016821345itions = (^ [P3:(nat > nat),N:nat]: (((! [I:nat]: ((((P3 @ I) != (zero_zero @ nat)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ I) & (((ord_less_eq @ nat) @ I) @ N))))) & (((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ (P3 @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ N)) = N)))))).
% 3.50/1.64 thf(fact_64_partitionsI, axiom, (! [P2:(nat > nat),N2:nat]: (((! [I3:nat]: ((((P2 @ I3) != (zero_zero @ nat)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ I3) & (((ord_less_eq @ nat) @ I3) @ N2))))) => ((((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ (P2 @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ N2)) = N2) => ((number2016821345itions @ P2) @ N2)))))).
% 3.50/1.64 thf(fact_65_partitionsE, axiom, (! [P2:(nat > nat),N2:nat]: ((((number2016821345itions @ P2) @ N2) => (~ (((! [I4:nat]: ((((P2 @ I4) != (zero_zero @ nat)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ I4) & (((ord_less_eq @ nat) @ I4) @ N2))))) => (((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ (P2 @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ N2)) != N2)))))))).
% 3.50/1.64 thf(fact_66_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((((ord_less_eq @ A) @ N2) @ (zero_zero @ A)) = (N2 = (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_67_mult__zero__left, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A4:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A4) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_68_mult__zero__right, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A4:A]: (((((times_times @ A) @ A4) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_69_mult__eq__0__iff, axiom, (! [A:$tType]: (((semiri1193490041visors @ A) => (! [A4:A,B4:A]: ((((((times_times @ A) @ A4) @ B4) = (zero_zero @ A)) = ((A4 = (zero_zero @ A)) | (B4 = (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_70_mult__cancel__left, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [C2:A,A4:A,B4:A]: ((((((times_times @ A) @ C2) @ A4) = (((times_times @ A) @ C2) @ B4)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 3.50/1.64 thf(fact_71_mult__cancel__right, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [A4:A,C2:A,B4:A]: ((((((times_times @ A) @ A4) @ C2) = (((times_times @ A) @ B4) @ C2)) = ((C2 = (zero_zero @ A)) | (A4 = B4))))))))).
% 3.50/1.64 thf(fact_72_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: (((((minus_minus @ A) @ A4) @ A4) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_73_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A]: (((((minus_minus @ A) @ A4) @ (zero_zero @ A)) = A4))))))).
% 3.50/1.64 thf(fact_74_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A4:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A4) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_75_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A4:A]: (((((minus_minus @ A) @ A4) @ (zero_zero @ A)) = A4))))))).
% 3.50/1.64 thf(fact_76_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A4:A]: (((((minus_minus @ A) @ A4) @ A4) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_77_bot__nat__0_Oextremum, axiom, (! [A4:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A4)))).
% 3.50/1.64 thf(fact_78_le0, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 3.50/1.64 thf(fact_79_diff__0__eq__0, axiom, (! [N2:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N2) = (zero_zero @ nat))))).
% 3.50/1.64 thf(fact_80_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 3.50/1.64 thf(fact_81_mult__is__0, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N2 = (zero_zero @ nat))))))).
% 3.50/1.64 thf(fact_82_mult__0__right, axiom, (! [M:nat]: (((((times_times @ nat) @ M) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 3.50/1.64 thf(fact_83_mult__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N2)) = ((M = N2) | (K = (zero_zero @ nat))))))).
% 3.50/1.64 thf(fact_84_mult__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: ((((((times_times @ nat) @ M) @ K) = (((times_times @ nat) @ N2) @ K)) = ((M = N2) | (K = (zero_zero @ nat))))))).
% 3.50/1.64 thf(fact_85_sum_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B)]: ((((((groups1340683514dd_sum @ B) @ A) @ (^ [Uu:B]: ((zero_zero @ A)))) @ A3) = (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_86_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A4) @ B4)) = (((ord_less_eq @ A) @ B4) @ A4)))))))).
% 3.50/1.64 thf(fact_87_mult__cancel__right2, axiom, (! [A:$tType]: (((ring_11004092258visors @ A) => (! [A4:A,C2:A]: ((((((times_times @ A) @ A4) @ C2) = C2) = ((C2 = (zero_zero @ A)) | (A4 = (one_one @ A)))))))))).
% 3.50/1.64 thf(fact_88_mult__cancel__right1, axiom, (! [A:$tType]: (((ring_11004092258visors @ A) => (! [C2:A,B4:A]: (((C2 = (((times_times @ A) @ B4) @ C2)) = ((C2 = (zero_zero @ A)) | (B4 = (one_one @ A)))))))))).
% 3.50/1.64 thf(fact_89_mult__cancel__left2, axiom, (! [A:$tType]: (((ring_11004092258visors @ A) => (! [C2:A,A4:A]: ((((((times_times @ A) @ C2) @ A4) = C2) = ((C2 = (zero_zero @ A)) | (A4 = (one_one @ A)))))))))).
% 3.50/1.64 thf(fact_90_mult__cancel__left1, axiom, (! [A:$tType]: (((ring_11004092258visors @ A) => (! [C2:A,B4:A]: (((C2 = (((times_times @ A) @ C2) @ B4)) = ((C2 = (zero_zero @ A)) | (B4 = (one_one @ A)))))))))).
% 3.50/1.64 thf(fact_91_diff__is__0__eq_H, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)))))).
% 3.50/1.64 thf(fact_92_diff__is__0__eq, axiom, (! [M:nat,N2:nat]: ((((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N2))))).
% 3.50/1.64 thf(fact_93_gr0, axiom, (((ord_less @ nat) @ (zero_zero @ nat)) @ (p @ k))).
% 3.50/1.64 thf(fact_94_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_95_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 3.50/1.64 thf(fact_96_mult__not__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A4:A,B4:A]: ((((((times_times @ A) @ A4) @ B4) != (zero_zero @ A)) => ((A4 != (zero_zero @ A)) & (B4 != (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_97_divisors__zero, axiom, (! [A:$tType]: (((semiri1193490041visors @ A) => (! [A4:A,B4:A]: ((((((times_times @ A) @ A4) @ B4) = (zero_zero @ A)) => ((A4 = (zero_zero @ A)) | (B4 = (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_98_no__zero__divisors, axiom, (! [A:$tType]: (((semiri1193490041visors @ A) => (! [A4:A,B4:A]: (((A4 != (zero_zero @ A)) => ((B4 != (zero_zero @ A)) => ((((times_times @ A) @ A4) @ B4) != (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_99_mult__left__cancel, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ C2) @ A4) = (((times_times @ A) @ C2) @ B4)) = (A4 = B4))))))))).
% 3.50/1.64 thf(fact_100_mult__right__cancel, axiom, (! [A:$tType]: (((semiri1923998003cancel @ A) => (! [C2:A,A4:A,B4:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A4) @ C2) = (((times_times @ A) @ B4) @ C2)) = (A4 = B4))))))))).
% 3.50/1.64 thf(fact_101_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [A6:A,B6:A]: (((((minus_minus @ A) @ A6) @ B6) = (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_102_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 3.50/1.64 thf(fact_103_lambda__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => ((^ [H2:A]: ((zero_zero @ A))) = ((times_times @ A) @ (zero_zero @ A))))))).
% 3.50/1.64 thf(fact_104_sum_Oneutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A3) => ((G @ X3) = (zero_zero @ A))))) => (((((groups1340683514dd_sum @ B) @ A) @ G) @ A3) = (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_105_sum_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),A3:(set @ B)]: (((((((groups1340683514dd_sum @ B) @ A) @ G) @ A3) != (zero_zero @ A)) => (~ ((! [A5:B]: (((((member @ B) @ A5) @ A3) => ((G @ A5) = (zero_zero @ A)))))))))))))).
% 3.50/1.64 thf(fact_106_bot__nat__0_Oextremum__uniqueI, axiom, (! [A4:nat]: (((((ord_less_eq @ nat) @ A4) @ (zero_zero @ nat)) => (A4 = (zero_zero @ nat)))))).
% 3.50/1.64 thf(fact_107_bot__nat__0_Oextremum__unique, axiom, (! [A4:nat]: (((((ord_less_eq @ nat) @ A4) @ (zero_zero @ nat)) = (A4 = (zero_zero @ nat)))))).
% 3.50/1.64 thf(fact_108_le__0__eq, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ N2) @ (zero_zero @ nat)) = (N2 = (zero_zero @ nat)))))).
% 3.50/1.64 thf(fact_109_less__eq__nat_Osimps_I1_J, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))).
% 3.50/1.64 thf(fact_110_diffs0__imp__equal, axiom, (! [M:nat,N2:nat]: ((((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N2) @ M) = (zero_zero @ nat)) => (M = N2)))))).
% 3.50/1.64 thf(fact_111_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 3.50/1.64 thf(fact_112_mult__0, axiom, (! [N2:nat]: (((((times_times @ nat) @ (zero_zero @ nat)) @ N2) = (zero_zero @ nat))))).
% 3.50/1.64 thf(fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom, (! [A:$tType]: (((ordere1490568538miring @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)))))))))).
% 3.50/1.64 thf(fact_114_zero__le__mult__iff, axiom, (! [A:$tType]: (((linord581940658strict @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A4) @ B4)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A))))))))))).
% 3.50/1.64 thf(fact_115_mult__nonneg__nonpos2, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) => ((((ord_less_eq @ A) @ B4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ B4) @ A4)) @ (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_116_mult__nonpos__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B4) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ B4)) @ (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_117_mult__nonneg__nonpos, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) => ((((ord_less_eq @ A) @ B4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ B4)) @ (zero_zero @ A)))))))))).
% 3.50/1.64 thf(fact_118_mult__nonneg__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B4) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A4) @ B4)))))))))).
% 3.50/1.64 thf(fact_119_split__mult__neg__le, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A4:A,B4:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4))) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ B4)) @ (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_120_mult__le__0__iff, axiom, (! [A:$tType]: (((linord581940658strict @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ B4)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)))))))))).
% 3.50/1.64 thf(fact_121_mult__right__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ C2)))))))))).
% 3.50/1.64 thf(fact_122_mult__right__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ C2)))))))))).
% 3.50/1.64 thf(fact_123_mult__left__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)))))))))).
% 3.50/1.64 thf(fact_124_mult__nonpos__nonpos, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B4) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A4) @ B4)))))))))).
% 3.50/1.64 thf(fact_125_mult__left__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A4)) @ (((times_times @ A) @ C2) @ B4)))))))))).
% 3.50/1.64 thf(fact_126_split__mult__pos__le, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A4:A,B4:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B4)) | ((((ord_less_eq @ A) @ A4) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B4) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A4) @ B4))))))))).
% 3.50/1.64 thf(fact_127_zero__le__square, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A4) @ A4)))))))).
% 3.50/1.64 thf(fact_128_mult__mono_H, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A4:A,B4:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ D2)))))))))))).
% 3.50/1.64 thf(fact_129_mult__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A4:A,B4:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ D2)))))))))))).
% 3.50/1.64 thf(fact_130_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A6:A,B6:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A6) @ B6)) @ (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_131_not__one__le__zero, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_132_zero__le__one, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 3.50/1.64 thf(fact_133_partitions__zero, axiom, (! [P2:(nat > nat)]: ((((number2016821345itions @ P2) @ (zero_zero @ nat)) = (P2 = (^ [I:nat]: ((zero_zero @ nat)))))))).
% 3.50/1.64 thf(fact_134_mult__left__le, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [C2:A,A4:A]: (((((ord_less_eq @ A) @ C2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A4) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ C2)) @ A4))))))))).
% 3.50/1.64 thf(fact_135_mult__le__one, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B4) => ((((ord_less_eq @ A) @ B4) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A4) @ B4)) @ (one_one @ A))))))))))).
% 3.50/1.64 thf(fact_136_mult__right__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ X) @ Y)) @ X)))))))))).
% 3.50/1.64 thf(fact_137_mult__left__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Y) @ X)) @ X)))))))))).
% 3.50/1.64 thf(fact_138_sum__nonpos, axiom, (! [B:$tType,A:$tType]: (((ordere216010020id_add @ A) => (! [A3:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A3) => (((ord_less_eq @ A) @ (F @ X3)) @ (zero_zero @ A))))) => (((ord_less_eq @ A) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3)) @ (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_139_sum__nonneg, axiom, (! [A:$tType,B:$tType]: (((ordere216010020id_add @ A) => (! [A3:(set @ B),F:(B > A)]: (((! [X3:B]: (((((member @ B) @ X3) @ A3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X3))))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((((groups1340683514dd_sum @ B) @ A) @ F) @ A3))))))))).
% 3.50/1.64 thf(fact_140_mult__eq__self__implies__10, axiom, (! [M:nat,N2:nat]: (((M = (((times_times @ nat) @ M) @ N2)) => ((N2 = (one_one @ nat)) | (M = (zero_zero @ nat))))))).
% 3.50/1.64 thf(fact_141_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ A4) @ B4) => (A4 = B4))))))))).
% 3.50/1.64 thf(fact_142_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ B6) @ A6) & (((ord_less_eq @ A) @ A6) @ B6))))))))).
