%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP026^2 : 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 : n016.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 08:54:57 AM UTC 2026
% Result : Theorem 33.81s 6.09s
% Output : Refutation 33.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP026^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n016.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:24:21 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP026^2 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer Algebra8 problem prob_1404__6432666_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : Algebra8/prob_1404__6432666_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 346 ( 118 unt; 60 typ; 0 def)
% 0.19/0.45 % Number of atoms : 769 ( 189 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.19/0.45 % Number of connectives : 5767 ( 66 ~; 11 |; 40 &;5270 @)
% 0.19/0.45 % ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 28 ( 9 avg)
% 0.19/0.45 % Number of types : 9 ( 8 usr)
% 0.19/0.45 % Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 54 ( 52 usr; 10 con; 0-9 aty)
% 0.19/0.45 % Number of variables : 978 ( 105 ^; 762 !; 29 ?; 978 :)
% 0.19/0.45 % ( 82 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.45 % SPC : TH1_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:31:27.197
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (13)
% 0.19/0.45 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext,type,
% 0.19/0.45 carrier_ext: $tType > $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Algebra7_OModule_OModule__ext,type,
% 0.19/0.45 module_ext: $tType > $tType > $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Algebra4_OaGroup_OaGroup__ext,type,
% 0.19/0.45 aGroup_ext: $tType > $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Algebra4_ORing_ORing__ext,type,
% 0.19/0.45 ring_ext: $tType > $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Product__Type_Ounit,type,
% 0.19/0.45 product_unit: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Set_Oset,type,
% 0.19/0.45 set: $tType > $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_t_Nat_Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_f,type,
% 0.19/0.45 f: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_e,type,
% 0.19/0.45 e: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_d,type,
% 0.19/0.45 d: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_c,type,
% 0.19/0.45 c: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_b,type,
% 0.19/0.45 b: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_tf_a,type,
% 0.19/0.45 a: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (47)
% 0.19/0.45 thf(sy_cl_HOL_Otype,type,
% 0.19/0.45 type:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.45 zero:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.45 order:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.45 group_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.19/0.45 wellorder:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.19/0.45 comm_monoid_diff:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.19/0.45 ordered_ab_group_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.19/0.45 cancel1352612707id_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.19/0.45 canoni770627133id_add:
% 0.19/0.45 !>[A: $tType] : $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra1_Ocarrier_Ocarrier,type,
% 0.19/0.45 carrier:
% 0.19/0.45 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ Z ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra4_ORing,type,
% 0.19/0.45 ring:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra4_OaGroup_Ozero,type,
% 0.19/0.45 zero2:
% 0.19/0.45 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra4_Oideal,type,
% 0.19/0.45 ideal:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_OAnnihilator,type,
% 0.19/0.45 annihilator:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_OHOM,type,
% 0.19/0.45 hom:
% 0.19/0.45 !>[B: $tType,More: $tType,A: $tType,More1: $tType,C: $tType,More2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ More ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ More1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ More2 ) ) ) > ( carrier_ext @ ( A > C ) @ ( aGroup_ext @ ( A > C ) @ ( module_ext @ ( A > C ) @ B @ product_unit ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_OModule,type,
% 0.19/0.45 module:
% 0.19/0.45 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Ofree__generator,type,
% 0.19/0.45 free_generator:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Ogenerator,type,
% 0.19/0.45 generator:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Ol__comb,type,
% 0.19/0.45 l_comb:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > nat > ( nat > R ) > ( nat > A ) > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Olinear__span,type,
% 0.19/0.45 linear_span:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Omdl,type,
% 0.19/0.45 mdl:
% 0.19/0.45 !>[A: $tType,B: $tType,M: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ product_unit ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Omisomorphic,type,
% 0.19/0.45 misomorphic:
% 0.19/0.45 !>[B: $tType,M: $tType,A: $tType,M1: $tType,C: $tType,M2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ M2 ) ) ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Oquotient__of__submodules,type,
% 0.19/0.45 quotie135653060odules:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ R ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Osmodule__ideal__coeff,type,
% 0.19/0.45 smodule_ideal_coeff:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra7_Osubmodule,type,
% 0.19/0.45 submodule:
% 0.19/0.45 !>[B: $tType,M: $tType,A: $tType,C: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_Ofgs,type,
% 0.19/0.45 algebr833503410le_fgs:
% 0.19/0.45 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_FuncSet_OPi,type,
% 0.19/0.45 pi:
% 0.19/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.45 minus_minus:
% 0.19/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.45 zero_zero:
% 0.19/0.45 !>[A: $tType] : A ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_OSuc,type,
% 0.19/0.45 suc: nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.19/0.45 bot_bot:
% 0.19/0.45 !>[A: $tType] : A ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.19/0.45 ord_less:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.45 ord_less_eq:
% 0.19/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect,type,
% 0.19/0.45 collect:
% 0.19/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_Oinsert,type,
% 0.19/0.45 insert:
% 0.19/0.45 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member,type,
% 0.19/0.45 member:
% 0.19/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_H,type,
% 0.19/0.45 h: set @ a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_H1,type,
% 0.19/0.45 h1: set @ a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_M,type,
% 0.19/0.45 m: carrier_ext @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_N,type,
% 0.19/0.45 n: carrier_ext @ e @ ( aGroup_ext @ e @ ( module_ext @ e @ b @ f ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_R,type,
% 0.19/0.45 r: carrier_ext @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_g,type,
% 0.19/0.45 g: nat > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_h,type,
% 0.19/0.45 h2: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_m,type,
% 0.19/0.45 m2: nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_t,type,
% 0.19/0.45 t: nat > b ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_x,type,
% 0.19/0.45 x: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_xa,type,
% 0.19/0.45 xa: nat ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (253)
% 0.19/0.45 thf(fact_0_sc__Ring,axiom,
% 0.19/0.45 ring @ b @ d @ r ).
% 0.19/0.45
% 0.19/0.45 % sc_Ring
% 0.19/0.45 thf(fact_1_submodule__subset1,axiom,
% 0.19/0.45 ! [H: set @ a,H2: a] :
% 0.19/0.45 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( ( member @ a @ H2 @ H )
% 0.19/0.45 => ( member @ a @ H2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % submodule_subset1
% 0.19/0.45 thf(fact_2_submodule__whole,axiom,
% 0.19/0.45 submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.45
% 0.19/0.45 % submodule_whole
% 0.19/0.45 thf(fact_3_submodule__subset,axiom,
% 0.19/0.45 ! [H: set @ a] :
% 0.19/0.45 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % submodule_subset
% 0.19/0.45 thf(fact_4_free__generator__sub,axiom,
% 0.19/0.45 ! [H: set @ a] :
% 0.19/0.45 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % free_generator_sub
% 0.19/0.45 thf(fact_5_elem__fgs,axiom,
% 0.19/0.45 ! [A2: set @ a,X: a] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ( member @ a @ X @ A2 )
% 0.19/0.45 => ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % elem_fgs
% 0.19/0.45 thf(fact_6_fgs__sub__carrier,axiom,
% 0.19/0.45 ! [A2: set @ a] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fgs_sub_carrier
% 0.19/0.45 thf(fact_7_fgs__mono,axiom,
% 0.19/0.45 ! [H: set @ a,J: set @ a,K: set @ a] :
% 0.19/0.45 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ a ) @ J @ K )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ J ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fgs_mono
% 0.19/0.45 thf(fact_8_fgs__submodule,axiom,
% 0.19/0.45 ! [A2: set @ a] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fgs_submodule
% 0.19/0.45 thf(fact_9_linear__comb__eq,axiom,
% 0.19/0.45 ! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ( member @ ( nat > b ) @ S
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 0.19/0.45 => ( ( member @ ( nat > a ) @ F
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( ( member @ ( nat > a ) @ G
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( ! [X2: nat] :
% 0.19/0.45 ( ( member @ nat @ X2
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 0.19/0.45 => ( ( F @ X2 )
% 0.19/0.45 = ( G @ X2 ) ) )
% 0.19/0.45 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
% 0.19/0.45 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % linear_comb_eq
% 0.19/0.45 thf(fact_10_linear__comb__eqTr,axiom,
% 0.19/0.45 ! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ( ( member @ ( nat > b ) @ S
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 0.19/0.45 & ( member @ ( nat > a ) @ F
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 & ( member @ ( nat > a ) @ G
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 & ! [X2: nat] :
% 0.19/0.45 ( ( member @ nat @ X2
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 0.19/0.45 => ( ( F @ X2 )
% 0.19/0.45 = ( G @ X2 ) ) ) )
% 0.19/0.45 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
% 0.19/0.45 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % linear_comb_eqTr
% 0.19/0.45 thf(fact_11_l__comb__mem,axiom,
% 0.19/0.45 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,M3: nat > a] :
% 0.19/0.45 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ( member @ ( nat > b ) @ S
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.45 => ( ( member @ ( nat > a ) @ M3
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % l_comb_mem
% 0.19/0.45 thf(fact_12_liear__comb__memTr,axiom,
% 0.19/0.45 ! [A3: set @ b,H: set @ a,N: nat] :
% 0.19/0.45 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ! [S2: nat > b,M4: nat > a] :
% 0.19/0.45 ( ( ( member @ ( nat > b ) @ S2
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.45 & ( member @ ( nat > a ) @ M4
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) ) )
% 0.19/0.45 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S2 @ M4 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % liear_comb_memTr
% 0.19/0.45 thf(fact_13_Ann__is__ideal,axiom,
% 0.19/0.45 ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ) ).
% 0.19/0.45
% 0.19/0.45 % Ann_is_ideal
% 0.19/0.45 thf(fact_14_mem__fgs__l__comb,axiom,
% 0.19/0.45 ! [K: set @ a,X: a] :
% 0.19/0.45 ( ( K
% 0.19/0.45 != ( bot_bot @ ( set @ a ) ) )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ a ) @ K @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.45 => ( ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) )
% 0.19/0.45 => ? [N2: nat,X2: nat > a] :
% 0.19/0.45 ( ( member @ ( nat > a ) @ X2
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : K ) )
% 0.19/0.45 & ? [Xa: nat > b] :
% 0.19/0.45 ( ( member @ ( nat > b ) @ Xa
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 0.19/0.45 & ( X
% 0.19/0.45 = ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mem_fgs_l_comb
% 0.19/0.45 thf(fact_15_Module__axioms,axiom,
% 0.19/0.45 module @ a @ b @ c @ d @ m @ r ).
% 0.19/0.45
% 0.19/0.45 % Module_axioms
% 0.19/0.45 thf(fact_16_quotient__of__submodules__is__ideal,axiom,
% 0.19/0.45 ! [P: set @ a,Q: set @ a] :
% 0.19/0.45 ( ( submodule @ b @ d @ a @ c @ r @ m @ P )
% 0.19/0.45 => ( ( submodule @ b @ d @ a @ c @ r @ m @ Q )
% 0.19/0.45 => ( ideal @ b @ d @ r @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ P @ Q ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % quotient_of_submodules_is_ideal
% 0.19/0.45 thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
% 0.19/0.45 ! [A3: set @ b] :
% 0.19/0.45 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.45 => ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % smodule_ideal_coeff_is_Submodule
% 0.19/0.45 thf(fact_18_mem__smodule__ideal__coeff,axiom,
% 0.19/0.45 ! [A3: set @ b,X: a] :
% 0.19/0.45 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.45 => ( ( member @ a @ X @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) )
% 0.19/0.45 => ? [N2: nat,X2: nat > b] :
% 0.19/0.45 ( ( member @ ( nat > b ) @ X2
% 0.19/0.45 @ ( pi @ nat @ b
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.45 & ? [Xa: nat > a] :
% 0.19/0.45 ( ( member @ ( nat > a ) @ Xa
% 0.19/0.45 @ ( pi @ nat @ a
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
% 0.19/0.45 & ( X
% 0.19/0.45 = ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mem_smodule_ideal_coeff
% 0.19/0.45 thf(fact_19_Module_Oelem__fgs,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A,X: A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ( member @ A @ X @ A2 )
% 0.19/0.45 => ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.elem_fgs
% 0.19/0.45 thf(fact_20_Module_Ofgs__mono,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,J: set @ A,K: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ J @ K )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ K @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ J ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.fgs_mono
% 0.19/0.45 thf(fact_21_Module_Ofgs__submodule,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.fgs_submodule
% 0.19/0.45 thf(fact_22_Module_Omem__fgs__l__comb,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),K: set @ A,X: A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( K
% 0.19/0.45 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ K @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) )
% 0.19/0.45 => ? [N2: nat,X2: nat > A] :
% 0.19/0.45 ( ( member @ ( nat > A ) @ X2
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : K ) )
% 0.19/0.45 & ? [Xa: nat > B] :
% 0.19/0.45 ( ( member @ ( nat > B ) @ Xa
% 0.19/0.45 @ ( pi @ nat @ B
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 0.19/0.45 & ( X
% 0.19/0.45 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.mem_fgs_l_comb
% 0.19/0.45 thf(fact_23_Module_Ofgs__sub__carrier,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.fgs_sub_carrier
% 0.19/0.45 thf(fact_24_misom__self,axiom,
% 0.19/0.45 misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m ).
% 0.19/0.45
% 0.19/0.45 % misom_self
% 0.19/0.45 thf(fact_25_HOM__is__module,axiom,
% 0.19/0.45 ! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
% 0.19/0.45 ( ( module @ E @ b @ F2 @ d @ N3 @ r )
% 0.19/0.45 => ( module @ ( a > E ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 ) @ r ) ) ).
% 0.19/0.45
% 0.19/0.45 % HOM_is_module
% 0.19/0.45 thf(fact_26_Suc__pred,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.19/0.45 = N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_pred
% 0.19/0.45 thf(fact_27_generator__sub__carrier,axiom,
% 0.19/0.45 ! [H: set @ a] :
% 0.19/0.45 ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % generator_sub_carrier
% 0.19/0.45 thf(fact_28_free__generator__generator,axiom,
% 0.19/0.45 ! [H: set @ a] :
% 0.19/0.45 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.45 => ( generator @ b @ d @ a @ c @ r @ m @ H ) ) ).
% 0.19/0.45
% 0.19/0.45 % free_generator_generator
% 0.19/0.45 thf(fact_29_Module_Ol__comb__mem,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,S: nat > B,N: nat,M3: nat > A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ideal @ B @ D @ R2 @ A3 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ( member @ ( nat > B ) @ S
% 0.19/0.45 @ ( pi @ nat @ B
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.45 => ( ( member @ ( nat > A ) @ M3
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ M3 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.l_comb_mem
% 0.19/0.45 thf(fact_30_Module_Oliear__comb__memTr,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,N: nat] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ideal @ B @ D @ R2 @ A3 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ! [S2: nat > B,M4: nat > A] :
% 0.19/0.45 ( ( ( member @ ( nat > B ) @ S2
% 0.19/0.45 @ ( pi @ nat @ B
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.45 & ( member @ ( nat > A ) @ M4
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) ) )
% 0.19/0.45 => ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S2 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.liear_comb_memTr
% 0.19/0.45 thf(fact_31_zero__less__diff,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M3 ) )
% 0.19/0.45 = ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % zero_less_diff
% 0.19/0.45 thf(fact_32_Module_Olinear__comb__eq,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ( member @ ( nat > B ) @ S
% 0.19/0.45 @ ( pi @ nat @ B
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 0.19/0.45 => ( ( member @ ( nat > A ) @ F
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( ( member @ ( nat > A ) @ G
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 => ( ! [X2: nat] :
% 0.19/0.45 ( ( member @ nat @ X2
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 0.19/0.45 => ( ( F @ X2 )
% 0.19/0.45 = ( G @ X2 ) ) )
% 0.19/0.45 => ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
% 0.19/0.45 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.linear_comb_eq
% 0.19/0.45 thf(fact_33_Module_Olinear__comb__eqTr,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 0.19/0.45 => ( ( ( member @ ( nat > B ) @ S
% 0.19/0.45 @ ( pi @ nat @ B
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 0.19/0.45 & ( member @ ( nat > A ) @ F
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 & ( member @ ( nat > A ) @ G
% 0.19/0.45 @ ( pi @ nat @ A
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.45 @ ^ [Uu: nat] : H ) )
% 0.19/0.45 & ! [X2: nat] :
% 0.19/0.45 ( ( member @ nat @ X2
% 0.19/0.45 @ ( collect @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 0.19/0.45 => ( ( F @ X2 )
% 0.19/0.45 = ( G @ X2 ) ) ) )
% 0.19/0.45 => ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
% 0.19/0.45 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.linear_comb_eqTr
% 0.19/0.45 thf(fact_34_diff__is__0__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ( minus_minus @ nat @ M3 @ N )
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 = ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_is_0_eq
% 0.19/0.45 thf(fact_35_diff__is__0__eq_H,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ( minus_minus @ nat @ M3 @ N )
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_is_0_eq'
% 0.19/0.45 thf(fact_36_old_Onat_Oinject,axiom,
% 0.19/0.45 ! [Nat: nat,Nat2: nat] :
% 0.19/0.45 ( ( ( suc @ Nat )
% 0.19/0.45 = ( suc @ Nat2 ) )
% 0.19/0.45 = ( Nat = Nat2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.inject
% 0.19/0.45 thf(fact_37_nat_Oinject,axiom,
% 0.19/0.45 ! [X22: nat,Y2: nat] :
% 0.19/0.45 ( ( ( suc @ X22 )
% 0.19/0.45 = ( suc @ Y2 ) )
% 0.19/0.45 = ( X22 = Y2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat.inject
% 0.19/0.45 thf(fact_38_misom__sym,axiom,
% 0.19/0.45 ! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
% 0.19/0.45 ( ( module @ E @ b @ F2 @ d @ N3 @ r )
% 0.19/0.45 => ( ( misomorphic @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 )
% 0.19/0.45 => ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ N3 @ m ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % misom_sym
% 0.19/0.45 thf(fact_39_Suc__le__mono,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M3 ) )
% 0.19/0.45 = ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_le_mono
% 0.19/0.45 thf(fact_40_bot__nat__0_Oextremum,axiom,
% 0.19/0.45 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.19/0.45
% 0.19/0.45 % bot_nat_0.extremum
% 0.19/0.45 thf(fact_41_le0,axiom,
% 0.19/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.45
% 0.19/0.45 % le0
% 0.19/0.45 thf(fact_42_Suc__less__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 0.19/0.45 = ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_less_eq
% 0.19/0.45 thf(fact_43_Suc__mono,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_mono
% 0.19/0.45 thf(fact_44_mem__Collect__eq,axiom,
% 0.19/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.19/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.19/0.45 = ( P @ A2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % mem_Collect_eq
% 0.19/0.45 thf(fact_45_Collect__mem__eq,axiom,
% 0.19/0.45 ! [A: $tType,A3: set @ A] :
% 0.19/0.45 ( ( collect @ A
% 0.19/0.45 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.19/0.45 = A3 ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_mem_eq
% 0.19/0.45 thf(fact_46_Collect__cong,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.45 ( ! [X2: A] :
% 0.19/0.45 ( ( P @ X2 )
% 0.19/0.45 = ( Q @ X2 ) )
% 0.19/0.45 => ( ( collect @ A @ P )
% 0.19/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Collect_cong
% 0.19/0.45 thf(fact_47_bot__nat__0_Onot__eq__extremum,axiom,
% 0.19/0.45 ! [A2: nat] :
% 0.19/0.45 ( ( A2
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % bot_nat_0.not_eq_extremum
% 0.19/0.45 thf(fact_48_less__nat__zero__code,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_nat_zero_code
% 0.19/0.45 thf(fact_49_neq0__conv,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( N
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % neq0_conv
% 0.19/0.45 thf(fact_50_diff__diff__cancel,axiom,
% 0.19/0.45 ! [I: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ I @ N )
% 0.19/0.45 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 0.19/0.45 = I ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_diff_cancel
% 0.19/0.45 thf(fact_51_Suc__diff__diff,axiom,
% 0.19/0.45 ! [M3: nat,N: nat,K2: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M3 ) @ N ) @ ( suc @ K2 ) )
% 0.19/0.45 = ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ N ) @ K2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_diff_diff
% 0.19/0.45 thf(fact_52_diff__Suc__Suc,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 0.19/0.45 = ( minus_minus @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_Suc_Suc
% 0.19/0.45 thf(fact_53_diff__self__eq__0,axiom,
% 0.19/0.45 ! [M3: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ M3 @ M3 )
% 0.19/0.45 = ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_self_eq_0
% 0.19/0.45 thf(fact_54_diff__0__eq__0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 = ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_0_eq_0
% 0.19/0.45 thf(fact_55_misom__trans,axiom,
% 0.19/0.45 ! [E: $tType,F2: $tType,H3: $tType,G2: $tType,L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ b @ H3 ) )] :
% 0.19/0.45 ( ( module @ E @ b @ F2 @ d @ L @ r )
% 0.19/0.45 => ( ( module @ G2 @ b @ H3 @ d @ N3 @ r )
% 0.19/0.45 => ( ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ L @ m )
% 0.19/0.45 => ( ( misomorphic @ b @ d @ a @ c @ G2 @ H3 @ r @ m @ N3 )
% 0.19/0.45 => ( misomorphic @ b @ d @ E @ F2 @ G2 @ H3 @ r @ L @ N3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % misom_trans
% 0.19/0.45 thf(fact_56_zero__less__Suc,axiom,
% 0.19/0.45 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % zero_less_Suc
% 0.19/0.45 thf(fact_57_less__Suc0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_Suc0
% 0.19/0.45 thf(fact_58_Module_OHOM__is__module,axiom,
% 0.19/0.45 ! [E: $tType,F2: $tType,C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
% 0.19/0.45 => ( module @ ( A > E ) @ B @ product_unit @ D @ ( hom @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 ) @ R2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.HOM_is_module
% 0.19/0.45 thf(fact_59_Module_Omisom__trans,axiom,
% 0.19/0.45 ! [C: $tType,A: $tType,E: $tType,F2: $tType,D: $tType,H3: $tType,B: $tType,G2: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ B @ H3 ) )] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( module @ E @ B @ F2 @ D @ L @ R2 )
% 0.19/0.45 => ( ( module @ G2 @ B @ H3 @ D @ N3 @ R2 )
% 0.19/0.45 => ( ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ L @ M5 )
% 0.19/0.45 => ( ( misomorphic @ B @ D @ A @ C @ G2 @ H3 @ R2 @ M5 @ N3 )
% 0.19/0.45 => ( misomorphic @ B @ D @ E @ F2 @ G2 @ H3 @ R2 @ L @ N3 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.misom_trans
% 0.19/0.45 thf(fact_60_Module_Omisom__self,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( misomorphic @ B @ D @ A @ C @ A @ C @ R2 @ M5 @ M5 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.misom_self
% 0.19/0.45 thf(fact_61_Module_Omisom__sym,axiom,
% 0.19/0.45 ! [E: $tType,F2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
% 0.19/0.45 => ( ( misomorphic @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 )
% 0.19/0.45 => ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ N3 @ M5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.misom_sym
% 0.19/0.45 thf(fact_62_Module_Ofree__generator__generator,axiom,
% 0.19/0.45 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.45 => ( generator @ B @ D @ A @ C @ R2 @ M5 @ H ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.free_generator_generator
% 0.19/0.45 thf(fact_63_Module_Ogenerator__sub__carrier,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.45 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.generator_sub_carrier
% 0.19/0.45 thf(fact_64_Module_OAnn__is__ideal,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ideal @ B @ D @ R2 @ ( annihilator @ B @ D @ A @ C @ R2 @ M5 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.Ann_is_ideal
% 0.19/0.45 thf(fact_65_measure__induct__rule,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( wellorder @ B )
% 0.19/0.45 => ! [F: A > B,P: A > $o,A2: A] :
% 0.19/0.45 ( ! [X2: A] :
% 0.19/0.45 ( ! [Y: A] :
% 0.19/0.45 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.19/0.45 => ( P @ Y ) )
% 0.19/0.45 => ( P @ X2 ) )
% 0.19/0.45 => ( P @ A2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % measure_induct_rule
% 0.19/0.45 thf(fact_66_measure__induct,axiom,
% 0.19/0.45 ! [B: $tType,A: $tType] :
% 0.19/0.45 ( ( wellorder @ B )
% 0.19/0.45 => ! [F: A > B,P: A > $o,A2: A] :
% 0.19/0.45 ( ! [X2: A] :
% 0.19/0.45 ( ! [Y: A] :
% 0.19/0.45 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 0.19/0.45 => ( P @ Y ) )
% 0.19/0.45 => ( P @ X2 ) )
% 0.19/0.45 => ( P @ A2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % measure_induct
% 0.19/0.45 thf(fact_67_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.45 ! [P: nat > $o,K2: nat,B2: nat] :
% 0.19/0.45 ( ( P @ K2 )
% 0.19/0.45 => ( ! [Y3: nat] :
% 0.19/0.45 ( ( P @ Y3 )
% 0.19/0.45 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 0.19/0.45 => ? [X2: nat] :
% 0.19/0.45 ( ( P @ X2 )
% 0.19/0.45 & ! [Y: nat] :
% 0.19/0.45 ( ( P @ Y )
% 0.19/0.45 => ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Nat.ex_has_greatest_nat
% 0.19/0.45 thf(fact_68_nat__le__linear,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 | ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_le_linear
% 0.19/0.45 thf(fact_69_le__antisym,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ N @ M3 )
% 0.19/0.45 => ( M3 = N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_antisym
% 0.19/0.45 thf(fact_70_le__trans,axiom,
% 0.19/0.45 ! [I: nat,J3: nat,K2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ I @ J3 )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 0.19/0.45 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_trans
% 0.19/0.45 thf(fact_71_n__not__Suc__n,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( N
% 0.19/0.45 != ( suc @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % n_not_Suc_n
% 0.19/0.45 thf(fact_72_Suc__inject,axiom,
% 0.19/0.45 ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( ( suc @ X )
% 0.19/0.45 = ( suc @ Y4 ) )
% 0.19/0.45 => ( X = Y4 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_inject
% 0.19/0.45 thf(fact_73_infinite__descent__measure,axiom,
% 0.19/0.45 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.19/0.45 ( ! [X2: A] :
% 0.19/0.45 ( ~ ( P @ X2 )
% 0.19/0.45 => ? [Y: A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.19/0.45 & ~ ( P @ Y ) ) )
% 0.19/0.45 => ( P @ X ) ) ).
% 0.19/0.45
% 0.19/0.45 % infinite_descent_measure
% 0.19/0.45 thf(fact_74_linorder__neqE__nat,axiom,
% 0.19/0.45 ! [X: nat,Y4: nat] :
% 0.19/0.45 ( ( X != Y4 )
% 0.19/0.45 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 0.19/0.45 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % linorder_neqE_nat
% 0.19/0.45 thf(fact_75_infinite__descent,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ! [N2: nat] :
% 0.19/0.45 ( ~ ( P @ N2 )
% 0.19/0.45 => ? [M4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M4 @ N2 )
% 0.19/0.45 & ~ ( P @ M4 ) ) )
% 0.19/0.45 => ( P @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % infinite_descent
% 0.19/0.45 thf(fact_76_nat__less__induct,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ! [N2: nat] :
% 0.19/0.45 ( ! [M4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M4 @ N2 )
% 0.19/0.45 => ( P @ M4 ) )
% 0.19/0.45 => ( P @ N2 ) )
% 0.19/0.45 => ( P @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_less_induct
% 0.19/0.45 thf(fact_77_less__irrefl__nat,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.45
% 0.19/0.45 % less_irrefl_nat
% 0.19/0.45 thf(fact_78_less__not__refl3,axiom,
% 0.19/0.45 ! [S: nat,T: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ S @ T )
% 0.19/0.45 => ( S != T ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl3
% 0.19/0.45 thf(fact_79_less__not__refl2,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ N @ M3 )
% 0.19/0.45 => ( M3 != N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl2
% 0.19/0.45 thf(fact_80_less__not__refl,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ N ) ).