% 3.50/1.64 thf(fact_143_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => ((((ord_less_eq @ A) @ C2) @ B4) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 3.50/1.64 thf(fact_144_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A4) @ B4))))))))).
% 3.50/1.64 thf(fact_145_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 3.50/1.64 thf(fact_146_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 3.50/1.64 thf(fact_147_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ A4) => (A4 = B4))))))))).
% 3.50/1.64 thf(fact_148_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((B4 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_149_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((A4 = B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_150_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) & (((ord_less_eq @ A) @ B6) @ A6))))))))).
% 3.50/1.64 thf(fact_151_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.50/1.64 thf(fact_152_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 3.50/1.64 thf(fact_153_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_154_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.50/1.64 thf(fact_155_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.50/1.64 thf(fact_156_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.50/1.64 thf(fact_157_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 3.50/1.64 thf(fact_158_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X2))))))))).
% 3.50/1.64 thf(fact_159_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B4:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A4) @ B4) => (((F @ B4) = C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C2)))))))))).
% 3.50/1.64 thf(fact_160_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((A4 = (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 3.50/1.64 thf(fact_161_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 3.50/1.64 thf(fact_162_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 3.50/1.64 thf(fact_163_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 3.50/1.64 thf(fact_164_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.50/1.64 thf(fact_165_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 3.50/1.64 thf(fact_166_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 3.50/1.64 thf(fact_167_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [A4:A,B4:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A4) @ B4)) @ C2) = (((times_times @ A) @ A4) @ (((times_times @ A) @ B4) @ C2))))))))).
% 3.50/1.64 thf(fact_168_mult_Oassoc, axiom, (! [A:$tType]: (((semigroup_mult @ A) => (! [A4:A,B4:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A4) @ B4)) @ C2) = (((times_times @ A) @ A4) @ (((times_times @ A) @ B4) @ C2))))))))).
% 3.50/1.64 thf(fact_169_mult_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((times_times @ A) = (^ [A6:A,B6:A]: ((((times_times @ A) @ B6) @ A6)))))))).
% 3.50/1.64 thf(fact_170_mult_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [B4:A,A4:A,C2:A]: (((((times_times @ A) @ B4) @ (((times_times @ A) @ A4) @ C2)) = (((times_times @ A) @ A4) @ (((times_times @ A) @ B4) @ C2))))))))).
% 3.50/1.64 thf(fact_171_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A4:A,B4:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A4) @ B4) = (((minus_minus @ A) @ C2) @ D2)) => ((A4 = B4) = (C2 = D2))))))))).
% 3.50/1.64 thf(fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A4:A,C2:A,B4:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ C2)) @ B4) = (((minus_minus @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ C2)))))))).
% 3.50/1.64 thf(fact_173_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 3.50/1.64 thf(fact_174_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = F) = ((F @ X) = Y))))).
% 3.50/1.64 thf(fact_175_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A4:A,C2:A,M:(A > B),B4:B,D2:B]: (((A4 != C2) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ A4) @ B4)) @ C2) @ D2) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M) @ C2) @ D2)) @ A4) @ B4)))))).
% 3.50/1.64 thf(fact_176_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z:A,X:A,F:(A > B),Y:B]: (((Z != X) => (((((((fun_upd @ A) @ B) @ F) @ X) @ Y) @ Z) = (F @ Z)))))).
% 3.50/1.64 thf(fact_177_fun__upd__same, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A]: ((((((((fun_upd @ B) @ A) @ F) @ X) @ Y) @ X) = Y)))).
% 3.50/1.64 thf(fact_178_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B,Y:A]: ((((F @ X) = Y) => ((((((fun_upd @ B) @ A) @ F) @ X) @ Y) = F))))).
% 3.50/1.64 thf(fact_179_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,G:(A > B),Z:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = (((((fun_upd @ A) @ B) @ G) @ X) @ Z)) => (Y = Z))))).
% 3.50/1.64 thf(fact_180_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F2:(A > B),A6:A,B6:B,X2:A]: (((((if @ B) @ (X2 = A6)) @ B6) @ (F2 @ X2)))))))).
% 3.50/1.64 thf(fact_181_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A,D2:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((((ord_less_eq @ A) @ D2) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ C2)) @ (((minus_minus @ A) @ B4) @ D2)))))))))).
% 3.50/1.64 thf(fact_182_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A4) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A4)) @ (((minus_minus @ A) @ C2) @ B4))))))))).
% 3.50/1.64 thf(fact_183_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B4) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ C2)) @ (((minus_minus @ A) @ B4) @ C2))))))))).
% 3.50/1.64 thf(fact_184_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A4) @ B4) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ A4) @ B4) = (((ord_less_eq @ A) @ C2) @ D2))))))))).
% 3.50/1.64 thf(fact_185_left__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A4:A,B4:A,C2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A4) @ C2)) @ (((times_times @ A) @ B4) @ C2))))))))).
% 3.50/1.64 thf(fact_186_right__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A4:A,B4:A,C2:A]: (((((times_times @ A) @ A4) @ (((minus_minus @ A) @ B4) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A4) @ B4)) @ (((times_times @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_187_left__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s1003936772cancel @ A) => (! [B4:A,C2:A,A4:A]: (((((times_times @ A) @ (((minus_minus @ A) @ B4) @ C2)) @ A4) = (((minus_minus @ A) @ (((times_times @ A) @ B4) @ A4)) @ (((times_times @ A) @ C2) @ A4))))))))).
% 3.50/1.64 thf(fact_188_right__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s1003936772cancel @ A) => (! [A4:A,B4:A,C2:A]: (((((times_times @ A) @ A4) @ (((minus_minus @ A) @ B4) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A4) @ B4)) @ (((times_times @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_189_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A4:A]: (((((times_times @ A) @ (one_one @ A)) @ A4) = A4))))))).
% 3.50/1.64 thf(fact_190_mult_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A4:A]: (((((times_times @ A) @ A4) @ (one_one @ A)) = A4))))))).
% 3.50/1.64 thf(fact_191_diff__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 3.50/1.64 thf(fact_192_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 3.50/1.64 thf(fact_193_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 3.50/1.64 thf(fact_194_partitions__remove1__bounds, axiom, (! [P2:(nat > nat),N2:nat,K:nat,I2:nat]: ((((number2016821345itions @ P2) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ (P2 @ K)) => ((((((((fun_upd @ nat) @ nat) @ P2) @ K) @ (((minus_minus @ nat) @ (P2 @ K)) @ (one_one @ nat))) @ I2) != (zero_zero @ nat)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ I2) & (((ord_less_eq @ nat) @ I2) @ (((minus_minus @ nat) @ N2) @ K))))))))).
% 3.50/1.64 thf(fact_195_nat__mult__eq__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N2)) = ((K = (zero_zero @ nat)) | (M = N2)))))).
% 3.50/1.64 thf(fact_196_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 3.50/1.64 thf(fact_197_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 3.50/1.64 thf(fact_198_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N2:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N2))) = (N2 = (zero_zero @ A))))))))).
% 3.50/1.64 thf(fact_199_bot__nat__0_Onot__eq__extremum, axiom, (! [A4:nat]: (((A4 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A4))))).
% 3.50/1.64 thf(fact_200_less__nat__zero__code, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))).
% 3.50/1.64 thf(fact_201_neq0__conv, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))).
% 3.50/1.64 thf(fact_202_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:A,B4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A4) @ B4)) = (((ord_less @ A) @ B4) @ A4)))))))).
% 3.50/1.64 thf(fact_203_zero__less__diff, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M)) = (((ord_less @ nat) @ M) @ N2))))).
% 3.50/1.64 thf(fact_204_nat__mult__less__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N2)))))).
% 3.50/1.64 thf(fact_205_mult__less__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N2)))))).
% 3.50/1.64 thf(fact_206_nat__0__less__mult__iff, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((times_times @ nat) @ M) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))).
% 3.50/1.64 thf(fact_207_less__one, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ (one_one @ nat)) = (N2 = (zero_zero @ nat)))))).
% 3.50/1.64 thf(fact_208_nat__mult__le__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N2)))))).
% 3.50/1.64 thf(fact_209_mult__le__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N2)))))).
% 3.50/1.64 thf(fact_210_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.50/1.64 thf(fact_211_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (A4 != B4)))))))).
% 3.50/1.64 thf(fact_212_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (A4 != B4)))))))).
% 3.50/1.64 thf(fact_213_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 3.50/1.64 thf(fact_214_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A,C2:A]: (((((ord_less @ A) @ B4) @ A4) => ((((ord_less @ A) @ C2) @ B4) => (((ord_less @ A) @ C2) @ A4))))))))).
% 3.50/1.64 thf(fact_215_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A4:A,B4:A]: (((! [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A]: (((P @ A5) @ A5))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A4) @ B4)))))))))).
% 3.50/1.64 thf(fact_216_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P4:(A > $o)]: ((? [X5:A]: ((P4 @ X5))))) = (^ [P5:(A > $o)]: ((? [N:A]: (((P5 @ N) & (! [M4:A]: (((((ord_less @ A) @ M4) @ N) => (~ ((P5 @ M4)))))))))))))))).
% 3.50/1.64 thf(fact_217_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.50/1.64 thf(fact_218_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_219_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A]: ((~ ((((ord_less @ A) @ A4) @ A4))))))))).
% 3.50/1.64 thf(fact_220_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.50/1.64 thf(fact_221_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 3.50/1.64 thf(fact_222_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 3.50/1.64 thf(fact_223_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 3.50/1.64 thf(fact_224_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y3:A]: (((((ord_less @ A) @ Y3) @ X3) => (P @ Y3)))) => (P @ X3)))) => (P @ A4)))))))).
% 3.50/1.64 thf(fact_225_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.50/1.64 thf(fact_226_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.50/1.64 thf(fact_227_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) => (~ ((((ord_less @ A) @ A4) @ B4)))))))))).
% 3.50/1.64 thf(fact_228_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((((ord_less @ A) @ A4) @ B4) => ((B4 = C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_229_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A4:A,B4:A,C2:A]: (((A4 = B4) => ((((ord_less @ A) @ B4) @ C2) => (((ord_less @ A) @ A4) @ C2))))))))).
% 3.50/1.64 thf(fact_230_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 3.50/1.64 thf(fact_231_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.50/1.64 thf(fact_232_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 3.50/1.64 thf(fact_233_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 3.50/1.64 thf(fact_234_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.50/1.64 thf(fact_235_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.50/1.64 thf(fact_236_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z4:A]: (((((ord_less @ A) @ X) @ Z4) & (((ord_less @ A) @ Z4) @ Y))))))))))).
% 3.50/1.64 thf(fact_237_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) => (~ ((((ord_less @ A) @ B4) @ A4)))))))))).
% 3.50/1.64 thf(fact_238_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.50/1.64 thf(fact_239_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.50/1.64 thf(fact_240_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 3.50/1.64 thf(fact_241_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y2:A]: ((((ord_less @ A) @ Y2) @ X))))))))).
% 3.50/1.64 thf(fact_242_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A4:A,B4:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A4) @ B4) => ((((ord_less @ C) @ (F @ B4)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ C) @ (F @ A4)) @ C2)))))))))).
% 3.50/1.64 thf(fact_243_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((((ord_less @ A) @ A4) @ (F @ B4)) => ((((ord_less @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 3.50/1.64 thf(fact_244_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,B4:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A4) @ B4) => (((F @ B4) = C2) => ((! [X3:A,Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ B) @ (F @ A4)) @ C2)))))))))).
% 3.50/1.64 thf(fact_245_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A4:A,F:(B > A),B4:B,C2:B]: (((A4 = (F @ B4)) => ((((ord_less @ B) @ B4) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less @ B) @ X3) @ Y2) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less @ A) @ A4) @ (F @ C2))))))))))).
% 3.50/1.64 thf(fact_246_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 3.50/1.64 thf(fact_247_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 3.50/1.64 thf(fact_248_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y3:A]: (((((ord_less @ nat) @ (V @ Y3)) @ (V @ X3)) & (~ ((P @ Y3))))))))) => (P @ X))))).
% 3.50/1.64 thf(fact_249_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A4:A]: (((! [X3:A]: (((! [Y3:A]: (((((ord_less @ B) @ (F @ Y3)) @ (F @ X3)) => (P @ Y3)))) => (P @ X3)))) => (P @ A4)))))))).
% 3.50/1.64 thf(fact_250_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 3.50/1.64 thf(fact_251_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M5:nat]: (((((ord_less @ nat) @ M5) @ N3) & (~ ((P @ M5))))))))) => (P @ N2))))).
% 3.50/1.64 thf(fact_252_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((! [M5:nat]: (((((ord_less @ nat) @ M5) @ N3) => (P @ M5)))) => (P @ N3)))) => (P @ N2))))).