% 0.19/0.45
% 0.19/0.45 % less_not_refl
% 0.19/0.45 thf(fact_81_nat__neq__iff,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( M3 != N )
% 0.19/0.45 = ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 | ( ord_less @ nat @ N @ M3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_neq_iff
% 0.19/0.45 thf(fact_82_diff__commute,axiom,
% 0.19/0.45 ! [I: nat,J3: nat,K2: nat] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J3 ) @ K2 )
% 0.19/0.45 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K2 ) @ J3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_commute
% 0.19/0.45 thf(fact_83_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 0.19/0.45 ! [A: $tType,C: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( ideal @ B @ D @ R2 @ A3 )
% 0.19/0.45 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.smodule_ideal_coeff_is_Submodule
% 0.19/0.45 thf(fact_84_Module_Oquotient__of__submodules__is__ideal,axiom,
% 0.19/0.45 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),P: set @ A,Q: set @ A] :
% 0.19/0.45 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.45 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ P )
% 0.19/0.45 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ Q )
% 0.19/0.45 => ( ideal @ B @ D @ R2 @ ( quotie135653060odules @ B @ D @ A @ C @ R2 @ M5 @ P @ Q ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Module.quotient_of_submodules_is_ideal
% 0.19/0.45 thf(fact_85_transitive__stepwise__le,axiom,
% 0.19/0.45 ! [M3: nat,N: nat,R2: nat > nat > $o] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ! [X2: nat] : ( R2 @ X2 @ X2 )
% 0.19/0.45 => ( ! [X2: nat,Y3: nat,Z2: nat] :
% 0.19/0.45 ( ( R2 @ X2 @ Y3 )
% 0.19/0.45 => ( ( R2 @ Y3 @ Z2 )
% 0.19/0.45 => ( R2 @ X2 @ Z2 ) ) )
% 0.19/0.45 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 0.19/0.45 => ( R2 @ M3 @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % transitive_stepwise_le
% 0.19/0.45 thf(fact_86_nat__induct__at__least,axiom,
% 0.19/0.45 ! [M3: nat,N: nat,P: nat > $o] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ( P @ M3 )
% 0.19/0.45 => ( ! [N2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N2 )
% 0.19/0.45 => ( ( P @ N2 )
% 0.19/0.45 => ( P @ ( suc @ N2 ) ) ) )
% 0.19/0.45 => ( P @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_induct_at_least
% 0.19/0.45 thf(fact_87_full__nat__induct,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ! [N2: nat] :
% 0.19/0.45 ( ! [M4: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N2 )
% 0.19/0.45 => ( P @ M4 ) )
% 0.19/0.45 => ( P @ N2 ) )
% 0.19/0.45 => ( P @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % full_nat_induct
% 0.19/0.45 thf(fact_88_not__less__eq__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ~ ( ord_less_eq @ nat @ M3 @ N ) )
% 0.19/0.45 = ( ord_less_eq @ nat @ ( suc @ N ) @ M3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_less_eq_eq
% 0.19/0.45 thf(fact_89_Suc__n__not__le__n,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_n_not_le_n
% 0.19/0.45 thf(fact_90_le__Suc__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.45 = ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 | ( M3
% 0.19/0.45 = ( suc @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_Suc_eq
% 0.19/0.45 thf(fact_91_Suc__le__D,axiom,
% 0.19/0.45 ! [N: nat,M6: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
% 0.19/0.45 => ? [M7: nat] :
% 0.19/0.45 ( M6
% 0.19/0.45 = ( suc @ M7 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_le_D
% 0.19/0.45 thf(fact_92_le__SucI,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M3 @ ( suc @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_SucI
% 0.19/0.45 thf(fact_93_le__SucE,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.45 => ( ~ ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( M3
% 0.19/0.45 = ( suc @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_SucE
% 0.19/0.45 thf(fact_94_Suc__leD,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_leD
% 0.19/0.45 thf(fact_95_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.19/0.45 ! [A2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( A2
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bot_nat_0.extremum_uniqueI
% 0.19/0.45 thf(fact_96_bot__nat__0_Oextremum__unique,axiom,
% 0.19/0.45 ! [A2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( A2
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bot_nat_0.extremum_unique
% 0.19/0.45 thf(fact_97_le__0__eq,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_0_eq
% 0.19/0.45 thf(fact_98_less__eq__nat_Osimps_I1_J,axiom,
% 0.19/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.19/0.45
% 0.19/0.45 % less_eq_nat.simps(1)
% 0.19/0.45 thf(fact_99_not0__implies__Suc,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( N
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ? [M7: nat] :
% 0.19/0.45 ( N
% 0.19/0.45 = ( suc @ M7 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not0_implies_Suc
% 0.19/0.45 thf(fact_100_old_Onat_Oinducts,axiom,
% 0.19/0.45 ! [P: nat > $o,Nat: nat] :
% 0.19/0.45 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( ! [Nat3: nat] :
% 0.19/0.45 ( ( P @ Nat3 )
% 0.19/0.45 => ( P @ ( suc @ Nat3 ) ) )
% 0.19/0.45 => ( P @ Nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.inducts
% 0.19/0.45 thf(fact_101_old_Onat_Oexhaust,axiom,
% 0.19/0.45 ! [Y4: nat] :
% 0.19/0.45 ( ( Y4
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ~ ! [Nat3: nat] :
% 0.19/0.45 ( Y4
% 0.19/0.45 != ( suc @ Nat3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.exhaust
% 0.19/0.45 thf(fact_102_Zero__not__Suc,axiom,
% 0.19/0.45 ! [M3: nat] :
% 0.19/0.45 ( ( zero_zero @ nat )
% 0.19/0.45 != ( suc @ M3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Zero_not_Suc
% 0.19/0.45 thf(fact_103_Zero__neq__Suc,axiom,
% 0.19/0.45 ! [M3: nat] :
% 0.19/0.45 ( ( zero_zero @ nat )
% 0.19/0.45 != ( suc @ M3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % Zero_neq_Suc
% 0.19/0.45 thf(fact_104_Suc__neq__Zero,axiom,
% 0.19/0.45 ! [M3: nat] :
% 0.19/0.45 ( ( suc @ M3 )
% 0.19/0.45 != ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_neq_Zero
% 0.19/0.45 thf(fact_105_zero__induct,axiom,
% 0.19/0.45 ! [P: nat > $o,K2: nat] :
% 0.19/0.45 ( ( P @ K2 )
% 0.19/0.45 => ( ! [N2: nat] :
% 0.19/0.45 ( ( P @ ( suc @ N2 ) )
% 0.19/0.45 => ( P @ N2 ) )
% 0.19/0.45 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % zero_induct
% 0.19/0.45 thf(fact_106_diff__induct,axiom,
% 0.19/0.45 ! [P: nat > nat > $o,M3: nat,N: nat] :
% 0.19/0.45 ( ! [X2: nat] : ( P @ X2 @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 0.19/0.45 => ( ! [X2: nat,Y3: nat] :
% 0.19/0.45 ( ( P @ X2 @ Y3 )
% 0.19/0.45 => ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
% 0.19/0.45 => ( P @ M3 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_induct
% 0.19/0.45 thf(fact_107_nat__induct,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( ! [N2: nat] :
% 0.19/0.45 ( ( P @ N2 )
% 0.19/0.45 => ( P @ ( suc @ N2 ) ) )
% 0.19/0.45 => ( P @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_induct
% 0.19/0.45 thf(fact_108_nat_OdiscI,axiom,
% 0.19/0.45 ! [Nat: nat,X22: nat] :
% 0.19/0.45 ( ( Nat
% 0.19/0.45 = ( suc @ X22 ) )
% 0.19/0.45 => ( Nat
% 0.19/0.45 != ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat.discI
% 0.19/0.45 thf(fact_109_old_Onat_Odistinct_I1_J,axiom,
% 0.19/0.45 ! [Nat2: nat] :
% 0.19/0.45 ( ( zero_zero @ nat )
% 0.19/0.45 != ( suc @ Nat2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.distinct(1)
% 0.19/0.45 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 0.19/0.45 ! [Nat4: nat] :
% 0.19/0.45 ( ( suc @ Nat4 )
% 0.19/0.45 != ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % old.nat.distinct(2)
% 0.19/0.45 thf(fact_111_nat_Odistinct_I1_J,axiom,
% 0.19/0.45 ! [X22: nat] :
% 0.19/0.45 ( ( zero_zero @ nat )
% 0.19/0.45 != ( suc @ X22 ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat.distinct(1)
% 0.19/0.45 thf(fact_112_less__mono__imp__le__mono,axiom,
% 0.19/0.45 ! [F: nat > nat,I: nat,J3: nat] :
% 0.19/0.45 ( ! [I2: nat,J4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ J4 )
% 0.19/0.45 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ I @ J3 )
% 0.19/0.45 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_mono_imp_le_mono
% 0.19/0.45 thf(fact_113_le__neq__implies__less,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.45 => ( ( M3 != N )
% 0.19/0.45 => ( ord_less @ nat @ M3 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_neq_implies_less
% 0.19/0.45 thf(fact_114_less__or__eq__imp__le,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 | ( M3 = N ) )
% 0.19/0.45 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_or_eq_imp_le
% 0.19/0.45 thf(fact_115_le__eq__less__or__eq,axiom,
% 0.19/0.45 ( ( ord_less_eq @ nat )
% 0.19/0.45 = ( ^ [M8: nat,N4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M8 @ N4 )
% 0.19/0.45 | ( M8 = N4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % le_eq_less_or_eq
% 0.19/0.45 thf(fact_116_less__imp__le__nat,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_imp_le_nat
% 0.19/0.45 thf(fact_117_nat__less__le,axiom,
% 0.19/0.45 ( ( ord_less @ nat )
% 0.19/0.45 = ( ^ [M8: nat,N4: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M8 @ N4 )
% 0.19/0.45 & ( M8 != N4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_less_le
% 0.19/0.45 thf(fact_118_not__less__less__Suc__eq,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ~ ( ord_less @ nat @ N @ M3 )
% 0.19/0.45 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 0.19/0.45 = ( N = M3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_less_less_Suc_eq
% 0.19/0.45 thf(fact_119_strict__inc__induct,axiom,
% 0.19/0.45 ! [I: nat,J3: nat,P: nat > $o] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J3 )
% 0.19/0.45 => ( ! [I2: nat] :
% 0.19/0.45 ( ( J3
% 0.19/0.45 = ( suc @ I2 ) )
% 0.19/0.45 => ( P @ I2 ) )
% 0.19/0.45 => ( ! [I2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ J3 )
% 0.19/0.45 => ( ( P @ ( suc @ I2 ) )
% 0.19/0.45 => ( P @ I2 ) ) )
% 0.19/0.45 => ( P @ I ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % strict_inc_induct
% 0.19/0.45 thf(fact_120_less__Suc__induct,axiom,
% 0.19/0.45 ! [I: nat,J3: nat,P: nat > nat > $o] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J3 )
% 0.19/0.45 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.19/0.45 => ( ! [I2: nat,J4: nat,K3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I2 @ J4 )
% 0.19/0.45 => ( ( ord_less @ nat @ J4 @ K3 )
% 0.19/0.45 => ( ( P @ I2 @ J4 )
% 0.19/0.45 => ( ( P @ J4 @ K3 )
% 0.19/0.45 => ( P @ I2 @ K3 ) ) ) ) )
% 0.19/0.45 => ( P @ I @ J3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_Suc_induct
% 0.19/0.45 thf(fact_121_less__trans__Suc,axiom,
% 0.19/0.45 ! [I: nat,J3: nat,K2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J3 )
% 0.19/0.45 => ( ( ord_less @ nat @ J3 @ K2 )
% 0.19/0.45 => ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_trans_Suc
% 0.19/0.45 thf(fact_122_Suc__less__SucD,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 0.19/0.45 => ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_less_SucD
% 0.19/0.45 thf(fact_123_less__antisym,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ~ ( ord_less @ nat @ N @ M3 )
% 0.19/0.45 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 0.19/0.45 => ( M3 = N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_antisym
% 0.19/0.45 thf(fact_124_Suc__less__eq2,axiom,
% 0.19/0.45 ! [N: nat,M3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( suc @ N ) @ M3 )
% 0.19/0.45 = ( ? [M9: nat] :
% 0.19/0.45 ( ( M3
% 0.19/0.45 = ( suc @ M9 ) )
% 0.19/0.45 & ( ord_less @ nat @ N @ M9 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_less_eq2
% 0.19/0.45 thf(fact_125_All__less__Suc,axiom,
% 0.19/0.45 ! [N: nat,P: nat > $o] :
% 0.19/0.45 ( ( ! [I3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.19/0.45 => ( P @ I3 ) ) )
% 0.19/0.45 = ( ( P @ N )
% 0.19/0.45 & ! [I3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I3 @ N )
% 0.19/0.45 => ( P @ I3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % All_less_Suc
% 0.19/0.45 thf(fact_126_not__less__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ~ ( ord_less @ nat @ M3 @ N ) )
% 0.19/0.45 = ( ord_less @ nat @ N @ ( suc @ M3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_less_eq
% 0.19/0.45 thf(fact_127_less__Suc__eq,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.45 = ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 | ( M3 = N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_Suc_eq
% 0.19/0.45 thf(fact_128_Ex__less__Suc,axiom,
% 0.19/0.45 ! [N: nat,P: nat > $o] :
% 0.19/0.45 ( ( ? [I3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.19/0.45 & ( P @ I3 ) ) )
% 0.19/0.45 = ( ( P @ N )
% 0.19/0.45 | ? [I3: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I3 @ N )
% 0.19/0.45 & ( P @ I3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Ex_less_Suc
% 0.19/0.45 thf(fact_129_less__SucI,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_SucI
% 0.19/0.45 thf(fact_130_less__SucE,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.45 => ( ~ ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( M3 = N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_SucE
% 0.19/0.45 thf(fact_131_Suc__lessI,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( ( ( suc @ M3 )
% 0.19/0.45 != N )
% 0.19/0.45 => ( ord_less @ nat @ ( suc @ M3 ) @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_lessI
% 0.19/0.45 thf(fact_132_Suc__lessE,axiom,
% 0.19/0.45 ! [I: nat,K2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( suc @ I ) @ K2 )
% 0.19/0.45 => ~ ! [J4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J4 )
% 0.19/0.45 => ( K2
% 0.19/0.45 != ( suc @ J4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_lessE
% 0.19/0.45 thf(fact_133_Suc__lessD,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( suc @ M3 ) @ N )
% 0.19/0.45 => ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_lessD
% 0.19/0.45 thf(fact_134_Nat_OlessE,axiom,
% 0.19/0.45 ! [I: nat,K2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ K2 )
% 0.19/0.45 => ( ( K2
% 0.19/0.45 != ( suc @ I ) )
% 0.19/0.45 => ~ ! [J4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ I @ J4 )
% 0.19/0.45 => ( K2
% 0.19/0.45 != ( suc @ J4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Nat.lessE
% 0.19/0.45 thf(fact_135_infinite__descent0__measure,axiom,
% 0.19/0.45 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.19/0.45 ( ! [X2: A] :
% 0.19/0.45 ( ( ( V @ X2 )
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 => ( P @ X2 ) )
% 0.19/0.45 => ( ! [X2: A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.19/0.45 => ( ~ ( P @ X2 )
% 0.19/0.45 => ? [Y: A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.19/0.45 & ~ ( P @ Y ) ) ) )
% 0.19/0.45 => ( P @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % infinite_descent0_measure
% 0.19/0.45 thf(fact_136_bot__nat__0_Oextremum__strict,axiom,
% 0.19/0.45 ! [A2: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % bot_nat_0.extremum_strict
% 0.19/0.45 thf(fact_137_infinite__descent0,axiom,
% 0.19/0.45 ! [P: nat > $o,N: nat] :
% 0.19/0.45 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.45 => ( ! [N2: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.19/0.45 => ( ~ ( P @ N2 )
% 0.19/0.45 => ? [M4: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M4 @ N2 )
% 0.19/0.45 & ~ ( P @ M4 ) ) ) )
% 0.19/0.45 => ( P @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % infinite_descent0
% 0.19/0.45 thf(fact_138_gr__implies__not0,axiom,
% 0.19/0.45 ! [M3: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.45 => ( N
% 0.19/0.45 != ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gr_implies_not0
% 0.19/0.45 thf(fact_139_less__zeroE,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_zeroE
% 0.19/0.45 thf(fact_140_not__less0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_less0
% 0.19/0.45 thf(fact_141_not__gr0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_gr0
% 0.19/0.46 thf(fact_142_gr0I,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( N
% 0.19/0.46 != ( zero_zero @ nat ) )
% 0.19/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr0I
% 0.19/0.46 thf(fact_143_diff__le__mono2,axiom,
% 0.19/0.46 ! [M3: nat,N: nat,L2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono2
% 0.19/0.46 thf(fact_144_le__diff__iff_H,axiom,
% 0.19/0.46 ! [A2: nat,C2: nat,B2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ A2 @ C2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff'
% 0.19/0.46 thf(fact_145_diff__le__self,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_self
% 0.19/0.46 thf(fact_146_diff__le__mono,axiom,
% 0.19/0.46 ! [M3: nat,N: nat,L2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ L2 ) @ ( minus_minus @ nat @ N @ L2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono
% 0.19/0.46 thf(fact_147_Nat_Odiff__diff__eq,axiom,
% 0.19/0.46 ! [K2: nat,M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 0.19/0.46 = ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_eq
% 0.19/0.46 thf(fact_148_le__diff__iff,axiom,
% 0.19/0.46 ! [K2: nat,M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M3 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff
% 0.19/0.46 thf(fact_149_eq__diff__iff,axiom,
% 0.19/0.46 ! [K2: nat,M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ( ( minus_minus @ nat @ M3 @ K2 )
% 0.19/0.46 = ( minus_minus @ nat @ N @ K2 ) )
% 0.19/0.46 = ( M3 = N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_diff_iff
% 0.19/0.46 thf(fact_150_zero__induct__lemma,axiom,
% 0.19/0.46 ! [P: nat > $o,K2: nat,I: nat] :
% 0.19/0.46 ( ( P @ K2 )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( P @ ( suc @ N2 ) )
% 0.19/0.46 => ( P @ N2 ) )
% 0.19/0.46 => ( P @ ( minus_minus @ nat @ K2 @ I ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_induct_lemma
% 0.19/0.46 thf(fact_151_minus__nat_Odiff__0,axiom,
% 0.19/0.46 ! [M3: nat] :
% 0.19/0.46 ( ( minus_minus @ nat @ M3 @ ( zero_zero @ nat ) )
% 0.19/0.46 = M3 ) ).
% 0.19/0.46
% 0.19/0.46 % minus_nat.diff_0
% 0.19/0.46 thf(fact_152_diffs0__imp__equal,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ( minus_minus @ nat @ M3 @ N )
% 0.19/0.46 = ( zero_zero @ nat ) )
% 0.19/0.46 => ( ( ( minus_minus @ nat @ N @ M3 )
% 0.19/0.46 = ( zero_zero @ nat ) )
% 0.19/0.46 => ( M3 = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diffs0_imp_equal
% 0.19/0.46 thf(fact_153_less__imp__diff__less,axiom,
% 0.19/0.46 ! [J3: nat,K2: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ J3 @ K2 )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ N ) @ K2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_diff_less
% 0.19/0.46 thf(fact_154_diff__less__mono2,axiom,
% 0.19/0.46 ! [M3: nat,N: nat,L2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ M3 @ L2 )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono2
% 0.19/0.46 thf(fact_155_Module_Osc__Ring,axiom,
% 0.19/0.46 ! [C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( ring @ B @ D @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.sc_Ring
% 0.19/0.46 thf(fact_156_Module_Omem__smodule__ideal__coeff,axiom,
% 0.19/0.46 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,X: A] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( ( ideal @ B @ D @ R2 @ A3 )
% 0.19/0.46 => ( ( member @ A @ X @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) )
% 0.19/0.46 => ? [N2: nat,X2: nat > B] :
% 0.19/0.46 ( ( member @ ( nat > B ) @ X2
% 0.19/0.46 @ ( pi @ nat @ B
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 & ? [Xa: nat > A] :
% 0.19/0.46 ( ( member @ ( nat > A ) @ Xa
% 0.19/0.46 @ ( pi @ nat @ A
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) )
% 0.19/0.46 & ( X
% 0.19/0.46 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.mem_smodule_ideal_coeff
% 0.19/0.46 thf(fact_157_lift__Suc__antimono__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N5: nat] :
% 0.19/0.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N @ N5 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ N5 ) @ ( F @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_antimono_le
% 0.19/0.46 thf(fact_158_lift__Suc__mono__le,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N5: nat] :
% 0.19/0.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ N @ N5 )
% 0.19/0.46 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_le
% 0.19/0.46 thf(fact_159_lift__Suc__mono__less__iff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,M3: nat] :
% 0.19/0.46 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.19/0.46 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M3 ) )
% 0.19/0.46 = ( ord_less @ nat @ N @ M3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_less_iff
% 0.19/0.46 thf(fact_160_lift__Suc__mono__less,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( order @ A )
% 0.19/0.46 => ! [F: nat > A,N: nat,N5: nat] :
% 0.19/0.46 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ N5 )
% 0.19/0.46 => ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lift_Suc_mono_less
% 0.19/0.46 thf(fact_161_le__imp__less__Suc,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.46 => ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_less_Suc
% 0.19/0.46 thf(fact_162_less__eq__Suc__le,axiom,
% 0.19/0.46 ( ( ord_less @ nat )
% 0.19/0.46 = ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_Suc_le
% 0.19/0.46 thf(fact_163_less__Suc__eq__le,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.46 = ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_Suc_eq_le
% 0.19/0.46 thf(fact_164_le__less__Suc__eq,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ M3 @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 0.19/0.46 = ( N = M3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_less_Suc_eq
% 0.19/0.46 thf(fact_165_Suc__le__lessD,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 0.19/0.46 => ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_lessD
% 0.19/0.46 thf(fact_166_inc__induct,axiom,
% 0.19/0.46 ! [I: nat,J3: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J3 )
% 0.19/0.46 => ( ( P @ J3 )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 0.19/0.46 => ( ( ord_less @ nat @ N2 @ J3 )
% 0.19/0.46 => ( ( P @ ( suc @ N2 ) )
% 0.19/0.46 => ( P @ N2 ) ) ) )
% 0.19/0.46 => ( P @ I ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % inc_induct
% 0.19/0.46 thf(fact_167_dec__induct,axiom,
% 0.19/0.46 ! [I: nat,J3: nat,P: nat > $o] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ J3 )
% 0.19/0.46 => ( ( P @ I )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 0.19/0.46 => ( ( ord_less @ nat @ N2 @ J3 )
% 0.19/0.46 => ( ( P @ N2 )
% 0.19/0.46 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.19/0.46 => ( P @ J3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dec_induct
% 0.19/0.46 thf(fact_168_Suc__le__eq,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 0.19/0.46 = ( ord_less @ nat @ M3 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_le_eq
% 0.19/0.46 thf(fact_169_Suc__leI,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ N )
% 0.19/0.46 => ( ord_less_eq @ nat @ ( suc @ M3 ) @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_leI
% 0.19/0.46 thf(fact_170_ex__least__nat__le,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ( P @ N )
% 0.19/0.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.19/0.46 => ? [K3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K3 @ N )
% 0.19/0.46 & ! [I4: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I4 @ K3 )
% 0.19/0.46 => ~ ( P @ I4 ) )
% 0.19/0.46 & ( P @ K3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_least_nat_le
% 0.19/0.46 thf(fact_171_less__Suc__eq__0__disj,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 0.19/0.46 = ( ( M3
% 0.19/0.46 = ( zero_zero @ nat ) )
% 0.19/0.46 | ? [J2: nat] :
% 0.19/0.46 ( ( M3
% 0.19/0.46 = ( suc @ J2 ) )
% 0.19/0.46 & ( ord_less @ nat @ J2 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_Suc_eq_0_disj
% 0.19/0.46 thf(fact_172_gr0__implies__Suc,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.46 => ? [M7: nat] :
% 0.19/0.46 ( N
% 0.19/0.46 = ( suc @ M7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr0_implies_Suc
% 0.19/0.46 thf(fact_173_All__less__Suc2,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o] :
% 0.19/0.46 ( ( ! [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.19/0.46 => ( P @ I3 ) ) )
% 0.19/0.46 = ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ N )
% 0.19/0.46 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % All_less_Suc2
% 0.19/0.46 thf(fact_174_gr0__conv__Suc,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.46 = ( ? [M8: nat] :
% 0.19/0.46 ( N
% 0.19/0.46 = ( suc @ M8 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr0_conv_Suc
% 0.19/0.46 thf(fact_175_Ex__less__Suc2,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o] :
% 0.19/0.46 ( ( ? [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.19/0.46 & ( P @ I3 ) ) )
% 0.19/0.46 = ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.46 | ? [I3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ I3 @ N )
% 0.19/0.46 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Ex_less_Suc2
% 0.19/0.46 thf(fact_176_Suc__diff__le,axiom,
% 0.19/0.46 ! [N: nat,M3: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ N @ M3 )
% 0.19/0.46 => ( ( minus_minus @ nat @ ( suc @ M3 ) @ N )
% 0.19/0.46 = ( suc @ ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_diff_le
% 0.19/0.46 thf(fact_177_diff__less__mono,axiom,
% 0.19/0.46 ! [A2: nat,B2: nat,C2: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ A2 @ B2 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ C2 @ A2 )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono
% 0.19/0.46 thf(fact_178_less__diff__iff,axiom,
% 0.19/0.46 ! [K2: nat,M3: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 0.19/0.46 => ( ( ord_less_eq @ nat @ K2 @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 0.19/0.46 = ( ord_less @ nat @ M3 @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_iff
% 0.19/0.46 thf(fact_179_diff__less__Suc,axiom,
% 0.19/0.46 ! [M3: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_Suc
% 0.19/0.46 thf(fact_180_Suc__diff__Suc,axiom,
% 0.19/0.46 ! [N: nat,M3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ N @ M3 )
% 0.19/0.46 => ( ( suc @ ( minus_minus @ nat @ M3 @ ( suc @ N ) ) )
% 0.19/0.46 = ( minus_minus @ nat @ M3 @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Suc_diff_Suc
% 0.19/0.46 thf(fact_181_diff__less,axiom,
% 0.19/0.46 ! [N: nat,M3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less
% 0.19/0.46 thf(fact_182_Module_Osubmodule__subset1,axiom,
% 0.19/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,H2: A] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.46 => ( ( member @ A @ H2 @ H )
% 0.19/0.46 => ( member @ A @ H2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.submodule_subset1
% 0.19/0.46 thf(fact_183_Module_Osubmodule__whole,axiom,
% 0.19/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.submodule_whole
% 0.19/0.46 thf(fact_184_ex__least__nat__less,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ( P @ N )
% 0.19/0.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.19/0.46 => ? [K3: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ K3 @ N )
% 0.19/0.46 & ! [I4: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ I4 @ K3 )
% 0.19/0.46 => ~ ( P @ I4 ) )
% 0.19/0.46 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_least_nat_less
% 0.19/0.46 thf(fact_185_diff__Suc__less,axiom,
% 0.19/0.46 ! [N: nat,I: nat] :
% 0.19/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_Suc_less
% 0.19/0.46 thf(fact_186_Module_Osubmodule__subset,axiom,
% 0.19/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.46 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.submodule_subset
% 0.19/0.46 thf(fact_187_Module_Ofree__generator__sub,axiom,
% 0.19/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 0.19/0.46 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 0.19/0.46 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 0.19/0.46 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Module.free_generator_sub
% 0.19/0.46 thf(fact_188_eSum__in__SubmoduleTr,axiom,
% 0.19/0.46 ! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 0.19/0.46 => ( ( ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : K ) )
% 0.19/0.46 & ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
% 0.19/0.46 => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
% 0.19/0.46 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eSum_in_SubmoduleTr
% 0.19/0.46 thf(fact_189_eSum__in__Submodule,axiom,
% 0.19/0.46 ! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 0.19/0.46 => ( ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : K ) )
% 0.19/0.46 => ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 0.19/0.46 => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
% 0.19/0.46 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eSum_in_Submodule
% 0.19/0.46 thf(fact_190_singleton__insert__inj__eq_H,axiom,
% 0.19/0.46 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 0.19/0.46 ( ( ( insert @ A @ A2 @ A3 )
% 0.19/0.46 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.46 = ( ( A2 = B2 )
% 0.19/0.46 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_insert_inj_eq'
% 0.19/0.46 thf(fact_191_singleton__insert__inj__eq,axiom,
% 0.19/0.46 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 0.19/0.46 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.46 = ( insert @ A @ A2 @ A3 ) )
% 0.19/0.46 = ( ( A2 = B2 )
% 0.19/0.46 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_insert_inj_eq
% 0.19/0.46 thf(fact_192_diff__gt__0__iff__gt,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A2: A,B2: A] :
% 0.19/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.19/0.46 = ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_gt_0_iff_gt
% 0.19/0.46 thf(fact_193_diff__ge__0__iff__ge,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( ordered_ab_group_add @ A )
% 0.19/0.46 => ! [A2: A,B2: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 0.19/0.46 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_ge_0_iff_ge
% 0.19/0.46 thf(fact_194_l__span__closed2Tr,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 & ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) )
% 0.19/0.46 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_closed2Tr
% 0.19/0.46 thf(fact_195_l__span__closed2,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 => ( ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) )
% 0.19/0.46 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_closed2
% 0.19/0.46 thf(fact_196_linear__comb0__2Tr,axiom,
% 0.19/0.46 ! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 & ( member @ ( nat > a ) @ M3
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
% 0.19/0.46 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
% 0.19/0.46 = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear_comb0_2Tr
% 0.19/0.46 thf(fact_197_module__inc__zero,axiom,
% 0.19/0.46 member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.46
% 0.19/0.46 % module_inc_zero
% 0.19/0.46 thf(fact_198_psubsetI,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.46 => ( ( A3 != B3 )
% 0.19/0.46 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % psubsetI
% 0.19/0.46 thf(fact_199_subsetI,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ! [X2: A] :
% 0.19/0.46 ( ( member @ A @ X2 @ A3 )
% 0.19/0.46 => ( member @ A @ X2 @ B3 ) )
% 0.19/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetI
% 0.19/0.46 thf(fact_200_subset__antisym,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.19/0.46 => ( A3 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_antisym
% 0.19/0.46 thf(fact_201_Diff__empty,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] :
% 0.19/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_empty
% 0.19/0.46 thf(fact_202_empty__Diff,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] :
% 0.19/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Diff
% 0.19/0.46 thf(fact_203_Diff__cancel,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] :
% 0.19/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_cancel
% 0.19/0.46 thf(fact_204_empty__iff,axiom,
% 0.19/0.46 ! [A: $tType,C2: A] :
% 0.19/0.46 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_iff
% 0.19/0.46 thf(fact_205_all__not__in__conv,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] :
% 0.19/0.46 ( ( ! [X3: A] :
% 0.19/0.46 ~ ( member @ A @ X3 @ A3 ) )
% 0.19/0.46 = ( A3
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % all_not_in_conv
% 0.19/0.46 thf(fact_206_Collect__empty__eq,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o] :
% 0.19/0.46 ( ( ( collect @ A @ P )
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.46 = ( ! [X3: A] :
% 0.19/0.46 ~ ( P @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_empty_eq
% 0.19/0.46 thf(fact_207_empty__Collect__eq,axiom,
% 0.19/0.46 ! [A: $tType,P: A > $o] :
% 0.19/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.19/0.46 = ( collect @ A @ P ) )
% 0.19/0.46 = ( ! [X3: A] :
% 0.19/0.46 ~ ( P @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Collect_eq
% 0.19/0.46 thf(fact_208_Diff__insert0,axiom,
% 0.19/0.46 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ~ ( member @ A @ X @ A3 )
% 0.19/0.46 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
% 0.19/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_insert0
% 0.19/0.46 thf(fact_209_insert__Diff1,axiom,
% 0.19/0.46 ! [A: $tType,X: A,B3: set @ A,A3: set @ A] :
% 0.19/0.46 ( ( member @ A @ X @ B3 )
% 0.19/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
% 0.19/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_Diff1
% 0.19/0.46 thf(fact_210_insertCI,axiom,
% 0.19/0.46 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 0.19/0.46 ( ( ~ ( member @ A @ A2 @ B3 )
% 0.19/0.46 => ( A2 = B2 ) )
% 0.19/0.46 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insertCI
% 0.19/0.46 thf(fact_211_insert__iff,axiom,
% 0.19/0.46 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 0.19/0.46 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 0.19/0.46 = ( ( A2 = B2 )
% 0.19/0.46 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_iff
% 0.19/0.46 thf(fact_212_insert__absorb2,axiom,
% 0.19/0.46 ! [A: $tType,X: A,A3: set @ A] :
% 0.19/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
% 0.19/0.46 = ( insert @ A @ X @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_absorb2
% 0.19/0.46 thf(fact_213_submodule__inc__0,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ H ) ) ).
% 0.19/0.46
% 0.19/0.46 % submodule_inc_0
% 0.19/0.46 thf(fact_214_mdl__carrier,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ H ) )
% 0.19/0.46 = H ) ) ).
% 0.19/0.46
% 0.19/0.46 % mdl_carrier
% 0.19/0.46 thf(fact_215_mdl__is__module,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ H ) @ r ) ) ).
% 0.19/0.46
% 0.19/0.46 % mdl_is_module
% 0.19/0.46 thf(fact_216_submodule__of__mdl,axiom,
% 0.19/0.46 ! [H: set @ a,N3: set @ a] :
% 0.19/0.46 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
% 0.19/0.46 => ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ N3 ) @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % submodule_of_mdl
% 0.19/0.46 thf(fact_217_h__in__linear__span,axiom,
% 0.19/0.46 ! [H: set @ a,H2: a] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( member @ a @ H2 @ H )
% 0.19/0.46 => ( member @ a @ H2 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % h_in_linear_span
% 0.19/0.46 thf(fact_218_l__span__cont__H,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_cont_H
% 0.19/0.46 thf(fact_219_l__span__l__span,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
% 0.19/0.46 = ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_l_span
% 0.19/0.46 thf(fact_220_l__span__gen__mono,axiom,
% 0.19/0.46 ! [K: set @ a,H: set @ a,A3: set @ b] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ K ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_gen_mono
% 0.19/0.46 thf(fact_221_linear__span__sub,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear_span_sub
% 0.19/0.46 thf(fact_222_l__span__sub__submodule,axiom,
% 0.19/0.46 ! [A3: set @ b,N3: set @ a,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
% 0.19/0.46 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ N3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_sub_submodule
% 0.19/0.46 thf(fact_223_fgs__generator,axiom,
% 0.19/0.46 ! [H: set @ a] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ H ) ) @ H ) ) ).
% 0.19/0.46
% 0.19/0.46 % fgs_generator
% 0.19/0.46 thf(fact_224_submodule__0,axiom,
% 0.19/0.46 submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % submodule_0
% 0.19/0.46 thf(fact_225_empty__fgs,axiom,
% 0.19/0.46 ( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
% 0.19/0.46 = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_fgs
% 0.19/0.46 thf(fact_226_le__zero__eq,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( canoni770627133id_add @ A )
% 0.19/0.46 => ! [N: A] :
% 0.19/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.46 = ( N
% 0.19/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_zero_eq
% 0.19/0.46 thf(fact_227_not__gr__zero,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( canoni770627133id_add @ A )
% 0.19/0.46 => ! [N: A] :
% 0.19/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.19/0.46 = ( N
% 0.19/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_gr_zero
% 0.19/0.46 thf(fact_228_diff__self,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.19/0.46 = ( zero_zero @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_self
% 0.19/0.46 thf(fact_229_diff__0__right,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( group_add @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.19/0.46 = A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_0_right
% 0.19/0.46 thf(fact_230_zero__diff,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( comm_monoid_diff @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.19/0.46 = ( zero_zero @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_diff
% 0.19/0.46 thf(fact_231_diff__zero,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel1352612707id_add @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.19/0.46 = A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_zero
% 0.19/0.46 thf(fact_232_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.19/0.46 ! [A: $tType] :
% 0.19/0.46 ( ( cancel1352612707id_add @ A )
% 0.19/0.46 => ! [A2: A] :
% 0.19/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.19/0.46 = ( zero_zero @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.19/0.46 thf(fact_233_Diff__eq__empty__iff,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.46 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_eq_empty_iff
% 0.19/0.46 thf(fact_234_subset__empty,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.46 = ( A3
% 0.19/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_empty
% 0.19/0.46 thf(fact_235_empty__subsetI,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % empty_subsetI
% 0.19/0.46 thf(fact_236_insert__subset,axiom,
% 0.19/0.46 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
% 0.19/0.46 = ( ( member @ A @ X @ B3 )
% 0.19/0.46 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_subset
% 0.19/0.46 thf(fact_237_insert__Diff__single,axiom,
% 0.19/0.46 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.46 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.46 = ( insert @ A @ A2 @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_Diff_single
% 0.19/0.46 thf(fact_238_singletonI,axiom,
% 0.19/0.46 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % singletonI
% 0.19/0.46 thf(fact_239_linear__span__subModule,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear_span_subModule
% 0.19/0.46 thf(fact_240_l__spanA__l__span,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
% 0.19/0.46 = ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_spanA_l_span
% 0.19/0.46 thf(fact_241_lin__span__coeff__mono,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lin_span_coeff_mono
% 0.19/0.46 thf(fact_242_linear__span__inc__0,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear_span_inc_0
% 0.19/0.46 thf(fact_243_generator__generator,axiom,
% 0.19/0.46 ! [H: set @ a,H1: set @ a] :
% 0.19/0.46 ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H1 ) )
% 0.19/0.46 => ( generator @ b @ d @ a @ c @ r @ m @ H1 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % generator_generator
% 0.19/0.46 thf(fact_244_l__span__zero,axiom,
% 0.19/0.46 ! [A3: set @ b] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
% 0.19/0.46 = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_zero
% 0.19/0.46 thf(fact_245_l__span__closed3,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 0.19/0.46 => ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 )
% 0.19/0.46 = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H )
% 0.19/0.46 = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_closed3
% 0.19/0.46 thf(fact_246_singleton__conv,axiom,
% 0.19/0.46 ! [A: $tType,A2: A] :
% 0.19/0.46 ( ( collect @ A
% 0.19/0.46 @ ^ [X3: A] : ( X3 = A2 ) )
% 0.19/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv
% 0.19/0.46 thf(fact_247_singleton__conv2,axiom,
% 0.19/0.46 ! [A: $tType,A2: A] :
% 0.19/0.46 ( ( collect @ A
% 0.19/0.46 @ ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 )
% 0.19/0.46 @ A2 ) )
% 0.19/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv2
% 0.19/0.46 thf(fact_248_l__comb__mem__linear__span,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 => ( ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : H ) )
% 0.19/0.46 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_comb_mem_linear_span
% 0.19/0.46 thf(fact_249_l__span__closed,axiom,
% 0.19/0.46 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 0.19/0.46 => ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 => ( ( member @ ( nat > a ) @ F
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) )
% 0.19/0.46 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % l_span_closed
% 0.19/0.46 thf(fact_250_linear__comb0__2,axiom,
% 0.19/0.46 ! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
% 0.19/0.46 ( ( ideal @ b @ d @ r @ A3 )
% 0.19/0.46 => ( ( member @ ( nat > b ) @ S
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : A3 ) )
% 0.19/0.46 => ( ( member @ ( nat > a ) @ M3
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 0.19/0.46 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
% 0.19/0.46 = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear_comb0_2
% 0.19/0.46 thf(fact_251_double__diff,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.19/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.19/0.46 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 0.19/0.46 = A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % double_diff
% 0.19/0.46 thf(fact_252_Diff__subset,axiom,
% 0.19/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Diff_subset
% 0.19/0.46
% 0.19/0.46 % Type constructors (11)
% 0.19/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.46 ! [A4: $tType,A5: $tType] :
% 0.19/0.46 ( ( order @ A5 )
% 0.19/0.46 => ( order @ ( A4 > A5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.19/0.46 canoni770627133id_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.19/0.46 cancel1352612707id_add @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.19/0.46 comm_monoid_diff @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.19/0.46 wellorder @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.19/0.46 order @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.19/0.46 zero @ nat ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.19/0.46 ! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
% 0.19/0.46 order @ $o ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_4,axiom,
% 0.19/0.46 wellorder @ product_unit ).
% 0.19/0.46
% 0.19/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_5,axiom,
% 0.19/0.46 order @ product_unit ).
% 0.19/0.46
% 0.19/0.46 % Conjectures (22)
% 0.19/0.46 thf(conj_0,hypothesis,
% 0.19/0.46 ( member @ ( nat > a ) @ g
% 0.19/0.46 @ ( pi @ nat @ a
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( insert @ a @ h2 @ h1 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_1,hypothesis,
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ X4 @ m2 )
% 0.19/0.46 => ! [Y: nat] :
% 0.19/0.46 ( ( ord_less_eq @ nat @ Y @ m2 )
% 0.19/0.46 => ( ( ( g @ X4 )
% 0.19/0.46 = ( g @ Y ) )
% 0.19/0.46 => ( X4 = Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_2,hypothesis,
% 0.19/0.46 ( member @ ( nat > b ) @ t
% 0.19/0.46 @ ( pi @ nat @ b
% 0.19/0.46 @ ( collect @ nat
% 0.19/0.46 @ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
% 0.19/0.46 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_3,hypothesis,
% 0.19/0.46 ( ( g @ m2 )
% 0.19/0.46 = h2 ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_4,hypothesis,
% 0.19/0.46 module @ e @ b @ f @ d @ n @ r ).
% 0.19/0.46
% 0.19/0.46 thf(conj_5,hypothesis,
% 0.19/0.46 free_generator @ b @ d @ a @ c @ r @ m @ h ).
% 0.19/0.46
% 0.19/0.46 thf(conj_6,hypothesis,
% 0.19/0.46 ord_less_eq @ ( set @ a ) @ h1 @ h ).
% 0.19/0.46
% 0.19/0.46 thf(conj_7,hypothesis,
% 0.19/0.46 $true ).
% 0.19/0.46
% 0.19/0.46 thf(conj_8,hypothesis,
% 0.19/0.46 ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_9,hypothesis,
% 0.19/0.46 ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_10,hypothesis,
% 0.19/0.46 ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_11,hypothesis,
% 0.19/0.46 ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_12,hypothesis,
% 0.19/0.46 member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_13,hypothesis,
% 0.19/0.46 ( x
% 0.19/0.46 = ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_14,hypothesis,
% 0.19/0.46 member @ a @ h2 @ h ).
% 0.19/0.46
% 0.19/0.46 thf(conj_15,hypothesis,
% 0.19/0.46 ~ ( member @ a @ h2 @ h1 ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_16,hypothesis,
% 0.19/0.46 ord_less @ nat @ ( zero_zero @ nat ) @ m2 ).
% 0.19/0.46
% 0.19/0.46 thf(conj_17,hypothesis,
% 0.19/0.46 member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_18,hypothesis,
% 0.19/0.46 submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_19,hypothesis,
% 0.19/0.46 ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_20,hypothesis,
% 0.19/0.46 ( ( ( g @ xa )
% 0.19/0.46 = h2 )
% 0.19/0.46 | ( member @ a @ ( g @ xa ) @ h1 ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(conj_21,conjecture,
% 0.19/0.46 member @ a @ ( g @ xa ) @ h1 ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 2.87/1.58 ---- Embedded in THF ---
% 2.87/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.87/1.58 %%% Generated on Tue Feb 24 08:24:22 EST 2026
% 2.87/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.87/1.58 %%% Logic specification used:
% 2.87/1.58 %%% thf(spec, logic, $$dhol).
% 2.87/1.58
% 2.87/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2DTF27230.p
% 2.87/1.58 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext, type, carrier_ext: ($tType > ($tType > $tType))).
% 2.87/1.58 thf(ty_t_Algebra7_OModule_OModule__ext, type, module_ext: ($tType > ($tType > ($tType > $tType)))).
% 2.87/1.58 thf(ty_t_Algebra4_OaGroup_OaGroup__ext, type, aGroup_ext: ($tType > ($tType > $tType))).
% 2.87/1.58 thf(ty_t_Algebra4_ORing_ORing__ext, type, ring_ext: ($tType > ($tType > $tType))).
% 2.87/1.58 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.87/1.58 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.87/1.58 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.87/1.58 thf(ty_tf_f, type, f: $tType).
% 2.87/1.58 thf(ty_tf_e, type, e: $tType).
% 2.87/1.58 thf(ty_tf_d, type, d: $tType).
% 2.87/1.58 thf(ty_tf_c, type, c: $tType).
% 2.87/1.58 thf(ty_tf_b, type, b: $tType).
% 2.87/1.58 thf(ty_tf_a, type, a: $tType).
% 2.87/1.58 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.87/1.58 thf(sy_c_Algebra1_Ocarrier_Ocarrier, type, carrier: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ Z) > (set @ A))))).
% 2.87/1.58 thf(sy_c_Algebra4_ORing, type, ring: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > $o)))).
% 2.87/1.58 thf(sy_c_Algebra4_OaGroup_Ozero, type, zero2: (!> [A:$tType,Z:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ Z)) > A)))).
% 2.87/1.58 thf(sy_c_Algebra4_Oideal, type, ideal: (!> [A:$tType,B:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ ((ring_ext @ A) @ B))) > ((set @ A) > $o))))).
% 2.87/1.58 thf(sy_c_Algebra7_OAnnihilator, type, annihilator: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > (set @ R)))))).
% 2.87/1.58 thf(sy_c_Algebra7_OHOM, type, hom: (!> [B:$tType,More:$tType,A:$tType,More1:$tType,C:$tType,More2:$tType]: ((((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ More))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ More1))) > (((carrier_ext @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ B) @ More2))) > ((carrier_ext @ (A > C)) @ ((aGroup_ext @ (A > C)) @ (((module_ext @ (A > C)) @ B) @ product_unit))))))))).
% 2.87/1.58 thf(sy_c_Algebra7_OModule, type, module: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) > (((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) > $o))))).
% 2.87/1.58 thf(sy_c_Algebra7_Ofree__generator, type, free_generator: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ A) > $o)))))).
% 2.87/1.58 thf(sy_c_Algebra7_Ogenerator, type, generator: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ A) > $o)))))).
% 2.87/1.58 thf(sy_c_Algebra7_Ol__comb, type, l_comb: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > (nat > ((nat > R) > ((nat > A) > A)))))))).
% 2.87/1.58 thf(sy_c_Algebra7_Olinear__span, type, linear_span: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ R) > ((set @ A) > (set @ A)))))))).
% 2.87/1.58 thf(sy_c_Algebra7_Omdl, type, mdl: (!> [A:$tType,B:$tType,M:$tType]: ((((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ M))) > ((set @ A) > ((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ product_unit)))))))).
% 2.87/1.58 thf(sy_c_Algebra7_Omisomorphic, type, misomorphic: (!> [B:$tType,M:$tType,A:$tType,M1:$tType,C:$tType,M2:$tType]: ((((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ M1))) > (((carrier_ext @ C) @ ((aGroup_ext @ C) @ (((module_ext @ C) @ B) @ M2))) > $o)))))).
% 2.87/1.58 thf(sy_c_Algebra7_Oquotient__of__submodules, type, quotie135653060odules: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ A) > ((set @ A) > (set @ R)))))))).
% 2.87/1.58 thf(sy_c_Algebra7_Osmodule__ideal__coeff, type, smodule_ideal_coeff: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ R) > (set @ A))))))).
% 2.87/1.58 thf(sy_c_Algebra7_Osubmodule, type, submodule: (!> [B:$tType,M:$tType,A:$tType,C:$tType]: ((((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) > ((set @ A) > $o)))))).
% 2.87/1.58 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_Ofgs, type, algebr833503410le_fgs: (!> [R:$tType,M:$tType,A:$tType,M1:$tType]: ((((carrier_ext @ R) @ ((aGroup_ext @ R) @ ((ring_ext @ R) @ M))) > (((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ R) @ M1))) > ((set @ A) > (set @ A))))))).
% 2.87/1.58 thf(sy_c_FuncSet_OPi, type, pi: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ (A > B))))))).
% 2.87/1.58 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.87/1.58 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.87/1.58 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.87/1.58 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.87/1.58 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.87/1.58 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.87/1.58 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.87/1.58 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.87/1.58 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.87/1.58 thf(sy_v_H, type, h: (set @ a)).
% 2.87/1.58 thf(sy_v_H1, type, h1: (set @ a)).
% 2.87/1.58 thf(sy_v_M, type, m: ((carrier_ext @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c)))).
% 2.87/1.58 thf(sy_v_N, type, n: ((carrier_ext @ e) @ ((aGroup_ext @ e) @ (((module_ext @ e) @ b) @ f)))).
% 2.87/1.58 thf(sy_v_R, type, r: ((carrier_ext @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d)))).
% 2.87/1.58 thf(sy_v_g, type, g: (nat > a)).
% 2.87/1.58 thf(sy_v_h, type, h2: a).
% 2.87/1.58 thf(sy_v_m, type, m2: nat).
% 2.87/1.58 thf(sy_v_t, type, t: (nat > b)).
% 2.87/1.58 thf(sy_v_x, type, x: a).
% 2.87/1.58 thf(sy_v_xa, type, xa: nat).
% 2.87/1.58 thf(fact_0_sc__Ring, axiom, (((ring @ b) @ d) @ r)).
% 2.87/1.58 thf(fact_1_submodule__subset1, axiom, (! [H:(set @ a),H2:a]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((member @ a) @ H2) @ H) => (((member @ a) @ H2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))))))).
% 2.87/1.58 thf(fact_2_submodule__whole, axiom, (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(fact_3_submodule__subset, axiom, (! [H:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))).
% 2.87/1.58 thf(fact_4_free__generator__sub, axiom, (! [H:(set @ a)]: (((((((((free_generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))).
% 2.87/1.58 thf(fact_5_elem__fgs, axiom, (! [A2:(set @ a),X:a]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ a) @ X) @ A2) => (((member @ a) @ X) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ A2))))))).
% 2.87/1.58 thf(fact_6_fgs__sub__carrier, axiom, (! [A2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((ord_less_eq @ (set @ a)) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ A2)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))).
% 2.87/1.58 thf(fact_7_fgs__mono, axiom, (! [H:(set @ a),J:(set @ a),K:(set @ a)]: (((((((((free_generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((ord_less_eq @ (set @ a)) @ J) @ K) => ((((ord_less_eq @ (set @ a)) @ K) @ H) => (((ord_less_eq @ (set @ a)) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ J)) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ K)))))))).
% 2.87/1.58 thf(fact_8_fgs__submodule, axiom, (! [A2:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ A2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ A2)))))).
% 2.87/1.58 thf(fact_9_linear__comb__eq, axiom, (! [H:(set @ a),S:(nat > b),N:nat,F:(nat > a),G:(nat > a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r))))) => ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => ((((member @ (nat > a)) @ G) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => ((! [X2:nat]: (((((member @ nat) @ X2) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) => ((F @ X2) = (G @ X2))))) => ((((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F) = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ G)))))))))).
% 2.87/1.58 thf(fact_10_linear__comb__eqTr, axiom, (! [H:(set @ a),S:(nat > b),N:nat,F:(nat > a),G:(nat > a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r))))) & ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) & ((((member @ (nat > a)) @ G) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) & (! [X2:nat]: (((((member @ nat) @ X2) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) => ((F @ X2) = (G @ X2)))))))) => ((((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F) = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ G))))))).
% 2.87/1.58 thf(fact_11_l__comb__mem, axiom, (! [A3:(set @ b),H:(set @ a),S:(nat > b),N:nat,M3:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > a)) @ M3) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ M3)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))))))))).
% 2.87/1.58 thf(fact_12_liear__comb__memTr, axiom, (! [A3:(set @ b),H:(set @ a),N:nat]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (! [S2:(nat > b),M4:(nat > a)]: ((((((member @ (nat > b)) @ S2) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) & (((member @ (nat > a)) @ M4) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H))))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S2) @ M4)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))))))).
% 2.87/1.58 thf(fact_13_Ann__is__ideal, axiom, ((((ideal @ b) @ d) @ r) @ ((((((annihilator @ b) @ d) @ a) @ c) @ r) @ m))).
% 2.87/1.58 thf(fact_14_mem__fgs__l__comb, axiom, (! [K:(set @ a),X:a]: (((K != (bot_bot @ (set @ a))) => ((((ord_less_eq @ (set @ a)) @ K) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ a) @ X) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ K)) => (? [N2:nat,X2:(nat > a)]: (((((member @ (nat > a)) @ X2) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: (K)))) & (? [Xa:(nat > b)]: (((((member @ (nat > b)) @ Xa) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r))))) & (X = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N2) @ Xa) @ X2)))))))))))))).
% 2.87/1.58 thf(fact_15_Module__axioms, axiom, ((((((module @ a) @ b) @ c) @ d) @ m) @ r)).
% 2.87/1.58 thf(fact_16_quotient__of__submodules__is__ideal, axiom, (! [P:(set @ a),Q:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ P) => ((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ Q) => ((((ideal @ b) @ d) @ r) @ ((((((((quotie135653060odules @ b) @ d) @ a) @ c) @ r) @ m) @ P) @ Q))))))).
% 2.87/1.58 thf(fact_17_smodule__ideal__coeff__is__Submodule, axiom, (! [A3:(set @ b)]: ((((((ideal @ b) @ d) @ r) @ A3) => (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ (((((((smodule_ideal_coeff @ b) @ d) @ a) @ c) @ r) @ m) @ A3)))))).
% 2.87/1.58 thf(fact_18_mem__smodule__ideal__coeff, axiom, (! [A3:(set @ b),X:a]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((member @ a) @ X) @ (((((((smodule_ideal_coeff @ b) @ d) @ a) @ c) @ r) @ m) @ A3)) => (? [N2:nat,X2:(nat > b)]: (((((member @ (nat > b)) @ X2) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: (A3)))) & (? [Xa:(nat > a)]: (((((member @ (nat > a)) @ Xa) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: ((((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))))) & (X = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N2) @ X2) @ Xa))))))))))))).
% 2.87/1.58 thf(fact_19_Module_Oelem__fgs, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A2:(set @ A),X:A]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ A2)))))))).
% 2.87/1.58 thf(fact_20_Module_Ofgs__mono, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A),J:(set @ A),K:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((free_generator @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => ((((ord_less_eq @ (set @ A)) @ J) @ K) => ((((ord_less_eq @ (set @ A)) @ K) @ H) => (((ord_less_eq @ (set @ A)) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ J)) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ K))))))))).
% 2.87/1.58 thf(fact_21_Module_Ofgs__submodule, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A2:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => (((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ A2))))))).
% 2.87/1.58 thf(fact_22_Module_Omem__fgs__l__comb, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),K:(set @ A),X:A]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((K != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ K) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => ((((member @ A) @ X) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ K)) => (? [N2:nat,X2:(nat > A)]: (((((member @ (nat > A)) @ X2) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: (K)))) & (? [Xa:(nat > B)]: (((((member @ (nat > B)) @ Xa) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: ((((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2))))) & (X = (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N2) @ Xa) @ X2))))))))))))))).
% 2.87/1.58 thf(fact_23_Module_Ofgs__sub__carrier, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A2:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => (((ord_less_eq @ (set @ A)) @ (((((((algebr833503410le_fgs @ B) @ D) @ A) @ C) @ R2) @ M5) @ A2)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))))).
% 2.87/1.58 thf(fact_24_misom__self, axiom, (((((((((misomorphic @ b) @ d) @ a) @ c) @ a) @ c) @ r) @ m) @ m)).
% 2.87/1.58 thf(fact_25_HOM__is__module, axiom, (! [E:$tType,F2:$tType,N3:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ b) @ F2)))]: ((((((((module @ E) @ b) @ F2) @ d) @ N3) @ r) => ((((((module @ (a > E)) @ b) @ product_unit) @ d) @ (((((((((hom @ b) @ d) @ a) @ c) @ E) @ F2) @ r) @ m) @ N3)) @ r))))).
% 2.87/1.58 thf(fact_26_Suc__pred, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) = N))))).
% 2.87/1.58 thf(fact_27_generator__sub__carrier, axiom, (! [H:(set @ a)]: (((((((((generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))).
% 2.87/1.58 thf(fact_28_free__generator__generator, axiom, (! [H:(set @ a)]: (((((((((free_generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((((((generator @ b) @ d) @ a) @ c) @ r) @ m) @ H))))).
% 2.87/1.58 thf(fact_29_Module_Ol__comb__mem, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:(set @ B),H:(set @ A),S:(nat > B),N:nat,M3:(nat > A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((ideal @ B) @ D) @ R2) @ A3) => ((((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => ((((member @ (nat > B)) @ S) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > A)) @ M3) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => (((member @ A) @ (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S) @ M3)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)))))))))).
% 2.87/1.58 thf(fact_30_Module_Oliear__comb__memTr, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:(set @ B),H:(set @ A),N:nat]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((ideal @ B) @ D) @ R2) @ A3) => ((((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => (! [S2:(nat > B),M4:(nat > A)]: ((((((member @ (nat > B)) @ S2) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) & (((member @ (nat > A)) @ M4) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H))))) => (((member @ A) @ (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S2) @ M4)) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))))))))).
% 2.87/1.58 thf(fact_31_zero__less__diff, axiom, (! [N:nat,M3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M3)) = (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_32_Module_Olinear__comb__eq, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A),S:(nat > B),N:nat,F:(nat > A),G:(nat > A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => ((((member @ (nat > B)) @ S) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2))))) => ((((member @ (nat > A)) @ F) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => ((((member @ (nat > A)) @ G) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => ((! [X2:nat]: (((((member @ nat) @ X2) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) => ((F @ X2) = (G @ X2))))) => ((((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S) @ F) = (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S) @ G))))))))))).
% 2.87/1.58 thf(fact_33_Module_Olinear__comb__eqTr, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A),S:(nat > B),N:nat,F:(nat > A),G:(nat > A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)) => (((((member @ (nat > B)) @ S) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))) @ R2))))) & ((((member @ (nat > A)) @ F) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) & ((((member @ (nat > A)) @ G) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) & (! [X2:nat]: (((((member @ nat) @ X2) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) => ((F @ X2) = (G @ X2)))))))) => ((((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S) @ F) = (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N) @ S) @ G)))))))).
% 2.87/1.58 thf(fact_34_diff__is__0__eq, axiom, (! [M3:nat,N:nat]: ((((((minus_minus @ nat) @ M3) @ N) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_35_diff__is__0__eq_H, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => ((((minus_minus @ nat) @ M3) @ N) = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_36_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.87/1.58 thf(fact_37_nat_Oinject, axiom, (! [X22:nat,Y2:nat]: ((((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))).
% 2.87/1.58 thf(fact_38_misom__sym, axiom, (! [E:$tType,F2:$tType,N3:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ b) @ F2)))]: ((((((((module @ E) @ b) @ F2) @ d) @ N3) @ r) => ((((((((((misomorphic @ b) @ d) @ a) @ c) @ E) @ F2) @ r) @ m) @ N3) => (((((((((misomorphic @ b) @ d) @ E) @ F2) @ a) @ c) @ r) @ N3) @ m)))))).
% 2.87/1.58 thf(fact_39_Suc__le__mono, axiom, (! [N:nat,M3:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M3)) = (((ord_less_eq @ nat) @ N) @ M3))))).
% 2.87/1.58 thf(fact_40_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.87/1.58 thf(fact_41_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.87/1.58 thf(fact_42_Suc__less__eq, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M3)) @ (suc @ N)) = (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_43_Suc__mono, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (((ord_less @ nat) @ (suc @ M3)) @ (suc @ N)))))).
% 2.87/1.58 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.87/1.58 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 2.87/1.58 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.87/1.58 thf(fact_47_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.87/1.58 thf(fact_48_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.87/1.58 thf(fact_49_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.87/1.58 thf(fact_50_diff__diff__cancel, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ N) => ((((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ N) @ I)) = I))))).
% 2.87/1.58 thf(fact_51_Suc__diff__diff, axiom, (! [M3:nat,N:nat,K2:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M3)) @ N)) @ (suc @ K2)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M3) @ N)) @ K2))))).
% 2.87/1.58 thf(fact_52_diff__Suc__Suc, axiom, (! [M3:nat,N:nat]: (((((minus_minus @ nat) @ (suc @ M3)) @ (suc @ N)) = (((minus_minus @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_53_diff__self__eq__0, axiom, (! [M3:nat]: (((((minus_minus @ nat) @ M3) @ M3) = (zero_zero @ nat))))).
% 2.87/1.58 thf(fact_54_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.87/1.58 thf(fact_55_misom__trans, axiom, (! [E:$tType,F2:$tType,H3:$tType,G2:$tType,L:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ b) @ F2))),N3:((carrier_ext @ G2) @ ((aGroup_ext @ G2) @ (((module_ext @ G2) @ b) @ H3)))]: ((((((((module @ E) @ b) @ F2) @ d) @ L) @ r) => (((((((module @ G2) @ b) @ H3) @ d) @ N3) @ r) => ((((((((((misomorphic @ b) @ d) @ E) @ F2) @ a) @ c) @ r) @ L) @ m) => ((((((((((misomorphic @ b) @ d) @ a) @ c) @ G2) @ H3) @ r) @ m) @ N3) => (((((((((misomorphic @ b) @ d) @ E) @ F2) @ G2) @ H3) @ r) @ L) @ N3)))))))).
% 2.87/1.58 thf(fact_56_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 2.87/1.58 thf(fact_57_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_58_Module_OHOM__is__module, axiom, (! [E:$tType,F2:$tType,C:$tType,A:$tType,D:$tType,B:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N3:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N3) @ R2) => ((((((module @ (A > E)) @ B) @ product_unit) @ D) @ (((((((((hom @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M5) @ N3)) @ R2)))))).
% 2.87/1.58 thf(fact_59_Module_Omisom__trans, axiom, (! [C:$tType,A:$tType,E:$tType,F2:$tType,D:$tType,H3:$tType,B:$tType,G2:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),L:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2))),N3:((carrier_ext @ G2) @ ((aGroup_ext @ G2) @ (((module_ext @ G2) @ B) @ H3)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ L) @ R2) => (((((((module @ G2) @ B) @ H3) @ D) @ N3) @ R2) => ((((((((((misomorphic @ B) @ D) @ E) @ F2) @ A) @ C) @ R2) @ L) @ M5) => ((((((((((misomorphic @ B) @ D) @ A) @ C) @ G2) @ H3) @ R2) @ M5) @ N3) => (((((((((misomorphic @ B) @ D) @ E) @ F2) @ G2) @ H3) @ R2) @ L) @ N3))))))))).
% 2.87/1.58 thf(fact_60_Module_Omisom__self, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((((((misomorphic @ B) @ D) @ A) @ C) @ A) @ C) @ R2) @ M5) @ M5))))).
% 2.87/1.58 thf(fact_61_Module_Omisom__sym, axiom, (! [E:$tType,F2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),N3:((carrier_ext @ E) @ ((aGroup_ext @ E) @ (((module_ext @ E) @ B) @ F2)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((((module @ E) @ B) @ F2) @ D) @ N3) @ R2) => ((((((((((misomorphic @ B) @ D) @ A) @ C) @ E) @ F2) @ R2) @ M5) @ N3) => (((((((((misomorphic @ B) @ D) @ E) @ F2) @ A) @ C) @ R2) @ N3) @ M5))))))).
% 2.87/1.58 thf(fact_62_Module_Ofree__generator__generator, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((free_generator @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => (((((((generator @ B) @ D) @ A) @ C) @ R2) @ M5) @ H)))))).
% 2.87/1.58 thf(fact_63_Module_Ogenerator__sub__carrier, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((generator @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => (((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))))).
% 2.87/1.58 thf(fact_64_Module_OAnn__is__ideal, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((ideal @ B) @ D) @ R2) @ ((((((annihilator @ B) @ D) @ A) @ C) @ R2) @ M5)))))).
% 2.87/1.58 thf(fact_65_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.87/1.58 thf(fact_66_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.87/1.58 thf(fact_67_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K2:nat,B2:nat]: (((P @ K2) => ((! [Y3:nat]: (((P @ Y3) => (((ord_less_eq @ nat) @ Y3) @ B2)))) => (? [X2:nat]: (((P @ X2) & (! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ X2)))))))))))).
% 2.87/1.58 thf(fact_68_nat__le__linear, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) | (((ord_less_eq @ nat) @ N) @ M3))))).
% 2.87/1.58 thf(fact_69_le__antisym, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => ((((ord_less_eq @ nat) @ N) @ M3) => (M3 = N)))))).
% 2.87/1.58 thf(fact_70_le__trans, axiom, (! [I:nat,J3:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J3) => ((((ord_less_eq @ nat) @ J3) @ K2) => (((ord_less_eq @ nat) @ I) @ K2)))))).
% 2.87/1.58 thf(fact_71_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.87/1.58 thf(fact_72_Suc__inject, axiom, (! [X:nat,Y4:nat]: ((((suc @ X) = (suc @ Y4)) => (X = Y4))))).
% 2.87/1.58 thf(fact_73_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y))))))))) => (P @ X))))).
% 2.87/1.58 thf(fact_74_linorder__neqE__nat, axiom, (! [X:nat,Y4:nat]: (((X != Y4) => ((~ ((((ord_less @ nat) @ X) @ Y4))) => (((ord_less @ nat) @ Y4) @ X)))))).
% 2.87/1.58 thf(fact_75_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M4:nat]: (((((ord_less @ nat) @ M4) @ N2) & (~ ((P @ M4))))))))) => (P @ N))))).
% 2.87/1.58 thf(fact_76_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M4:nat]: (((((ord_less @ nat) @ M4) @ N2) => (P @ M4)))) => (P @ N2)))) => (P @ N))))).
% 2.87/1.58 thf(fact_77_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.87/1.58 thf(fact_78_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.87/1.58 thf(fact_79_less__not__refl2, axiom, (! [N:nat,M3:nat]: (((((ord_less @ nat) @ N) @ M3) => (M3 != N))))).
% 2.87/1.58 thf(fact_80_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.87/1.58 thf(fact_81_nat__neq__iff, axiom, (! [M3:nat,N:nat]: (((M3 != N) = ((((ord_less @ nat) @ M3) @ N) | (((ord_less @ nat) @ N) @ M3)))))).
% 2.87/1.58 thf(fact_82_diff__commute, axiom, (! [I:nat,J3:nat,K2:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J3)) @ K2) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K2)) @ J3))))).
% 2.87/1.58 thf(fact_83_Module_Osmodule__ideal__coeff__is__Submodule, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:(set @ B)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((ideal @ B) @ D) @ R2) @ A3) => (((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ (((((((smodule_ideal_coeff @ B) @ D) @ A) @ C) @ R2) @ M5) @ A3))))))).
% 2.87/1.58 thf(fact_84_Module_Oquotient__of__submodules__is__ideal, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),P:(set @ A),Q:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ P) => ((((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ Q) => ((((ideal @ B) @ D) @ R2) @ ((((((((quotie135653060odules @ B) @ D) @ A) @ C) @ R2) @ M5) @ P) @ Q)))))))).
% 2.87/1.58 thf(fact_85_transitive__stepwise__le, axiom, (! [M3:nat,N:nat,R2:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M3) @ N) => ((! [X2:nat]: (((R2 @ X2) @ X2))) => ((! [X2:nat,Y3:nat,Z2:nat]: ((((R2 @ X2) @ Y3) => (((R2 @ Y3) @ Z2) => ((R2 @ X2) @ Z2))))) => ((! [N2:nat]: (((R2 @ N2) @ (suc @ N2)))) => ((R2 @ M3) @ N)))))))).
% 2.87/1.58 thf(fact_86_nat__induct__at__least, axiom, (! [M3:nat,N:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M3) @ N) => ((P @ M3) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ M3) @ N2) => ((P @ N2) => (P @ (suc @ N2)))))) => (P @ N))))))).
% 2.87/1.58 thf(fact_87_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M4:nat]: (((((ord_less_eq @ nat) @ (suc @ M4)) @ N2) => (P @ M4)))) => (P @ N2)))) => (P @ N))))).
% 2.87/1.58 thf(fact_88_not__less__eq__eq, axiom, (! [M3:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M3) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M3))))).
% 2.87/1.58 thf(fact_89_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.87/1.58 thf(fact_90_le__Suc__eq, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M3) @ N) | (M3 = (suc @ N))))))).
% 2.87/1.58 thf(fact_91_Suc__le__D, axiom, (! [N:nat,M6:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M6) => (? [M7:nat]: ((M6 = (suc @ M7)))))))).
% 2.87/1.58 thf(fact_92_le__SucI, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => (((ord_less_eq @ nat) @ M3) @ (suc @ N)))))).
% 2.87/1.58 thf(fact_93_le__SucE, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M3) @ N))) => (M3 = (suc @ N))))))).
% 2.87/1.58 thf(fact_94_Suc__leD, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N) => (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_95_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_96_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_97_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_98_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.87/1.58 thf(fact_99_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M7:nat]: ((N = (suc @ M7)))))))).
% 2.87/1.58 thf(fact_100_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat3:nat]: (((P @ Nat3) => (P @ (suc @ Nat3))))) => (P @ Nat)))))).
% 2.87/1.58 thf(fact_101_old_Onat_Oexhaust, axiom, (! [Y4:nat]: (((Y4 != (zero_zero @ nat)) => (~ ((! [Nat3:nat]: ((Y4 != (suc @ Nat3)))))))))).
% 2.87/1.58 thf(fact_102_Zero__not__Suc, axiom, (! [M3:nat]: (((zero_zero @ nat) != (suc @ M3))))).
% 2.87/1.58 thf(fact_103_Zero__neq__Suc, axiom, (! [M3:nat]: (((zero_zero @ nat) != (suc @ M3))))).
% 2.87/1.58 thf(fact_104_Suc__neq__Zero, axiom, (! [M3:nat]: (((suc @ M3) != (zero_zero @ nat))))).
% 2.87/1.58 thf(fact_105_zero__induct, axiom, (! [P:(nat > $o),K2:nat]: (((P @ K2) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (zero_zero @ nat))))))).
% 2.87/1.58 thf(fact_106_diff__induct, axiom, (! [P:(nat > (nat > $o)),M3:nat,N:nat]: (((! [X2:nat]: (((P @ X2) @ (zero_zero @ nat)))) => ((! [Y3:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y3)))) => ((! [X2:nat,Y3:nat]: ((((P @ X2) @ Y3) => ((P @ (suc @ X2)) @ (suc @ Y3))))) => ((P @ M3) @ N))))))).
% 2.87/1.58 thf(fact_107_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((P @ N2) => (P @ (suc @ N2))))) => (P @ N)))))).
% 2.87/1.58 thf(fact_108_nat_OdiscI, axiom, (! [Nat:nat,X22:nat]: (((Nat = (suc @ X22)) => (Nat != (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_109_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.87/1.58 thf(fact_110_old_Onat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((suc @ Nat4) != (zero_zero @ nat))))).
% 2.87/1.58 thf(fact_111_nat_Odistinct_I1_J, axiom, (! [X22:nat]: (((zero_zero @ nat) != (suc @ X22))))).
% 2.87/1.58 thf(fact_112_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J3:nat]: (((! [I2:nat,J4:nat]: (((((ord_less @ nat) @ I2) @ J4) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J4))))) => ((((ord_less_eq @ nat) @ I) @ J3) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J3))))))).
% 2.87/1.58 thf(fact_113_le__neq__implies__less, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => ((M3 != N) => (((ord_less @ nat) @ M3) @ N)))))).
% 2.87/1.58 thf(fact_114_less__or__eq__imp__le, axiom, (! [M3:nat,N:nat]: ((((((ord_less @ nat) @ M3) @ N) | (M3 = N)) => (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_115_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M8:nat,N4:nat]: (((((ord_less @ nat) @ M8) @ N4) | (M8 = N4)))))).
% 2.87/1.58 thf(fact_116_less__imp__le__nat, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_117_nat__less__le, axiom, ((ord_less @ nat) = (^ [M8:nat,N4:nat]: (((((ord_less_eq @ nat) @ M8) @ N4) & (M8 != N4)))))).
% 2.87/1.58 thf(fact_118_not__less__less__Suc__eq, axiom, (! [N:nat,M3:nat]: (((~ ((((ord_less @ nat) @ N) @ M3))) => ((((ord_less @ nat) @ N) @ (suc @ M3)) = (N = M3)))))).
% 2.87/1.58 thf(fact_119_strict__inc__induct, axiom, (! [I:nat,J3:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I) @ J3) => ((! [I2:nat]: (((J3 = (suc @ I2)) => (P @ I2)))) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ J3) => ((P @ (suc @ I2)) => (P @ I2))))) => (P @ I))))))).
% 2.87/1.58 thf(fact_120_less__Suc__induct, axiom, (! [I:nat,J3:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I) @ J3) => ((! [I2:nat]: (((P @ I2) @ (suc @ I2)))) => ((! [I2:nat,J4:nat,K3:nat]: (((((ord_less @ nat) @ I2) @ J4) => ((((ord_less @ nat) @ J4) @ K3) => (((P @ I2) @ J4) => (((P @ J4) @ K3) => ((P @ I2) @ K3))))))) => ((P @ I) @ J3))))))).
% 2.87/1.58 thf(fact_121_less__trans__Suc, axiom, (! [I:nat,J3:nat,K2:nat]: (((((ord_less @ nat) @ I) @ J3) => ((((ord_less @ nat) @ J3) @ K2) => (((ord_less @ nat) @ (suc @ I)) @ K2)))))).
% 2.87/1.58 thf(fact_122_Suc__less__SucD, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M3)) @ (suc @ N)) => (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_123_less__antisym, axiom, (! [N:nat,M3:nat]: (((~ ((((ord_less @ nat) @ N) @ M3))) => ((((ord_less @ nat) @ N) @ (suc @ M3)) => (M3 = N)))))).
% 2.87/1.58 thf(fact_124_Suc__less__eq2, axiom, (! [N:nat,M3:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M3) = (? [M9:nat]: (((M3 = (suc @ M9)) & (((ord_less @ nat) @ N) @ M9)))))))).
% 2.87/1.58 thf(fact_125_All__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) => (P @ I3)))) = ((P @ N) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => (P @ I3))))))))).
% 2.87/1.58 thf(fact_126_not__less__eq, axiom, (! [M3:nat,N:nat]: (((~ ((((ord_less @ nat) @ M3) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M3)))))).