% 3.50/1.64 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 3.50/1.64 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 3.50/1.64 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 3.50/1.64 thf(tcon_fun___Groups_Ominus, axiom, (! [A7:$tType,A8:$tType]: (((minus @ A8) => (minus @ (A7 > A8)))))).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel, axiom, (semiri1923998003cancel @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, axiom, (semiri1193490041visors @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, axiom, (ordere216010020id_add @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel, axiom, (comm_s1003936772cancel @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring, axiom, (ordere1490568538miring @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0, axiom, (ordered_semiring_0 @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult, axiom, (ab_semigroup_mult @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Oordered__semiring, axiom, (ordered_semiring @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Osemigroup__mult, axiom, (semigroup_mult @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Ozero__neq__one, axiom, (zero_neq_one @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Omonoid__mult, axiom, (monoid_mult @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Osemiring__0, axiom, (semiring_0 @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, (no_top @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Omult__zero, axiom, (mult_zero @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ominus_4, axiom, (minus @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 3.50/1.64 thf(tcon_Nat_Onat___Rings_Odvd, axiom, (dvd @ nat)).
% 3.50/1.64 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 3.50/1.64 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 3.50/1.64 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 3.50/1.64 thf(tcon_Set_Oset___Groups_Ominus_8, axiom, (! [A7:$tType]: ((minus @ (set @ A7))))).
% 3.50/1.64 thf(tcon_HOL_Obool___Orderings_Opreorder_9, axiom, (preorder @ $o)).
% 3.50/1.64 thf(tcon_HOL_Obool___Orderings_Olinorder_10, axiom, (linorder @ $o)).
% 3.50/1.64 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, (order @ $o)).
% 3.50/1.64 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, (ord @ $o)).
% 3.50/1.64 thf(tcon_HOL_Obool___Groups_Ominus_13, axiom, (minus @ $o)).
% 3.50/1.64 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.50/1.64 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 3.50/1.64 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 3.50/1.64 thf(conj_0, conjecture, ((((minus_minus @ nat) @ ((((groups1340683514dd_sum @ nat) @ nat) @ (^ [I:nat]: ((((times_times @ nat) @ (p @ I)) @ I)))) @ ((set_ord_atMost @ nat) @ n))) @ k) = (((minus_minus @ nat) @ n) @ k))).
% 3.50/1.64 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.37uXGZ2q2e/DTF2DTF25510.p
% 3.50/1.64 ------------------------
% 3.50/1.69 ---- Cleaned THF ---
% 3.50/1.69 thf(ty_t_Set_Oset,type,
% 3.50/1.69 set: $tType > $tType ).
% 3.50/1.69
% 3.50/1.69 thf(ty_t_Nat_Onat,type,
% 3.50/1.69 nat: $tType ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_HOL_Otype,type,
% 3.50/1.69 type:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Odvd,type,
% 3.50/1.69 dvd:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Oone,type,
% 3.50/1.69 one:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oring,type,
% 3.50/1.69 ring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ozero,type,
% 3.50/1.69 zero:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ominus,type,
% 3.50/1.69 minus:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Oord,type,
% 3.50/1.69 ord:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Num_Oneg__numeral,type,
% 3.50/1.69 neg_numeral:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Oorder,type,
% 3.50/1.69 order:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Ocomm__ring,type,
% 3.50/1.69 comm_ring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Omult__zero,type,
% 3.50/1.69 mult_zero:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ogroup__add,type,
% 3.50/1.69 group_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Ono__bot,type,
% 3.50/1.69 no_bot:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Ono__top,type,
% 3.50/1.69 no_top:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Osemiring__0,type,
% 3.50/1.69 semiring_0:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Omonoid__mult,type,
% 3.50/1.69 monoid_mult:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Olinorder,type,
% 3.50/1.69 linorder:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Opreorder,type,
% 3.50/1.69 preorder:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oordered__ring,type,
% 3.50/1.69 ordered_ring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Ozero__neq__one,type,
% 3.50/1.69 zero_neq_one:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Oab__group__add,type,
% 3.50/1.69 ab_group_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Owellorder,type,
% 3.50/1.69 wellorder:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Osemigroup__mult,type,
% 3.50/1.69 semigroup_mult:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Orderings_Odense__order,type,
% 3.50/1.69 dense_order:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Olinordered__idom,type,
% 3.50/1.69 linordered_idom:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Olinordered__ring,type,
% 3.50/1.69 linordered_ring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 3.50/1.69 comm_monoid_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oordered__semiring,type,
% 3.50/1.69 ordered_semiring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 3.50/1.69 comm_monoid_diff:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 3.50/1.69 comm_monoid_mult:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 3.50/1.69 ab_semigroup_mult:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oordered__semiring__0,type,
% 3.50/1.69 ordered_semiring_0:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.50/1.69 ordered_ab_group_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oordered__comm__semiring,type,
% 3.50/1.69 ordere1490568538miring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 3.50/1.69 comm_s1003936772cancel:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 3.50/1.69 linord581940658strict:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 3.50/1.69 cancel1352612707id_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 3.50/1.69 ring_11004092258visors:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 3.50/1.69 cancel146912293up_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 3.50/1.69 ordere216010020id_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 3.50/1.69 semiri1193490041visors:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 3.50/1.69 linord1659791738miring:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.50/1.69 canoni770627133id_add:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 3.50/1.69 semiri1923998003cancel:
% 3.50/1.69 !>[A: $tType] : $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Fun_Ofun__upd,type,
% 3.50/1.69 fun_upd:
% 3.50/1.69 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.50/1.69 minus_minus:
% 3.50/1.69 !>[A: $tType] : ( A > A > A ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Groups_Oone__class_Oone,type,
% 3.50/1.69 one_one:
% 3.50/1.69 !>[A: $tType] : A ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 3.50/1.69 times_times:
% 3.50/1.69 !>[A: $tType] : ( A > A > A ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.50/1.69 zero_zero:
% 3.50/1.69 !>[A: $tType] : A ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 3.50/1.69 groups1340683514dd_sum:
% 3.50/1.69 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_If,type,
% 3.50/1.69 if:
% 3.50/1.69 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 3.50/1.69 number2016821345itions: ( nat > nat ) > nat > $o ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.50/1.69 ord_less:
% 3.50/1.69 !>[A: $tType] : ( A > A > $o ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.50/1.69 ord_less_eq:
% 3.50/1.69 !>[A: $tType] : ( A > A > $o ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Set_OCollect,type,
% 3.50/1.69 collect:
% 3.50/1.69 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 3.50/1.69 set_ord_atMost:
% 3.50/1.69 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_c_member,type,
% 3.50/1.69 member:
% 3.50/1.69 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.50/1.69
% 3.50/1.69 thf(sy_v_k,type,
% 3.50/1.69 k: nat ).
% 3.50/1.69
% 3.50/1.69 thf(sy_v_n,type,
% 3.50/1.69 n: nat ).
% 3.50/1.69
% 3.50/1.69 thf(sy_v_p,type,
% 3.50/1.69 p: nat > nat ).
% 3.50/1.69
% 3.50/1.69 thf(fact_0_partitions,axiom,
% 3.50/1.69 number2016821345itions @ p @ n ).
% 3.50/1.69
% 3.50/1.69 thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 3.50/1.69 ord_less_eq @ nat @ k @ n ).
% 3.50/1.69
% 3.50/1.69 thf(fact_2_atMost__eq__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ( set_ord_atMost @ A @ X )
% 3.50/1.69 = ( set_ord_atMost @ A @ Y ) )
% 3.50/1.69 = ( X = Y ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_3_minus__apply,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( minus @ B )
% 3.50/1.69 => ( ( minus_minus @ ( A > B ) )
% 3.50/1.69 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062,axiom,
% 3.50/1.69 ( ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
% 3.50/1.69 = ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ n ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_5_calculation,axiom,
% 3.50/1.69 ( ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
% 3.50/1.69 = ( minus_minus @ nat
% 3.50/1.69 @ ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ n ) )
% 3.50/1.69 @ k ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_6_sum__subtractf,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ab_group_add @ A )
% 3.50/1.69 => ! [F: B > A,G: B > A,A3: set @ B] :
% 3.50/1.69 ( ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [X2: B] : ( minus_minus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
% 3.50/1.69 @ A3 )
% 3.50/1.69 = ( minus_minus @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( groups1340683514dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_7_sum__product,axiom,
% 3.50/1.69 ! [B: $tType,C: $tType,A: $tType] :
% 3.50/1.69 ( ( semiring_0 @ B )
% 3.50/1.69 => ! [F: A > B,A3: set @ A,G: C > B,B3: set @ C] :
% 3.50/1.69 ( ( times_times @ B @ ( groups1340683514dd_sum @ A @ B @ F @ A3 ) @ ( groups1340683514dd_sum @ C @ B @ G @ B3 ) )
% 3.50/1.69 = ( groups1340683514dd_sum @ A @ B
% 3.50/1.69 @ ^ [I: A] :
% 3.50/1.69 ( groups1340683514dd_sum @ C @ B
% 3.50/1.69 @ ^ [J: C] : ( times_times @ B @ ( F @ I ) @ ( G @ J ) )
% 3.50/1.69 @ B3 )
% 3.50/1.69 @ A3 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_8_sum__distrib__left,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( semiring_0 @ A )
% 3.50/1.69 => ! [R: A,F: B > A,A3: set @ B] :
% 3.50/1.69 ( ( times_times @ A @ R @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) )
% 3.50/1.69 = ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [N: B] : ( times_times @ A @ R @ ( F @ N ) )
% 3.50/1.69 @ A3 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_9_sum__distrib__right,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( semiring_0 @ A )
% 3.50/1.69 => ! [F: B > A,A3: set @ B,R: A] :
% 3.50/1.69 ( ( times_times @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ R )
% 3.50/1.69 = ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R )
% 3.50/1.69 @ A3 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_10_diff__mult__distrib,axiom,
% 3.50/1.69 ! [M: nat,N2: nat,K: nat] :
% 3.50/1.69 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
% 3.50/1.69 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_11_diff__mult__distrib2,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
% 3.50/1.69 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_12_inf__period_I2_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ( comm_ring @ A )
% 3.50/1.69 & ( dvd @ A ) )
% 3.50/1.69 => ! [P: A > $o,D: A,Q: A > $o] :
% 3.50/1.69 ( ! [X3: A,K2: A] :
% 3.50/1.69 ( ( P @ X3 )
% 3.50/1.69 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 3.50/1.69 => ( ! [X3: A,K2: A] :
% 3.50/1.69 ( ( Q @ X3 )
% 3.50/1.69 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 3.50/1.69 => ! [X4: A,K3: A] :
% 3.50/1.69 ( ( ( P @ X4 )
% 3.50/1.69 | ( Q @ X4 ) )
% 3.50/1.69 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
% 3.50/1.69 | ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_13_atMost__subset__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
% 3.50/1.69 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_14_atMost__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ! [I2: A,K: A] :
% 3.50/1.69 ( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K ) )
% 3.50/1.69 = ( ord_less_eq @ A @ I2 @ K ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_15_diff__diff__cancel,axiom,
% 3.50/1.69 ! [I2: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ I2 @ N2 )
% 3.50/1.69 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I2 ) )
% 3.50/1.69 = I2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_16_nat__mult__eq__1__iff,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ( times_times @ nat @ M @ N2 )
% 3.50/1.69 = ( one_one @ nat ) )
% 3.50/1.69 = ( ( M
% 3.50/1.69 = ( one_one @ nat ) )
% 3.50/1.69 & ( N2
% 3.50/1.69 = ( one_one @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_17_nat__1__eq__mult__iff,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ( one_one @ nat )
% 3.50/1.69 = ( times_times @ nat @ M @ N2 ) )
% 3.50/1.69 = ( ( M
% 3.50/1.69 = ( one_one @ nat ) )
% 3.50/1.69 & ( N2
% 3.50/1.69 = ( one_one @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_18_le__refl,axiom,
% 3.50/1.69 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_19_le__trans,axiom,
% 3.50/1.69 ! [I2: nat,J2: nat,K: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ J2 @ K )
% 3.50/1.69 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_20_eq__imp__le,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( M = N2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_21_le__antisym,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ M @ N2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ N2 @ M )
% 3.50/1.69 => ( M = N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_22_nat__le__linear,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ M @ N2 )
% 3.50/1.69 | ( ord_less_eq @ nat @ N2 @ M ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
% 3.50/1.69 ! [P: nat > $o,K: nat,B4: nat] :
% 3.50/1.69 ( ( P @ K )
% 3.50/1.69 => ( ! [Y2: nat] :
% 3.50/1.69 ( ( P @ Y2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ Y2 @ B4 ) )
% 3.50/1.69 => ? [X3: nat] :
% 3.50/1.69 ( ( P @ X3 )
% 3.50/1.69 & ! [Y3: nat] :
% 3.50/1.69 ( ( P @ Y3 )
% 3.50/1.69 => ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_24_bounded__Max__nat,axiom,
% 3.50/1.69 ! [P: nat > $o,X: nat,M2: nat] :
% 3.50/1.69 ( ( P @ X )
% 3.50/1.69 => ( ! [X3: nat] :
% 3.50/1.69 ( ( P @ X3 )
% 3.50/1.69 => ( ord_less_eq @ nat @ X3 @ M2 ) )
% 3.50/1.69 => ~ ! [M3: nat] :
% 3.50/1.69 ( ( P @ M3 )
% 3.50/1.69 => ~ ! [X4: nat] :
% 3.50/1.69 ( ( P @ X4 )
% 3.50/1.69 => ( ord_less_eq @ nat @ X4 @ M3 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_25_partitions__bounds,axiom,
% 3.50/1.69 ! [P2: nat > nat,N2: nat,I2: nat] :
% 3.50/1.69 ( ( number2016821345itions @ P2 @ N2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( P2 @ I2 ) @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_26_diff__le__mono2,axiom,
% 3.50/1.69 ! [M: nat,N2: nat,L: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ M @ N2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_27_le__diff__iff_H,axiom,
% 3.50/1.69 ! [A4: nat,C2: nat,B4: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ A4 @ C2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ B4 @ C2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A4 ) @ ( minus_minus @ nat @ C2 @ B4 ) )
% 3.50/1.69 = ( ord_less_eq @ nat @ B4 @ A4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_28_diff__le__self,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_29_diff__le__mono,axiom,
% 3.50/1.69 ! [M: nat,N2: nat,L: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ M @ N2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_30_Nat_Odiff__diff__eq,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ K @ M )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ K @ N2 )
% 3.50/1.69 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 3.50/1.69 = ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_31_le__diff__iff,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ K @ M )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ K @ N2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
% 3.50/1.69 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_32_eq__diff__iff,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ K @ M )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ K @ N2 )
% 3.50/1.69 => ( ( ( minus_minus @ nat @ M @ K )
% 3.50/1.69 = ( minus_minus @ nat @ N2 @ K ) )
% 3.50/1.69 = ( M = N2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_33_mult__le__mono2,axiom,
% 3.50/1.69 ! [I2: nat,J2: nat,K: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_34_mult__le__mono1,axiom,
% 3.50/1.69 ! [I2: nat,J2: nat,K: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_35_mult__le__mono,axiom,
% 3.50/1.69 ! [I2: nat,J2: nat,K: nat,L: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ I2 @ J2 )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ K @ L )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_36_le__square,axiom,
% 3.50/1.69 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_37_le__cube,axiom,
% 3.50/1.69 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_38_nat__mult__1__right,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
% 3.50/1.69 = N2 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_39_nat__mult__1,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
% 3.50/1.69 = N2 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_40_sum__mono,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ordere216010020id_add @ A )
% 3.50/1.69 => ! [K4: set @ B,F: B > A,G: B > A] :
% 3.50/1.69 ( ! [I3: B] :
% 3.50/1.69 ( ( member @ B @ I3 @ K4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ K4 ) @ ( groups1340683514dd_sum @ B @ A @ G @ K4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_41_atMost__def,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ( ( set_ord_atMost @ A )
% 3.50/1.69 = ( ^ [U: A] :
% 3.50/1.69 ( collect @ A
% 3.50/1.69 @ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_42_sum__subtractf__nat,axiom,
% 3.50/1.69 ! [A: $tType,A3: set @ A,G: A > nat,F: A > nat] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ( member @ A @ X3 @ A3 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ A @ nat
% 3.50/1.69 @ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 3.50/1.69 @ A3 )
% 3.50/1.69 = ( minus_minus @ nat @ ( groups1340683514dd_sum @ A @ nat @ F @ A3 ) @ ( groups1340683514dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_43_partitions__parts__bounded,axiom,
% 3.50/1.69 ! [P2: nat > nat,N2: nat] :
% 3.50/1.69 ( ( number2016821345itions @ P2 @ N2 )
% 3.50/1.69 => ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ P2 @ ( set_ord_atMost @ nat @ N2 ) ) @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_44_fun__diff__def,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( minus @ B )
% 3.50/1.69 => ( ( minus_minus @ ( A > B ) )
% 3.50/1.69 = ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_45_mem__Collect__eq,axiom,
% 3.50/1.69 ! [A: $tType,A4: A,P: A > $o] :
% 3.50/1.69 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 3.50/1.69 = ( P @ A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_46_Collect__mem__eq,axiom,
% 3.50/1.69 ! [A: $tType,A3: set @ A] :
% 3.50/1.69 ( ( collect @ A
% 3.50/1.69 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 3.50/1.69 = A3 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_47_Collect__cong,axiom,
% 3.50/1.69 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ( P @ X3 )
% 3.50/1.69 = ( Q @ X3 ) )
% 3.50/1.69 => ( ( collect @ A @ P )
% 3.50/1.69 = ( collect @ A @ Q ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_48_ext,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ( F @ X3 )
% 3.50/1.69 = ( G @ X3 ) )
% 3.50/1.69 => ( F = G ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_49_sum_Oreindex__bij__witness,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType,C: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [S: set @ B,I2: C > B,J2: B > C,T: set @ C,H: C > A,G: B > A] :
% 3.50/1.69 ( ! [A5: B] :
% 3.50/1.69 ( ( member @ B @ A5 @ S )
% 3.50/1.69 => ( ( I2 @ ( J2 @ A5 ) )
% 3.50/1.69 = A5 ) )
% 3.50/1.69 => ( ! [A5: B] :
% 3.50/1.69 ( ( member @ B @ A5 @ S )
% 3.50/1.69 => ( member @ C @ ( J2 @ A5 ) @ T ) )
% 3.50/1.69 => ( ! [B5: C] :
% 3.50/1.69 ( ( member @ C @ B5 @ T )
% 3.50/1.69 => ( ( J2 @ ( I2 @ B5 ) )
% 3.50/1.69 = B5 ) )
% 3.50/1.69 => ( ! [B5: C] :
% 3.50/1.69 ( ( member @ C @ B5 @ T )
% 3.50/1.69 => ( member @ B @ ( I2 @ B5 ) @ S ) )
% 3.50/1.69 => ( ! [A5: B] :
% 3.50/1.69 ( ( member @ B @ A5 @ S )
% 3.50/1.69 => ( ( H @ ( J2 @ A5 ) )
% 3.50/1.69 = ( G @ A5 ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ B @ A @ G @ S )
% 3.50/1.69 = ( groups1340683514dd_sum @ C @ A @ H @ T ) ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_50_sum_Oeq__general__inverses,axiom,
% 3.50/1.69 ! [C: $tType,A: $tType,B: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [B3: set @ B,K: B > C,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 3.50/1.69 ( ! [Y2: B] :
% 3.50/1.69 ( ( member @ B @ Y2 @ B3 )
% 3.50/1.69 => ( ( member @ C @ ( K @ Y2 ) @ A3 )
% 3.50/1.69 & ( ( H @ ( K @ Y2 ) )
% 3.50/1.69 = Y2 ) ) )
% 3.50/1.69 => ( ! [X3: C] :
% 3.50/1.69 ( ( member @ C @ X3 @ A3 )
% 3.50/1.69 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 3.50/1.69 & ( ( K @ ( H @ X3 ) )
% 3.50/1.69 = X3 )
% 3.50/1.69 & ( ( Gamma @ ( H @ X3 ) )
% 3.50/1.69 = ( Phi @ X3 ) ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 3.50/1.69 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_51_sum_Oeq__general,axiom,
% 3.50/1.69 ! [C: $tType,A: $tType,B: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [B3: set @ B,A3: set @ C,H: C > B,Gamma: B > A,Phi: C > A] :
% 3.50/1.69 ( ! [Y2: B] :
% 3.50/1.69 ( ( member @ B @ Y2 @ B3 )
% 3.50/1.69 => ? [X4: C] :
% 3.50/1.69 ( ( member @ C @ X4 @ A3 )
% 3.50/1.69 & ( ( H @ X4 )
% 3.50/1.69 = Y2 )
% 3.50/1.69 & ! [Ya: C] :
% 3.50/1.69 ( ( ( member @ C @ Ya @ A3 )
% 3.50/1.69 & ( ( H @ Ya )
% 3.50/1.69 = Y2 ) )
% 3.50/1.69 => ( Ya = X4 ) ) ) )
% 3.50/1.69 => ( ! [X3: C] :
% 3.50/1.69 ( ( member @ C @ X3 @ A3 )
% 3.50/1.69 => ( ( member @ B @ ( H @ X3 ) @ B3 )
% 3.50/1.69 & ( ( Gamma @ ( H @ X3 ) )
% 3.50/1.69 = ( Phi @ X3 ) ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ C @ A @ Phi @ A3 )
% 3.50/1.69 = ( groups1340683514dd_sum @ B @ A @ Gamma @ B3 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_52_sum_Ocong,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [A3: set @ B,B3: set @ B,G: B > A,H: B > A] :
% 3.50/1.69 ( ( A3 = B3 )
% 3.50/1.69 => ( ! [X3: B] :
% 3.50/1.69 ( ( member @ B @ X3 @ B3 )
% 3.50/1.69 => ( ( G @ X3 )
% 3.50/1.69 = ( H @ X3 ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 3.50/1.69 = ( groups1340683514dd_sum @ B @ A @ H @ B3 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_53_diff__commute,axiom,
% 3.50/1.69 ! [I2: nat,J2: nat,K: nat] :
% 3.50/1.69 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ K )
% 3.50/1.69 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_54_sum_Oswap,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,C: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [G: B > C > A,B3: set @ C,A3: set @ B] :
% 3.50/1.69 ( ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [I: B] : ( groups1340683514dd_sum @ C @ A @ ( G @ I ) @ B3 )
% 3.50/1.69 @ A3 )
% 3.50/1.69 = ( groups1340683514dd_sum @ C @ A
% 3.50/1.69 @ ^ [J: C] :
% 3.50/1.69 ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [I: B] : ( G @ I @ J )
% 3.50/1.69 @ A3 )
% 3.50/1.69 @ B3 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_55_inf__period_I1_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ( comm_ring @ A )
% 3.50/1.69 & ( dvd @ A ) )
% 3.50/1.69 => ! [P: A > $o,D: A,Q: A > $o] :
% 3.50/1.69 ( ! [X3: A,K2: A] :
% 3.50/1.69 ( ( P @ X3 )
% 3.50/1.69 = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 3.50/1.69 => ( ! [X3: A,K2: A] :
% 3.50/1.69 ( ( Q @ X3 )
% 3.50/1.69 = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
% 3.50/1.69 => ! [X4: A,K3: A] :
% 3.50/1.69 ( ( ( P @ X4 )
% 3.50/1.69 & ( Q @ X4 ) )
% 3.50/1.69 = ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) )
% 3.50/1.69 & ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K3 @ D ) ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_56_mult_Oright__neutral,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( monoid_mult @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ A4 @ ( one_one @ A ) )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_57_mult_Oleft__neutral,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( monoid_mult @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ ( one_one @ A ) @ A4 )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_58_fun__upd__upd,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
% 3.50/1.69 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
% 3.50/1.69 = ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_59_fun__upd__triv,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 3.50/1.69 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 3.50/1.69 = F ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_60_fun__upd__apply,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( fun_upd @ B @ A )
% 3.50/1.69 = ( ^ [F2: B > A,X2: B,Y4: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_61_order__refl,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_62_lambda__one,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( monoid_mult @ A )
% 3.50/1.69 => ( ( ^ [X2: A] : X2 )
% 3.50/1.69 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_63_partitions__def,axiom,
% 3.50/1.69 ( number2016821345itions
% 3.50/1.69 = ( ^ [P3: nat > nat,N: nat] :
% 3.50/1.69 ( ! [I: nat] :
% 3.50/1.69 ( ( ( P3 @ I )
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I )
% 3.50/1.69 & ( ord_less_eq @ nat @ I @ N ) ) )
% 3.50/1.69 & ( ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( P3 @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ N ) )
% 3.50/1.69 = N ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_64_partitionsI,axiom,
% 3.50/1.69 ! [P2: nat > nat,N2: nat] :
% 3.50/1.69 ( ! [I3: nat] :
% 3.50/1.69 ( ( ( P2 @ I3 )
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I3 )
% 3.50/1.69 & ( ord_less_eq @ nat @ I3 @ N2 ) ) )
% 3.50/1.69 => ( ( ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ N2 ) )
% 3.50/1.69 = N2 )
% 3.50/1.69 => ( number2016821345itions @ P2 @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_65_partitionsE,axiom,
% 3.50/1.69 ! [P2: nat > nat,N2: nat] :
% 3.50/1.69 ( ( number2016821345itions @ P2 @ N2 )
% 3.50/1.69 => ~ ( ! [I4: nat] :
% 3.50/1.69 ( ( ( P2 @ I4 )
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I4 )
% 3.50/1.69 & ( ord_less_eq @ nat @ I4 @ N2 ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( P2 @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ N2 ) )
% 3.50/1.69 != N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_66_le__zero__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( canoni770627133id_add @ A )
% 3.50/1.69 => ! [N2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
% 3.50/1.69 = ( N2
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_67_mult__zero__left,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( mult_zero @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_68_mult__zero__right,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( mult_zero @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_69_mult__eq__0__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1193490041visors @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ A4 @ B4 )
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 = ( ( A4
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( B4
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_70_mult__cancel__left,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1923998003cancel @ A )
% 3.50/1.69 => ! [C2: A,A4: A,B4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ C2 @ A4 )
% 3.50/1.69 = ( times_times @ A @ C2 @ B4 ) )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_71_mult__cancel__right,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1923998003cancel @ A )
% 3.50/1.69 => ! [A4: A,C2: A,B4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ A4 @ C2 )
% 3.50/1.69 = ( times_times @ A @ B4 @ C2 ) )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_72_diff__self,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( group_add @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ A4 @ A4 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_73_diff__0__right,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( group_add @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_74_zero__diff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( comm_monoid_diff @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_75_diff__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( cancel1352612707id_add @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_76_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( cancel1352612707id_add @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ A4 @ A4 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_77_bot__nat__0_Oextremum,axiom,
% 3.50/1.69 ! [A4: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A4 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_78_le0,axiom,
% 3.50/1.69 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_79_diff__0__eq__0,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_80_diff__self__eq__0,axiom,
% 3.50/1.69 ! [M: nat] :
% 3.50/1.69 ( ( minus_minus @ nat @ M @ M )
% 3.50/1.69 = ( zero_zero @ nat ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_81_mult__is__0,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ( times_times @ nat @ M @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 = ( ( M
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 | ( N2
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_82_mult__0__right,axiom,
% 3.50/1.69 ! [M: nat] :
% 3.50/1.69 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 3.50/1.69 = ( zero_zero @ nat ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_83_mult__cancel1,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ( times_times @ nat @ K @ M )