% 2.87/1.58 thf(fact_127_less__Suc__eq, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ (suc @ N)) = ((((ord_less @ nat) @ M3) @ N) | (M3 = N)))))).
% 2.87/1.58 thf(fact_128_Ex__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) & (P @ I3)))) = ((P @ N) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N) & (P @ I3))))))))).
% 2.87/1.58 thf(fact_129_less__SucI, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (((ord_less @ nat) @ M3) @ (suc @ N)))))).
% 2.87/1.58 thf(fact_130_less__SucE, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M3) @ N))) => (M3 = N)))))).
% 2.87/1.58 thf(fact_131_Suc__lessI, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (((suc @ M3) != N) => (((ord_less @ nat) @ (suc @ M3)) @ N)))))).
% 2.87/1.58 thf(fact_132_Suc__lessE, axiom, (! [I:nat,K2:nat]: (((((ord_less @ nat) @ (suc @ I)) @ K2) => (~ ((! [J4:nat]: (((((ord_less @ nat) @ I) @ J4) => (K2 != (suc @ J4))))))))))).
% 2.87/1.58 thf(fact_133_Suc__lessD, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M3)) @ N) => (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_134_Nat_OlessE, axiom, (! [I:nat,K2:nat]: (((((ord_less @ nat) @ I) @ K2) => ((K2 != (suc @ I)) => (~ ((! [J4:nat]: (((((ord_less @ nat) @ I) @ J4) => (K2 != (suc @ J4)))))))))))).
% 2.87/1.58 thf(fact_135_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X:A]: (((! [X2:A]: ((((V @ X2) = (zero_zero @ nat)) => (P @ X2)))) => ((! [X2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X2)) => ((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y)))))))))) => (P @ X)))))).
% 2.87/1.58 thf(fact_136_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.87/1.58 thf(fact_137_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M4:nat]: (((((ord_less @ nat) @ M4) @ N2) & (~ ((P @ M4)))))))))) => (P @ N)))))).
% 2.87/1.58 thf(fact_138_gr__implies__not0, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (N != (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_139_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.87/1.58 thf(fact_140_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.87/1.58 thf(fact_141_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.87/1.58 thf(fact_142_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.87/1.58 thf(fact_143_diff__le__mono2, axiom, (! [M3:nat,N:nat,L2:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L2) @ N)) @ (((minus_minus @ nat) @ L2) @ M3)))))).
% 2.87/1.58 thf(fact_144_le__diff__iff_H, axiom, (! [A2:nat,C2:nat,B2:nat]: (((((ord_less_eq @ nat) @ A2) @ C2) => ((((ord_less_eq @ nat) @ B2) @ C2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C2) @ A2)) @ (((minus_minus @ nat) @ C2) @ B2)) = (((ord_less_eq @ nat) @ B2) @ A2))))))).
% 2.87/1.58 thf(fact_145_diff__le__self, axiom, (! [M3:nat,N:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M3) @ N)) @ M3)))).
% 2.87/1.58 thf(fact_146_diff__le__mono, axiom, (! [M3:nat,N:nat,L2:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M3) @ L2)) @ (((minus_minus @ nat) @ N) @ L2)))))).
% 2.87/1.58 thf(fact_147_Nat_Odiff__diff__eq, axiom, (! [K2:nat,M3:nat,N:nat]: (((((ord_less_eq @ nat) @ K2) @ M3) => ((((ord_less_eq @ nat) @ K2) @ N) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M3) @ K2)) @ (((minus_minus @ nat) @ N) @ K2)) = (((minus_minus @ nat) @ M3) @ N))))))).
% 2.87/1.58 thf(fact_148_le__diff__iff, axiom, (! [K2:nat,M3:nat,N:nat]: (((((ord_less_eq @ nat) @ K2) @ M3) => ((((ord_less_eq @ nat) @ K2) @ N) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M3) @ K2)) @ (((minus_minus @ nat) @ N) @ K2)) = (((ord_less_eq @ nat) @ M3) @ N))))))).
% 2.87/1.58 thf(fact_149_eq__diff__iff, axiom, (! [K2:nat,M3:nat,N:nat]: (((((ord_less_eq @ nat) @ K2) @ M3) => ((((ord_less_eq @ nat) @ K2) @ N) => (((((minus_minus @ nat) @ M3) @ K2) = (((minus_minus @ nat) @ N) @ K2)) = (M3 = N))))))).
% 2.87/1.58 thf(fact_150_zero__induct__lemma, axiom, (! [P:(nat > $o),K2:nat,I:nat]: (((P @ K2) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (((minus_minus @ nat) @ K2) @ I))))))).
% 2.87/1.58 thf(fact_151_minus__nat_Odiff__0, axiom, (! [M3:nat]: (((((minus_minus @ nat) @ M3) @ (zero_zero @ nat)) = M3)))).
% 2.87/1.58 thf(fact_152_diffs0__imp__equal, axiom, (! [M3:nat,N:nat]: ((((((minus_minus @ nat) @ M3) @ N) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N) @ M3) = (zero_zero @ nat)) => (M3 = N)))))).
% 2.87/1.58 thf(fact_153_less__imp__diff__less, axiom, (! [J3:nat,K2:nat,N:nat]: (((((ord_less @ nat) @ J3) @ K2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J3) @ N)) @ K2))))).
% 2.87/1.58 thf(fact_154_diff__less__mono2, axiom, (! [M3:nat,N:nat,L2:nat]: (((((ord_less @ nat) @ M3) @ N) => ((((ord_less @ nat) @ M3) @ L2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L2) @ N)) @ (((minus_minus @ nat) @ L2) @ M3))))))).
% 2.87/1.58 thf(fact_155_Module_Osc__Ring, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((ring @ B) @ D) @ R2))))).
% 2.87/1.58 thf(fact_156_Module_Omem__smodule__ideal__coeff, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),A3:(set @ B),X:A]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((ideal @ B) @ D) @ R2) @ A3) => ((((member @ A) @ X) @ (((((((smodule_ideal_coeff @ B) @ D) @ A) @ C) @ R2) @ M5) @ A3)) => (? [N2:nat,X2:(nat > B)]: (((((member @ (nat > B)) @ X2) @ ((((pi @ nat) @ B) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: (A3)))) & (? [Xa:(nat > A)]: (((((member @ (nat > A)) @ Xa) @ ((((pi @ nat) @ A) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N2))))) @ (^ [Uu:nat]: ((((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))) & (X = (((((((((l_comb @ B) @ D) @ A) @ C) @ R2) @ M5) @ N2) @ X2) @ Xa)))))))))))))).
% 2.87/1.58 thf(fact_157_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N5:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N2))) @ (F @ N2)))) => ((((ord_less_eq @ nat) @ N) @ N5) => (((ord_less_eq @ A) @ (F @ N5)) @ (F @ N)))))))))).
% 2.87/1.58 thf(fact_158_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N5:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less_eq @ nat) @ N) @ N5) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N5)))))))))).
% 2.87/1.58 thf(fact_159_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M3:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M3)) = (((ord_less @ nat) @ N) @ M3))))))))).
% 2.87/1.58 thf(fact_160_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N5:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ nat) @ N) @ N5) => (((ord_less @ A) @ (F @ N)) @ (F @ N5)))))))))).
% 2.87/1.58 thf(fact_161_le__imp__less__Suc, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => (((ord_less @ nat) @ M3) @ (suc @ N)))))).
% 2.87/1.58 thf(fact_162_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N4:nat]: (((ord_less_eq @ nat) @ (suc @ N4)))))).
% 2.87/1.58 thf(fact_163_less__Suc__eq__le, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ (suc @ N)) = (((ord_less_eq @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_164_le__less__Suc__eq, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ M3) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M3)) = (N = M3)))))).
% 2.87/1.58 thf(fact_165_Suc__le__lessD, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N) => (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_166_inc__induct, axiom, (! [I:nat,J3:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J3) => ((P @ J3) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((ord_less @ nat) @ N2) @ J3) => ((P @ (suc @ N2)) => (P @ N2)))))) => (P @ I))))))).
% 2.87/1.58 thf(fact_167_dec__induct, axiom, (! [I:nat,J3:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J3) => ((P @ I) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((ord_less @ nat) @ N2) @ J3) => ((P @ N2) => (P @ (suc @ N2))))))) => (P @ J3))))))).
% 2.87/1.58 thf(fact_168_Suc__le__eq, axiom, (! [M3:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N) = (((ord_less @ nat) @ M3) @ N))))).
% 2.87/1.58 thf(fact_169_Suc__leI, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ N) => (((ord_less_eq @ nat) @ (suc @ M3)) @ N))))).
% 2.87/1.58 thf(fact_170_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K3:nat]: (((((ord_less_eq @ nat) @ K3) @ N) & ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K3) => (~ ((P @ I4)))))) & (P @ K3)))))))))).
% 2.87/1.58 thf(fact_171_less__Suc__eq__0__disj, axiom, (! [M3:nat,N:nat]: (((((ord_less @ nat) @ M3) @ (suc @ N)) = ((M3 = (zero_zero @ nat)) | (? [J2:nat]: (((M3 = (suc @ J2)) & (((ord_less @ nat) @ J2) @ N))))))))).
% 2.87/1.58 thf(fact_172_gr0__implies__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (? [M7:nat]: ((N = (suc @ M7)))))))).
% 2.87/1.58 thf(fact_173_All__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) => (P @ I3)))) = ((P @ (zero_zero @ nat)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => (P @ (suc @ I3)))))))))).
% 2.87/1.58 thf(fact_174_gr0__conv__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) = (? [M8:nat]: ((N = (suc @ M8)))))))).
% 2.87/1.58 thf(fact_175_Ex__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) & (P @ I3)))) = ((P @ (zero_zero @ nat)) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N) & (P @ (suc @ I3)))))))))).
% 2.87/1.58 thf(fact_176_Suc__diff__le, axiom, (! [N:nat,M3:nat]: (((((ord_less_eq @ nat) @ N) @ M3) => ((((minus_minus @ nat) @ (suc @ M3)) @ N) = (suc @ (((minus_minus @ nat) @ M3) @ N))))))).
% 2.87/1.58 thf(fact_177_diff__less__mono, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((ord_less @ nat) @ A2) @ B2) => ((((ord_less_eq @ nat) @ C2) @ A2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A2) @ C2)) @ (((minus_minus @ nat) @ B2) @ C2))))))).
% 2.87/1.58 thf(fact_178_less__diff__iff, axiom, (! [K2:nat,M3:nat,N:nat]: (((((ord_less_eq @ nat) @ K2) @ M3) => ((((ord_less_eq @ nat) @ K2) @ N) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M3) @ K2)) @ (((minus_minus @ nat) @ N) @ K2)) = (((ord_less @ nat) @ M3) @ N))))))).
% 2.87/1.58 thf(fact_179_diff__less__Suc, axiom, (! [M3:nat,N:nat]: ((((ord_less @ nat) @ (((minus_minus @ nat) @ M3) @ N)) @ (suc @ M3))))).
% 2.87/1.58 thf(fact_180_Suc__diff__Suc, axiom, (! [N:nat,M3:nat]: (((((ord_less @ nat) @ N) @ M3) => ((suc @ (((minus_minus @ nat) @ M3) @ (suc @ N))) = (((minus_minus @ nat) @ M3) @ N)))))).
% 2.87/1.58 thf(fact_181_diff__less, axiom, (! [N:nat,M3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M3) => (((ord_less @ nat) @ (((minus_minus @ nat) @ M3) @ N)) @ M3)))))).
% 2.87/1.58 thf(fact_182_Module_Osubmodule__subset1, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A),H2:A]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => ((((member @ A) @ H2) @ H) => (((member @ A) @ H2) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)))))))).
% 2.87/1.58 thf(fact_183_Module_Osubmodule__whole, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D)))]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => (((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5)))))).
% 2.87/1.58 thf(fact_184_ex__least__nat__less, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K3:nat]: (((((ord_less @ nat) @ K3) @ N) & ((! [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ K3) => (~ ((P @ I4)))))) & (P @ (suc @ K3))))))))))).
% 2.87/1.58 thf(fact_185_diff__Suc__less, axiom, (! [N:nat,I:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (((minus_minus @ nat) @ N) @ (suc @ I))) @ N))))).
% 2.87/1.58 thf(fact_186_Module_Osubmodule__subset, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((submodule @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => (((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))))).
% 2.87/1.58 thf(fact_187_Module_Ofree__generator__sub, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,M5:((carrier_ext @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))),R2:((carrier_ext @ B) @ ((aGroup_ext @ B) @ ((ring_ext @ B) @ D))),H:(set @ A)]: ((((((((module @ A) @ B) @ C) @ D) @ M5) @ R2) => ((((((((free_generator @ B) @ D) @ A) @ C) @ R2) @ M5) @ H) => (((ord_less_eq @ (set @ A)) @ H) @ (((carrier @ A) @ ((aGroup_ext @ A) @ (((module_ext @ A) @ B) @ C))) @ M5))))))).
% 2.87/1.58 thf(fact_188_eSum__in__SubmoduleTr, axiom, (! [H:(set @ a),K:(set @ a),F:(nat > a),N:nat,S:(nat > b)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((ord_less_eq @ (set @ a)) @ K) @ H) => (((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (K)))) & (((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)))))) => ((((((((((l_comb @ b) @ d) @ a) @ product_unit) @ r) @ (((((mdl @ a) @ b) @ c) @ m) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ K))) @ N) @ S) @ F) = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F)))))))).
% 2.87/1.58 thf(fact_189_eSum__in__Submodule, axiom, (! [H:(set @ a),K:(set @ a),F:(nat > a),N:nat,S:(nat > b)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((ord_less_eq @ (set @ a)) @ K) @ H) => ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (K)))) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r))))) => ((((((((((l_comb @ b) @ d) @ a) @ product_unit) @ r) @ (((((mdl @ a) @ b) @ c) @ m) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ K))) @ N) @ S) @ F) = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F))))))))).
% 2.87/1.58 thf(fact_190_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.87/1.58 thf(fact_191_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A3:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A3)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.87/1.58 thf(fact_192_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))).
% 2.87/1.58 thf(fact_193_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.87/1.58 thf(fact_194_l__span__closed2Tr, axiom, (! [A3:(set @ b),H:(set @ a),S:(nat > b),N:nat,F:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) & (((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H)))))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H)))))))).
% 2.87/1.58 thf(fact_195_l__span__closed2, axiom, (! [A3:(set @ b),H:(set @ a),S:(nat > b),N:nat,F:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H))))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))))).
% 2.87/1.58 thf(fact_196_linear__comb0__2Tr, axiom, (! [A3:(set @ b),S:(nat > b),N:nat,M3:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => (((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) & (((member @ (nat > a)) @ M3) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a)))))))) => ((((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ M3) = (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m))))))).
% 2.87/1.58 thf(fact_197_module__inc__zero, axiom, (((member @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(fact_198_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ (set @ A)) @ A3) @ B3)))))).
% 2.87/1.58 thf(fact_199_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.87/1.58 thf(fact_200_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 2.87/1.58 thf(fact_201_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.87/1.58 thf(fact_202_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 2.87/1.58 thf(fact_203_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 2.87/1.58 thf(fact_204_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.87/1.58 thf(fact_205_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.87/1.58 thf(fact_206_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.87/1.58 thf(fact_207_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.87/1.58 thf(fact_208_Diff__insert0, axiom, (! [A:$tType,X:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X) @ A3))) => ((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X) @ B3)) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.87/1.58 thf(fact_209_insert__Diff1, axiom, (! [A:$tType,X:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ X) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.87/1.58 thf(fact_210_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 2.87/1.58 thf(fact_211_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) = ((A2 = B2) | (((member @ A) @ A2) @ A3)))))).
% 2.87/1.58 thf(fact_212_insert__absorb2, axiom, (! [A:$tType,X:A,A3:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A3)) = (((insert @ A) @ X) @ A3))))).
% 2.87/1.58 thf(fact_213_submodule__inc__0, axiom, (! [H:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((member @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ H))))).
% 2.87/1.58 thf(fact_214_mdl__carrier, axiom, (! [H:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ product_unit))) @ (((((mdl @ a) @ b) @ c) @ m) @ H)) = H))))).
% 2.87/1.58 thf(fact_215_mdl__is__module, axiom, (! [H:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((((module @ a) @ b) @ product_unit) @ d) @ (((((mdl @ a) @ b) @ c) @ m) @ H)) @ r))))).
% 2.87/1.58 thf(fact_216_submodule__of__mdl, axiom, (! [H:(set @ a),N3:(set @ a)]: (((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ N3) => ((((ord_less_eq @ (set @ a)) @ H) @ N3) => (((((((submodule @ b) @ d) @ a) @ product_unit) @ r) @ (((((mdl @ a) @ b) @ c) @ m) @ N3)) @ H))))))).
% 2.87/1.58 thf(fact_217_h__in__linear__span, axiom, (! [H:(set @ a),H2:a]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ a) @ H2) @ H) => (((member @ a) @ H2) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H))))))).
% 2.87/1.58 thf(fact_218_l__span__cont__H, axiom, (! [H:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((ord_less_eq @ (set @ a)) @ H) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H)))))).
% 2.87/1.58 thf(fact_219_l__span__l__span, axiom, (! [H:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H)) = ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H)))))).
% 2.87/1.58 thf(fact_220_l__span__gen__mono, axiom, (! [K:(set @ a),H:(set @ a),A3:(set @ b)]: (((((ord_less_eq @ (set @ a)) @ K) @ H) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((ideal @ b) @ d) @ r) @ A3) => (((ord_less_eq @ (set @ a)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ K)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H)))))))).
% 2.87/1.58 thf(fact_221_linear__span__sub, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((ord_less_eq @ (set @ a)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))))))).
% 2.87/1.58 thf(fact_222_l__span__sub__submodule, axiom, (! [A3:(set @ b),N3:(set @ a),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ N3) => ((((ord_less_eq @ (set @ a)) @ H) @ N3) => (((ord_less_eq @ (set @ a)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H)) @ N3))))))).
% 2.87/1.58 thf(fact_223_fgs__generator, axiom, (! [H:(set @ a)]: (((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((generator @ b) @ d) @ a) @ product_unit) @ r) @ (((((mdl @ a) @ b) @ c) @ m) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ H))) @ H))))).
% 2.87/1.58 thf(fact_224_submodule__0, axiom, (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ (((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a))))).
% 2.87/1.58 thf(fact_225_empty__fgs, axiom, ((((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ (bot_bot @ (set @ a))) = (((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a))))).
% 2.87/1.58 thf(fact_226_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.87/1.58 thf(fact_227_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.87/1.58 thf(fact_228_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.87/1.58 thf(fact_229_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.87/1.58 thf(fact_230_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.87/1.58 thf(fact_231_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.87/1.58 thf(fact_232_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.87/1.58 thf(fact_233_Diff__eq__empty__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.87/1.58 thf(fact_234_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.87/1.58 thf(fact_235_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.87/1.58 thf(fact_236_insert__subset, axiom, (! [A:$tType,X:A,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A3)) @ B3) = ((((member @ A) @ X) @ B3) & (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))).
% 2.87/1.58 thf(fact_237_insert__Diff__single, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A3))))).
% 2.87/1.58 thf(fact_238_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.87/1.58 thf(fact_239_linear__span__subModule, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))).
% 2.87/1.58 thf(fact_240_l__spanA__l__span, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H)) = ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))).
% 2.87/1.58 thf(fact_241_lin__span__coeff__mono, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((ord_less_eq @ (set @ a)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H))))))).
% 2.87/1.58 thf(fact_242_linear__span__inc__0, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((member @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))).
% 2.87/1.58 thf(fact_243_generator__generator, axiom, (! [H:(set @ a),H1:(set @ a)]: (((((((((generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => ((((ord_less_eq @ (set @ a)) @ H1) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((ord_less_eq @ (set @ a)) @ H) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)) @ H1)) => (((((((generator @ b) @ d) @ a) @ c) @ r) @ m) @ H1))))))).
% 2.87/1.58 thf(fact_244_l__span__zero, axiom, (! [A3:(set @ b)]: ((((((ideal @ b) @ d) @ r) @ A3) => (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ (((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a)))) = (((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a)))))))).
% 2.87/1.58 thf(fact_245_l__span__closed3, axiom, (! [A3:(set @ b),H:(set @ a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((((((generator @ b) @ d) @ a) @ c) @ r) @ m) @ H) => (((((((((smodule_ideal_coeff @ b) @ d) @ a) @ c) @ r) @ m) @ A3) = (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H) = (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)))))))).
% 2.87/1.58 thf(fact_246_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X3:A]: ((X3 = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.87/1.58 thf(fact_247_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.87/1.58 thf(fact_248_l__comb__mem__linear__span, axiom, (! [A3:(set @ b),H:(set @ a),S:(nat > b),N:nat,F:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (H)))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))))).
% 2.87/1.58 thf(fact_249_l__span__closed, axiom, (! [A3:(set @ b),H:(set @ a),S:(nat > b),N:nat,F:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((ord_less_eq @ (set @ a)) @ H) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m)) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > a)) @ F) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))) => (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ F)) @ ((((((((linear_span @ b) @ d) @ a) @ c) @ r) @ m) @ A3) @ H))))))))).
% 2.87/1.58 thf(fact_250_linear__comb0__2, axiom, (! [A3:(set @ b),S:(nat > b),N:nat,M3:(nat > a)]: ((((((ideal @ b) @ d) @ r) @ A3) => ((((member @ (nat > b)) @ S) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: (A3)))) => ((((member @ (nat > a)) @ M3) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [J2:nat]: ((((ord_less_eq @ nat) @ J2) @ N))))) @ (^ [Uu:nat]: ((((insert @ a) @ (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)) @ (bot_bot @ (set @ a))))))) => ((((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ N) @ S) @ M3) = (((zero2 @ a) @ (((module_ext @ a) @ b) @ c)) @ m)))))))).
% 2.87/1.58 thf(fact_251_double__diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => ((((minus_minus @ (set @ A)) @ B3) @ (((minus_minus @ (set @ A)) @ C3) @ A3)) = A3)))))).
% 2.87/1.58 thf(fact_252_Diff__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ A3)))).
% 2.87/1.58 thf(tcon_fun___Orderings_Oorder, axiom, (! [A4:$tType,A5:$tType]: (((order @ A5) => (order @ (A4 > A5)))))).
% 2.87/1.58 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.87/1.58 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 2.87/1.58 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.87/1.58 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.87/1.58 thf(tcon_Nat_Onat___Orderings_Oorder_1, axiom, (order @ nat)).
% 2.87/1.58 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.87/1.58 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A4:$tType]: ((order @ (set @ A4))))).
% 2.87/1.58 thf(tcon_HOL_Obool___Orderings_Oorder_3, axiom, (order @ $o)).
% 2.87/1.58 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_4, axiom, (wellorder @ product_unit)).
% 2.87/1.58 thf(tcon_Product__Type_Ounit___Orderings_Oorder_5, axiom, (order @ product_unit)).
% 2.87/1.58 thf(conj_0, hypothesis, (((member @ (nat > a)) @ g) @ ((((pi @ nat) @ a) @ ((collect @ nat) @ (^ [K4:nat]: ((((ord_less_eq @ nat) @ K4) @ m2))))) @ (^ [Uu:nat]: ((((insert @ a) @ h2) @ h1)))))).
% 2.87/1.58 thf(conj_1, hypothesis, (! [X4:nat]: (((((ord_less_eq @ nat) @ X4) @ m2) => (! [Y:nat]: (((((ord_less_eq @ nat) @ Y) @ m2) => (((g @ X4) = (g @ Y)) => (X4 = Y))))))))).
% 2.87/1.58 thf(conj_2, hypothesis, (((member @ (nat > b)) @ t) @ ((((pi @ nat) @ b) @ ((collect @ nat) @ (^ [K4:nat]: ((((ord_less_eq @ nat) @ K4) @ m2))))) @ (^ [Uu:nat]: ((((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r)))))).
% 2.87/1.58 thf(conj_3, hypothesis, ((g @ m2) = h2)).
% 2.87/1.58 thf(conj_4, hypothesis, ((((((module @ e) @ b) @ f) @ d) @ n) @ r)).
% 2.87/1.58 thf(conj_5, hypothesis, (((((((free_generator @ b) @ d) @ a) @ c) @ r) @ m) @ h)).
% 2.87/1.58 thf(conj_6, hypothesis, (((ord_less_eq @ (set @ a)) @ h1) @ h)).
% 2.87/1.58 thf(conj_7, hypothesis, $true).
% 2.87/1.58 thf(conj_8, hypothesis, ((((ideal @ b) @ d) @ r) @ (((carrier @ b) @ ((aGroup_ext @ b) @ ((ring_ext @ b) @ d))) @ r))).
% 2.87/1.58 thf(conj_9, hypothesis, (((ord_less_eq @ (set @ a)) @ h) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(conj_10, hypothesis, (((ord_less_eq @ (set @ a)) @ h1) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(conj_11, hypothesis, (((ord_less_eq @ (set @ a)) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ h1)) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(conj_12, hypothesis, (((member @ a) @ (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ m2) @ t) @ g)) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ (((insert @ a) @ h2) @ h1)))).
% 2.87/1.58 thf(conj_13, hypothesis, (x = (((((((((l_comb @ b) @ d) @ a) @ c) @ r) @ m) @ m2) @ t) @ g))).
% 2.87/1.58 thf(conj_14, hypothesis, (((member @ a) @ h2) @ h)).
% 2.87/1.58 thf(conj_15, hypothesis, (~ ((((member @ a) @ h2) @ h1)))).
% 2.87/1.58 thf(conj_16, hypothesis, (((ord_less @ nat) @ (zero_zero @ nat)) @ m2)).
% 2.87/1.58 thf(conj_17, hypothesis, (((member @ a) @ h2) @ (((carrier @ a) @ ((aGroup_ext @ a) @ (((module_ext @ a) @ b) @ c))) @ m))).
% 2.87/1.58 thf(conj_18, hypothesis, (((((((submodule @ b) @ d) @ a) @ c) @ r) @ m) @ (((((((algebr833503410le_fgs @ b) @ d) @ a) @ c) @ r) @ m) @ (((insert @ a) @ h2) @ (bot_bot @ (set @ a)))))).
% 2.87/1.58 thf(conj_19, hypothesis, (((ord_less_eq @ nat) @ xa) @ (((minus_minus @ nat) @ m2) @ (suc @ (zero_zero @ nat))))).
% 2.87/1.58 thf(conj_20, hypothesis, (((g @ xa) = h2) | (((member @ a) @ (g @ xa)) @ h1))).
% 2.87/1.58 thf(conj_21, conjecture, (((member @ a) @ (g @ xa)) @ h1)).
% 2.87/1.58 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2DTF27230.p
% 2.87/1.58 ------------------------
% 3.15/1.62 ---- Cleaned THF ---
% 3.15/1.62 thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext,type,
% 3.15/1.62 carrier_ext: $tType > $tType > $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Algebra7_OModule_OModule__ext,type,
% 3.15/1.62 module_ext: $tType > $tType > $tType > $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Algebra4_OaGroup_OaGroup__ext,type,
% 3.15/1.62 aGroup_ext: $tType > $tType > $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Algebra4_ORing_ORing__ext,type,
% 3.15/1.62 ring_ext: $tType > $tType > $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Product__Type_Ounit,type,
% 3.15/1.62 product_unit: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Set_Oset,type,
% 3.15/1.62 set: $tType > $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_t_Nat_Onat,type,
% 3.15/1.62 nat: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_f,type,
% 3.15/1.62 f: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_e,type,
% 3.15/1.62 e: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_d,type,
% 3.15/1.62 d: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_c,type,
% 3.15/1.62 c: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_b,type,
% 3.15/1.62 b: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(ty_tf_a,type,
% 3.15/1.62 a: $tType ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_HOL_Otype,type,
% 3.15/1.62 type:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Ozero,type,
% 3.15/1.62 zero:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Orderings_Oorder,type,
% 3.15/1.62 order:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Ogroup__add,type,
% 3.15/1.62 group_add:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Orderings_Owellorder,type,
% 3.15/1.62 wellorder:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 3.15/1.62 comm_monoid_diff:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.15/1.62 ordered_ab_group_add:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 3.15/1.62 cancel1352612707id_add:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.15/1.62 canoni770627133id_add:
% 3.15/1.62 !>[A: $tType] : $o ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra1_Ocarrier_Ocarrier,type,
% 3.15/1.62 carrier:
% 3.15/1.62 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ Z ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra4_ORing,type,
% 3.15/1.62 ring:
% 3.15/1.62 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra4_OaGroup_Ozero,type,
% 3.15/1.62 zero2:
% 3.15/1.62 !>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra4_Oideal,type,
% 3.15/1.62 ideal:
% 3.15/1.62 !>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_OAnnihilator,type,
% 3.15/1.62 annihilator:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_OHOM,type,
% 3.15/1.62 hom:
% 3.15/1.62 !>[B: $tType,More: $tType,A: $tType,More1: $tType,C: $tType,More2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ More ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ More1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ More2 ) ) ) > ( carrier_ext @ ( A > C ) @ ( aGroup_ext @ ( A > C ) @ ( module_ext @ ( A > C ) @ B @ product_unit ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_OModule,type,
% 3.15/1.62 module:
% 3.15/1.62 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Ofree__generator,type,
% 3.15/1.62 free_generator:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Ogenerator,type,
% 3.15/1.62 generator:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Ol__comb,type,
% 3.15/1.62 l_comb:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > nat > ( nat > R ) > ( nat > A ) > A ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Olinear__span,type,
% 3.15/1.62 linear_span:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Omdl,type,
% 3.15/1.62 mdl:
% 3.15/1.62 !>[A: $tType,B: $tType,M: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ product_unit ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Omisomorphic,type,
% 3.15/1.62 misomorphic:
% 3.15/1.62 !>[B: $tType,M: $tType,A: $tType,M1: $tType,C: $tType,M2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ M2 ) ) ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Oquotient__of__submodules,type,
% 3.15/1.62 quotie135653060odules:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ R ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Osmodule__ideal__coeff,type,
% 3.15/1.62 smodule_ideal_coeff:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra7_Osubmodule,type,
% 3.15/1.62 submodule:
% 3.15/1.62 !>[B: $tType,M: $tType,A: $tType,C: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( set @ A ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_Ofgs,type,
% 3.15/1.62 algebr833503410le_fgs:
% 3.15/1.62 !>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_FuncSet_OPi,type,
% 3.15/1.62 pi:
% 3.15/1.62 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.15/1.62 minus_minus:
% 3.15/1.62 !>[A: $tType] : ( A > A > A ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.15/1.62 zero_zero:
% 3.15/1.62 !>[A: $tType] : A ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Nat_OSuc,type,
% 3.15/1.62 suc: nat > nat ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.15/1.62 bot_bot:
% 3.15/1.62 !>[A: $tType] : A ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.15/1.62 ord_less:
% 3.15/1.62 !>[A: $tType] : ( A > A > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.15/1.62 ord_less_eq:
% 3.15/1.62 !>[A: $tType] : ( A > A > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Set_OCollect,type,
% 3.15/1.62 collect:
% 3.15/1.62 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_Set_Oinsert,type,
% 3.15/1.62 insert:
% 3.15/1.62 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_c_member,type,
% 3.15/1.62 member:
% 3.15/1.62 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_H,type,
% 3.15/1.62 h: set @ a ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_H1,type,
% 3.15/1.62 h1: set @ a ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_M,type,
% 3.15/1.62 m: carrier_ext @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_N,type,
% 3.15/1.62 n: carrier_ext @ e @ ( aGroup_ext @ e @ ( module_ext @ e @ b @ f ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_R,type,
% 3.15/1.62 r: carrier_ext @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_g,type,
% 3.15/1.62 g: nat > a ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_h,type,
% 3.15/1.62 h2: a ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_m,type,
% 3.15/1.62 m2: nat ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_t,type,
% 3.15/1.62 t: nat > b ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_x,type,
% 3.15/1.62 x: a ).
% 3.15/1.62
% 3.15/1.62 thf(sy_v_xa,type,
% 3.15/1.62 xa: nat ).
% 3.15/1.62
% 3.15/1.62 thf(fact_0_sc__Ring,axiom,
% 3.15/1.62 ring @ b @ d @ r ).
% 3.15/1.62
% 3.15/1.62 thf(fact_1_submodule__subset1,axiom,
% 3.15/1.62 ! [H: set @ a,H2: a] :
% 3.15/1.62 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( ( member @ a @ H2 @ H )
% 3.15/1.62 => ( member @ a @ H2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_2_submodule__whole,axiom,
% 3.15/1.62 submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_3_submodule__subset,axiom,
% 3.15/1.62 ! [H: set @ a] :
% 3.15/1.62 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_4_free__generator__sub,axiom,
% 3.15/1.62 ! [H: set @ a] :
% 3.15/1.62 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_5_elem__fgs,axiom,
% 3.15/1.62 ! [A2: set @ a,X: a] :
% 3.15/1.62 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ( member @ a @ X @ A2 )
% 3.15/1.62 => ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_6_fgs__sub__carrier,axiom,
% 3.15/1.62 ! [A2: set @ a] :
% 3.15/1.62 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_7_fgs__mono,axiom,
% 3.15/1.62 ! [H: set @ a,J: set @ a,K: set @ a] :
% 3.15/1.62 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ a ) @ J @ K )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ J ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_8_fgs__submodule,axiom,
% 3.15/1.62 ! [A2: set @ a] :
% 3.15/1.62 ( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_9_linear__comb__eq,axiom,
% 3.15/1.62 ! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.15/1.62 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ( member @ ( nat > b ) @ S
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 3.15/1.62 => ( ( member @ ( nat > a ) @ F
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( ( member @ ( nat > a ) @ G
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( ! [X2: nat] :
% 3.15/1.62 ( ( member @ nat @ X2
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 3.15/1.62 => ( ( F @ X2 )
% 3.15/1.62 = ( G @ X2 ) ) )
% 3.15/1.62 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
% 3.15/1.62 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_10_linear__comb__eqTr,axiom,
% 3.15/1.62 ! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
% 3.15/1.62 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ( ( member @ ( nat > b ) @ S
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 3.15/1.62 & ( member @ ( nat > a ) @ F
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 & ( member @ ( nat > a ) @ G
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 & ! [X2: nat] :
% 3.15/1.62 ( ( member @ nat @ X2
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 3.15/1.62 => ( ( F @ X2 )
% 3.15/1.62 = ( G @ X2 ) ) ) )
% 3.15/1.62 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
% 3.15/1.62 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_11_l__comb__mem,axiom,
% 3.15/1.62 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,M3: nat > a] :
% 3.15/1.62 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ( member @ ( nat > b ) @ S
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 => ( ( member @ ( nat > a ) @ M3
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_12_liear__comb__memTr,axiom,
% 3.15/1.62 ! [A3: set @ b,H: set @ a,N: nat] :
% 3.15/1.62 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ! [S2: nat > b,M4: nat > a] :
% 3.15/1.62 ( ( ( member @ ( nat > b ) @ S2
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 & ( member @ ( nat > a ) @ M4
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) ) )
% 3.15/1.62 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S2 @ M4 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_13_Ann__is__ideal,axiom,
% 3.15/1.62 ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_14_mem__fgs__l__comb,axiom,
% 3.15/1.62 ! [K: set @ a,X: a] :
% 3.15/1.62 ( ( K
% 3.15/1.62 != ( bot_bot @ ( set @ a ) ) )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ a ) @ K @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.62 => ( ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) )
% 3.15/1.62 => ? [N2: nat,X2: nat > a] :
% 3.15/1.62 ( ( member @ ( nat > a ) @ X2
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : K ) )
% 3.15/1.62 & ? [Xa: nat > b] :
% 3.15/1.62 ( ( member @ ( nat > b ) @ Xa
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 3.15/1.62 & ( X
% 3.15/1.62 = ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_15_Module__axioms,axiom,
% 3.15/1.62 module @ a @ b @ c @ d @ m @ r ).