% 3.50/1.69 = ( times_times @ nat @ K @ N2 ) )
% 3.50/1.69 = ( ( M = N2 )
% 3.50/1.69 | ( K
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_84_mult__cancel2,axiom,
% 3.50/1.69 ! [M: nat,K: nat,N2: nat] :
% 3.50/1.69 ( ( ( times_times @ nat @ M @ K )
% 3.50/1.69 = ( times_times @ nat @ N2 @ K ) )
% 3.50/1.69 = ( ( M = N2 )
% 3.50/1.69 | ( K
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_85_sum_Oneutral__const,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [A3: set @ B] :
% 3.50/1.69 ( ( groups1340683514dd_sum @ B @ A
% 3.50/1.69 @ ^ [Uu: B] : ( zero_zero @ A )
% 3.50/1.69 @ A3 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_86_diff__ge__0__iff__ge,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
% 3.50/1.69 = ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_87_mult__cancel__right2,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring_11004092258visors @ A )
% 3.50/1.69 => ! [A4: A,C2: A] :
% 3.50/1.69 ( ( ( times_times @ A @ A4 @ C2 )
% 3.50/1.69 = C2 )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( A4
% 3.50/1.69 = ( one_one @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_88_mult__cancel__right1,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring_11004092258visors @ A )
% 3.50/1.69 => ! [C2: A,B4: A] :
% 3.50/1.69 ( ( C2
% 3.50/1.69 = ( times_times @ A @ B4 @ C2 ) )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( B4
% 3.50/1.69 = ( one_one @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_89_mult__cancel__left2,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring_11004092258visors @ A )
% 3.50/1.69 => ! [C2: A,A4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ C2 @ A4 )
% 3.50/1.69 = C2 )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( A4
% 3.50/1.69 = ( one_one @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_90_mult__cancel__left1,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring_11004092258visors @ A )
% 3.50/1.69 => ! [C2: A,B4: A] :
% 3.50/1.69 ( ( C2
% 3.50/1.69 = ( times_times @ A @ C2 @ B4 ) )
% 3.50/1.69 = ( ( C2
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( B4
% 3.50/1.69 = ( one_one @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_91_diff__is__0__eq_H,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ M @ N2 )
% 3.50/1.69 => ( ( minus_minus @ nat @ M @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_92_diff__is__0__eq,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ( minus_minus @ nat @ M @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 = ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_93_gr0,axiom,
% 3.50/1.69 ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_94_zero__reorient,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( zero @ A )
% 3.50/1.69 => ! [X: A] :
% 3.50/1.69 ( ( ( zero_zero @ A )
% 3.50/1.69 = X )
% 3.50/1.69 = ( X
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_95_zero__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( canoni770627133id_add @ A )
% 3.50/1.69 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_96_mult__not__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( mult_zero @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ A4 @ B4 )
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ( ( A4
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 & ( B4
% 3.50/1.69 != ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_97_divisors__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1193490041visors @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ( times_times @ A @ A4 @ B4 )
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 => ( ( A4
% 3.50/1.69 = ( zero_zero @ A ) )
% 3.50/1.69 | ( B4
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_98_no__zero__divisors,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1193490041visors @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( A4
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ( ( B4
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ( ( times_times @ A @ A4 @ B4 )
% 3.50/1.69 != ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_99_mult__left__cancel,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1923998003cancel @ A )
% 3.50/1.69 => ! [C2: A,A4: A,B4: A] :
% 3.50/1.69 ( ( C2
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ( ( ( times_times @ A @ C2 @ A4 )
% 3.50/1.69 = ( times_times @ A @ C2 @ B4 ) )
% 3.50/1.69 = ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_100_mult__right__cancel,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semiri1923998003cancel @ A )
% 3.50/1.69 => ! [C2: A,A4: A,B4: A] :
% 3.50/1.69 ( ( C2
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ( ( ( times_times @ A @ A4 @ C2 )
% 3.50/1.69 = ( times_times @ A @ B4 @ C2 ) )
% 3.50/1.69 = ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_101_eq__iff__diff__eq__0,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( group_add @ A )
% 3.50/1.69 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.50/1.69 = ( ^ [A6: A,B6: A] :
% 3.50/1.69 ( ( minus_minus @ A @ A6 @ B6 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_102_zero__neq__one,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( zero_neq_one @ A )
% 3.50/1.69 => ( ( zero_zero @ A )
% 3.50/1.69 != ( one_one @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_103_lambda__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( mult_zero @ A )
% 3.50/1.69 => ( ( ^ [H2: A] : ( zero_zero @ A ) )
% 3.50/1.69 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_104_sum_Oneutral,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [A3: set @ B,G: B > A] :
% 3.50/1.69 ( ! [X3: B] :
% 3.50/1.69 ( ( member @ B @ X3 @ A3 )
% 3.50/1.69 => ( ( G @ X3 )
% 3.50/1.69 = ( zero_zero @ A ) ) )
% 3.50/1.69 => ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_105_sum_Onot__neutral__contains__not__neutral,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( comm_monoid_add @ A )
% 3.50/1.69 => ! [G: B > A,A3: set @ B] :
% 3.50/1.69 ( ( ( groups1340683514dd_sum @ B @ A @ G @ A3 )
% 3.50/1.69 != ( zero_zero @ A ) )
% 3.50/1.69 => ~ ! [A5: B] :
% 3.50/1.69 ( ( member @ B @ A5 @ A3 )
% 3.50/1.69 => ( ( G @ A5 )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_106_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.50/1.69 ! [A4: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
% 3.50/1.69 => ( A4
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_107_bot__nat__0_Oextremum__unique,axiom,
% 3.50/1.69 ! [A4: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ A4 @ ( zero_zero @ nat ) )
% 3.50/1.69 = ( A4
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_108_le__0__eq,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
% 3.50/1.69 = ( N2
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_109_less__eq__nat_Osimps_I1_J,axiom,
% 3.50/1.69 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_110_diffs0__imp__equal,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ( minus_minus @ nat @ M @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 => ( ( ( minus_minus @ nat @ N2 @ M )
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 => ( M = N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_111_minus__nat_Odiff__0,axiom,
% 3.50/1.69 ! [M: nat] :
% 3.50/1.69 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 3.50/1.69 = M ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_112_mult__0,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.50/1.69 = ( zero_zero @ nat ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordere1490568538miring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_114_zero__le__mult__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord581940658strict @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) )
% 3.50/1.69 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 3.50/1.69 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_115_mult__nonneg__nonpos2,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring_0 @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ B4 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_116_mult__nonpos__nonneg,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring_0 @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_117_mult__nonneg__nonpos,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring_0 @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_118_mult__nonneg__nonneg,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring_0 @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_119_split__mult__neg__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring_0 @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 3.50/1.69 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_120_mult__le__0__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord581940658strict @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( zero_zero @ A ) )
% 3.50/1.69 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) )
% 3.50/1.69 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_121_mult__right__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_122_mult__right__mono__neg,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ring @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_123_mult__left__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_124_mult__nonpos__nonpos,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ring @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_125_mult__left__mono__neg,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ring @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_126_split__mult__pos__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ring @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) )
% 3.50/1.69 | ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
% 3.50/1.69 & ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_127_zero__le__square,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linordered_ring @ A )
% 3.50/1.69 => ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_128_mult__mono_H,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A,D2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_129_mult__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_semiring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A,D2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ C2 @ D2 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ D2 ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_130_le__iff__diff__le__0,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ( ( ord_less_eq @ A )
% 3.50/1.69 = ( ^ [A6: A,B6: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_131_not__one__le__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_132_zero__le__one,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_133_partitions__zero,axiom,
% 3.50/1.69 ! [P2: nat > nat] :
% 3.50/1.69 ( ( number2016821345itions @ P2 @ ( zero_zero @ nat ) )
% 3.50/1.69 = ( P2
% 3.50/1.69 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_134_mult__left__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ! [C2: A,A4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ A4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_135_mult__le__one,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ ( one_one @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B4 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_136_mult__right__le__one__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linordered_idom @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 3.50/1.69 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_137_mult__left__le__one__le,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linordered_idom @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 3.50/1.69 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 3.50/1.69 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_138_sum__nonpos,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( ordere216010020id_add @ A )
% 3.50/1.69 => ! [A3: set @ B,F: B > A] :
% 3.50/1.69 ( ! [X3: B] :
% 3.50/1.69 ( ( member @ B @ X3 @ A3 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_139_sum__nonneg,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ordere216010020id_add @ A )
% 3.50/1.69 => ! [A3: set @ B,F: B > A] :
% 3.50/1.69 ( ! [X3: B] :
% 3.50/1.69 ( ( member @ B @ X3 @ A3 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups1340683514dd_sum @ B @ A @ F @ A3 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_140_mult__eq__self__implies__10,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( M
% 3.50/1.69 = ( times_times @ nat @ M @ N2 ) )
% 3.50/1.69 => ( ( N2
% 3.50/1.69 = ( one_one @ nat ) )
% 3.50/1.69 | ( M
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_141_dual__order_Oantisym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [B4: A,A4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_142_dual__order_Oeq__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.50/1.69 = ( ^ [A6: A,B6: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B6 @ A6 )
% 3.50/1.69 & ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_143_dual__order_Otrans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 3.50/1.69 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_144_linorder__wlog,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [P: A > A > $o,A4: A,B4: A] :
% 3.50/1.69 ( ! [A5: A,B5: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.50/1.69 => ( P @ A5 @ B5 ) )
% 3.50/1.69 => ( ! [A5: A,B5: A] :
% 3.50/1.69 ( ( P @ B5 @ A5 )
% 3.50/1.69 => ( P @ A5 @ B5 ) )
% 3.50/1.69 => ( P @ A4 @ B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_145_dual__order_Orefl,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_146_order__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A,Z: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.50/1.69 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_147_order__class_Oorder_Oantisym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( A4 = B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_148_ord__le__eq__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( B4 = C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_149_ord__eq__le__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( A4 = B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_150_order__class_Oorder_Oeq__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.50/1.69 = ( ^ [A6: A,B6: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A6 @ B6 )
% 3.50/1.69 & ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_151_antisym__conv,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [Y: A,X: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ Y @ X )
% 3.50/1.69 => ( ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 = ( X = Y ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_152_le__cases3,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A,Z: A] :
% 3.50/1.69 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 3.50/1.69 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 3.50/1.69 => ( ( ( ord_less_eq @ A @ X @ Z )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 3.50/1.69 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.50/1.69 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 3.50/1.69 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 3.50/1.69 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_153_order_Otrans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_154_le__cases,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_155_eq__refl,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( X = Y )