% 3.15/1.62
% 3.15/1.62 thf(fact_16_quotient__of__submodules__is__ideal,axiom,
% 3.15/1.62 ! [P: set @ a,Q: set @ a] :
% 3.15/1.62 ( ( submodule @ b @ d @ a @ c @ r @ m @ P )
% 3.15/1.62 => ( ( submodule @ b @ d @ a @ c @ r @ m @ Q )
% 3.15/1.62 => ( ideal @ b @ d @ r @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ P @ Q ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
% 3.15/1.62 ! [A3: set @ b] :
% 3.15/1.62 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.62 => ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_18_mem__smodule__ideal__coeff,axiom,
% 3.15/1.62 ! [A3: set @ b,X: a] :
% 3.15/1.62 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.62 => ( ( member @ a @ X @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) )
% 3.15/1.62 => ? [N2: nat,X2: nat > b] :
% 3.15/1.62 ( ( member @ ( nat > b ) @ X2
% 3.15/1.62 @ ( pi @ nat @ b
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 & ? [Xa: nat > a] :
% 3.15/1.62 ( ( member @ ( nat > a ) @ Xa
% 3.15/1.62 @ ( pi @ nat @ a
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
% 3.15/1.62 & ( X
% 3.15/1.62 = ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_19_Module_Oelem__fgs,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A,X: A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ( member @ A @ X @ A2 )
% 3.15/1.62 => ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_20_Module_Ofgs__mono,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,J: set @ A,K: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ J @ K )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ K @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ J ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_21_Module_Ofgs__submodule,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_22_Module_Omem__fgs__l__comb,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),K: set @ A,X: A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( K
% 3.15/1.62 != ( bot_bot @ ( set @ A ) ) )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ K @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) )
% 3.15/1.62 => ? [N2: nat,X2: nat > A] :
% 3.15/1.62 ( ( member @ ( nat > A ) @ X2
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : K ) )
% 3.15/1.62 & ? [Xa: nat > B] :
% 3.15/1.62 ( ( member @ ( nat > B ) @ Xa
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 3.15/1.62 & ( X
% 3.15/1.62 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_23_Module_Ofgs__sub__carrier,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_24_misom__self,axiom,
% 3.15/1.62 misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m ).
% 3.15/1.62
% 3.15/1.62 thf(fact_25_HOM__is__module,axiom,
% 3.15/1.62 ! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
% 3.15/1.62 ( ( module @ E @ b @ F2 @ d @ N3 @ r )
% 3.15/1.62 => ( module @ ( a > E ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 ) @ r ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_26_Suc__pred,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.62 => ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
% 3.15/1.62 = N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_27_generator__sub__carrier,axiom,
% 3.15/1.62 ! [H: set @ a] :
% 3.15/1.62 ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_28_free__generator__generator,axiom,
% 3.15/1.62 ! [H: set @ a] :
% 3.15/1.62 ( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.62 => ( generator @ b @ d @ a @ c @ r @ m @ H ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_29_Module_Ol__comb__mem,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,S: nat > B,N: nat,M3: nat > A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ideal @ B @ D @ R2 @ A3 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ( member @ ( nat > B ) @ S
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 => ( ( member @ ( nat > A ) @ M3
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ M3 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_30_Module_Oliear__comb__memTr,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,N: nat] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ideal @ B @ D @ R2 @ A3 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ! [S2: nat > B,M4: nat > A] :
% 3.15/1.62 ( ( ( member @ ( nat > B ) @ S2
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 & ( member @ ( nat > A ) @ M4
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) ) )
% 3.15/1.62 => ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S2 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_31_zero__less__diff,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M3 ) )
% 3.15/1.62 = ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_32_Module_Olinear__comb__eq,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ( member @ ( nat > B ) @ S
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 3.15/1.62 => ( ( member @ ( nat > A ) @ F
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( ( member @ ( nat > A ) @ G
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 => ( ! [X2: nat] :
% 3.15/1.62 ( ( member @ nat @ X2
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 3.15/1.62 => ( ( F @ X2 )
% 3.15/1.62 = ( G @ X2 ) ) )
% 3.15/1.62 => ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
% 3.15/1.62 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_33_Module_Olinear__comb__eqTr,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
% 3.15/1.62 => ( ( ( member @ ( nat > B ) @ S
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
% 3.15/1.62 & ( member @ ( nat > A ) @ F
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 & ( member @ ( nat > A ) @ G
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.62 @ ^ [Uu: nat] : H ) )
% 3.15/1.62 & ! [X2: nat] :
% 3.15/1.62 ( ( member @ nat @ X2
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
% 3.15/1.62 => ( ( F @ X2 )
% 3.15/1.62 = ( G @ X2 ) ) ) )
% 3.15/1.62 => ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
% 3.15/1.62 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_34_diff__is__0__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ( minus_minus @ nat @ M3 @ N )
% 3.15/1.62 = ( zero_zero @ nat ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_35_diff__is__0__eq_H,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ( minus_minus @ nat @ M3 @ N )
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_36_old_Onat_Oinject,axiom,
% 3.15/1.62 ! [Nat: nat,Nat2: nat] :
% 3.15/1.62 ( ( ( suc @ Nat )
% 3.15/1.62 = ( suc @ Nat2 ) )
% 3.15/1.62 = ( Nat = Nat2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_37_nat_Oinject,axiom,
% 3.15/1.62 ! [X22: nat,Y2: nat] :
% 3.15/1.62 ( ( ( suc @ X22 )
% 3.15/1.62 = ( suc @ Y2 ) )
% 3.15/1.62 = ( X22 = Y2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_38_misom__sym,axiom,
% 3.15/1.62 ! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
% 3.15/1.62 ( ( module @ E @ b @ F2 @ d @ N3 @ r )
% 3.15/1.62 => ( ( misomorphic @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 )
% 3.15/1.62 => ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ N3 @ m ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_39_Suc__le__mono,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M3 ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_40_bot__nat__0_Oextremum,axiom,
% 3.15/1.62 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_41_le0,axiom,
% 3.15/1.62 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_42_Suc__less__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 3.15/1.62 = ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_43_Suc__mono,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_44_mem__Collect__eq,axiom,
% 3.15/1.62 ! [A: $tType,A2: A,P: A > $o] :
% 3.15/1.62 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.15/1.62 = ( P @ A2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_45_Collect__mem__eq,axiom,
% 3.15/1.62 ! [A: $tType,A3: set @ A] :
% 3.15/1.62 ( ( collect @ A
% 3.15/1.62 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 3.15/1.62 = A3 ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_46_Collect__cong,axiom,
% 3.15/1.62 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.62 ( ! [X2: A] :
% 3.15/1.62 ( ( P @ X2 )
% 3.15/1.62 = ( Q @ X2 ) )
% 3.15/1.62 => ( ( collect @ A @ P )
% 3.15/1.62 = ( collect @ A @ Q ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_47_bot__nat__0_Onot__eq__extremum,axiom,
% 3.15/1.62 ! [A2: nat] :
% 3.15/1.62 ( ( A2
% 3.15/1.62 != ( zero_zero @ nat ) )
% 3.15/1.62 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_48_less__nat__zero__code,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_49_neq0__conv,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( N
% 3.15/1.62 != ( zero_zero @ nat ) )
% 3.15/1.62 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_50_diff__diff__cancel,axiom,
% 3.15/1.62 ! [I: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ N )
% 3.15/1.62 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
% 3.15/1.62 = I ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_51_Suc__diff__diff,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,K2: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M3 ) @ N ) @ ( suc @ K2 ) )
% 3.15/1.62 = ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ N ) @ K2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_52_diff__Suc__Suc,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 3.15/1.62 = ( minus_minus @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_53_diff__self__eq__0,axiom,
% 3.15/1.62 ! [M3: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ M3 @ M3 )
% 3.15/1.62 = ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_54_diff__0__eq__0,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.62 = ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_55_misom__trans,axiom,
% 3.15/1.62 ! [E: $tType,F2: $tType,H3: $tType,G2: $tType,L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ b @ H3 ) )] :
% 3.15/1.62 ( ( module @ E @ b @ F2 @ d @ L @ r )
% 3.15/1.62 => ( ( module @ G2 @ b @ H3 @ d @ N3 @ r )
% 3.15/1.62 => ( ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ L @ m )
% 3.15/1.62 => ( ( misomorphic @ b @ d @ a @ c @ G2 @ H3 @ r @ m @ N3 )
% 3.15/1.62 => ( misomorphic @ b @ d @ E @ F2 @ G2 @ H3 @ r @ L @ N3 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_56_zero__less__Suc,axiom,
% 3.15/1.62 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_57_less__Suc0,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 3.15/1.62 = ( N
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_58_Module_OHOM__is__module,axiom,
% 3.15/1.62 ! [E: $tType,F2: $tType,C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
% 3.15/1.62 => ( module @ ( A > E ) @ B @ product_unit @ D @ ( hom @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 ) @ R2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_59_Module_Omisom__trans,axiom,
% 3.15/1.62 ! [C: $tType,A: $tType,E: $tType,F2: $tType,D: $tType,H3: $tType,B: $tType,G2: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ B @ H3 ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( module @ E @ B @ F2 @ D @ L @ R2 )
% 3.15/1.62 => ( ( module @ G2 @ B @ H3 @ D @ N3 @ R2 )
% 3.15/1.62 => ( ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ L @ M5 )
% 3.15/1.62 => ( ( misomorphic @ B @ D @ A @ C @ G2 @ H3 @ R2 @ M5 @ N3 )
% 3.15/1.62 => ( misomorphic @ B @ D @ E @ F2 @ G2 @ H3 @ R2 @ L @ N3 ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_60_Module_Omisom__self,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( misomorphic @ B @ D @ A @ C @ A @ C @ R2 @ M5 @ M5 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_61_Module_Omisom__sym,axiom,
% 3.15/1.62 ! [E: $tType,F2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
% 3.15/1.62 => ( ( misomorphic @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 )
% 3.15/1.62 => ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ N3 @ M5 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_62_Module_Ofree__generator__generator,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.62 => ( generator @ B @ D @ A @ C @ R2 @ M5 @ H ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_63_Module_Ogenerator__sub__carrier,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.62 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_64_Module_OAnn__is__ideal,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ideal @ B @ D @ R2 @ ( annihilator @ B @ D @ A @ C @ R2 @ M5 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_65_measure__induct__rule,axiom,
% 3.15/1.62 ! [B: $tType,A: $tType] :
% 3.15/1.62 ( ( wellorder @ B )
% 3.15/1.62 => ! [F: A > B,P: A > $o,A2: A] :
% 3.15/1.62 ( ! [X2: A] :
% 3.15/1.62 ( ! [Y: A] :
% 3.15/1.62 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 3.15/1.62 => ( P @ Y ) )
% 3.15/1.62 => ( P @ X2 ) )
% 3.15/1.62 => ( P @ A2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_66_measure__induct,axiom,
% 3.15/1.62 ! [B: $tType,A: $tType] :
% 3.15/1.62 ( ( wellorder @ B )
% 3.15/1.62 => ! [F: A > B,P: A > $o,A2: A] :
% 3.15/1.62 ( ! [X2: A] :
% 3.15/1.62 ( ! [Y: A] :
% 3.15/1.62 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
% 3.15/1.62 => ( P @ Y ) )
% 3.15/1.62 => ( P @ X2 ) )
% 3.15/1.62 => ( P @ A2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_67_Nat_Oex__has__greatest__nat,axiom,
% 3.15/1.62 ! [P: nat > $o,K2: nat,B2: nat] :
% 3.15/1.62 ( ( P @ K2 )
% 3.15/1.62 => ( ! [Y3: nat] :
% 3.15/1.62 ( ( P @ Y3 )
% 3.15/1.62 => ( ord_less_eq @ nat @ Y3 @ B2 ) )
% 3.15/1.62 => ? [X2: nat] :
% 3.15/1.62 ( ( P @ X2 )
% 3.15/1.62 & ! [Y: nat] :
% 3.15/1.62 ( ( P @ Y )
% 3.15/1.62 => ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_68_nat__le__linear,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 | ( ord_less_eq @ nat @ N @ M3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_69_le__antisym,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ N @ M3 )
% 3.15/1.62 => ( M3 = N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_70_le__trans,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,K2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ J3 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 3.15/1.62 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_71_n__not__Suc__n,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( N
% 3.15/1.62 != ( suc @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_72_Suc__inject,axiom,
% 3.15/1.62 ! [X: nat,Y4: nat] :
% 3.15/1.62 ( ( ( suc @ X )
% 3.15/1.62 = ( suc @ Y4 ) )
% 3.15/1.62 => ( X = Y4 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_73_infinite__descent__measure,axiom,
% 3.15/1.62 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 3.15/1.62 ( ! [X2: A] :
% 3.15/1.62 ( ~ ( P @ X2 )
% 3.15/1.62 => ? [Y: A] :
% 3.15/1.62 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 3.15/1.62 & ~ ( P @ Y ) ) )
% 3.15/1.62 => ( P @ X ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_74_linorder__neqE__nat,axiom,
% 3.15/1.62 ! [X: nat,Y4: nat] :
% 3.15/1.62 ( ( X != Y4 )
% 3.15/1.62 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 3.15/1.62 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_75_infinite__descent,axiom,
% 3.15/1.62 ! [P: nat > $o,N: nat] :
% 3.15/1.62 ( ! [N2: nat] :
% 3.15/1.62 ( ~ ( P @ N2 )
% 3.15/1.62 => ? [M4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M4 @ N2 )
% 3.15/1.62 & ~ ( P @ M4 ) ) )
% 3.15/1.62 => ( P @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_76_nat__less__induct,axiom,
% 3.15/1.62 ! [P: nat > $o,N: nat] :
% 3.15/1.62 ( ! [N2: nat] :
% 3.15/1.62 ( ! [M4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M4 @ N2 )
% 3.15/1.62 => ( P @ M4 ) )
% 3.15/1.62 => ( P @ N2 ) )
% 3.15/1.62 => ( P @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_77_less__irrefl__nat,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ N @ N ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_78_less__not__refl3,axiom,
% 3.15/1.62 ! [S: nat,T: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ S @ T )
% 3.15/1.62 => ( S != T ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_79_less__not__refl2,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ N @ M3 )
% 3.15/1.62 => ( M3 != N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_80_less__not__refl,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ N @ N ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_81_nat__neq__iff,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( M3 != N )
% 3.15/1.62 = ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 | ( ord_less @ nat @ N @ M3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_82_diff__commute,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,K2: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J3 ) @ K2 )
% 3.15/1.62 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K2 ) @ J3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_83_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
% 3.15/1.62 ! [A: $tType,C: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ideal @ B @ D @ R2 @ A3 )
% 3.15/1.62 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_84_Module_Oquotient__of__submodules__is__ideal,axiom,
% 3.15/1.62 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),P: set @ A,Q: set @ A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ P )
% 3.15/1.62 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ Q )
% 3.15/1.62 => ( ideal @ B @ D @ R2 @ ( quotie135653060odules @ B @ D @ A @ C @ R2 @ M5 @ P @ Q ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_85_transitive__stepwise__le,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,R2: nat > nat > $o] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ! [X2: nat] : ( R2 @ X2 @ X2 )
% 3.15/1.62 => ( ! [X2: nat,Y3: nat,Z2: nat] :
% 3.15/1.62 ( ( R2 @ X2 @ Y3 )
% 3.15/1.62 => ( ( R2 @ Y3 @ Z2 )
% 3.15/1.62 => ( R2 @ X2 @ Z2 ) ) )
% 3.15/1.62 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 3.15/1.62 => ( R2 @ M3 @ N ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_86_nat__induct__at__least,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,P: nat > $o] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ( P @ M3 )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N2 )
% 3.15/1.62 => ( ( P @ N2 )
% 3.15/1.62 => ( P @ ( suc @ N2 ) ) ) )
% 3.15/1.62 => ( P @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_87_full__nat__induct,axiom,
% 3.15/1.62 ! [P: nat > $o,N: nat] :
% 3.15/1.62 ( ! [N2: nat] :
% 3.15/1.62 ( ! [M4: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N2 )
% 3.15/1.62 => ( P @ M4 ) )
% 3.15/1.62 => ( P @ N2 ) )
% 3.15/1.62 => ( P @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_88_not__less__eq__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ~ ( ord_less_eq @ nat @ M3 @ N ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ ( suc @ N ) @ M3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_89_Suc__n__not__le__n,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_90_le__Suc__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.62 = ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 | ( M3
% 3.15/1.62 = ( suc @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_91_Suc__le__D,axiom,
% 3.15/1.62 ! [N: nat,M6: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
% 3.15/1.62 => ? [M7: nat] :
% 3.15/1.62 ( M6
% 3.15/1.62 = ( suc @ M7 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_92_le__SucI,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less_eq @ nat @ M3 @ ( suc @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_93_le__SucE,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.62 => ( ~ ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( M3
% 3.15/1.62 = ( suc @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_94_Suc__leD,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 3.15/1.62 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_95_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.15/1.62 ! [A2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 3.15/1.62 => ( A2
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_96_bot__nat__0_Oextremum__unique,axiom,
% 3.15/1.62 ! [A2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 3.15/1.62 = ( A2
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_97_le__0__eq,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 3.15/1.62 = ( N
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_98_less__eq__nat_Osimps_I1_J,axiom,
% 3.15/1.62 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_99_not0__implies__Suc,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( N
% 3.15/1.62 != ( zero_zero @ nat ) )
% 3.15/1.62 => ? [M7: nat] :
% 3.15/1.62 ( N
% 3.15/1.62 = ( suc @ M7 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_100_old_Onat_Oinducts,axiom,
% 3.15/1.62 ! [P: nat > $o,Nat: nat] :
% 3.15/1.62 ( ( P @ ( zero_zero @ nat ) )
% 3.15/1.62 => ( ! [Nat3: nat] :
% 3.15/1.62 ( ( P @ Nat3 )
% 3.15/1.62 => ( P @ ( suc @ Nat3 ) ) )
% 3.15/1.62 => ( P @ Nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_101_old_Onat_Oexhaust,axiom,
% 3.15/1.62 ! [Y4: nat] :
% 3.15/1.62 ( ( Y4
% 3.15/1.62 != ( zero_zero @ nat ) )
% 3.15/1.62 => ~ ! [Nat3: nat] :
% 3.15/1.62 ( Y4
% 3.15/1.62 != ( suc @ Nat3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_102_Zero__not__Suc,axiom,
% 3.15/1.62 ! [M3: nat] :
% 3.15/1.62 ( ( zero_zero @ nat )
% 3.15/1.62 != ( suc @ M3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_103_Zero__neq__Suc,axiom,
% 3.15/1.62 ! [M3: nat] :
% 3.15/1.62 ( ( zero_zero @ nat )
% 3.15/1.62 != ( suc @ M3 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_104_Suc__neq__Zero,axiom,
% 3.15/1.62 ! [M3: nat] :
% 3.15/1.62 ( ( suc @ M3 )
% 3.15/1.62 != ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_105_zero__induct,axiom,
% 3.15/1.62 ! [P: nat > $o,K2: nat] :
% 3.15/1.62 ( ( P @ K2 )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( P @ ( suc @ N2 ) )
% 3.15/1.62 => ( P @ N2 ) )
% 3.15/1.62 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_106_diff__induct,axiom,
% 3.15/1.62 ! [P: nat > nat > $o,M3: nat,N: nat] :
% 3.15/1.62 ( ! [X2: nat] : ( P @ X2 @ ( zero_zero @ nat ) )
% 3.15/1.62 => ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
% 3.15/1.62 => ( ! [X2: nat,Y3: nat] :
% 3.15/1.62 ( ( P @ X2 @ Y3 )
% 3.15/1.62 => ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
% 3.15/1.62 => ( P @ M3 @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_107_nat__induct,axiom,
% 3.15/1.62 ! [P: nat > $o,N: nat] :
% 3.15/1.62 ( ( P @ ( zero_zero @ nat ) )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( P @ N2 )
% 3.15/1.62 => ( P @ ( suc @ N2 ) ) )
% 3.15/1.62 => ( P @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_108_nat_OdiscI,axiom,
% 3.15/1.62 ! [Nat: nat,X22: nat] :
% 3.15/1.62 ( ( Nat
% 3.15/1.62 = ( suc @ X22 ) )
% 3.15/1.62 => ( Nat
% 3.15/1.62 != ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_109_old_Onat_Odistinct_I1_J,axiom,
% 3.15/1.62 ! [Nat2: nat] :
% 3.15/1.62 ( ( zero_zero @ nat )
% 3.15/1.62 != ( suc @ Nat2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 3.15/1.62 ! [Nat4: nat] :
% 3.15/1.62 ( ( suc @ Nat4 )
% 3.15/1.62 != ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_111_nat_Odistinct_I1_J,axiom,
% 3.15/1.62 ! [X22: nat] :
% 3.15/1.62 ( ( zero_zero @ nat )
% 3.15/1.62 != ( suc @ X22 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_112_less__mono__imp__le__mono,axiom,
% 3.15/1.62 ! [F: nat > nat,I: nat,J3: nat] :
% 3.15/1.62 ( ! [I2: nat,J4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I2 @ J4 )
% 3.15/1.62 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ I @ J3 )
% 3.15/1.62 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_113_le__neq__implies__less,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ( M3 != N )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_114_less__or__eq__imp__le,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 | ( M3 = N ) )
% 3.15/1.62 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_115_le__eq__less__or__eq,axiom,
% 3.15/1.62 ( ( ord_less_eq @ nat )
% 3.15/1.62 = ( ^ [M8: nat,N4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M8 @ N4 )
% 3.15/1.62 | ( M8 = N4 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_116_less__imp__le__nat,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_117_nat__less__le,axiom,
% 3.15/1.62 ( ( ord_less @ nat )
% 3.15/1.62 = ( ^ [M8: nat,N4: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M8 @ N4 )
% 3.15/1.62 & ( M8 != N4 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_118_not__less__less__Suc__eq,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ~ ( ord_less @ nat @ N @ M3 )
% 3.15/1.62 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 3.15/1.62 = ( N = M3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_119_strict__inc__induct,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,P: nat > $o] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ J3 )
% 3.15/1.62 => ( ! [I2: nat] :
% 3.15/1.62 ( ( J3
% 3.15/1.62 = ( suc @ I2 ) )
% 3.15/1.62 => ( P @ I2 ) )
% 3.15/1.62 => ( ! [I2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I2 @ J3 )
% 3.15/1.62 => ( ( P @ ( suc @ I2 ) )
% 3.15/1.62 => ( P @ I2 ) ) )
% 3.15/1.62 => ( P @ I ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_120_less__Suc__induct,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,P: nat > nat > $o] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ J3 )
% 3.15/1.62 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 3.15/1.62 => ( ! [I2: nat,J4: nat,K3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I2 @ J4 )
% 3.15/1.62 => ( ( ord_less @ nat @ J4 @ K3 )
% 3.15/1.62 => ( ( P @ I2 @ J4 )
% 3.15/1.62 => ( ( P @ J4 @ K3 )
% 3.15/1.62 => ( P @ I2 @ K3 ) ) ) ) )
% 3.15/1.62 => ( P @ I @ J3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_121_less__trans__Suc,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,K2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ J3 )
% 3.15/1.62 => ( ( ord_less @ nat @ J3 @ K2 )
% 3.15/1.62 => ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_122_Suc__less__SucD,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_123_less__antisym,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ~ ( ord_less @ nat @ N @ M3 )
% 3.15/1.62 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 3.15/1.62 => ( M3 = N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_124_Suc__less__eq2,axiom,
% 3.15/1.62 ! [N: nat,M3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( suc @ N ) @ M3 )
% 3.15/1.62 = ( ? [M9: nat] :
% 3.15/1.62 ( ( M3
% 3.15/1.62 = ( suc @ M9 ) )
% 3.15/1.62 & ( ord_less @ nat @ N @ M9 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_125_All__less__Suc,axiom,
% 3.15/1.62 ! [N: nat,P: nat > $o] :
% 3.15/1.62 ( ( ! [I3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 3.15/1.62 => ( P @ I3 ) ) )
% 3.15/1.62 = ( ( P @ N )
% 3.15/1.62 & ! [I3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I3 @ N )
% 3.15/1.62 => ( P @ I3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_126_not__less__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ~ ( ord_less @ nat @ M3 @ N ) )
% 3.15/1.62 = ( ord_less @ nat @ N @ ( suc @ M3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_127_less__Suc__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.62 = ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 | ( M3 = N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_128_Ex__less__Suc,axiom,
% 3.15/1.62 ! [N: nat,P: nat > $o] :
% 3.15/1.62 ( ( ? [I3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 3.15/1.62 & ( P @ I3 ) ) )
% 3.15/1.62 = ( ( P @ N )
% 3.15/1.62 | ? [I3: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I3 @ N )
% 3.15/1.62 & ( P @ I3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_129_less__SucI,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_130_less__SucE,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.62 => ( ~ ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( M3 = N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_131_Suc__lessI,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( ( ( suc @ M3 )
% 3.15/1.62 != N )
% 3.15/1.62 => ( ord_less @ nat @ ( suc @ M3 ) @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_132_Suc__lessE,axiom,
% 3.15/1.62 ! [I: nat,K2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( suc @ I ) @ K2 )
% 3.15/1.62 => ~ ! [J4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ J4 )
% 3.15/1.62 => ( K2
% 3.15/1.62 != ( suc @ J4 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_133_Suc__lessD,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( suc @ M3 ) @ N )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_134_Nat_OlessE,axiom,
% 3.15/1.62 ! [I: nat,K2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ K2 )
% 3.15/1.62 => ( ( K2
% 3.15/1.62 != ( suc @ I ) )
% 3.15/1.62 => ~ ! [J4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ I @ J4 )
% 3.15/1.62 => ( K2
% 3.15/1.62 != ( suc @ J4 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_135_infinite__descent0__measure,axiom,
% 3.15/1.62 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 3.15/1.62 ( ! [X2: A] :
% 3.15/1.62 ( ( ( V @ X2 )
% 3.15/1.62 = ( zero_zero @ nat ) )
% 3.15/1.62 => ( P @ X2 ) )
% 3.15/1.62 => ( ! [X2: A] :
% 3.15/1.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 3.15/1.62 => ( ~ ( P @ X2 )
% 3.15/1.62 => ? [Y: A] :
% 3.15/1.62 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 3.15/1.62 & ~ ( P @ Y ) ) ) )
% 3.15/1.62 => ( P @ X ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_136_bot__nat__0_Oextremum__strict,axiom,
% 3.15/1.62 ! [A2: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_137_infinite__descent0,axiom,
% 3.15/1.62 ! [P: nat > $o,N: nat] :
% 3.15/1.62 ( ( P @ ( zero_zero @ nat ) )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.15/1.62 => ( ~ ( P @ N2 )
% 3.15/1.62 => ? [M4: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M4 @ N2 )
% 3.15/1.62 & ~ ( P @ M4 ) ) ) )
% 3.15/1.62 => ( P @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_138_gr__implies__not0,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( N
% 3.15/1.62 != ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_139_less__zeroE,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_140_not__less0,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_141_not__gr0,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 3.15/1.62 = ( N
% 3.15/1.62 = ( zero_zero @ nat ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_142_gr0I,axiom,
% 3.15/1.62 ! [N: nat] :
% 3.15/1.62 ( ( N
% 3.15/1.62 != ( zero_zero @ nat ) )
% 3.15/1.62 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_143_diff__le__mono2,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,L2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_144_le__diff__iff_H,axiom,
% 3.15/1.62 ! [A2: nat,C2: nat,B2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ A2 @ C2 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ B2 @ C2 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_145_diff__le__self,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_146_diff__le__mono,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,L2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ L2 ) @ ( minus_minus @ nat @ N @ L2 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_147_Nat_Odiff__diff__eq,axiom,
% 3.15/1.62 ! [K2: nat,M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ K2 @ N )
% 3.15/1.62 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 3.15/1.62 = ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_148_le__diff__iff,axiom,
% 3.15/1.62 ! [K2: nat,M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ K2 @ N )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ M3 @ N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_149_eq__diff__iff,axiom,
% 3.15/1.62 ! [K2: nat,M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ K2 @ N )
% 3.15/1.62 => ( ( ( minus_minus @ nat @ M3 @ K2 )
% 3.15/1.62 = ( minus_minus @ nat @ N @ K2 ) )
% 3.15/1.62 = ( M3 = N ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_150_zero__induct__lemma,axiom,
% 3.15/1.62 ! [P: nat > $o,K2: nat,I: nat] :
% 3.15/1.62 ( ( P @ K2 )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( P @ ( suc @ N2 ) )
% 3.15/1.62 => ( P @ N2 ) )
% 3.15/1.62 => ( P @ ( minus_minus @ nat @ K2 @ I ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_151_minus__nat_Odiff__0,axiom,
% 3.15/1.62 ! [M3: nat] :
% 3.15/1.62 ( ( minus_minus @ nat @ M3 @ ( zero_zero @ nat ) )
% 3.15/1.62 = M3 ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_152_diffs0__imp__equal,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ( minus_minus @ nat @ M3 @ N )
% 3.15/1.62 = ( zero_zero @ nat ) )
% 3.15/1.62 => ( ( ( minus_minus @ nat @ N @ M3 )
% 3.15/1.62 = ( zero_zero @ nat ) )
% 3.15/1.62 => ( M3 = N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_153_less__imp__diff__less,axiom,
% 3.15/1.62 ! [J3: nat,K2: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ J3 @ K2 )
% 3.15/1.62 => ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ N ) @ K2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_154_diff__less__mono2,axiom,
% 3.15/1.62 ! [M3: nat,N: nat,L2: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.62 => ( ( ord_less @ nat @ M3 @ L2 )
% 3.15/1.62 => ( ord_less @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_155_Module_Osc__Ring,axiom,
% 3.15/1.62 ! [C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ring @ B @ D @ R2 ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_156_Module_Omem__smodule__ideal__coeff,axiom,
% 3.15/1.62 ! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,X: A] :
% 3.15/1.62 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.62 => ( ( ideal @ B @ D @ R2 @ A3 )
% 3.15/1.62 => ( ( member @ A @ X @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) )
% 3.15/1.62 => ? [N2: nat,X2: nat > B] :
% 3.15/1.62 ( ( member @ ( nat > B ) @ X2
% 3.15/1.62 @ ( pi @ nat @ B
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.62 & ? [Xa: nat > A] :
% 3.15/1.62 ( ( member @ ( nat > A ) @ Xa
% 3.15/1.62 @ ( pi @ nat @ A
% 3.15/1.62 @ ( collect @ nat
% 3.15/1.62 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
% 3.15/1.62 @ ^ [Uu: nat] : ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) )
% 3.15/1.62 & ( X
% 3.15/1.62 = ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_157_lift__Suc__antimono__le,axiom,
% 3.15/1.62 ! [A: $tType] :
% 3.15/1.62 ( ( order @ A )
% 3.15/1.62 => ! [F: nat > A,N: nat,N5: nat] :
% 3.15/1.62 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ N @ N5 )
% 3.15/1.62 => ( ord_less_eq @ A @ ( F @ N5 ) @ ( F @ N ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_158_lift__Suc__mono__le,axiom,
% 3.15/1.62 ! [A: $tType] :
% 3.15/1.62 ( ( order @ A )
% 3.15/1.62 => ! [F: nat > A,N: nat,N5: nat] :
% 3.15/1.62 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.15/1.62 => ( ( ord_less_eq @ nat @ N @ N5 )
% 3.15/1.62 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_159_lift__Suc__mono__less__iff,axiom,
% 3.15/1.62 ! [A: $tType] :
% 3.15/1.62 ( ( order @ A )
% 3.15/1.62 => ! [F: nat > A,N: nat,M3: nat] :
% 3.15/1.62 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.15/1.62 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M3 ) )
% 3.15/1.62 = ( ord_less @ nat @ N @ M3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_160_lift__Suc__mono__less,axiom,
% 3.15/1.62 ! [A: $tType] :
% 3.15/1.62 ( ( order @ A )
% 3.15/1.62 => ! [F: nat > A,N: nat,N5: nat] :
% 3.15/1.62 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 3.15/1.62 => ( ( ord_less @ nat @ N @ N5 )
% 3.15/1.62 => ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_161_le__imp__less__Suc,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_162_less__eq__Suc__le,axiom,
% 3.15/1.62 ( ( ord_less @ nat )
% 3.15/1.62 = ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_163_less__Suc__eq__le,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.62 = ( ord_less_eq @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_164_le__less__Suc__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ M3 @ N )
% 3.15/1.62 => ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
% 3.15/1.62 = ( N = M3 ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_165_Suc__le__lessD,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 3.15/1.62 => ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_166_inc__induct,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,P: nat > $o] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ J3 )
% 3.15/1.62 => ( ( P @ J3 )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ N2 )
% 3.15/1.62 => ( ( ord_less @ nat @ N2 @ J3 )
% 3.15/1.62 => ( ( P @ ( suc @ N2 ) )
% 3.15/1.62 => ( P @ N2 ) ) ) )
% 3.15/1.62 => ( P @ I ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_167_dec__induct,axiom,
% 3.15/1.62 ! [I: nat,J3: nat,P: nat > $o] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ J3 )
% 3.15/1.62 => ( ( P @ I )
% 3.15/1.62 => ( ! [N2: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ I @ N2 )
% 3.15/1.62 => ( ( ord_less @ nat @ N2 @ J3 )
% 3.15/1.62 => ( ( P @ N2 )
% 3.15/1.62 => ( P @ ( suc @ N2 ) ) ) ) )
% 3.15/1.62 => ( P @ J3 ) ) ) ) ).
% 3.15/1.62
% 3.15/1.62 thf(fact_168_Suc__le__eq,axiom,
% 3.15/1.62 ! [M3: nat,N: nat] :
% 3.15/1.62 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
% 3.15/1.63 = ( ord_less @ nat @ M3 @ N ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_169_Suc__leI,axiom,
% 3.15/1.63 ! [M3: nat,N: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ M3 @ N )
% 3.15/1.63 => ( ord_less_eq @ nat @ ( suc @ M3 ) @ N ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_170_ex__least__nat__le,axiom,
% 3.15/1.63 ! [P: nat > $o,N: nat] :
% 3.15/1.63 ( ( P @ N )
% 3.15/1.63 => ( ~ ( P @ ( zero_zero @ nat ) )
% 3.15/1.63 => ? [K3: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ K3 @ N )
% 3.15/1.63 & ! [I4: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ I4 @ K3 )
% 3.15/1.63 => ~ ( P @ I4 ) )
% 3.15/1.63 & ( P @ K3 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_171_less__Suc__eq__0__disj,axiom,
% 3.15/1.63 ! [M3: nat,N: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ M3 @ ( suc @ N ) )
% 3.15/1.63 = ( ( M3
% 3.15/1.63 = ( zero_zero @ nat ) )
% 3.15/1.63 | ? [J2: nat] :
% 3.15/1.63 ( ( M3
% 3.15/1.63 = ( suc @ J2 ) )
% 3.15/1.63 & ( ord_less @ nat @ J2 @ N ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_172_gr0__implies__Suc,axiom,
% 3.15/1.63 ! [N: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.63 => ? [M7: nat] :
% 3.15/1.63 ( N
% 3.15/1.63 = ( suc @ M7 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_173_All__less__Suc2,axiom,
% 3.15/1.63 ! [N: nat,P: nat > $o] :
% 3.15/1.63 ( ( ! [I3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 3.15/1.63 => ( P @ I3 ) ) )
% 3.15/1.63 = ( ( P @ ( zero_zero @ nat ) )
% 3.15/1.63 & ! [I3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ I3 @ N )
% 3.15/1.63 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_174_gr0__conv__Suc,axiom,
% 3.15/1.63 ! [N: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.63 = ( ? [M8: nat] :
% 3.15/1.63 ( N
% 3.15/1.63 = ( suc @ M8 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_175_Ex__less__Suc2,axiom,
% 3.15/1.63 ! [N: nat,P: nat > $o] :
% 3.15/1.63 ( ( ? [I3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 3.15/1.63 & ( P @ I3 ) ) )
% 3.15/1.63 = ( ( P @ ( zero_zero @ nat ) )
% 3.15/1.63 | ? [I3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ I3 @ N )
% 3.15/1.63 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_176_Suc__diff__le,axiom,
% 3.15/1.63 ! [N: nat,M3: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ N @ M3 )
% 3.15/1.63 => ( ( minus_minus @ nat @ ( suc @ M3 ) @ N )
% 3.15/1.63 = ( suc @ ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_177_diff__less__mono,axiom,
% 3.15/1.63 ! [A2: nat,B2: nat,C2: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ A2 @ B2 )
% 3.15/1.63 => ( ( ord_less_eq @ nat @ C2 @ A2 )
% 3.15/1.63 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_178_less__diff__iff,axiom,
% 3.15/1.63 ! [K2: nat,M3: nat,N: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ K2 @ M3 )
% 3.15/1.63 => ( ( ord_less_eq @ nat @ K2 @ N )
% 3.15/1.63 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
% 3.15/1.63 = ( ord_less @ nat @ M3 @ N ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_179_diff__less__Suc,axiom,
% 3.15/1.63 ! [M3: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_180_Suc__diff__Suc,axiom,
% 3.15/1.63 ! [N: nat,M3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ N @ M3 )
% 3.15/1.63 => ( ( suc @ ( minus_minus @ nat @ M3 @ ( suc @ N ) ) )
% 3.15/1.63 = ( minus_minus @ nat @ M3 @ N ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_181_diff__less,axiom,
% 3.15/1.63 ! [N: nat,M3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.63 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
% 3.15/1.63 => ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_182_Module_Osubmodule__subset1,axiom,
% 3.15/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,H2: A] :
% 3.15/1.63 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.63 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.63 => ( ( member @ A @ H2 @ H )
% 3.15/1.63 => ( member @ A @ H2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_183_Module_Osubmodule__whole,axiom,
% 3.15/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
% 3.15/1.63 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.63 => ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_184_ex__least__nat__less,axiom,
% 3.15/1.63 ! [P: nat > $o,N: nat] :
% 3.15/1.63 ( ( P @ N )
% 3.15/1.63 => ( ~ ( P @ ( zero_zero @ nat ) )
% 3.15/1.63 => ? [K3: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ K3 @ N )
% 3.15/1.63 & ! [I4: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ I4 @ K3 )
% 3.15/1.63 => ~ ( P @ I4 ) )
% 3.15/1.63 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_185_diff__Suc__less,axiom,
% 3.15/1.63 ! [N: nat,I: nat] :
% 3.15/1.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 3.15/1.63 => ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_186_Module_Osubmodule__subset,axiom,
% 3.15/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 3.15/1.63 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.63 => ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.63 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_187_Module_Ofree__generator__sub,axiom,
% 3.15/1.63 ! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
% 3.15/1.63 ( ( module @ A @ B @ C @ D @ M5 @ R2 )
% 3.15/1.63 => ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
% 3.15/1.63 => ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_188_eSum__in__SubmoduleTr,axiom,
% 3.15/1.63 ! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 3.15/1.63 => ( ( ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : K ) )
% 3.15/1.63 & ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
% 3.15/1.63 => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
% 3.15/1.63 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_189_eSum__in__Submodule,axiom,
% 3.15/1.63 ! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 3.15/1.63 => ( ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : K ) )
% 3.15/1.63 => ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
% 3.15/1.63 => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
% 3.15/1.63 = ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_190_singleton__insert__inj__eq_H,axiom,
% 3.15/1.63 ! [A: $tType,A2: A,A3: set @ A,B2: A] :
% 3.15/1.63 ( ( ( insert @ A @ A2 @ A3 )
% 3.15/1.63 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.63 = ( ( A2 = B2 )
% 3.15/1.63 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_191_singleton__insert__inj__eq,axiom,
% 3.15/1.63 ! [A: $tType,B2: A,A2: A,A3: set @ A] :
% 3.15/1.63 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.63 = ( insert @ A @ A2 @ A3 ) )
% 3.15/1.63 = ( ( A2 = B2 )
% 3.15/1.63 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_192_diff__gt__0__iff__gt,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( ordered_ab_group_add @ A )
% 3.15/1.63 => ! [A2: A,B2: A] :
% 3.15/1.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 3.15/1.63 = ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_193_diff__ge__0__iff__ge,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( ordered_ab_group_add @ A )
% 3.15/1.63 => ! [A2: A,B2: A] :
% 3.15/1.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
% 3.15/1.63 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_194_l__span__closed2Tr,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 & ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) )
% 3.15/1.63 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_195_l__span__closed2,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 => ( ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) )
% 3.15/1.63 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_196_linear__comb0__2Tr,axiom,
% 3.15/1.63 ! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 & ( member @ ( nat > a ) @ M3
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
% 3.15/1.63 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
% 3.15/1.63 = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_197_module__inc__zero,axiom,
% 3.15/1.63 member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_198_psubsetI,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.15/1.63 => ( ( A3 != B3 )
% 3.15/1.63 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_199_subsetI,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ! [X2: A] :
% 3.15/1.63 ( ( member @ A @ X2 @ A3 )
% 3.15/1.63 => ( member @ A @ X2 @ B3 ) )
% 3.15/1.63 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_200_subset__antisym,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 3.15/1.63 => ( A3 = B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_201_Diff__empty,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] :
% 3.15/1.63 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.63 = A3 ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_202_empty__Diff,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] :
% 3.15/1.63 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_203_Diff__cancel,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] :
% 3.15/1.63 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_204_empty__iff,axiom,
% 3.15/1.63 ! [A: $tType,C2: A] :
% 3.15/1.63 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_205_all__not__in__conv,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] :
% 3.15/1.63 ( ( ! [X3: A] :
% 3.15/1.63 ~ ( member @ A @ X3 @ A3 ) )
% 3.15/1.63 = ( A3
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_206_Collect__empty__eq,axiom,
% 3.15/1.63 ! [A: $tType,P: A > $o] :
% 3.15/1.63 ( ( ( collect @ A @ P )
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.63 = ( ! [X3: A] :
% 3.15/1.63 ~ ( P @ X3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_207_empty__Collect__eq,axiom,
% 3.15/1.63 ! [A: $tType,P: A > $o] :
% 3.15/1.63 ( ( ( bot_bot @ ( set @ A ) )
% 3.15/1.63 = ( collect @ A @ P ) )
% 3.15/1.63 = ( ! [X3: A] :
% 3.15/1.63 ~ ( P @ X3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_208_Diff__insert0,axiom,
% 3.15/1.63 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ~ ( member @ A @ X @ A3 )
% 3.15/1.63 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
% 3.15/1.63 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_209_insert__Diff1,axiom,
% 3.15/1.63 ! [A: $tType,X: A,B3: set @ A,A3: set @ A] :
% 3.15/1.63 ( ( member @ A @ X @ B3 )
% 3.15/1.63 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
% 3.15/1.63 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_210_insertCI,axiom,
% 3.15/1.63 ! [A: $tType,A2: A,B3: set @ A,B2: A] :
% 3.15/1.63 ( ( ~ ( member @ A @ A2 @ B3 )
% 3.15/1.63 => ( A2 = B2 ) )
% 3.15/1.63 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_211_insert__iff,axiom,
% 3.15/1.63 ! [A: $tType,A2: A,B2: A,A3: set @ A] :
% 3.15/1.63 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
% 3.15/1.63 = ( ( A2 = B2 )
% 3.15/1.63 | ( member @ A @ A2 @ A3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_212_insert__absorb2,axiom,
% 3.15/1.63 ! [A: $tType,X: A,A3: set @ A] :
% 3.15/1.63 ( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
% 3.15/1.63 = ( insert @ A @ X @ A3 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_213_submodule__inc__0,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ H ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_214_mdl__carrier,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ H ) )
% 3.15/1.63 = H ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_215_mdl__is__module,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ H ) @ r ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_216_submodule__of__mdl,axiom,
% 3.15/1.63 ! [H: set @ a,N3: set @ a] :
% 3.15/1.63 ( ( submodule @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
% 3.15/1.63 => ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ N3 ) @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_217_h__in__linear__span,axiom,
% 3.15/1.63 ! [H: set @ a,H2: a] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( member @ a @ H2 @ H )
% 3.15/1.63 => ( member @ a @ H2 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_218_l__span__cont__H,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_219_l__span__l__span,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
% 3.15/1.63 = ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_220_l__span__gen__mono,axiom,
% 3.15/1.63 ! [K: set @ a,H: set @ a,A3: set @ b] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ K @ H )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ K ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_221_linear__span__sub,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_222_l__span__sub__submodule,axiom,
% 3.15/1.63 ! [A3: set @ b,N3: set @ a,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
% 3.15/1.63 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ N3 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_223_fgs__generator,axiom,
% 3.15/1.63 ! [H: set @ a] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ H ) ) @ H ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_224_submodule__0,axiom,
% 3.15/1.63 submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_225_empty__fgs,axiom,
% 3.15/1.63 ( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
% 3.15/1.63 = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_226_le__zero__eq,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( canoni770627133id_add @ A )
% 3.15/1.63 => ! [N: A] :
% 3.15/1.63 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.15/1.63 = ( N
% 3.15/1.63 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_227_not__gr__zero,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( canoni770627133id_add @ A )
% 3.15/1.63 => ! [N: A] :
% 3.15/1.63 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 3.15/1.63 = ( N
% 3.15/1.63 = ( zero_zero @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_228_diff__self,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( group_add @ A )
% 3.15/1.63 => ! [A2: A] :
% 3.15/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 3.15/1.63 = ( zero_zero @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_229_diff__0__right,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( group_add @ A )
% 3.15/1.63 => ! [A2: A] :
% 3.15/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 3.15/1.63 = A2 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_230_zero__diff,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( comm_monoid_diff @ A )
% 3.15/1.63 => ! [A2: A] :
% 3.15/1.63 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 3.15/1.63 = ( zero_zero @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_231_diff__zero,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( cancel1352612707id_add @ A )
% 3.15/1.63 => ! [A2: A] :
% 3.15/1.63 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 3.15/1.63 = A2 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_232_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 3.15/1.63 ! [A: $tType] :
% 3.15/1.63 ( ( cancel1352612707id_add @ A )
% 3.15/1.63 => ! [A2: A] :
% 3.15/1.63 ( ( minus_minus @ A @ A2 @ A2 )
% 3.15/1.63 = ( zero_zero @ A ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_233_Diff__eq__empty__iff,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.63 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_234_subset__empty,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.63 = ( A3
% 3.15/1.63 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_235_empty__subsetI,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_236_insert__subset,axiom,
% 3.15/1.63 ! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
% 3.15/1.63 = ( ( member @ A @ X @ B3 )
% 3.15/1.63 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_237_insert__Diff__single,axiom,
% 3.15/1.63 ! [A: $tType,A2: A,A3: set @ A] :
% 3.15/1.63 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.15/1.63 = ( insert @ A @ A2 @ A3 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_238_singletonI,axiom,
% 3.15/1.63 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_239_linear__span__subModule,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_240_l__spanA__l__span,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
% 3.15/1.63 = ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_241_lin__span__coeff__mono,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_242_linear__span__inc__0,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_243_generator__generator,axiom,
% 3.15/1.63 ! [H: set @ a,H1: set @ a] :
% 3.15/1.63 ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H1 ) )
% 3.15/1.63 => ( generator @ b @ d @ a @ c @ r @ m @ H1 ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_244_l__span__zero,axiom,
% 3.15/1.63 ! [A3: set @ b] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
% 3.15/1.63 = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_245_l__span__closed3,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( generator @ b @ d @ a @ c @ r @ m @ H )
% 3.15/1.63 => ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 )
% 3.15/1.63 = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H )
% 3.15/1.63 = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_246_singleton__conv,axiom,
% 3.15/1.63 ! [A: $tType,A2: A] :
% 3.15/1.63 ( ( collect @ A
% 3.15/1.63 @ ^ [X3: A] : ( X3 = A2 ) )
% 3.15/1.63 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_247_singleton__conv2,axiom,
% 3.15/1.63 ! [A: $tType,A2: A] :
% 3.15/1.63 ( ( collect @ A
% 3.15/1.63 @ ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 )
% 3.15/1.63 @ A2 ) )
% 3.15/1.63 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_248_l__comb__mem__linear__span,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 => ( ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : H ) )
% 3.15/1.63 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_249_l__span__closed,axiom,
% 3.15/1.63 ! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
% 3.15/1.63 => ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 => ( ( member @ ( nat > a ) @ F
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) )
% 3.15/1.63 => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_250_linear__comb0__2,axiom,
% 3.15/1.63 ! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
% 3.15/1.63 ( ( ideal @ b @ d @ r @ A3 )
% 3.15/1.63 => ( ( member @ ( nat > b ) @ S
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : A3 ) )
% 3.15/1.63 => ( ( member @ ( nat > a ) @ M3
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
% 3.15/1.63 => ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
% 3.15/1.63 = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_251_double__diff,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.15/1.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.15/1.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 3.15/1.63 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 3.15/1.63 = A3 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_252_Diff__subset,axiom,
% 3.15/1.63 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.15/1.63 ! [A4: $tType,A5: $tType] :
% 3.15/1.63 ( ( order @ A5 )
% 3.15/1.63 => ( order @ ( A4 > A5 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.15/1.63 canoni770627133id_add @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 3.15/1.63 cancel1352612707id_add @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 3.15/1.63 comm_monoid_diff @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.15/1.63 wellorder @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 3.15/1.63 order @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 3.15/1.63 zero @ nat ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 3.15/1.63 ! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
% 3.15/1.63 order @ $o ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_4,axiom,
% 3.15/1.63 wellorder @ product_unit ).
% 3.15/1.63
% 3.15/1.63 thf(tcon_Product__Type_Ounit___Orderings_Oorder_5,axiom,
% 3.15/1.63 order @ product_unit ).
% 3.15/1.63
% 3.15/1.63 thf(conj_0,hypothesis,
% 3.15/1.63 ( member @ ( nat > a ) @ g
% 3.15/1.63 @ ( pi @ nat @ a
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( insert @ a @ h2 @ h1 ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_1,hypothesis,
% 3.15/1.63 ! [X4: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ X4 @ m2 )
% 3.15/1.63 => ! [Y: nat] :
% 3.15/1.63 ( ( ord_less_eq @ nat @ Y @ m2 )
% 3.15/1.63 => ( ( ( g @ X4 )
% 3.15/1.63 = ( g @ Y ) )
% 3.15/1.63 => ( X4 = Y ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_2,hypothesis,
% 3.15/1.63 ( member @ ( nat > b ) @ t
% 3.15/1.63 @ ( pi @ nat @ b
% 3.15/1.63 @ ( collect @ nat
% 3.15/1.63 @ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
% 3.15/1.63 @ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_3,hypothesis,
% 3.15/1.63 ( ( g @ m2 )
% 3.15/1.63 = h2 ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_4,hypothesis,
% 3.15/1.63 module @ e @ b @ f @ d @ n @ r ).
% 3.15/1.63
% 3.15/1.63 thf(conj_5,hypothesis,
% 3.15/1.63 free_generator @ b @ d @ a @ c @ r @ m @ h ).
% 3.15/1.63
% 3.15/1.63 thf(conj_6,hypothesis,
% 3.15/1.63 ord_less_eq @ ( set @ a ) @ h1 @ h ).
% 3.15/1.63
% 3.15/1.63 thf(conj_7,hypothesis,
% 3.15/1.63 $true ).
% 3.15/1.63
% 3.15/1.63 thf(conj_8,hypothesis,
% 3.15/1.63 ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_9,hypothesis,
% 3.15/1.63 ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_10,hypothesis,
% 3.15/1.63 ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_11,hypothesis,
% 3.15/1.63 ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_12,hypothesis,
% 3.15/1.63 member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_13,hypothesis,
% 3.15/1.63 ( x
% 3.15/1.63 = ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_14,hypothesis,
% 3.15/1.63 member @ a @ h2 @ h ).
% 3.15/1.63
% 3.15/1.63 thf(conj_15,hypothesis,
% 3.15/1.63 ~ ( member @ a @ h2 @ h1 ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_16,hypothesis,
% 3.15/1.63 ord_less @ nat @ ( zero_zero @ nat ) @ m2 ).
% 3.15/1.63
% 3.15/1.63 thf(conj_17,hypothesis,
% 3.15/1.63 member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_18,hypothesis,
% 3.15/1.63 submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_19,hypothesis,
% 3.15/1.63 ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_20,hypothesis,
% 3.15/1.63 ( ( ( g @ xa )
% 3.15/1.63 = h2 )
% 3.15/1.63 | ( member @ a @ ( g @ xa ) @ h1 ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(conj_21,conjecture,
% 3.15/1.63 member @ a @ ( g @ xa ) @ h1 ).
% 3.15/1.63 ------------------------
% 3.15/1.66 % (27425)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_27230 for (2999ds/4Mi)
% 3.15/1.66 % (27424)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/183Mi)
% 3.15/1.66 % (27426)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_27230 for (2999ds/27Mi)
% 3.15/1.66 % (27427)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/2Mi)
% 3.15/1.66 % (27428)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_27230 for (2999ds/2Mi)
% 3.15/1.66 % (27425)Instruction limit reached!
% 3.15/1.66 % (27425)------------------------------
% 3.15/1.66 % (27425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.66 % (27425)Termination reason: Unknown
% 3.15/1.66 % (27425)Termination phase: shuffling
% 3.15/1.66
% 3.15/1.66 % (27425)Memory used [KB]: 1407
% 3.15/1.66 % (27425)Time elapsed: 0.003 s
% 3.15/1.66 % (27425)Instructions burned: 5 (million)
% 3.15/1.66 % (27425)------------------------------
% 3.15/1.66 % (27425)------------------------------
% 3.15/1.66 % (27427)Instruction limit reached!
% 3.15/1.66 % (27427)------------------------------
% 3.15/1.66 % (27427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.66 % (27427)Termination reason: Unknown
% 3.15/1.66 % (27427)Termination phase: shuffling
% 3.15/1.66
% 3.15/1.66 % (27427)Memory used [KB]: 1407
% 3.15/1.66 % (27427)Time elapsed: 0.002 s
% 3.15/1.66 % (27427)Instructions burned: 3 (million)
% 3.15/1.66 % (27427)------------------------------
% 3.15/1.66 % (27427)------------------------------
% 3.15/1.66 % (27428)Instruction limit reached!
% 3.15/1.66 % (27428)------------------------------
% 3.15/1.66 % (27428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.66 % (27428)Termination reason: Unknown
% 3.15/1.66 % (27428)Termination phase: shuffling
% 3.15/1.66
% 3.15/1.66 % (27428)Memory used [KB]: 1407
% 3.15/1.66 % (27428)Time elapsed: 0.002 s
% 3.15/1.66 % (27428)Instructions burned: 3 (million)
% 3.15/1.66 % (27428)------------------------------
% 3.15/1.66 % (27428)------------------------------
% 3.15/1.67 % (27430)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/18Mi)
% 3.15/1.67 % (27426)Instruction limit reached!
% 3.15/1.67 % (27426)------------------------------
% 3.15/1.67 % (27426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.67 % (27426)Termination reason: Unknown
% 3.15/1.67 % (27426)Termination phase: shuffling
% 3.15/1.67
% 3.15/1.67 % (27426)Memory used [KB]: 1791
% 3.15/1.67 % (27426)Time elapsed: 0.011 s
% 3.15/1.67 % (27426)Instructions burned: 28 (million)
% 3.15/1.67 % (27426)------------------------------
% 3.15/1.67 % (27426)------------------------------
% 3.15/1.67 % (27430)Instruction limit reached!
% 3.15/1.67 % (27430)------------------------------
% 3.15/1.67 % (27430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.67 % (27430)Termination reason: Unknown
% 3.15/1.67 % (27430)Termination phase: shuffling
% 3.15/1.67
% 3.15/1.67 % (27429)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_27230 for (2999ds/275Mi)
% 3.15/1.67 % (27430)Memory used [KB]: 1535
% 3.15/1.67 % (27430)Time elapsed: 0.008 s
% 3.15/1.67 % (27430)Instructions burned: 18 (million)
% 3.15/1.67 % (27430)------------------------------
% 3.15/1.67 % (27430)------------------------------
% 3.15/1.67 % (27434)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/3Mi)
% 3.15/1.67 % (27435)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_27230 for (2999ds/37Mi)
% 3.15/1.67 % (27436)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_27230 for (2999ds/15Mi)
% 3.15/1.67 % (27434)Instruction limit reached!
% 3.15/1.67 % (27434)------------------------------
% 3.15/1.67 % (27434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.67 % (27434)Termination reason: Unknown
% 3.15/1.67 % (27434)Termination phase: shuffling
% 3.15/1.67
% 3.15/1.67 % (27434)Memory used [KB]: 1407
% 3.15/1.67 % (27434)Time elapsed: 0.003 s
% 3.15/1.67 % (27434)Instructions burned: 5 (million)
% 3.15/1.67 % (27434)------------------------------
% 3.15/1.67 % (27434)------------------------------
% 3.15/1.68 % (27436)Instruction limit reached!
% 3.15/1.68 % (27436)------------------------------
% 3.15/1.68 % (27436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.68 % (27436)Termination reason: Unknown
% 3.15/1.68 % (27436)Termination phase: shuffling
% 3.15/1.68
% 3.15/1.68 % (27436)Memory used [KB]: 1535
% 3.15/1.68 % (27436)Time elapsed: 0.007 s
% 3.15/1.68 % (27436)Instructions burned: 15 (million)
% 3.15/1.68 % (27436)------------------------------
% 3.15/1.68 % (27436)------------------------------
% 3.15/1.68 % (27437)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_27230 for (2999ds/3Mi)
% 3.15/1.68 % (27438)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_27230 for (2999ds/1041Mi)
% 3.15/1.68 % (27437)Instruction limit reached!
% 3.15/1.68 % (27437)------------------------------
% 3.15/1.68 % (27437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.68 % (27437)Termination reason: Unknown
% 3.15/1.68 % (27437)Termination phase: shuffling
% 3.15/1.68
% 3.15/1.68 % (27437)Memory used [KB]: 1407
% 3.15/1.68 % (27437)Time elapsed: 0.003 s
% 3.15/1.68 % (27437)Instructions burned: 4 (million)
% 3.15/1.68 % (27437)------------------------------
% 3.15/1.68 % (27437)------------------------------
% 3.15/1.68 % (27439)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/7Mi)
% 3.15/1.69 % (27435)Instruction limit reached!
% 3.15/1.69 % (27435)------------------------------
% 3.15/1.69 % (27435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.69 % (27435)Termination reason: Unknown
% 3.15/1.69 % (27435)Termination phase: shuffling
% 3.15/1.69
% 3.15/1.69 % (27435)Memory used [KB]: 2046
% 3.15/1.69 % (27435)Time elapsed: 0.013 s
% 3.15/1.69 % (27435)Instructions burned: 37 (million)
% 3.15/1.69 % (27435)------------------------------
% 3.15/1.69 % (27435)------------------------------
% 3.15/1.69 % (27439)Instruction limit reached!
% 3.15/1.69 % (27439)------------------------------
% 3.15/1.69 % (27439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.69 % (27439)Termination reason: Unknown
% 3.15/1.69 % (27439)Termination phase: shuffling
% 3.15/1.69
% 3.15/1.69 % (27439)Memory used [KB]: 1407
% 3.15/1.69 % (27439)Time elapsed: 0.004 s
% 3.15/1.69 % (27439)Instructions burned: 8 (million)
% 3.15/1.69 % (27439)------------------------------
% 3.15/1.69 % (27439)------------------------------
% 3.15/1.69 % (27440)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/16Mi)
% 3.15/1.69 % (27441)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/3Mi)
% 3.15/1.70 % (27442)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/3Mi)
% 3.15/1.70 % (27441)Instruction limit reached!
% 3.15/1.70 % (27441)------------------------------
% 3.15/1.70 % (27441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.70 % (27441)Termination reason: Unknown
% 3.15/1.70 % (27441)Termination phase: shuffling
% 3.15/1.70
% 3.15/1.70 % (27441)Memory used [KB]: 1407
% 3.15/1.70 % (27441)Time elapsed: 0.003 s
% 3.15/1.70 % (27441)Instructions burned: 3 (million)
% 3.15/1.70 % (27441)------------------------------
% 3.15/1.70 % (27441)------------------------------
% 3.15/1.70 % (27440)Instruction limit reached!
% 3.15/1.70 % (27440)------------------------------
% 3.15/1.70 % (27440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.70 % (27440)Termination reason: Unknown
% 3.15/1.70 % (27440)Termination phase: shuffling
% 3.15/1.70
% 3.15/1.70 % (27440)Memory used [KB]: 1535
% 3.15/1.70 % (27440)Time elapsed: 0.008 s
% 3.15/1.70 % (27440)Instructions burned: 18 (million)
% 3.15/1.70 % (27440)------------------------------
% 3.15/1.70 % (27440)------------------------------
% 3.20/1.70 % (27442)Instruction limit reached!
% 3.20/1.70 % (27442)------------------------------
% 3.20/1.70 % (27442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.70 % (27442)Termination reason: Unknown
% 3.20/1.70 % (27442)Termination phase: shuffling
% 3.20/1.70
% 3.20/1.70 % (27442)Memory used [KB]: 1407
% 3.20/1.70 % (27442)Time elapsed: 0.003 s
% 3.20/1.70 % (27442)Instructions burned: 3 (million)
% 3.20/1.70 % (27442)------------------------------
% 3.20/1.70 % (27442)------------------------------
% 3.20/1.70 % (27443)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/7Mi)
% 3.20/1.70 % (27443)Instruction limit reached!
% 3.20/1.70 % (27443)------------------------------
% 3.20/1.70 % (27443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.70 % (27443)Termination reason: Unknown
% 3.20/1.70 % (27443)Termination phase: shuffling
% 3.20/1.70
% 3.20/1.70 % (27443)Memory used [KB]: 1535
% 3.20/1.70 % (27443)Time elapsed: 0.004 s
% 3.20/1.70 % (27443)Instructions burned: 9 (million)
% 3.20/1.70 % (27443)------------------------------
% 3.20/1.70 % (27443)------------------------------
% 3.20/1.71 % (27445)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/4Mi)
% 3.20/1.71 % (27444)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/3Mi)
% 3.20/1.71 % (27446)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/18Mi)
% 3.20/1.71 % (27445)Instruction limit reached!
% 3.20/1.71 % (27445)------------------------------
% 3.20/1.71 % (27445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (27445)Termination reason: Unknown
% 3.20/1.71 % (27445)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (27445)Memory used [KB]: 1407
% 3.20/1.71 % (27445)Time elapsed: 0.003 s
% 3.20/1.71 % (27445)Instructions burned: 4 (million)
% 3.20/1.71 % (27445)------------------------------
% 3.20/1.71 % (27445)------------------------------
% 3.20/1.71 % (27444)Instruction limit reached!
% 3.20/1.71 % (27444)------------------------------
% 3.20/1.71 % (27444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (27444)Termination reason: Unknown
% 3.20/1.71 % (27444)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (27444)Memory used [KB]: 1407
% 3.20/1.71 % (27444)Time elapsed: 0.003 s
% 3.20/1.71 % (27444)Instructions burned: 5 (million)
% 3.20/1.71 % (27444)------------------------------
% 3.20/1.71 % (27444)------------------------------
% 3.20/1.71 % (27447)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/710Mi)
% 3.20/1.71 % (27446)Instruction limit reached!
% 3.20/1.71 % (27446)------------------------------
% 3.20/1.71 % (27446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (27446)Termination reason: Unknown
% 3.20/1.71 % (27446)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (27446)Memory used [KB]: 1535
% 3.20/1.71 % (27446)Time elapsed: 0.008 s
% 3.20/1.71 % (27446)Instructions burned: 18 (million)
% 3.20/1.71 % (27446)------------------------------
% 3.20/1.71 % (27446)------------------------------
% 3.20/1.72 % (27450)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/6Mi)
% 3.20/1.72 % (27424)Instruction limit reached!
% 3.20/1.72 % (27424)------------------------------
% 3.20/1.72 % (27424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.72 % (27424)Termination reason: Unknown
% 3.20/1.72 % (27424)Termination phase: Saturation
% 3.20/1.72
% 3.20/1.72 % (27424)Memory used [KB]: 7036
% 3.20/1.72 % (27424)Time elapsed: 0.057 s
% 3.20/1.72 % (27424)Instructions burned: 183 (million)
% 3.20/1.72 % (27424)------------------------------
% 3.20/1.72 % (27424)------------------------------
% 3.20/1.72 % (27451)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/902Mi)
% 3.20/1.72 % (27450)Instruction limit reached!
% 3.20/1.72 % (27450)------------------------------
% 3.20/1.72 % (27450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.72 % (27450)Termination reason: Unknown
% 3.20/1.72 % (27450)Termination phase: shuffling
% 3.20/1.72
% 3.20/1.72 % (27450)Memory used [KB]: 1407
% 3.20/1.72 % (27450)Time elapsed: 0.004 s
% 3.20/1.72 % (27450)Instructions burned: 7 (million)
% 3.20/1.72 % (27450)------------------------------
% 3.20/1.72 % (27450)------------------------------
% 3.20/1.73 % (27452)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/21Mi)
% 3.20/1.73 % (27453)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/5Mi)
% 3.20/1.74 % (27453)Instruction limit reached!
% 3.20/1.74 % (27453)------------------------------
% 3.20/1.74 % (27453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.74 % (27453)Termination reason: Unknown
% 3.20/1.74 % (27453)Termination phase: shuffling
% 3.20/1.74
% 3.20/1.74 % (27453)Memory used [KB]: 1407
% 3.20/1.74 % (27453)Time elapsed: 0.004 s
% 3.20/1.74 % (27453)Instructions burned: 6 (million)
% 3.20/1.74 % (27453)------------------------------
% 3.20/1.74 % (27453)------------------------------
% 3.20/1.74 % (27461)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/6Mi)
% 3.20/1.74 % (27452)Instruction limit reached!
% 3.20/1.74 % (27452)------------------------------
% 3.20/1.74 % (27452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.74 % (27452)Termination reason: Unknown
% 3.20/1.74 % (27452)Termination phase: shuffling
% 3.20/1.74
% 3.20/1.74 % (27452)Memory used [KB]: 1663
% 3.20/1.74 % (27452)Time elapsed: 0.013 s
% 3.20/1.74 % (27452)Instructions burned: 22 (million)
% 3.20/1.74 % (27452)------------------------------
% 3.20/1.74 % (27452)------------------------------
% 3.20/1.74 % (27461)Instruction limit reached!