% 3.50/1.69 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_156_linear,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_157_antisym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X @ Y )
% 3.50/1.69 => ( ( ord_less_eq @ A @ Y @ X )
% 3.50/1.69 => ( X = Y ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_158_eq__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 3.50/1.69 = ( ^ [X2: A,Y4: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.50/1.69 & ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_159_ord__le__eq__subst,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( ord @ B )
% 3.50/1.69 & ( ord @ A ) )
% 3.50/1.69 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ( F @ B4 )
% 3.50/1.69 = C2 )
% 3.50/1.69 => ( ! [X3: A,Y2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_160_ord__eq__le__subst,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( ord @ B )
% 3.50/1.69 & ( ord @ A ) )
% 3.50/1.69 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 3.50/1.69 ( ( A4
% 3.50/1.69 = ( F @ B4 ) )
% 3.50/1.69 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 3.50/1.69 => ( ! [X3: B,Y2: B] :
% 3.50/1.69 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_161_order__subst2,axiom,
% 3.50/1.69 ! [A: $tType,C: $tType] :
% 3.50/1.69 ( ( ( order @ C )
% 3.50/1.69 & ( order @ A ) )
% 3.50/1.69 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
% 3.50/1.69 => ( ! [X3: A,Y2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_162_order__subst1,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( order @ B )
% 3.50/1.69 & ( order @ A ) )
% 3.50/1.69 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
% 3.50/1.69 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 3.50/1.69 => ( ! [X3: B,Y2: B] :
% 3.50/1.69 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_163_le__fun__def,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( ord @ B )
% 3.50/1.69 => ( ( ord_less_eq @ ( A > B ) )
% 3.50/1.69 = ( ^ [F2: A > B,G2: A > B] :
% 3.50/1.69 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_164_le__funI,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( ord @ B )
% 3.50/1.69 => ! [F: A > B,G: A > B] :
% 3.50/1.69 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 3.50/1.69 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_165_le__funE,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( ord @ B )
% 3.50/1.69 => ! [F: A > B,G: A > B,X: A] :
% 3.50/1.69 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.50/1.69 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_166_le__funD,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( ord @ B )
% 3.50/1.69 => ! [F: A > B,G: A > B,X: A] :
% 3.50/1.69 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.50/1.69 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_167_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ab_semigroup_mult @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
% 3.50/1.69 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_168_mult_Oassoc,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( semigroup_mult @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 )
% 3.50/1.69 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_169_mult_Ocommute,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ab_semigroup_mult @ A )
% 3.50/1.69 => ( ( times_times @ A )
% 3.50/1.69 = ( ^ [A6: A,B6: A] : ( times_times @ A @ B6 @ A6 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_170_mult_Oleft__commute,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ab_semigroup_mult @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ B4 @ ( times_times @ A @ A4 @ C2 ) )
% 3.50/1.69 = ( times_times @ A @ A4 @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_171_diff__eq__diff__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A,D2: A] :
% 3.50/1.69 ( ( ( minus_minus @ A @ A4 @ B4 )
% 3.50/1.69 = ( minus_minus @ A @ C2 @ D2 ) )
% 3.50/1.69 => ( ( A4 = B4 )
% 3.50/1.69 = ( C2 = D2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( cancel146912293up_add @ A )
% 3.50/1.69 => ! [A4: A,C2: A,B4: A] :
% 3.50/1.69 ( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B4 )
% 3.50/1.69 = ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_173_one__reorient,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( one @ A )
% 3.50/1.69 => ! [X: A] :
% 3.50/1.69 ( ( ( one_one @ A )
% 3.50/1.69 = X )
% 3.50/1.69 = ( X
% 3.50/1.69 = ( one_one @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_174_fun__upd__idem__iff,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 3.50/1.69 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 3.50/1.69 = F )
% 3.50/1.69 = ( ( F @ X )
% 3.50/1.69 = Y ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_175_fun__upd__twist,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,A4: A,C2: A,M: A > B,B4: B,D2: B] :
% 3.50/1.69 ( ( A4 != C2 )
% 3.50/1.69 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A4 @ B4 ) @ C2 @ D2 )
% 3.50/1.69 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D2 ) @ A4 @ B4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_176_fun__upd__other,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
% 3.50/1.69 ( ( Z != X )
% 3.50/1.69 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
% 3.50/1.69 = ( F @ Z ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_177_fun__upd__same,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 3.50/1.69 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 3.50/1.69 = Y ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_178_fun__upd__idem,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 3.50/1.69 ( ( ( F @ X )
% 3.50/1.69 = Y )
% 3.50/1.69 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 3.50/1.69 = F ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_179_fun__upd__eqD,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
% 3.50/1.69 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 3.50/1.69 = ( fun_upd @ A @ B @ G @ X @ Z ) )
% 3.50/1.69 => ( Y = Z ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_180_fun__upd__def,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( fun_upd @ A @ B )
% 3.50/1.69 = ( ^ [F2: A > B,A6: A,B6: B,X2: A] : ( if @ B @ ( X2 = A6 ) @ B6 @ ( F2 @ X2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_181_diff__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A,D2: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_182_diff__left__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_183_diff__right__mono,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 => ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_184_diff__eq__diff__less__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A,D2: A] :
% 3.50/1.69 ( ( ( minus_minus @ A @ A4 @ B4 )
% 3.50/1.69 = ( minus_minus @ A @ C2 @ D2 ) )
% 3.50/1.69 => ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.50/1.69 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_185_left__diff__distrib,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 )
% 3.50/1.69 = ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_186_right__diff__distrib,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ring @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
% 3.50/1.69 = ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_187_left__diff__distrib_H,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( comm_s1003936772cancel @ A )
% 3.50/1.69 => ! [B4: A,C2: A,A4: A] :
% 3.50/1.69 ( ( times_times @ A @ ( minus_minus @ A @ B4 @ C2 ) @ A4 )
% 3.50/1.69 = ( minus_minus @ A @ ( times_times @ A @ B4 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_188_right__diff__distrib_H,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( comm_s1003936772cancel @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B4 @ C2 ) )
% 3.50/1.69 = ( minus_minus @ A @ ( times_times @ A @ A4 @ B4 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_189_comm__monoid__mult__class_Omult__1,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( comm_monoid_mult @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ ( one_one @ A ) @ A4 )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_190_mult_Ocomm__neutral,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( comm_monoid_mult @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ( ( times_times @ A @ A4 @ ( one_one @ A ) )
% 3.50/1.69 = A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_191_diff__numeral__special_I9_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( neg_numeral @ A )
% 3.50/1.69 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 3.50/1.69 = ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_192_subsetI,axiom,
% 3.50/1.69 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ( member @ A @ X3 @ A3 )
% 3.50/1.69 => ( member @ A @ X3 @ B3 ) )
% 3.50/1.69 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_193_subset__antisym,axiom,
% 3.50/1.69 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.50/1.69 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.50/1.69 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 3.50/1.69 => ( A3 = B3 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_194_partitions__remove1__bounds,axiom,
% 3.50/1.69 ! [P2: nat > nat,N2: nat,K: nat,I2: nat] :
% 3.50/1.69 ( ( number2016821345itions @ P2 @ N2 )
% 3.50/1.69 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P2 @ K ) )
% 3.50/1.69 => ( ( ( fun_upd @ nat @ nat @ P2 @ K @ ( minus_minus @ nat @ ( P2 @ K ) @ ( one_one @ nat ) ) @ I2 )
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ I2 )
% 3.50/1.69 & ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_195_nat__mult__eq__cancel__disj,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ( times_times @ nat @ K @ M )
% 3.50/1.69 = ( times_times @ nat @ K @ N2 ) )
% 3.50/1.69 = ( ( K
% 3.50/1.69 = ( zero_zero @ nat ) )
% 3.50/1.69 | ( M = N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_196_le__numeral__extra_I4_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_197_le__numeral__extra_I3_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_198_not__gr__zero,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( canoni770627133id_add @ A )
% 3.50/1.69 => ! [N2: A] :
% 3.50/1.69 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
% 3.50/1.69 = ( N2
% 3.50/1.69 = ( zero_zero @ A ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_199_bot__nat__0_Onot__eq__extremum,axiom,
% 3.50/1.69 ! [A4: nat] :
% 3.50/1.69 ( ( A4
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_200_less__nat__zero__code,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_201_neq0__conv,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( N2
% 3.50/1.69 != ( zero_zero @ nat ) )
% 3.50/1.69 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_202_diff__gt__0__iff__gt,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ordered_ab_group_add @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B4 ) )
% 3.50/1.69 = ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_203_zero__less__diff,axiom,
% 3.50/1.69 ! [N2: nat,M: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
% 3.50/1.69 = ( ord_less @ nat @ M @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_204_nat__mult__less__cancel__disj,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
% 3.50/1.69 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 3.50/1.69 & ( ord_less @ nat @ M @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_205_mult__less__cancel2,axiom,
% 3.50/1.69 ! [M: nat,K: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
% 3.50/1.69 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 3.50/1.69 & ( ord_less @ nat @ M @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_206_nat__0__less__mult__iff,axiom,
% 3.50/1.69 ! [M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
% 3.50/1.69 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 3.50/1.69 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_207_less__one,axiom,
% 3.50/1.69 ! [N2: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
% 3.50/1.69 = ( N2
% 3.50/1.69 = ( zero_zero @ nat ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_208_nat__mult__le__cancel__disj,axiom,
% 3.50/1.69 ! [K: nat,M: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
% 3.50/1.69 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 3.50/1.69 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_209_mult__le__cancel2,axiom,
% 3.50/1.69 ! [M: nat,K: nat,N2: nat] :
% 3.50/1.69 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
% 3.50/1.69 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 3.50/1.69 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_210_linorder__neqE__linordered__idom,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linordered_idom @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( X != Y )
% 3.50/1.69 => ( ~ ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [B4: A,A4: A] :
% 3.50/1.69 ( ( ord_less @ A @ B4 @ A4 )
% 3.50/1.69 => ( A4 != B4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_212_order_Ostrict__implies__not__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ( A4 != B4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_213_not__less__iff__gr__or__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.50/1.69 = ( ( ord_less @ A @ Y @ X )
% 3.50/1.69 | ( X = Y ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_214_dual__order_Ostrict__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [B4: A,A4: A,C2: A] :
% 3.50/1.69 ( ( ord_less @ A @ B4 @ A4 )
% 3.50/1.69 => ( ( ord_less @ A @ C2 @ B4 )
% 3.50/1.69 => ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_215_linorder__less__wlog,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [P: A > A > $o,A4: A,B4: A] :
% 3.50/1.69 ( ! [A5: A,B5: A] :
% 3.50/1.69 ( ( ord_less @ A @ A5 @ B5 )
% 3.50/1.69 => ( P @ A5 @ B5 ) )
% 3.50/1.69 => ( ! [A5: A] : ( P @ A5 @ A5 )
% 3.50/1.69 => ( ! [A5: A,B5: A] :
% 3.50/1.69 ( ( P @ B5 @ A5 )
% 3.50/1.69 => ( P @ A5 @ B5 ) )
% 3.50/1.69 => ( P @ A4 @ B4 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_216_exists__least__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( wellorder @ A )
% 3.50/1.69 => ( ( ^ [P4: A > $o] :
% 3.50/1.69 ? [X5: A] : ( P4 @ X5 ) )
% 3.50/1.69 = ( ^ [P5: A > $o] :
% 3.50/1.69 ? [N: A] :
% 3.50/1.69 ( ( P5 @ N )
% 3.50/1.69 & ! [M4: A] :
% 3.50/1.69 ( ( ord_less @ A @ M4 @ N )
% 3.50/1.69 => ~ ( P5 @ M4 ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_217_less__imp__not__less,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_218_order_Ostrict__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less @ A @ B4 @ C2 )
% 3.50/1.69 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_219_dual__order_Oirrefl,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A] :
% 3.50/1.69 ~ ( ord_less @ A @ A4 @ A4 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_220_linorder__cases,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ~ ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( ( X != Y )
% 3.50/1.69 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_221_less__imp__triv,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A,P: $o] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( ( ord_less @ A @ Y @ X )
% 3.50/1.69 => P ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_222_less__imp__not__eq2,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( Y != X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_223_antisym__conv3,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [Y: A,X: A] :