% 3.20/1.74 % (27461)------------------------------
% 3.20/1.74 % (27461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.74 % (27461)Termination reason: Unknown
% 3.20/1.74 % (27461)Termination phase: shuffling
% 3.20/1.74
% 3.20/1.74 % (27461)Memory used [KB]: 1407
% 3.20/1.74 % (27461)Time elapsed: 0.005 s
% 3.20/1.74 % (27461)Instructions burned: 6 (million)
% 3.20/1.74 % (27461)------------------------------
% 3.20/1.74 % (27461)------------------------------
% 3.20/1.75 % (27462)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/377Mi)
% 3.20/1.75 % (27467)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_27230 for (2998ds/19Mi)
% 3.20/1.76 % (27466)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_27230 for (2999ds/779Mi)
% 3.20/1.76 % (27467)Instruction limit reached!
% 3.20/1.76 % (27467)------------------------------
% 3.20/1.76 % (27467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.76 % (27467)Termination reason: Unknown
% 3.20/1.76 % (27467)Termination phase: shuffling
% 3.20/1.76
% 3.20/1.76 % (27467)Memory used [KB]: 1663
% 3.20/1.76 % (27467)Time elapsed: 0.007 s
% 3.20/1.76 % (27467)Instructions burned: 20 (million)
% 3.20/1.76 % (27467)------------------------------
% 3.20/1.76 % (27467)------------------------------
% 3.20/1.77 % (27472)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27230 for (2998ds/879Mi)
% 3.20/1.80 % (27429)Instruction limit reached!
% 3.20/1.80 % (27429)------------------------------
% 3.20/1.80 % (27429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.80 % (27429)Termination reason: Unknown
% 3.20/1.80 % (27429)Termination phase: Saturation
% 3.20/1.80
% 3.20/1.80 % (27429)Memory used [KB]: 7419
% 3.20/1.80 % (27429)Time elapsed: 0.133 s
% 3.20/1.80 % (27429)Instructions burned: 277 (million)
% 3.20/1.80 % (27429)------------------------------
% 3.20/1.80 % (27429)------------------------------
% 3.20/1.81 % (27482)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27230 for (2998ds/17Mi)
% 3.20/1.82 % (27482)Instruction limit reached!
% 3.20/1.82 % (27482)------------------------------
% 3.20/1.82 % (27482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.82 % (27482)Termination reason: Unknown
% 3.20/1.82 % (27482)Termination phase: shuffling
% 3.20/1.82
% 3.20/1.82 % (27482)Memory used [KB]: 1535
% 3.20/1.82 % (27482)Time elapsed: 0.007 s
% 3.20/1.82 % (27482)Instructions burned: 17 (million)
% 3.20/1.82 % (27482)------------------------------
% 3.20/1.82 % (27482)------------------------------
% 3.20/1.83 % (27483)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2998ds/3Mi)
% 3.20/1.83 % (27483)Instruction limit reached!
% 3.20/1.83 % (27483)------------------------------
% 3.20/1.83 % (27483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.83 % (27483)Termination reason: Unknown
% 3.20/1.83 % (27483)Termination phase: shuffling
% 3.20/1.83
% 3.20/1.83 % (27483)Memory used [KB]: 1407
% 3.20/1.83 % (27483)Time elapsed: 0.002 s
% 3.20/1.83 % (27483)Instructions burned: 3 (million)
% 3.20/1.83 % (27483)------------------------------
% 3.20/1.83 % (27483)------------------------------
% 3.20/1.84 % (27484)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_27230 for (2998ds/30Mi)
% 3.20/1.85 % (27484)Instruction limit reached!
% 3.20/1.85 % (27484)------------------------------
% 3.20/1.85 % (27484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.85 % (27484)Termination reason: Unknown
% 3.20/1.85 % (27484)Termination phase: shuffling
% 3.20/1.85
% 3.20/1.85 % (27484)Memory used [KB]: 1791
% 3.20/1.85 % (27484)Time elapsed: 0.012 s
% 3.20/1.85 % (27484)Instructions burned: 31 (million)
% 3.20/1.85 % (27484)------------------------------
% 3.20/1.85 % (27484)------------------------------
% 3.20/1.86 % (27485)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/127Mi)
% 3.20/1.88 % (27462)Instruction limit reached!
% 3.20/1.88 % (27462)------------------------------
% 3.20/1.88 % (27462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.88 % (27462)Termination reason: Unknown
% 3.20/1.88 % (27462)Termination phase: Saturation
% 3.20/1.88
% 3.20/1.88 % (27462)Memory used [KB]: 7931
% 3.20/1.88 % (27462)Time elapsed: 0.137 s
% 3.20/1.88 % (27462)Instructions burned: 379 (million)
% 3.20/1.88 % (27462)------------------------------
% 3.20/1.88 % (27462)------------------------------
% 3.20/1.89 % (27486)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/100Mi)
% 3.20/1.90 % (27485)Instruction limit reached!
% 3.20/1.90 % (27485)------------------------------
% 3.20/1.90 % (27485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.90 % (27485)Termination reason: Unknown
% 3.20/1.90 % (27485)Termination phase: Saturation
% 3.20/1.90
% 3.20/1.90 % (27485)Memory used [KB]: 6780
% 3.20/1.90 % (27485)Time elapsed: 0.038 s
% 3.20/1.90 % (27485)Instructions burned: 128 (million)
% 3.20/1.90 % (27485)------------------------------
% 3.20/1.90 % (27485)------------------------------
% 3.20/1.91 % (27487)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/3Mi)
% 3.20/1.91 % (27487)Instruction limit reached!
% 3.20/1.91 % (27487)------------------------------
% 3.20/1.91 % (27487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.91 % (27487)Termination reason: Unknown
% 3.20/1.91 % (27487)Termination phase: shuffling
% 3.20/1.91
% 3.20/1.91 % (27487)Memory used [KB]: 1407
% 3.20/1.91 % (27487)Time elapsed: 0.002 s
% 3.20/1.91 % (27487)Instructions burned: 4 (million)
% 3.20/1.91 % (27487)------------------------------
% 3.20/1.91 % (27487)------------------------------
% 3.20/1.92 % (27488)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/20Mi)
% 3.20/1.92 % (27486)Instruction limit reached!
% 3.20/1.92 % (27486)------------------------------
% 3.20/1.92 % (27486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.92 % (27486)Termination reason: Unknown
% 3.20/1.92 % (27486)Termination phase: Property scanning
% 3.20/1.92
% 3.20/1.92 % (27486)Memory used [KB]: 2302
% 3.20/1.92 % (27486)Time elapsed: 0.028 s
% 3.20/1.92 % (27486)Instructions burned: 103 (million)
% 3.20/1.92 % (27486)------------------------------
% 3.20/1.92 % (27486)------------------------------
% 3.20/1.92 % (27488)Instruction limit reached!
% 3.20/1.92 % (27488)------------------------------
% 3.20/1.92 % (27488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.92 % (27488)Termination reason: Unknown
% 3.20/1.92 % (27488)Termination phase: shuffling
% 3.20/1.92
% 3.20/1.92 % (27488)Memory used [KB]: 1663
% 3.20/1.92 % (27488)Time elapsed: 0.008 s
% 3.20/1.92 % (27488)Instructions burned: 22 (million)
% 3.20/1.92 % (27488)------------------------------
% 3.20/1.92 % (27488)------------------------------
% 3.20/1.93 % (27489)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/86Mi)
% 3.20/1.93 % (27490)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_27230 for (2997ds/107Mi)
% 3.20/1.96 % (27489)Instruction limit reached!
% 3.20/1.96 % (27489)------------------------------
% 3.20/1.96 % (27489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.96 % (27489)Termination reason: Unknown
% 3.20/1.96 % (27489)Termination phase: Property scanning
% 3.20/1.96
% 3.20/1.96 % (27489)Memory used [KB]: 2174
% 3.20/1.96 % (27489)Time elapsed: 0.027 s
% 3.20/1.96 % (27489)Instructions burned: 86 (million)
% 3.20/1.96 % (27489)------------------------------
% 3.20/1.96 % (27489)------------------------------
% 3.20/1.96 % (27447)Instruction limit reached!
% 3.20/1.96 % (27447)------------------------------
% 3.20/1.96 % (27447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.96 % (27447)Termination reason: Unknown
% 3.20/1.96 % (27447)Termination phase: Saturation
% 3.20/1.96
% 3.20/1.96 % (27447)Memory used [KB]: 8315
% 3.20/1.96 % (27447)Time elapsed: 0.256 s
% 3.20/1.96 % (27447)Instructions burned: 712 (million)
% 3.20/1.96 % (27447)------------------------------
% 3.20/1.96 % (27447)------------------------------
% 3.20/1.97 % (27490)Instruction limit reached!
% 3.20/1.97 % (27490)------------------------------
% 3.20/1.97 % (27490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.97 % (27490)Termination reason: Unknown
% 3.20/1.97 % (27490)Termination phase: Property scanning
% 3.20/1.97
% 3.20/1.97 % (27490)Memory used [KB]: 2302
% 3.20/1.97 % (27490)Time elapsed: 0.033 s
% 3.20/1.97 % (27490)Instructions burned: 109 (million)
% 3.20/1.97 % (27490)------------------------------
% 3.20/1.97 % (27490)------------------------------
% 3.20/1.97 % (27501)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/39Mi)
% 3.20/1.97 % (27505)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/448Mi)
% 3.20/1.97 % (27506)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/46Mi)
% 3.20/1.98 % (27501)Instruction limit reached!
% 3.20/1.98 % (27501)------------------------------
% 3.20/1.98 % (27501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.98 % (27501)Termination reason: Unknown
% 3.20/1.98 % (27501)Termination phase: shuffling
% 3.20/1.98
% 3.20/1.98 % (27501)Memory used [KB]: 2174
% 3.20/1.98 % (27501)Time elapsed: 0.012 s
% 3.20/1.98 % (27501)Instructions burned: 40 (million)
% 3.20/1.98 % (27501)------------------------------
% 3.20/1.98 % (27501)------------------------------
% 3.20/1.99 % (27511)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/98Mi)
% 3.20/1.99 % (27506)Instruction limit reached!
% 3.20/1.99 % (27506)------------------------------
% 3.20/1.99 % (27506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.99 % (27506)Termination reason: Unknown
% 3.20/1.99 % (27506)Termination phase: Property scanning
% 3.20/1.99
% 3.20/1.99 % (27506)Memory used [KB]: 1918
% 3.20/1.99 % (27506)Time elapsed: 0.016 s
% 3.20/1.99 % (27506)Instructions burned: 46 (million)
% 3.20/1.99 % (27506)------------------------------
% 3.20/1.99 % (27506)------------------------------
% 3.20/2.00 % (27517)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/507Mi)
% 3.20/2.01 % (27466)Instruction limit reached!
% 3.20/2.01 % (27466)------------------------------
% 3.20/2.01 % (27466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/2.01 % (27466)Termination reason: Unknown
% 3.20/2.01 % (27466)Termination phase: Saturation
% 3.20/2.01
% 3.20/2.01 % (27466)Memory used [KB]: 8187
% 3.20/2.01 % (27466)Time elapsed: 0.277 s
% 3.20/2.01 % (27466)Instructions burned: 781 (million)
% 3.20/2.01 % (27466)------------------------------
% 3.20/2.01 % (27466)------------------------------
% 5.66/2.01 % (27451)Instruction limit reached!
% 5.66/2.01 % (27451)------------------------------
% 5.66/2.01 % (27451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.01 % (27451)Termination reason: Unknown
% 5.66/2.01 % (27451)Termination phase: Saturation
% 5.66/2.01
% 5.66/2.01 % (27451)Memory used [KB]: 8571
% 5.66/2.01 % (27451)Time elapsed: 0.295 s
% 5.66/2.01 % (27451)Instructions burned: 905 (million)
% 5.66/2.01 % (27451)------------------------------
% 5.66/2.01 % (27451)------------------------------
% 5.66/2.01 % (27511)Instruction limit reached!
% 5.66/2.01 % (27511)------------------------------
% 5.66/2.01 % (27511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.01 % (27511)Termination reason: Unknown
% 5.66/2.01 % (27511)Termination phase: Property scanning
% 5.66/2.01
% 5.66/2.01 % (27511)Memory used [KB]: 2302
% 5.66/2.01 % (27511)Time elapsed: 0.028 s
% 5.66/2.01 % (27511)Instructions burned: 100 (million)
% 5.66/2.01 % (27511)------------------------------
% 5.66/2.01 % (27511)------------------------------
% 5.66/2.02 % (27524)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/149Mi)
% 5.66/2.02 % (27527)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/39Mi)
% 5.66/2.02 % (27530)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/6Mi)
% 5.66/2.02 % (27530)Instruction limit reached!
% 5.66/2.02 % (27530)------------------------------
% 5.66/2.02 % (27530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.02 % (27530)Termination reason: Unknown
% 5.66/2.02 % (27530)Termination phase: shuffling
% 5.66/2.02
% 5.66/2.02 % (27530)Memory used [KB]: 1535
% 5.66/2.02 % (27530)Time elapsed: 0.003 s
% 5.66/2.02 % (27530)Instructions burned: 7 (million)
% 5.66/2.02 % (27530)------------------------------
% 5.66/2.02 % (27530)------------------------------
% 5.66/2.03 % (27438)Instruction limit reached!
% 5.66/2.03 % (27438)------------------------------
% 5.66/2.03 % (27438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.03 % (27438)Termination reason: Unknown
% 5.66/2.03 % (27438)Termination phase: Saturation
% 5.66/2.03
% 5.66/2.03 % (27438)Memory used [KB]: 9850
% 5.66/2.03 % (27438)Time elapsed: 0.347 s
% 5.66/2.03 % (27438)Instructions burned: 1042 (million)
% 5.66/2.03 % (27438)------------------------------
% 5.66/2.03 % (27438)------------------------------
% 5.66/2.03 % (27536)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/1087Mi)
% 5.66/2.03 % (27527)Instruction limit reached!
% 5.66/2.03 % (27527)------------------------------
% 5.66/2.03 % (27527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.03 % (27527)Termination reason: Unknown
% 5.66/2.03 % (27527)Termination phase: Property scanning
% 5.66/2.03
% 5.66/2.03 % (27527)Memory used [KB]: 1918
% 5.66/2.03 % (27527)Time elapsed: 0.014 s
% 5.66/2.03 % (27527)Instructions burned: 39 (million)
% 5.66/2.03 % (27527)------------------------------
% 5.66/2.03 % (27527)------------------------------
% 5.66/2.04 % (27538)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_27230 for (2996ds/47Mi)
% 5.66/2.04 % (27541)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/44Mi)
% 5.66/2.05 % (27538)Instruction limit reached!
% 5.66/2.05 % (27538)------------------------------
% 5.66/2.05 % (27538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.05 % (27538)Termination reason: Unknown
% 5.66/2.05 % (27538)Termination phase: Property scanning
% 5.66/2.05
% 5.66/2.05 % (27538)Memory used [KB]: 1918
% 5.66/2.05 % (27538)Time elapsed: 0.016 s
% 5.66/2.05 % (27538)Instructions burned: 47 (million)
% 5.66/2.05 % (27538)------------------------------
% 5.66/2.05 % (27538)------------------------------
% 5.66/2.06 % (27541)Instruction limit reached!
% 5.66/2.06 % (27541)------------------------------
% 5.66/2.06 % (27541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.06 % (27541)Termination reason: Unknown
% 5.66/2.06 % (27541)Termination phase: Property scanning
% 5.66/2.06
% 5.66/2.06 % (27541)Memory used [KB]: 2174
% 5.66/2.06 % (27541)Time elapsed: 0.014 s
% 5.66/2.06 % (27541)Instructions burned: 46 (million)
% 5.66/2.06 % (27541)------------------------------
% 5.66/2.06 % (27541)------------------------------
% 5.66/2.06 % (27472)Instruction limit reached!
% 5.66/2.06 % (27472)------------------------------
% 5.66/2.06 % (27472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.06 % (27472)Termination reason: Unknown
% 5.66/2.06 % (27472)Termination phase: Saturation
% 5.66/2.06
% 5.66/2.06 % (27472)Memory used [KB]: 8315
% 5.66/2.06 % (27472)Time elapsed: 0.288 s
% 5.66/2.06 % (27472)Instructions burned: 881 (million)
% 5.66/2.06 % (27472)------------------------------
% 5.66/2.06 % (27472)------------------------------
% 5.66/2.06 % (27546)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/432Mi)
% 5.66/2.06 % (27524)Instruction limit reached!
% 5.66/2.06 % (27524)------------------------------
% 5.66/2.06 % (27524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.06 % (27524)Termination reason: Unknown
% 5.66/2.06 % (27524)Termination phase: Saturation
% 5.66/2.06
% 5.66/2.06 % (27524)Memory used [KB]: 6908
% 5.66/2.06 % (27524)Time elapsed: 0.046 s
% 5.66/2.06 % (27524)Instructions burned: 149 (million)
% 5.66/2.06 % (27524)------------------------------
% 5.66/2.06 % (27524)------------------------------
% 5.66/2.07 % (27548)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/62Mi)
% 5.66/2.07 % (27549)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/31Mi)
% 5.66/2.07 % (27551)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/96Mi)
% 5.66/2.08 % (27549)Instruction limit reached!
% 5.66/2.08 % (27549)------------------------------
% 5.66/2.08 % (27549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.08 % (27549)Termination reason: Unknown
% 5.66/2.08 % (27549)Termination phase: shuffling
% 5.66/2.08
% 5.66/2.08 % (27549)Memory used [KB]: 1791
% 5.66/2.08 % (27549)Time elapsed: 0.012 s
% 5.66/2.08 % (27549)Instructions burned: 31 (million)
% 5.66/2.08 % (27549)------------------------------
% 5.66/2.08 % (27549)------------------------------
% 5.66/2.09 % (27548)Instruction limit reached!
% 5.66/2.09 % (27548)------------------------------
% 5.66/2.09 % (27548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.09 % (27548)Termination reason: Unknown
% 5.66/2.09 % (27548)Termination phase: Property scanning
% 5.66/2.09
% 5.66/2.09 % (27548)Memory used [KB]: 2174
% 5.66/2.09 % (27548)Time elapsed: 0.019 s
% 5.66/2.09 % (27548)Instructions burned: 65 (million)
% 5.66/2.09 % (27548)------------------------------
% 5.66/2.09 % (27548)------------------------------
% 5.66/2.09 % (27553)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/453Mi)
% 5.66/2.09 % (27555)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/71Mi)
% 5.66/2.10 % (27551)Instruction limit reached!
% 5.66/2.10 % (27551)------------------------------
% 5.66/2.10 % (27551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.10 % (27551)Termination reason: Unknown
% 5.66/2.10 % (27551)Termination phase: Property scanning
% 5.66/2.10
% 5.66/2.10 % (27551)Memory used [KB]: 2174
% 5.66/2.10 % (27551)Time elapsed: 0.030 s
% 5.66/2.10 % (27551)Instructions burned: 97 (million)
% 5.66/2.10 % (27551)------------------------------
% 5.66/2.10 % (27551)------------------------------
% 5.66/2.11 % (27505)Instruction limit reached!
% 5.66/2.11 % (27505)------------------------------
% 5.66/2.11 % (27505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.66/2.11 % (27505)Termination reason: Unknown
% 5.66/2.11 % (27505)Termination phase: Saturation
% 5.66/2.11
% 5.66/2.11 % (27505)Memory used [KB]: 8187
% 5.66/2.11 % (27505)Time elapsed: 0.157 s
% 5.66/2.11 % (27505)Instructions burned: 449 (million)
% 5.66/2.11 % (27505)------------------------------
% 5.66/2.11 % (27505)------------------------------
% 5.66/2.11 % (27563)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/83Mi)
% 6.21/2.12 % (27565)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/11Mi)
% 6.21/2.12 % (27555)Instruction limit reached!
% 6.21/2.12 % (27555)------------------------------
% 6.21/2.12 % (27555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.12 % (27555)Termination reason: Unknown
% 6.21/2.12 % (27555)Termination phase: Property scanning
% 6.21/2.12
% 6.21/2.12 % (27555)Memory used [KB]: 2174
% 6.21/2.12 % (27555)Time elapsed: 0.023 s
% 6.21/2.12 % (27555)Instructions burned: 71 (million)
% 6.21/2.12 % (27555)------------------------------
% 6.21/2.12 % (27555)------------------------------
% 6.21/2.12 % (27565)Instruction limit reached!
% 6.21/2.12 % (27565)------------------------------
% 6.21/2.12 % (27565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.12 % (27565)Termination reason: Unknown
% 6.21/2.12 % (27565)Termination phase: shuffling
% 6.21/2.12
% 6.21/2.12 % (27565)Memory used [KB]: 1663
% 6.21/2.12 % (27565)Time elapsed: 0.005 s
% 6.21/2.12 % (27565)Instructions burned: 14 (million)
% 6.21/2.12 % (27565)------------------------------
% 6.21/2.12 % (27565)------------------------------
% 6.21/2.13 % (27568)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/41Mi)
% 6.21/2.13 % (27569)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/6Mi)
% 6.21/2.13 % (27569)Instruction limit reached!
% 6.21/2.13 % (27569)------------------------------
% 6.21/2.13 % (27569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.13 % (27569)Termination reason: Unknown
% 6.21/2.13 % (27569)Termination phase: shuffling
% 6.21/2.13
% 6.21/2.13 % (27569)Memory used [KB]: 1407
% 6.21/2.13 % (27569)Time elapsed: 0.003 s
% 6.21/2.13 % (27569)Instructions burned: 6 (million)
% 6.21/2.13 % (27569)------------------------------
% 6.21/2.13 % (27569)------------------------------
% 6.21/2.14 % (27563)Instruction limit reached!
% 6.21/2.14 % (27563)------------------------------
% 6.21/2.14 % (27563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.14 % (27563)Termination reason: Unknown
% 6.21/2.14 % (27563)Termination phase: Property scanning
% 6.21/2.14
% 6.21/2.14 % (27563)Memory used [KB]: 2174
% 6.21/2.14 % (27563)Time elapsed: 0.027 s
% 6.21/2.14 % (27563)Instructions burned: 85 (million)
% 6.21/2.14 % (27563)------------------------------
% 6.21/2.14 % (27563)------------------------------
% 6.21/2.14 % (27568)Instruction limit reached!
% 6.21/2.14 % (27568)------------------------------
% 6.21/2.14 % (27568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.14 % (27568)Termination reason: Unknown
% 6.21/2.14 % (27568)Termination phase: shuffling
% 6.21/2.14
% 6.21/2.14 % (27568)Memory used [KB]: 2174
% 6.21/2.14 % (27568)Time elapsed: 0.013 s
% 6.21/2.14 % (27568)Instructions burned: 44 (million)
% 6.21/2.14 % (27568)------------------------------
% 6.21/2.14 % (27568)------------------------------
% 6.21/2.14 % (27575)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/150Mi)
% 6.21/2.15 % (27580)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/343Mi)
% 6.21/2.15 % (27581)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_27230 for (2995ds/109Mi)
% 6.21/2.15 % (27517)Instruction limit reached!
% 6.21/2.15 % (27517)------------------------------
% 6.21/2.15 % (27517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.15 % (27517)Termination reason: Unknown
% 6.21/2.15 % (27517)Termination phase: Saturation
% 6.21/2.15
% 6.21/2.15 % (27517)Memory used [KB]: 8059
% 6.21/2.15 % (27517)Time elapsed: 0.154 s
% 6.21/2.15 % (27517)Instructions burned: 507 (million)
% 6.21/2.15 % (27517)------------------------------
% 6.21/2.15 % (27517)------------------------------
% 6.21/2.16 % (27583)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_27230 for (2994ds/253Mi)
% 6.21/2.18 % (27581)Instruction limit reached!
% 6.21/2.18 % (27581)------------------------------
% 6.21/2.18 % (27581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.18 % (27581)Termination reason: Unknown
% 6.21/2.18 % (27581)Termination phase: Property scanning
% 6.21/2.18
% 6.21/2.18 % (27581)Memory used [KB]: 2174
% 6.21/2.18 % (27581)Time elapsed: 0.033 s
% 6.21/2.18 % (27581)Instructions burned: 109 (million)
% 6.21/2.18 % (27581)------------------------------
% 6.21/2.18 % (27581)------------------------------
% 6.21/2.18 % (27575)Instruction limit reached!
% 6.21/2.18 % (27575)------------------------------
% 6.21/2.18 % (27575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.18 % (27575)Termination reason: Unknown
% 6.21/2.18 % (27575)Termination phase: Saturation
% 6.21/2.18
% 6.21/2.18 % (27575)Memory used [KB]: 7036
% 6.21/2.18 % (27575)Time elapsed: 0.046 s
% 6.21/2.18 % (27575)Instructions burned: 151 (million)
% 6.21/2.18 % (27575)------------------------------
% 6.21/2.18 % (27575)------------------------------
% 6.21/2.19 % (27603)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_27230 for (2994ds/140Mi)
% 6.21/2.19 % (27546)Instruction limit reached!
% 6.21/2.19 % (27546)------------------------------
% 6.21/2.19 % (27546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.19 % (27546)Termination reason: Unknown
% 6.21/2.19 % (27604)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_27230 for (2994ds/614Mi)
% 6.21/2.19 % (27546)Termination phase: Saturation
% 6.21/2.19
% 6.21/2.19 % (27546)Memory used [KB]: 4221
% 6.21/2.19 % (27546)Time elapsed: 0.133 s
% 6.21/2.19 % (27546)Instructions burned: 434 (million)
% 6.21/2.19 % (27546)------------------------------
% 6.21/2.19 % (27546)------------------------------
% 6.61/2.20 % (27605)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_27230 for (2994ds/202Mi)
% 6.61/2.22 % (27553)Instruction limit reached!
% 6.61/2.22 % (27553)------------------------------
% 6.61/2.22 % (27553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.22 % (27553)Termination reason: Unknown
% 6.61/2.22 % (27553)Termination phase: Saturation
% 6.61/2.22
% 6.61/2.22 % (27553)Memory used [KB]: 7803
% 6.61/2.22 % (27553)Time elapsed: 0.133 s
% 6.61/2.22 % (27553)Instructions burned: 455 (million)
% 6.61/2.22 % (27553)------------------------------
% 6.61/2.22 % (27553)------------------------------
% 6.61/2.23 % (27625)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/339Mi)
% 6.61/2.23 % (27603)Instruction limit reached!
% 6.61/2.23 % (27603)------------------------------
% 6.61/2.23 % (27603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.23 % (27603)Termination reason: Unknown
% 6.61/2.23 % (27603)Termination phase: Saturation
% 6.61/2.23
% 6.61/2.23 % (27603)Memory used [KB]: 6908
% 6.61/2.23 % (27603)Time elapsed: 0.045 s
% 6.61/2.23 % (27603)Instructions burned: 142 (million)
% 6.61/2.23 % (27603)------------------------------
% 6.61/2.23 % (27603)------------------------------
% 6.61/2.24 % (27583)Instruction limit reached!
% 6.61/2.24 % (27583)------------------------------
% 6.61/2.24 % (27583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.24 % (27583)Termination reason: Unknown
% 6.61/2.24 % (27583)Termination phase: Saturation
% 6.61/2.24
% 6.61/2.24 % (27583)Memory used [KB]: 7931
% 6.61/2.24 % (27583)Time elapsed: 0.101 s
% 6.61/2.24 % (27583)Instructions burned: 255 (million)
% 6.61/2.24 % (27583)------------------------------
% 6.61/2.24 % (27583)------------------------------
% 6.61/2.24 % (27631)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/283Mi)
% 6.61/2.25 % (27632)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/134Mi)
% 6.61/2.25 % (27580)Instruction limit reached!
% 6.61/2.25 % (27580)------------------------------
% 6.61/2.25 % (27580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.25 % (27580)Termination reason: Unknown
% 6.61/2.25 % (27580)Termination phase: Saturation
% 6.61/2.25
% 6.61/2.25 % (27580)Memory used [KB]: 7675
% 6.61/2.25 % (27580)Time elapsed: 0.130 s
% 6.61/2.25 % (27580)Instructions burned: 344 (million)
% 6.61/2.25 % (27580)------------------------------
% 6.61/2.25 % (27580)------------------------------
% 6.61/2.26 % (27644)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/298Mi)
% 6.61/2.27 % (27605)Instruction limit reached!
% 6.61/2.27 % (27605)------------------------------
% 6.61/2.27 % (27605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.27 % (27605)Termination reason: Unknown
% 6.61/2.27 % (27605)Termination phase: Saturation
% 6.61/2.27
% 6.61/2.27 % (27605)Memory used [KB]: 7803
% 6.61/2.27 % (27605)Time elapsed: 0.063 s
% 6.61/2.27 % (27605)Instructions burned: 202 (million)
% 6.61/2.27 % (27605)------------------------------
% 6.61/2.27 % (27605)------------------------------
% 6.61/2.28 % (27649)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/192Mi)
% 6.61/2.29 % (27632)Instruction limit reached!
% 6.61/2.29 % (27632)------------------------------
% 6.61/2.29 % (27632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.61/2.29 % (27632)Termination reason: Unknown
% 6.61/2.29 % (27632)Termination phase: Saturation
% 6.61/2.29
% 6.61/2.29 % (27632)Memory used [KB]: 6780
% 6.61/2.29 % (27632)Time elapsed: 0.040 s
% 6.61/2.29 % (27632)Instructions burned: 134 (million)
% 6.61/2.29 % (27632)------------------------------
% 6.61/2.29 % (27632)------------------------------
% 7.13/2.30 % (27656)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_27230 for (2993ds/258Mi)
% 7.13/2.33 % (27631)Instruction limit reached!
% 7.13/2.33 % (27631)------------------------------
% 7.13/2.33 % (27631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.33 % (27631)Termination reason: Unknown
% 7.13/2.33 % (27631)Termination phase: Saturation
% 7.13/2.33
% 7.13/2.33 % (27631)Memory used [KB]: 7547
% 7.13/2.33 % (27631)Time elapsed: 0.085 s
% 7.13/2.33 % (27631)Instructions burned: 284 (million)
% 7.13/2.33 % (27631)------------------------------
% 7.13/2.33 % (27631)------------------------------
% 7.13/2.33 % (27649)Instruction limit reached!
% 7.13/2.33 % (27649)------------------------------
% 7.13/2.33 % (27649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.33 % (27649)Termination reason: Unknown
% 7.13/2.33 % (27649)Termination phase: Saturation
% 7.13/2.33
% 7.13/2.33 % (27649)Memory used [KB]: 7931
% 7.13/2.33 % (27649)Time elapsed: 0.060 s
% 7.13/2.33 % (27649)Instructions burned: 192 (million)
% 7.13/2.33 % (27649)------------------------------
% 7.13/2.33 % (27649)------------------------------
% 7.13/2.34 % (27671)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/90Mi)
% 7.13/2.34 % (27625)Instruction limit reached!
% 7.13/2.34 % (27625)------------------------------
% 7.13/2.34 % (27625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.34 % (27625)Termination reason: Unknown
% 7.13/2.34 % (27625)Termination phase: Saturation
% 7.13/2.34
% 7.13/2.34 % (27625)Memory used [KB]: 7547
% 7.13/2.34 % (27625)Time elapsed: 0.107 s
% 7.13/2.34 % (27625)Instructions burned: 339 (million)
% 7.13/2.34 % (27625)------------------------------
% 7.13/2.34 % (27625)------------------------------
% 7.13/2.34 % (27674)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/231Mi)
% 7.13/2.35 % (27644)Instruction limit reached!
% 7.13/2.35 % (27644)------------------------------
% 7.13/2.35 % (27644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.35 % (27644)Termination reason: Unknown
% 7.13/2.35 % (27644)Termination phase: Saturation
% 7.13/2.35
% 7.13/2.35 % (27644)Memory used [KB]: 7803
% 7.13/2.35 % (27644)Time elapsed: 0.083 s
% 7.13/2.35 % (27644)Instructions burned: 300 (million)
% 7.13/2.35 % (27644)------------------------------
% 7.13/2.35 % (27644)------------------------------
% 7.13/2.35 % (27676)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/458Mi)
% 7.13/2.36 % (27679)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/31Mi)
% 7.13/2.36 % (27671)Instruction limit reached!
% 7.13/2.36 % (27671)------------------------------
% 7.13/2.36 % (27671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.36 % (27671)Termination reason: Unknown
% 7.13/2.36 % (27671)Termination phase: Function definition elimination
% 7.13/2.36
% 7.13/2.36 % (27671)Memory used [KB]: 2174
% 7.13/2.36 % (27671)Time elapsed: 0.029 s
% 7.13/2.36 % (27671)Instructions burned: 92 (million)
% 7.13/2.36 % (27671)------------------------------
% 7.13/2.36 % (27671)------------------------------
% 7.13/2.37 % (27679)Instruction limit reached!
% 7.13/2.37 % (27679)------------------------------
% 7.13/2.37 % (27679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.37 % (27679)Termination reason: Unknown
% 7.13/2.37 % (27679)Termination phase: shuffling
% 7.13/2.37
% 7.13/2.37 % (27679)Memory used [KB]: 1918
% 7.13/2.37 % (27679)Time elapsed: 0.012 s
% 7.13/2.37 % (27679)Instructions burned: 32 (million)
% 7.13/2.37 % (27679)------------------------------
% 7.13/2.37 % (27679)------------------------------
% 7.13/2.37 % (27684)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/297Mi)
% 7.13/2.38 % (27685)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/987Mi)
% 7.13/2.38 % (27656)Instruction limit reached!
% 7.13/2.38 % (27656)------------------------------
% 7.13/2.38 % (27656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.38 % (27656)Termination reason: Unknown
% 7.13/2.38 % (27656)Termination phase: Saturation
% 7.13/2.38
% 7.13/2.38 % (27656)Memory used [KB]: 7803
% 7.13/2.38 % (27656)Time elapsed: 0.080 s
% 7.13/2.38 % (27656)Instructions burned: 260 (million)
% 7.13/2.38 % (27656)------------------------------
% 7.13/2.38 % (27656)------------------------------
% 7.13/2.38 % (27536)Instruction limit reached!