% 3.50/1.69 ( ~ ( ord_less @ A @ Y @ X )
% 3.50/1.69 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.50/1.69 = ( X = Y ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_224_less__induct,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( wellorder @ A )
% 3.50/1.69 => ! [P: A > $o,A4: A] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ! [Y3: A] :
% 3.50/1.69 ( ( ord_less @ A @ Y3 @ X3 )
% 3.50/1.69 => ( P @ Y3 ) )
% 3.50/1.69 => ( P @ X3 ) )
% 3.50/1.69 => ( P @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_225_less__not__sym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_226_less__imp__not__eq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( X != Y ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_227_dual__order_Oasym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [B4: A,A4: A] :
% 3.50/1.69 ( ( ord_less @ A @ B4 @ A4 )
% 3.50/1.69 => ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_228_ord__less__eq__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( B4 = C2 )
% 3.50/1.69 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_229_ord__eq__less__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( ord @ A )
% 3.50/1.69 => ! [A4: A,B4: A,C2: A] :
% 3.50/1.69 ( ( A4 = B4 )
% 3.50/1.69 => ( ( ord_less @ A @ B4 @ C2 )
% 3.50/1.69 => ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_230_less__irrefl,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A] :
% 3.50/1.69 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_231_less__linear,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 | ( X = Y )
% 3.50/1.69 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_232_less__trans,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A,Z: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( ( ord_less @ A @ Y @ Z )
% 3.50/1.69 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_233_less__asym_H,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_234_less__asym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( preorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_235_less__imp__neq,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( X != Y ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_236_dense,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( dense_order @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ? [Z4: A] :
% 3.50/1.69 ( ( ord_less @ A @ X @ Z4 )
% 3.50/1.69 & ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_237_order_Oasym,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( order @ A )
% 3.50/1.69 => ! [A4: A,B4: A] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_238_neq__iff,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( X != Y )
% 3.50/1.69 = ( ( ord_less @ A @ X @ Y )
% 3.50/1.69 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_239_neqE,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linorder @ A )
% 3.50/1.69 => ! [X: A,Y: A] :
% 3.50/1.69 ( ( X != Y )
% 3.50/1.69 => ( ~ ( ord_less @ A @ X @ Y )
% 3.50/1.69 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_240_gt__ex,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( no_top @ A )
% 3.50/1.69 => ! [X: A] :
% 3.50/1.69 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_241_lt__ex,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( no_bot @ A )
% 3.50/1.69 => ! [X: A] :
% 3.50/1.69 ? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_242_order__less__subst2,axiom,
% 3.50/1.69 ! [A: $tType,C: $tType] :
% 3.50/1.69 ( ( ( order @ C )
% 3.50/1.69 & ( order @ A ) )
% 3.50/1.69 => ! [A4: A,B4: A,F: A > C,C2: C] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
% 3.50/1.69 => ( ! [X3: A,Y2: A] :
% 3.50/1.69 ( ( ord_less @ A @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_243_order__less__subst1,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( order @ B )
% 3.50/1.69 & ( order @ A ) )
% 3.50/1.69 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ ( F @ B4 ) )
% 3.50/1.69 => ( ( ord_less @ B @ B4 @ C2 )
% 3.50/1.69 => ( ! [X3: B,Y2: B] :
% 3.50/1.69 ( ( ord_less @ B @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_244_ord__less__eq__subst,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( ord @ B )
% 3.50/1.69 & ( ord @ A ) )
% 3.50/1.69 => ! [A4: A,B4: A,F: A > B,C2: B] :
% 3.50/1.69 ( ( ord_less @ A @ A4 @ B4 )
% 3.50/1.69 => ( ( ( F @ B4 )
% 3.50/1.69 = C2 )
% 3.50/1.69 => ( ! [X3: A,Y2: A] :
% 3.50/1.69 ( ( ord_less @ A @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_245_ord__eq__less__subst,axiom,
% 3.50/1.69 ! [A: $tType,B: $tType] :
% 3.50/1.69 ( ( ( ord @ B )
% 3.50/1.69 & ( ord @ A ) )
% 3.50/1.69 => ! [A4: A,F: B > A,B4: B,C2: B] :
% 3.50/1.69 ( ( A4
% 3.50/1.69 = ( F @ B4 ) )
% 3.50/1.69 => ( ( ord_less @ B @ B4 @ C2 )
% 3.50/1.69 => ( ! [X3: B,Y2: B] :
% 3.50/1.69 ( ( ord_less @ B @ X3 @ Y2 )
% 3.50/1.69 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 3.50/1.69 => ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_246_less__numeral__extra_I3_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_247_less__numeral__extra_I4_J,axiom,
% 3.50/1.69 ! [A: $tType] :
% 3.50/1.69 ( ( linord1659791738miring @ A )
% 3.50/1.69 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_248_infinite__descent__measure,axiom,
% 3.50/1.69 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ~ ( P @ X3 )
% 3.50/1.69 => ? [Y3: A] :
% 3.50/1.69 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X3 ) )
% 3.50/1.69 & ~ ( P @ Y3 ) ) )
% 3.50/1.69 => ( P @ X ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_249_measure__induct__rule,axiom,
% 3.50/1.69 ! [B: $tType,A: $tType] :
% 3.50/1.69 ( ( wellorder @ B )
% 3.50/1.69 => ! [F: A > B,P: A > $o,A4: A] :
% 3.50/1.69 ( ! [X3: A] :
% 3.50/1.69 ( ! [Y3: A] :
% 3.50/1.69 ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
% 3.50/1.69 => ( P @ Y3 ) )
% 3.50/1.69 => ( P @ X3 ) )
% 3.50/1.69 => ( P @ A4 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_250_linorder__neqE__nat,axiom,
% 3.50/1.69 ! [X: nat,Y: nat] :
% 3.50/1.69 ( ( X != Y )
% 3.50/1.69 => ( ~ ( ord_less @ nat @ X @ Y )
% 3.50/1.69 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_251_infinite__descent,axiom,
% 3.50/1.69 ! [P: nat > $o,N2: nat] :
% 3.50/1.69 ( ! [N3: nat] :
% 3.50/1.69 ( ~ ( P @ N3 )
% 3.50/1.69 => ? [M5: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ M5 @ N3 )
% 3.50/1.69 & ~ ( P @ M5 ) ) )
% 3.50/1.69 => ( P @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(fact_252_nat__less__induct,axiom,
% 3.50/1.69 ! [P: nat > $o,N2: nat] :
% 3.50/1.69 ( ! [N3: nat] :
% 3.50/1.69 ( ! [M5: nat] :
% 3.50/1.69 ( ( ord_less @ nat @ M5 @ N3 )
% 3.50/1.69 => ( P @ M5 ) )
% 3.50/1.69 => ( P @ N3 ) )
% 3.50/1.69 => ( P @ N2 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.50/1.69 ! [A7: $tType,A8: $tType] :
% 3.50/1.69 ( ( preorder @ A8 )
% 3.50/1.69 => ( preorder @ ( A7 > A8 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.50/1.69 ! [A7: $tType,A8: $tType] :
% 3.50/1.69 ( ( order @ A8 )
% 3.50/1.69 => ( order @ ( A7 > A8 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_fun___Orderings_Oord,axiom,
% 3.50/1.69 ! [A7: $tType,A8: $tType] :
% 3.50/1.69 ( ( ord @ A8 )
% 3.50/1.69 => ( ord @ ( A7 > A8 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_fun___Groups_Ominus,axiom,
% 3.50/1.69 ! [A7: $tType,A8: $tType] :
% 3.50/1.69 ( ( minus @ A8 )
% 3.50/1.69 => ( minus @ ( A7 > A8 ) ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 3.50/1.69 semiri1923998003cancel @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.50/1.69 canoni770627133id_add @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 3.50/1.69 linord1659791738miring @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
% 3.50/1.69 semiri1193490041visors @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
% 3.50/1.69 ordere216010020id_add @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 3.50/1.69 cancel146912293up_add @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 3.50/1.69 cancel1352612707id_add @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel,axiom,
% 3.50/1.69 comm_s1003936772cancel @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
% 3.50/1.69 ordere1490568538miring @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
% 3.50/1.69 ordered_semiring_0 @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 3.50/1.69 ab_semigroup_mult @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 3.50/1.69 comm_monoid_mult @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 3.50/1.69 comm_monoid_diff @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
% 3.50/1.69 ordered_semiring @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 3.50/1.69 comm_monoid_add @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 3.50/1.69 semigroup_mult @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.50/1.69 wellorder @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 3.50/1.69 zero_neq_one @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 3.50/1.69 preorder @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.50/1.69 linorder @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 3.50/1.69 monoid_mult @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Osemiring__0,axiom,
% 3.50/1.69 semiring_0 @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.50/1.69 no_top @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
% 3.50/1.69 mult_zero @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 3.50/1.69 order @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 3.50/1.69 ord @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
% 3.50/1.69 minus @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 3.50/1.69 zero @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 3.50/1.69 one @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Nat_Onat___Rings_Odvd,axiom,
% 3.50/1.69 dvd @ nat ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 3.50/1.69 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 3.50/1.69 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 3.50/1.69 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
% 3.50/1.69 ! [A7: $tType] : ( minus @ ( set @ A7 ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 3.50/1.69 preorder @ $o ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
% 3.50/1.69 linorder @ $o ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 3.50/1.69 order @ $o ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 3.50/1.69 ord @ $o ).
% 3.50/1.69
% 3.50/1.69 thf(tcon_HOL_Obool___Groups_Ominus_13,axiom,
% 3.50/1.69 minus @ $o ).
% 3.50/1.69
% 3.50/1.69 thf(help_If_3_1_T,axiom,
% 3.50/1.69 ! [P: $o] :
% 3.50/1.69 ( ( P = $true )
% 3.50/1.69 | ( P = $false ) ) ).
% 3.50/1.69
% 3.50/1.69 thf(help_If_2_1_T,axiom,
% 3.50/1.69 ! [A: $tType,X: A,Y: A] :
% 3.50/1.69 ( ( if @ A @ $false @ X @ Y )
% 3.50/1.69 = Y ) ).
% 3.50/1.69
% 3.50/1.69 thf(help_If_1_1_T,axiom,
% 3.50/1.69 ! [A: $tType,X: A,Y: A] :
% 3.50/1.69 ( ( if @ A @ $true @ X @ Y )
% 3.50/1.69 = X ) ).
% 3.50/1.69
% 3.50/1.69 thf(conj_0,conjecture,
% 3.50/1.69 ( ( minus_minus @ nat
% 3.50/1.69 @ ( groups1340683514dd_sum @ nat @ nat
% 3.50/1.69 @ ^ [I: nat] : ( times_times @ nat @ ( p @ I ) @ I )
% 3.50/1.69 @ ( set_ord_atMost @ nat @ n ) )
% 3.50/1.69 @ k )
% 3.50/1.69 = ( minus_minus @ nat @ n @ k ) ) ).
% 3.50/1.69 ------------------------
% 3.63/1.73 % (25692)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_25510 for (2999ds/275Mi)
% 3.63/1.73 % (25691)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_25510 for (2999ds/2Mi)
% 3.63/1.73 % (25690)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25510 for (2999ds/2Mi)
% 3.63/1.73 % (25687)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25510 for (2999ds/183Mi)
% 3.63/1.73 % (25689)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_25510 for (2999ds/27Mi)
% 3.63/1.73 % (25688)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_25510 for (2999ds/4Mi)
% 3.63/1.73 % (25693)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25510 for (2999ds/18Mi)
% 3.63/1.73 % (25690)Instruction limit reached!
% 3.63/1.73 % (25690)------------------------------
% 3.63/1.73 % (25690)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.73 % (25690)Termination reason: Unknown
% 3.63/1.73 % (25690)Termination phase: shuffling
% 3.63/1.73 % (25691)Instruction limit reached!
% 3.63/1.73 % (25691)------------------------------
% 3.63/1.73 % (25691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.73 % (25691)Termination reason: Unknown
% 3.63/1.73 % (25691)Termination phase: shuffling
% 3.63/1.73
% 3.63/1.73 % (25691)Memory used [KB]: 1407
% 3.63/1.73 % (25691)Time elapsed: 0.003 s
% 3.63/1.73 % (25691)Instructions burned: 3 (million)
% 3.63/1.73 % (25691)------------------------------
% 3.63/1.73 % (25691)------------------------------
% 3.63/1.73
% 3.63/1.73 % (25690)Memory used [KB]: 1407
% 3.63/1.73 % (25690)Time elapsed: 0.003 s
% 3.63/1.73 % (25690)Instructions burned: 2 (million)
% 3.63/1.73 % (25690)------------------------------
% 3.63/1.73 % (25690)------------------------------
% 3.63/1.73 % (25688)Instruction limit reached!
% 3.63/1.73 % (25688)------------------------------
% 3.63/1.73 % (25688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.73 % (25688)Termination reason: Unknown
% 3.63/1.73 % (25688)Termination phase: shuffling
% 3.63/1.73
% 3.63/1.73 % (25688)Memory used [KB]: 1407
% 3.63/1.73 % (25688)Time elapsed: 0.004 s
% 3.63/1.73 % (25688)Instructions burned: 4 (million)
% 3.63/1.73 % (25688)------------------------------
% 3.63/1.73 % (25688)------------------------------
% 3.63/1.74 % (25693)Instruction limit reached!
% 3.63/1.74 % (25693)------------------------------
% 3.63/1.74 % (25693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.74 % (25693)Termination reason: Unknown
% 3.63/1.74 % (25693)Termination phase: shuffling
% 3.63/1.74
% 3.63/1.74 % (25693)Memory used [KB]: 1791
% 3.63/1.74 % (25693)Time elapsed: 0.012 s
% 3.63/1.74 % (25693)Instructions burned: 19 (million)
% 3.63/1.74 % (25693)------------------------------
% 3.63/1.74 % (25693)------------------------------
% 3.63/1.74 % (25689)Instruction limit reached!
% 3.63/1.74 % (25689)------------------------------
% 3.63/1.74 % (25689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.74 % (25689)Termination reason: Unknown
% 3.63/1.74 % (25689)Termination phase: Property scanning
% 3.63/1.74
% 3.63/1.74 % (25689)Memory used [KB]: 1791
% 3.63/1.74 % (25689)Time elapsed: 0.017 s
% 3.63/1.74 % (25689)Instructions burned: 28 (million)
% 3.63/1.74 % (25689)------------------------------
% 3.63/1.74 % (25689)------------------------------
% 3.63/1.75 % (25695)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_25510 for (2999ds/37Mi)
% 3.63/1.75 % (25694)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25510 for (2999ds/3Mi)
% 3.63/1.75 % (25696)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_25510 for (2999ds/15Mi)
% 3.63/1.75 % (25694)Instruction limit reached!
% 3.63/1.75 % (25694)------------------------------
% 3.63/1.75 % (25694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.75 % (25694)Termination reason: Unknown
% 3.63/1.75 % (25694)Termination phase: shuffling
% 3.63/1.75
% 3.63/1.75 % (25694)Memory used [KB]: 1407
% 3.63/1.75 % (25694)Time elapsed: 0.004 s
% 3.63/1.75 % (25694)Instructions burned: 4 (million)
% 3.63/1.75 % (25694)------------------------------
% 3.63/1.75 % (25694)------------------------------
% 3.63/1.76 % (25696)Instruction limit reached!
% 3.63/1.76 % (25696)------------------------------
% 3.63/1.76 % (25696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.76 % (25696)Termination reason: Unknown
% 3.63/1.76 % (25696)Termination phase: shuffling
% 3.63/1.76
% 3.63/1.76 % (25696)Memory used [KB]: 1663
% 3.63/1.76 % (25696)Time elapsed: 0.010 s
% 3.63/1.76 % (25696)Instructions burned: 15 (million)
% 3.63/1.76 % (25696)------------------------------
% 3.63/1.76 % (25696)------------------------------
% 3.63/1.76 % (25697)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_25510 for (2999ds/3Mi)
% 3.63/1.76 % (25697)Instruction limit reached!
% 3.63/1.76 % (25697)------------------------------
% 3.63/1.76 % (25697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.76 % (25697)Termination reason: Unknown
% 3.63/1.76 % (25697)Termination phase: shuffling
% 3.63/1.76
% 3.63/1.76 % (25697)Memory used [KB]: 1407
% 3.63/1.76 % (25697)Time elapsed: 0.004 s
% 3.63/1.76 % (25697)Instructions burned: 4 (million)
% 3.63/1.76 % (25697)------------------------------
% 3.63/1.76 % (25697)------------------------------
% 3.63/1.76 % (25698)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_25510 for (2999ds/1041Mi)
% 3.63/1.77 % (25692)First to succeed.
% 3.63/1.77 % (25695)Instruction limit reached!
% 3.63/1.77 % (25695)------------------------------
% 3.63/1.77 % (25695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.77 % (25695)Termination reason: Unknown
% 3.63/1.77 % (25695)Termination phase: shuffling
% 3.63/1.77
% 3.63/1.77 % (25695)Memory used [KB]: 2174
% 3.63/1.77 % (25695)Time elapsed: 0.021 s
% 3.63/1.77 % (25695)Instructions burned: 37 (million)
% 3.63/1.77 % (25695)------------------------------
% 3.63/1.77 % (25695)------------------------------
% 3.63/1.77 % (25699)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25510 for (2999ds/7Mi)
% 3.63/1.77 % (25692)Refutation found. Thanks to Tanya!
% 3.63/1.77 % SZS status Theorem for DTF2THF_25510
% 3.63/1.77 % SZS output start Proof for DTF2THF_25510
% 3.63/1.77 thf(type_def_6, type, set: ($tType) > $tType).
% 3.63/1.77 thf(type_def_7, type, nat: $tType).
% 3.63/1.77 thf(func_def_0, type, set: ($tType) > $tType).
% 3.63/1.77 thf(func_def_1, type, nat: $tType).
% 3.63/1.77 thf(func_def_2, type, type: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_3, type, dvd: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_4, type, one: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_5, type, ring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_6, type, zero: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_7, type, minus: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_9, type, neg_numeral: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_10, type, order: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_11, type, comm_ring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_12, type, mult_zero: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_13, type, group_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_14, type, no_bot: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_15, type, no_top: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_16, type, semiring_0: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_17, type, monoid_mult: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_18, type, linorder: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_19, type, preorder: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_20, type, ordered_ring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_21, type, zero_neq_one: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_22, type, ab_group_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_23, type, wellorder: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_24, type, semigroup_mult: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_25, type, dense_order: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_26, type, linordered_idom: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_27, type, linordered_ring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_28, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_29, type, ordered_semiring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_30, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_31, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_32, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_33, type, ordered_semiring_0: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_34, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_35, type, ordere1490568538miring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_36, type, comm_s1003936772cancel: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_37, type, linord581940658strict: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_38, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_39, type, ring_11004092258visors: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_40, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_41, type, ordere216010020id_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_42, type, semiri1193490041visors: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_43, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_44, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_45, type, semiri1923998003cancel: !>[X0: $tType]:($o)).
% 3.63/1.77 thf(func_def_46, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 3.63/1.77 thf(func_def_47, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.63/1.77 thf(func_def_48, type, one_one: !>[X0: $tType]:(X0)).
% 3.63/1.77 thf(func_def_49, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.63/1.77 thf(func_def_50, type, zero_zero: !>[X0: $tType]:(X0)).
% 3.63/1.77 thf(func_def_51, type, groups1340683514dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 3.63/1.77 thf(func_def_52, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.63/1.77 thf(func_def_53, type, number2016821345itions: (nat > nat) > nat > $o).
% 3.63/1.77 thf(func_def_54, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.63/1.77 thf(func_def_55, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.63/1.77 thf(func_def_56, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.63/1.77 thf(func_def_57, type, set_ord_atMost: !>[X0: $tType]:(X0 > set @ X0)).
% 3.63/1.77 thf(func_def_58, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.63/1.77 thf(func_def_59, type, k: nat).
% 3.63/1.77 thf(func_def_60, type, n: nat).
% 3.63/1.77 thf(func_def_61, type, p: nat > nat).
% 3.63/1.77 thf(func_def_76, type, sK0: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.63/1.77 thf(func_def_77, type, sK1: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_78, type, sK2: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_79, type, sK3: nat > (nat > nat) > nat).
% 3.63/1.77 thf(func_def_80, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_81, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_82, type, sK6: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_83, type, sK7: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_84, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_85, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_86, type, sK10: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 3.63/1.77 thf(func_def_87, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_88, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_89, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_90, type, sK14: (nat > $o) > nat).
% 3.63/1.77 thf(func_def_91, type, sK15: !>[X0: $tType]:((X0 > nat) > (X0 > nat) > set @ X0 > X0)).
% 3.63/1.77 thf(func_def_92, type, sK16: (nat > $o) > nat).
% 3.63/1.77 thf(func_def_93, type, sK17: nat > (nat > $o) > nat).
% 3.63/1.77 thf(func_def_94, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_95, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_96, type, sK20: (nat > $o) > nat).
% 3.63/1.77 thf(func_def_97, type, sK21: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_98, type, sK22: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_99, type, sK23: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_100, type, sK24: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.63/1.77 thf(func_def_101, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.63/1.77 thf(func_def_102, type, sK26: (nat > $o) > nat).
% 3.63/1.77 thf(func_def_103, type, sK27: nat > (nat > $o) > nat).
% 3.63/1.77 thf(func_def_105, type, ph29: !>[X0: $tType]:(X0)).
% 3.63/1.77 thf(f1345,plain,(
% 3.63/1.77 $false),
% 3.63/1.77 inference(subsumption_resolution,[status(thm)],[f1334,f1146])).
% 3.63/1.77 thf(f1146,plain,(
% 3.63/1.77 ((number2016821345itions @ p @ n) = $true)),
% 3.63/1.77 inference(cnf_transformation,[status(thm)],[f560])).
% 3.63/1.77 thf(f560,plain,(
% 3.63/1.77 ((number2016821345itions @ p @ n) = $true)),
% 3.63/1.77 inference(fool_elimination,[status(thm)],[f559])).
% 3.63/1.77 thf(f559,plain,(
% 3.63/1.77 (number2016821345itions @ p @ n)),
% 3.63/1.77 inference(rectify,[status(thm)],[f1])).
% 3.63/1.77 thf(f1,axiom,(
% 3.63/1.77 (number2016821345itions @ p @ n)),
% 3.63/1.77 file('/export/starexec/sandbox/tmp/tmp.37uXGZ2q2e/DTF2THF_25510.p',fact_0_partitions)).
% 3.63/1.77 thf(f1334,plain,(
% 3.63/1.77 ((number2016821345itions @ p @ n) != $true)),
% 3.63/1.77 inference(trivial_inequality_removal,[status(thm)],[f1331])).
% 3.63/1.77 thf(f1331,plain,(
% 3.63/1.77 ((minus_minus @ nat @ n @ k) != (minus_minus @ nat @ n @ k)) | ((number2016821345itions @ p @ n) != $true)),
% 3.63/1.77 inference(superposition,[status(thm)],[f1270,f1167])).
% 3.63/1.77 thf(f1167,plain,(
% 3.63/1.77 ( ! [X0 : nat > nat,X1 : nat] : (((groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ X1)) = X1) | ($true != (number2016821345itions @ X0 @ X1))) )),
% 3.63/1.77 inference(cnf_transformation,[status(thm)],[f908])).
% 3.63/1.77 thf(f908,plain,(
% 3.63/1.77 ! [X0 : nat > nat,X1 : nat] : (($true != (number2016821345itions @ X0 @ X1)) | (((groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ X1)) = X1) & ! [X2 : nat] : ((((ord_less_eq @ nat @ X2 @ X1) = $true) & ((ord_less_eq @ nat @ one_one @ nat @ X2) = $true)) | (zero_zero @ nat = (X0 @ X2)))))),
% 3.63/1.77 inference(ennf_transformation,[status(thm)],[f778])).
% 3.63/1.77 thf(f778,plain,(
% 3.63/1.77 ! [X0 : nat > nat,X1 : nat] : (($true = (number2016821345itions @ X0 @ X1)) => ~(! [X2 : nat] : ((zero_zero @ nat != (X0 @ X2)) => (((ord_less_eq @ nat @ X2 @ X1) = $true) & ((ord_less_eq @ nat @ one_one @ nat @ X2) = $true))) => ((groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ X1)) != X1)))),
% 3.63/1.77 inference(fool_elimination,[status(thm)],[f777])).
% 3.63/1.77 thf(f777,plain,(
% 3.63/1.77 ! [X0 : nat > nat,X1 : nat] : ((number2016821345itions @ X0 @ X1) => ~(! [X2 : nat] : ((zero_zero @ nat != (X0 @ X2)) => ((ord_less_eq @ nat @ one_one @ nat @ X2) & (ord_less_eq @ nat @ X2 @ X1))) => (X1 != (groups1340683514dd_sum @ nat @ nat @ (^[X3 : nat] : (times_times @ nat @ (X0 @ X3) @ X3)) @ (set_ord_atMost @ nat @ X1)))))),
% 3.63/1.77 inference(rectify,[status(thm)],[f66])).
% 3.63/1.77 thf(f66,axiom,(
% 3.63/1.77 ! [X33 : nat > nat,X17 : nat] : ((number2016821345itions @ X33 @ X17) => ~(! [X53 : nat] : ((zero_zero @ nat != (X33 @ X53)) => ((ord_less_eq @ nat @ one_one @ nat @ X53) & (ord_less_eq @ nat @ X53 @ X17))) => (X17 != (groups1340683514dd_sum @ nat @ nat @ (^[X7 : nat] : (times_times @ nat @ (X33 @ X7) @ X7)) @ (set_ord_atMost @ nat @ X17)))))),
% 3.63/1.77 file('/export/starexec/sandbox/tmp/tmp.37uXGZ2q2e/DTF2THF_25510.p',fact_65_partitionsE)).
% 3.63/1.77 thf(f1270,plain,(
% 3.63/1.77 ((minus_minus @ nat @ n @ k) != (minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ n)) @ k))),
% 3.63/1.77 inference(cnf_transformation,[status(thm)],[f857])).
% 3.63/1.77 thf(f857,plain,(
% 3.63/1.77 ((minus_minus @ nat @ n @ k) != (minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ n)) @ k))),
% 3.63/1.77 inference(flattening,[status(thm)],[f638])).
% 3.63/1.77 thf(f638,plain,(
% 3.63/1.77 ~((minus_minus @ nat @ n @ k) = (minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[Y0 : nat]: (times_times @ nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost @ nat @ n)) @ k))),
% 3.63/1.77 inference(fool_elimination,[status(thm)],[f637])).
% 3.63/1.77 thf(f637,plain,(
% 3.63/1.77 ~((minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[X0 : nat] : (times_times @ nat @ (p @ X0) @ X0)) @ (set_ord_atMost @ nat @ n)) @ k) = (minus_minus @ nat @ n @ k))),
% 3.63/1.77 inference(rectify,[status(thm)],[f301])).
% 3.63/1.77 thf(f301,negated_conjecture,(
% 3.63/1.77 ~((minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[X7 : nat] : (times_times @ nat @ (p @ X7) @ X7)) @ (set_ord_atMost @ nat @ n)) @ k) = (minus_minus @ nat @ n @ k))),
% 3.63/1.77 inference(negated_conjecture,[status(cth)],[f300])).
% 3.63/1.77 thf(f300,conjecture,(
% 3.63/1.77 ((minus_minus @ nat @ (groups1340683514dd_sum @ nat @ nat @ (^[X7 : nat] : (times_times @ nat @ (p @ X7) @ X7)) @ (set_ord_atMost @ nat @ n)) @ k) = (minus_minus @ nat @ n @ k))),
% 3.63/1.77 file('/export/starexec/sandbox/tmp/tmp.37uXGZ2q2e/DTF2THF_25510.p',conj_0)).
% 3.63/1.77 % SZS output end Proof for DTF2THF_25510
% 3.63/1.77 % (25692)------------------------------
% 3.63/1.77 % (25692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.63/1.77 % (25692)Termination reason: Refutation
% 3.63/1.77
% 3.63/1.77 % (25692)Memory used [KB]: 6780
% 3.63/1.77 % (25692)Time elapsed: 0.041 s
% 3.63/1.77 % (25692)Instructions burned: 70 (million)
% 3.63/1.77 % (25692)------------------------------
% 3.63/1.77 % (25692)------------------------------
% 3.63/1.77 % (25686)Success in time 0.072 s
%------------------------------------------------------------------------------