% 7.13/2.38 % (27536)------------------------------
% 7.13/2.38 % (27536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.38 % (27536)Termination reason: Unknown
% 7.13/2.38 % (27536)Termination phase: Saturation
% 7.13/2.38
% 7.13/2.38 % (27536)Memory used [KB]: 8699
% 7.13/2.38 % (27536)Time elapsed: 0.350 s
% 7.13/2.38 % (27536)Instructions burned: 1087 (million)
% 7.13/2.38 % (27536)------------------------------
% 7.13/2.38 % (27536)------------------------------
% 7.13/2.39 % (27604)Instruction limit reached!
% 7.13/2.39 % (27604)------------------------------
% 7.13/2.39 % (27604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.13/2.39 % (27604)Termination reason: Unknown
% 7.13/2.39 % (27604)Termination phase: Saturation
% 7.13/2.39
% 7.13/2.39 % (27604)Memory used [KB]: 8443
% 7.13/2.39 % (27604)Time elapsed: 0.194 s
% 7.13/2.39 % (27604)Instructions burned: 616 (million)
% 7.13/2.39 % (27604)------------------------------
% 7.13/2.39 % (27604)------------------------------
% 7.13/2.39 % (27686)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/182Mi)
% 7.13/2.39 % (27687)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/182Mi)
% 8.21/2.41 % (27688)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/154Mi)
% 8.21/2.41 % (27674)Instruction limit reached!
% 8.21/2.41 % (27674)------------------------------
% 8.21/2.41 % (27674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.41 % (27674)Termination reason: Unknown
% 8.21/2.41 % (27674)Termination phase: Saturation
% 8.21/2.41
% 8.21/2.41 % (27674)Memory used [KB]: 7291
% 8.21/2.41 % (27674)Time elapsed: 0.068 s
% 8.21/2.41 % (27674)Instructions burned: 231 (million)
% 8.21/2.41 % (27674)------------------------------
% 8.21/2.41 % (27674)------------------------------
% 8.21/2.42 % (27695)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_27230 for (2992ds/79Mi)
% 8.21/2.44 % (27686)Instruction limit reached!
% 8.21/2.44 % (27686)------------------------------
% 8.21/2.44 % (27686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.44 % (27686)Termination reason: Unknown
% 8.21/2.44 % (27686)Termination phase: Saturation
% 8.21/2.44
% 8.21/2.44 % (27686)Memory used [KB]: 7036
% 8.21/2.44 % (27686)Time elapsed: 0.056 s
% 8.21/2.44 % (27686)Instructions burned: 184 (million)
% 8.21/2.44 % (27686)------------------------------
% 8.21/2.44 % (27686)------------------------------
% 8.21/2.44 % (27695)Instruction limit reached!
% 8.21/2.44 % (27695)------------------------------
% 8.21/2.44 % (27695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.44 % (27695)Termination reason: Unknown
% 8.21/2.44 % (27695)Termination phase: Property scanning
% 8.21/2.44
% 8.21/2.44 % (27695)Memory used [KB]: 2174
% 8.21/2.44 % (27695)Time elapsed: 0.025 s
% 8.21/2.44 % (27695)Instructions burned: 80 (million)
% 8.21/2.44 % (27695)------------------------------
% 8.21/2.44 % (27695)------------------------------
% 8.21/2.45 % (27687)Instruction limit reached!
% 8.21/2.45 % (27687)------------------------------
% 8.21/2.45 % (27687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.45 % (27687)Termination reason: Unknown
% 8.21/2.45 % (27687)Termination phase: Saturation
% 8.21/2.45
% 8.21/2.45 % (27687)Memory used [KB]: 7164
% 8.21/2.45 % (27687)Time elapsed: 0.057 s
% 8.21/2.45 % (27687)Instructions burned: 183 (million)
% 8.21/2.45 % (27687)------------------------------
% 8.21/2.45 % (27687)------------------------------
% 8.21/2.45 % (27706)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/216Mi)
% 8.21/2.45 % (27707)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/114Mi)
% 8.21/2.46 % (27709)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/922Mi)
% 8.21/2.46 % (27684)Instruction limit reached!
% 8.21/2.46 % (27684)------------------------------
% 8.21/2.46 % (27684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.46 % (27684)Termination reason: Unknown
% 8.21/2.46 % (27684)Termination phase: Saturation
% 8.21/2.46
% 8.21/2.46 % (27684)Memory used [KB]: 7291
% 8.21/2.46 % (27684)Time elapsed: 0.086 s
% 8.21/2.46 % (27684)Instructions burned: 298 (million)
% 8.21/2.46 % (27684)------------------------------
% 8.21/2.46 % (27684)------------------------------
% 8.21/2.47 % (27712)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/182Mi)
% 8.21/2.48 % (27676)Refutation not found, non-redundant clauses discarded% (27676)------------------------------
% 8.21/2.48 % (27676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.48 % (27676)Termination reason: Refutation not found, non-redundant clauses discarded
% 8.21/2.48
% 8.21/2.48 % (27676)Memory used [KB]: 7931
% 8.21/2.48 % (27676)Time elapsed: 0.136 s
% 8.21/2.48 % (27676)Instructions burned: 426 (million)
% 8.21/2.48 % (27676)------------------------------
% 8.21/2.48 % (27676)------------------------------
% 8.21/2.49 % (27688)Instruction limit reached!
% 8.21/2.49 % (27688)------------------------------
% 8.21/2.49 % (27688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.49 % (27688)Termination reason: Unknown
% 8.21/2.49 % (27688)Termination phase: Saturation
% 8.21/2.49
% 8.21/2.49 % (27688)Memory used [KB]: 2430
% 8.21/2.49 % (27688)Time elapsed: 0.082 s
% 8.21/2.49 % (27688)Instructions burned: 155 (million)
% 8.21/2.49 % (27688)------------------------------
% 8.21/2.49 % (27688)------------------------------
% 8.21/2.49 % (27707)Instruction limit reached!
% 8.21/2.49 % (27707)------------------------------
% 8.21/2.49 % (27707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.21/2.49 % (27707)Termination reason: Unknown
% 8.21/2.49 % (27707)Termination phase: Saturation
% 8.21/2.49
% 8.21/2.49 % (27707)Memory used [KB]: 6780
% 8.21/2.49 % (27707)Time elapsed: 0.035 s
% 8.21/2.49 % (27707)Instructions burned: 116 (million)
% 8.21/2.49 % (27707)------------------------------
% 8.21/2.49 % (27707)------------------------------
% 8.21/2.49 % (27721)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/164Mi)
% 8.82/2.50 % (27723)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/50Mi)
% 8.82/2.50 % (27724)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/368Mi)
% 8.82/2.51 % (27706)Instruction limit reached!
% 8.82/2.51 % (27706)------------------------------
% 8.82/2.51 % (27706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.51 % (27706)Termination reason: Unknown
% 8.82/2.51 % (27706)Termination phase: Saturation
% 8.82/2.51
% 8.82/2.51 % (27706)Memory used [KB]: 7291
% 8.82/2.51 % (27706)Time elapsed: 0.060 s
% 8.82/2.51 % (27706)Instructions burned: 218 (million)
% 8.82/2.51 % (27706)------------------------------
% 8.82/2.51 % (27706)------------------------------
% 8.82/2.51 % (27723)Instruction limit reached!
% 8.82/2.51 % (27723)------------------------------
% 8.82/2.51 % (27723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.51 % (27723)Termination reason: Unknown
% 8.82/2.51 % (27723)Termination phase: Property scanning
% 8.82/2.51
% 8.82/2.51 % (27723)Memory used [KB]: 2174
% 8.82/2.51 % (27723)Time elapsed: 0.016 s
% 8.82/2.51 % (27723)Instructions burned: 51 (million)
% 8.82/2.51 % (27723)------------------------------
% 8.82/2.51 % (27723)------------------------------
% 8.82/2.52 % (27712)Instruction limit reached!
% 8.82/2.52 % (27712)------------------------------
% 8.82/2.52 % (27712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.52 % (27712)Termination reason: Unknown
% 8.82/2.52 % (27712)Termination phase: Saturation
% 8.82/2.52
% 8.82/2.52 % (27712)Memory used [KB]: 6908
% 8.82/2.52 % (27712)Time elapsed: 0.050 s
% 8.82/2.52 % (27712)Instructions burned: 184 (million)
% 8.82/2.52 % (27712)------------------------------
% 8.82/2.52 % (27712)------------------------------
% 8.82/2.52 % (27731)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/1021Mi)
% 8.82/2.52 % (27732)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/267Mi)
% 8.82/2.53 % (27736)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/204Mi)
% 8.82/2.54 % (27721)Instruction limit reached!
% 8.82/2.54 % (27721)------------------------------
% 8.82/2.54 % (27721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.54 % (27721)Termination reason: Unknown
% 8.82/2.54 % (27721)Termination phase: Saturation
% 8.82/2.54
% 8.82/2.54 % (27721)Memory used [KB]: 2302
% 8.82/2.54 % (27721)Time elapsed: 0.045 s
% 8.82/2.54 % (27721)Instructions burned: 164 (million)
% 8.82/2.54 % (27721)------------------------------
% 8.82/2.54 % (27721)------------------------------
% 8.82/2.55 % (27743)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_27230 for (2991ds/26Mi)
% 8.82/2.55 % (27743)Instruction limit reached!
% 8.82/2.55 % (27743)------------------------------
% 8.82/2.55 % (27743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.55 % (27743)Termination reason: Unknown
% 8.82/2.55 % (27743)Termination phase: shuffling
% 8.82/2.55
% 8.82/2.55 % (27743)Memory used [KB]: 1791
% 8.82/2.55 % (27743)Time elapsed: 0.009 s
% 8.82/2.55 % (27743)Instructions burned: 27 (million)
% 8.82/2.55 % (27743)------------------------------
% 8.82/2.55 % (27743)------------------------------
% 8.82/2.56 % (27754)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_27230 for (2990ds/53Mi)
% 8.82/2.58 % (27754)Instruction limit reached!
% 8.82/2.58 % (27754)------------------------------
% 8.82/2.58 % (27754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.58 % (27754)Termination reason: Unknown
% 8.82/2.58 % (27754)Termination phase: Property scanning
% 8.82/2.58
% 8.82/2.58 % (27754)Memory used [KB]: 2174
% 8.82/2.58 % (27754)Time elapsed: 0.016 s
% 8.82/2.58 % (27754)Instructions burned: 56 (million)
% 8.82/2.58 % (27754)------------------------------
% 8.82/2.58 % (27754)------------------------------
% 8.82/2.59 % (27769)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_27230 for (2990ds/4436Mi)
% 8.82/2.59 % (27736)Instruction limit reached!
% 8.82/2.59 % (27736)------------------------------
% 8.82/2.59 % (27736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.59 % (27736)Termination reason: Unknown
% 8.82/2.59 % (27736)Termination phase: Saturation
% 8.82/2.59
% 8.82/2.59 % (27736)Memory used [KB]: 6908
% 8.82/2.59 % (27736)Time elapsed: 0.064 s
% 8.82/2.59 % (27736)Instructions burned: 205 (million)
% 8.82/2.59 % (27736)------------------------------
% 8.82/2.59 % (27736)------------------------------
% 9.15/2.60 % (27775)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_27230 for (2990ds/1539Mi)
% 9.15/2.61 % (27724)Instruction limit reached!
% 9.15/2.61 % (27724)------------------------------
% 9.15/2.61 % (27724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.15/2.61 % (27724)Termination reason: Unknown
% 9.15/2.61 % (27724)Termination phase: Saturation
% 9.15/2.61
% 9.15/2.61 % (27724)Memory used [KB]: 7419
% 9.15/2.61 % (27724)Time elapsed: 0.110 s
% 9.15/2.61 % (27724)Instructions burned: 368 (million)
% 9.15/2.61 % (27724)------------------------------
% 9.15/2.61 % (27724)------------------------------
% 9.15/2.61 % (27732)Instruction limit reached!
% 9.15/2.61 % (27732)------------------------------
% 9.15/2.61 % (27732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.15/2.61 % (27732)Termination reason: Unknown
% 9.15/2.61 % (27732)Termination phase: Saturation
% 9.15/2.61
% 9.15/2.61 % (27732)Memory used [KB]: 7547
% 9.15/2.61 % (27732)Time elapsed: 0.088 s
% 9.15/2.61 % (27732)Instructions burned: 268 (million)
% 9.15/2.61 % (27732)------------------------------
% 9.15/2.61 % (27732)------------------------------
% 9.15/2.62 % (27779)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_27230 for (2990ds/408Mi)
% 9.15/2.62 % (27781)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_27230 for (2990ds/5205Mi)
% 9.15/2.68 % (27685)Instruction limit reached!
% 9.15/2.68 % (27685)------------------------------
% 9.15/2.68 % (27685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.15/2.68 % (27685)Termination reason: Unknown
% 9.15/2.68 % (27685)Termination phase: Saturation
% 9.15/2.68
% 9.15/2.68 % (27685)Memory used [KB]: 8699
% 9.15/2.68 % (27685)Time elapsed: 0.305 s
% 9.15/2.68 % (27685)Instructions burned: 989 (million)
% 9.15/2.68 % (27685)------------------------------
% 9.15/2.68 % (27685)------------------------------
% 9.15/2.69 % (27783)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_27230 for (2989ds/1913Mi)
% 9.68/2.74 % (27779)Instruction limit reached!
% 9.68/2.74 % (27779)------------------------------
% 9.68/2.74 % (27779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.68/2.74 % (27779)Termination reason: Unknown
% 9.68/2.74 % (27779)Termination phase: Saturation
% 9.68/2.74
% 9.68/2.74 % (27779)Memory used [KB]: 8315
% 9.68/2.74 % (27779)Time elapsed: 0.128 s
% 9.68/2.74 % (27779)Instructions burned: 408 (million)
% 9.68/2.74 % (27779)------------------------------
% 9.68/2.74 % (27779)------------------------------
% 9.68/2.75 % (27784)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_27230 for (2988ds/4350Mi)
% 9.68/2.76 % (27709)Instruction limit reached!
% 9.68/2.76 % (27709)------------------------------
% 9.68/2.76 % (27709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.68/2.76 % (27709)Termination reason: Unknown
% 9.68/2.76 % (27709)Termination phase: Saturation
% 9.68/2.76
% 9.68/2.76 % (27709)Memory used [KB]: 8955
% 9.68/2.76 % (27709)Time elapsed: 0.300 s
% 9.68/2.76 % (27709)Instructions burned: 924 (million)
% 9.68/2.76 % (27709)------------------------------
% 9.68/2.76 % (27709)------------------------------
% 9.68/2.77 % (27785)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_27230 for (2988ds/68Mi)
% 9.68/2.79 % (27785)Instruction limit reached!
% 9.68/2.79 % (27785)------------------------------
% 9.68/2.79 % (27785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.68/2.79 % (27785)Termination reason: Unknown
% 9.68/2.79 % (27785)Termination phase: Preprocessing 2
% 9.68/2.79
% 9.68/2.79 % (27785)Memory used [KB]: 1918
% 9.68/2.79 % (27785)Time elapsed: 0.022 s
% 9.68/2.79 % (27785)Instructions burned: 70 (million)
% 9.68/2.79 % (27785)------------------------------
% 9.68/2.79 % (27785)------------------------------
% 10.15/2.80 % (27786)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_27230 for (2988ds/358Mi)
% 10.15/2.85 % (27731)Instruction limit reached!
% 10.15/2.85 % (27731)------------------------------
% 10.15/2.85 % (27731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.15/2.85 % (27731)Termination reason: Unknown
% 10.15/2.85 % (27731)Termination phase: Saturation
% 10.15/2.85
% 10.15/2.85 % (27731)Memory used [KB]: 8827
% 10.15/2.85 % (27731)Time elapsed: 0.329 s
% 10.15/2.85 % (27731)Instructions burned: 1024 (million)
% 10.15/2.85 % (27731)------------------------------
% 10.15/2.85 % (27731)------------------------------
% 10.15/2.86 % (27787)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27230 for (2987ds/31Mi)
% 10.15/2.87 % (27787)Instruction limit reached!
% 10.15/2.87 % (27787)------------------------------
% 10.15/2.87 % (27787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.15/2.87 % (27787)Termination reason: Unknown
% 10.15/2.87 % (27787)Termination phase: shuffling
% 10.15/2.87
% 10.15/2.87 % (27787)Memory used [KB]: 1791
% 10.15/2.87 % (27787)Time elapsed: 0.012 s
% 10.15/2.87 % (27787)Instructions burned: 31 (million)
% 10.15/2.87 % (27787)------------------------------
% 10.15/2.87 % (27787)------------------------------
% 10.15/2.88 % (27788)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_27230 for (2987ds/279Mi)
% 10.52/2.91 % (27786)Instruction limit reached!
% 10.52/2.91 % (27786)------------------------------
% 10.52/2.91 % (27786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.52/2.91 % (27786)Termination reason: Unknown
% 10.52/2.91 % (27786)Termination phase: Saturation
% 10.52/2.91
% 10.52/2.91 % (27786)Memory used [KB]: 7547
% 10.52/2.91 % (27786)Time elapsed: 0.136 s
% 10.52/2.91 % (27786)Instructions burned: 359 (million)
% 10.52/2.91 % (27786)------------------------------
% 10.52/2.91 % (27786)------------------------------
% 10.52/2.92 % (27789)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_27230 for (2987ds/258Mi)
% 10.52/2.93 % (27788)Refutation not found, incomplete strategy
% 10.52/2.93 % (27788)------------------------------
% 10.52/2.93 % (27788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.52/2.93 % (27788)Termination reason: Refutation not found, incomplete strategy
% 10.52/2.93
% 10.52/2.93
% 10.52/2.93 % (27788)Memory used [KB]: 7164
% 10.52/2.93 % (27788)Time elapsed: 0.050 s
% 10.52/2.93 % (27788)Instructions burned: 180 (million)
% 10.52/2.93 % (27788)------------------------------
% 10.52/2.93 % (27788)------------------------------
% 10.52/2.94 % (27790)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_27230 for (2986ds/26Mi)
% 10.52/2.94 % (27790)Instruction limit reached!
% 10.52/2.94 % (27790)------------------------------
% 10.52/2.94 % (27790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.52/2.94 % (27790)Termination reason: Unknown
% 10.52/2.94 % (27790)Termination phase: shuffling
% 10.52/2.94
% 10.52/2.94 % (27790)Memory used [KB]: 1791
% 10.52/2.94 % (27790)Time elapsed: 0.008 s
% 10.52/2.94 % (27790)Instructions burned: 26 (million)
% 10.52/2.94 % (27790)------------------------------
% 10.52/2.94 % (27790)------------------------------
% 10.52/2.95 % (27791)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_27230 for (2986ds/995Mi)
% 10.52/3.00 % (27789)Instruction limit reached!
% 10.52/3.00 % (27789)------------------------------
% 10.52/3.00 % (27789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.52/3.00 % (27789)Termination reason: Unknown
% 10.52/3.00 % (27789)Termination phase: Saturation
% 10.52/3.00
% 10.52/3.00 % (27789)Memory used [KB]: 7547
% 10.52/3.00 % (27789)Time elapsed: 0.082 s
% 10.52/3.00 % (27789)Instructions burned: 261 (million)
% 10.52/3.00 % (27789)------------------------------
% 10.52/3.00 % (27789)------------------------------
% 10.52/3.01 % (27792)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_27230 for (2986ds/373Mi)
% 12.68/3.09 % (27775)Instruction limit reached!
% 12.68/3.09 % (27775)------------------------------
% 12.68/3.09 % (27775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.68/3.09 % (27775)Termination reason: Unknown
% 12.68/3.09 % (27775)Termination phase: Saturation
% 12.68/3.09
% 12.68/3.09 % (27775)Memory used [KB]: 10234
% 12.68/3.09 % (27775)Time elapsed: 0.516 s
% 12.68/3.09 % (27775)Instructions burned: 1540 (million)
% 12.68/3.09 % (27775)------------------------------
% 12.68/3.09 % (27775)------------------------------
% 12.68/3.10 % (27783)Instruction limit reached!
% 12.68/3.10 % (27783)------------------------------
% 12.68/3.10 % (27783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.68/3.10 % (27783)Termination reason: Unknown
% 12.68/3.10 % (27783)Termination phase: Saturation
% 12.68/3.10
% 12.68/3.10 % (27783)Memory used [KB]: 2814
% 12.68/3.10 % (27783)Time elapsed: 0.435 s
% 12.68/3.10 % (27783)Instructions burned: 1914 (million)
% 12.68/3.10 % (27783)------------------------------
% 12.68/3.10 % (27783)------------------------------
% 12.68/3.10 % (27793)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_27230 for (2985ds/253Mi)
% 12.68/3.11 % (27794)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_27230 for (2985ds/832Mi)
% 13.27/3.13 % (27792)Instruction limit reached!
% 13.27/3.13 % (27792)------------------------------
% 13.27/3.13 % (27792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.27/3.13 % (27792)Termination reason: Unknown
% 13.27/3.13 % (27792)Termination phase: Saturation
% 13.27/3.13
% 13.27/3.13 % (27792)Memory used [KB]: 7547
% 13.27/3.13 % (27792)Time elapsed: 0.122 s
% 13.27/3.13 % (27792)Instructions burned: 373 (million)
% 13.27/3.13 % (27792)------------------------------
% 13.27/3.13 % (27792)------------------------------
% 13.27/3.14 % (27795)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_27230 for (2984ds/668Mi)
% 13.27/3.17 % (27793)Instruction limit reached!
% 13.27/3.17 % (27793)------------------------------
% 13.27/3.17 % (27793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.27/3.17 % (27793)Termination reason: Unknown
% 13.27/3.17 % (27793)Termination phase: Saturation
% 13.27/3.17
% 13.27/3.17 % (27793)Memory used [KB]: 7675
% 13.27/3.17 % (27793)Time elapsed: 0.071 s
% 13.27/3.17 % (27793)Instructions burned: 255 (million)
% 13.27/3.17 % (27793)------------------------------
% 13.27/3.17 % (27793)------------------------------
% 13.27/3.18 % (27796)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_27230 for (2984ds/40Mi)
% 13.27/3.19 % (27796)Instruction limit reached!
% 13.27/3.19 % (27796)------------------------------
% 13.27/3.19 % (27796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.27/3.19 % (27796)Termination reason: Unknown
% 13.27/3.19 % (27796)Termination phase: shuffling
% 13.27/3.19
% 13.27/3.19 % (27796)Memory used [KB]: 2174
% 13.27/3.19 % (27796)Time elapsed: 0.013 s
% 13.27/3.19 % (27796)Instructions burned: 41 (million)
% 13.27/3.19 % (27796)------------------------------
% 13.27/3.19 % (27796)------------------------------
% 13.87/3.20 % (27797)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_27230 for (2984ds/294Mi)
% 13.87/3.26 % (27791)Instruction limit reached!
% 13.87/3.26 % (27791)------------------------------
% 13.87/3.26 % (27791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.87/3.26 % (27791)Termination reason: Unknown
% 13.87/3.26 % (27791)Termination phase: Saturation
% 13.87/3.26
% 13.87/3.26 % (27791)Memory used [KB]: 9850
% 13.87/3.26 % (27791)Time elapsed: 0.303 s
% 13.87/3.26 % (27791)Instructions burned: 996 (million)
% 13.87/3.26 % (27791)------------------------------
% 13.87/3.26 % (27791)------------------------------
% 13.87/3.27 % (27798)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_27230 for (2983ds/187Mi)
% 13.87/3.28 % (27797)Instruction limit reached!
% 13.87/3.28 % (27797)------------------------------
% 13.87/3.28 % (27797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.87/3.28 % (27797)Termination reason: Unknown
% 13.87/3.28 % (27797)Termination phase: Saturation
% 13.87/3.28
% 13.87/3.28 % (27797)Memory used [KB]: 7419
% 13.87/3.28 % (27797)Time elapsed: 0.082 s
% 13.87/3.28 % (27797)Instructions burned: 296 (million)
% 13.87/3.28 % (27797)------------------------------
% 13.87/3.28 % (27797)------------------------------
% 13.87/3.29 % (27799)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_27230 for (2983ds/371Mi)
% 14.30/3.32 % (27798)Instruction limit reached!
% 14.30/3.32 % (27798)------------------------------
% 14.30/3.32 % (27798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.30/3.32 % (27798)Termination reason: Unknown
% 14.30/3.32 % (27798)Termination phase: Saturation
% 14.30/3.32
% 14.30/3.32 % (27798)Memory used [KB]: 7036
% 14.30/3.32 % (27798)Time elapsed: 0.060 s
% 14.30/3.32 % (27798)Instructions burned: 187 (million)
% 14.30/3.32 % (27798)------------------------------
% 14.30/3.32 % (27798)------------------------------
% 14.30/3.33 % (27800)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_27230 for (2983ds/1827Mi)
% 14.30/3.34 % (27795)Instruction limit reached!
% 14.30/3.34 % (27795)------------------------------
% 14.30/3.34 % (27795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.30/3.34 % (27795)Termination reason: Unknown
% 14.30/3.34 % (27795)Termination phase: Saturation
% 14.30/3.34
% 14.30/3.34 % (27795)Memory used [KB]: 14200
% 14.30/3.34 % (27795)Time elapsed: 0.201 s
% 14.30/3.34 % (27795)Instructions burned: 669 (million)
% 14.30/3.34 % (27795)------------------------------
% 14.30/3.34 % (27795)------------------------------
% 14.30/3.35 % (27801)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_27230 for (2982ds/70Mi)
% 14.30/3.37 % (27801)Instruction limit reached!
% 14.30/3.37 % (27801)------------------------------
% 14.30/3.37 % (27801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.30/3.37 % (27801)Termination reason: Unknown
% 14.30/3.37 % (27801)Termination phase: Property scanning
% 14.30/3.37
% 14.30/3.37 % (27801)Memory used [KB]: 2174
% 14.30/3.37 % (27801)Time elapsed: 0.020 s
% 14.30/3.37 % (27801)Instructions burned: 71 (million)
% 14.30/3.37 % (27801)------------------------------
% 14.30/3.37 % (27801)------------------------------
% 14.30/3.37 % (27794)Instruction limit reached!
% 14.30/3.37 % (27794)------------------------------
% 14.30/3.37 % (27794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.30/3.37 % (27794)Termination reason: Unknown
% 14.30/3.37 % (27794)Termination phase: Saturation
% 14.30/3.37
% 14.30/3.37 % (27794)Memory used [KB]: 8187
% 14.30/3.37 % (27794)Time elapsed: 0.260 s
% 14.30/3.37 % (27794)Instructions burned: 834 (million)
% 14.30/3.37 % (27794)------------------------------
% 14.30/3.37 % (27794)------------------------------
% 14.30/3.38 % (27803)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_27230 for (2982ds/932Mi)
% 14.30/3.39 % (27802)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_27230 for (2982ds/316Mi)
% 15.19/3.42 % (27799)Instruction limit reached!
% 15.19/3.42 % (27799)------------------------------
% 15.19/3.42 % (27799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.19/3.42 % (27799)Termination reason: Unknown
% 15.19/3.42 % (27799)Termination phase: Saturation
% 15.19/3.42
% 15.19/3.42 % (27799)Memory used [KB]: 7803
% 15.19/3.42 % (27799)Time elapsed: 0.122 s
% 15.19/3.42 % (27799)Instructions burned: 373 (million)
% 15.19/3.42 % (27799)------------------------------
% 15.19/3.42 % (27799)------------------------------
% 15.19/3.42 % (27804)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_27230 for (2982ds/4714Mi)
% 15.19/3.47 % (27802)Refutation not found, incomplete strategy
% 15.19/3.47 % (27802)------------------------------
% 15.19/3.47 % (27802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.19/3.47 % (27802)Termination reason: Refutation not found, incomplete strategy
% 15.19/3.47
% 15.19/3.47
% 15.19/3.47 % (27802)Memory used [KB]: 7419
% 15.19/3.47 % (27802)Time elapsed: 0.088 s
% 15.19/3.47 % (27802)Instructions burned: 206 (million)
% 15.19/3.47 % (27802)------------------------------
% 15.19/3.47 % (27802)------------------------------
% 15.19/3.48 % (27805)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_27230 for (2981ds/8250Mi)
% 15.19/3.49 % (27804)Refutation not found, incomplete strategy
% 15.19/3.49 % (27804)------------------------------
% 15.19/3.49 % (27804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.19/3.49 % (27804)Termination reason: Refutation not found, incomplete strategy
% 15.19/3.49
% 15.19/3.49
% 15.19/3.49 % (27804)Memory used [KB]: 7675
% 15.19/3.49 % (27804)Time elapsed: 0.066 s
% 15.19/3.49 % (27804)Instructions burned: 237 (million)
% 15.19/3.49 % (27804)------------------------------
% 15.19/3.49 % (27804)------------------------------
% 15.79/3.50 % (27806)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_27230 for (2981ds/770Mi)
% 16.09/3.67 % (27803)Instruction limit reached!
% 16.09/3.67 % (27803)------------------------------
% 16.09/3.67 % (27803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.09/3.67 % (27803)Termination reason: Unknown
% 16.09/3.67 % (27803)Termination phase: Saturation
% 16.09/3.67
% 16.09/3.67 % (27803)Memory used [KB]: 9722
% 16.09/3.67 % (27803)Time elapsed: 0.288 s
% 16.09/3.67 % (27803)Instructions burned: 933 (million)
% 16.09/3.67 % (27803)------------------------------
% 16.09/3.67 % (27803)------------------------------
% 16.09/3.68 % (27807)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_27230 for (2979ds/794Mi)
% 16.69/3.73 % (27806)Instruction limit reached!
% 16.69/3.73 % (27806)------------------------------
% 16.69/3.73 % (27806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.69/3.73 % (27806)Termination reason: Unknown
% 16.69/3.73 % (27806)Termination phase: Saturation
% 16.69/3.73
% 16.69/3.73 % (27806)Memory used [KB]: 8699
% 16.69/3.73 % (27806)Time elapsed: 0.233 s
% 16.69/3.73 % (27806)Instructions burned: 770 (million)
% 16.69/3.73 % (27806)------------------------------
% 16.69/3.73 % (27806)------------------------------
% 16.69/3.74 % (27808)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_27230 for (2978ds/464Mi)
% 17.23/3.89 % (27808)Instruction limit reached!
% 17.23/3.89 % (27808)------------------------------
% 17.23/3.89 % (27808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.23/3.89 % (27808)Termination reason: Unknown
% 17.23/3.89 % (27808)Termination phase: Saturation
% 17.23/3.89
% 17.23/3.89 % (27808)Memory used [KB]: 7547
% 17.23/3.89 % (27808)Time elapsed: 0.148 s
% 17.23/3.89 % (27808)Instructions burned: 466 (million)
% 17.23/3.89 % (27808)------------------------------
% 17.23/3.89 % (27808)------------------------------
% 17.69/3.90 % (27809)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_27230 for (2977ds/1376Mi)
% 17.69/3.92 % (27807)Instruction limit reached!
% 17.69/3.92 % (27807)------------------------------
% 17.69/3.92 % (27807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.69/3.92 % (27807)Termination reason: Unknown
% 17.69/3.92 % (27807)Termination phase: Saturation
% 17.69/3.92
% 17.69/3.92 % (27807)Memory used [KB]: 6268
% 17.69/3.92 % (27807)Time elapsed: 0.248 s
% 17.69/3.92 % (27807)Instructions burned: 794 (million)
% 17.69/3.92 % (27807)------------------------------
% 17.69/3.92 % (27807)------------------------------
% 17.69/3.93 % (27800)Instruction limit reached!
% 17.69/3.93 % (27800)------------------------------
% 17.69/3.93 % (27800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.69/3.93 % (27800)Termination reason: Unknown
% 17.69/3.93 % (27800)Termination phase: Saturation
% 17.69/3.93
% 17.69/3.93 % (27800)Memory used [KB]: 8827
% 17.69/3.93 % (27800)Time elapsed: 0.594 s
% 17.69/3.93 % (27800)Instructions burned: 1829 (million)
% 17.69/3.93 % (27800)------------------------------
% 17.69/3.93 % (27800)------------------------------
% 17.69/3.93 % (27810)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_27230 for (2977ds/906Mi)
% 17.69/3.94 % (27811)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_27230 for (2977ds/434Mi)
% 17.69/4.01 % (27769)Instruction limit reached!
% 17.69/4.01 % (27769)------------------------------
% 17.69/4.01 % (27769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.69/4.01 % (27769)Termination reason: Unknown
% 17.69/4.01 % (27769)Termination phase: Saturation
% 17.69/4.01
% 17.69/4.01 % (27769)Memory used [KB]: 13304
% 17.69/4.01 % (27769)Time elapsed: 1.424 s
% 17.69/4.01 % (27769)Instructions burned: 4438 (million)
% 17.69/4.01 % (27769)------------------------------
% 17.69/4.01 % (27769)------------------------------
% 18.27/4.02 % (27812)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_27230 for (2976ds/865Mi)
% 18.27/4.07 % (27811)Instruction limit reached!
% 18.27/4.07 % (27811)------------------------------
% 18.27/4.07 % (27811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.27/4.07 % (27811)Termination reason: Unknown
% 18.27/4.07 % (27811)Termination phase: Saturation
% 18.27/4.07
% 18.27/4.07 % (27811)Memory used [KB]: 7803
% 18.27/4.07 % (27811)Time elapsed: 0.135 s
% 18.27/4.07 % (27811)Instructions burned: 434 (million)
% 18.27/4.07 % (27811)------------------------------
% 18.27/4.07 % (27811)------------------------------
% 18.27/4.08 % (27813)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_27230 for (2975ds/593Mi)
% 18.27/4.09 % (27784)Instruction limit reached!
% 18.27/4.09 % (27784)------------------------------
% 18.27/4.09 % (27784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.27/4.09 % (27784)Termination reason: Unknown
% 18.27/4.09 % (27784)Termination phase: Saturation
% 18.27/4.09
% 18.27/4.09 % (27784)Memory used [KB]: 12537
% 18.27/4.09 % (27784)Time elapsed: 1.365 s
% 18.27/4.09 % (27784)Instructions burned: 4351 (million)
% 18.27/4.09 % (27784)------------------------------
% 18.27/4.09 % (27784)------------------------------
% 18.27/4.10 % (27814)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_27230 for (2975ds/443Mi)
% 20.60/4.24 % (27814)Instruction limit reached!
% 20.60/4.24 % (27814)------------------------------
% 20.60/4.24 % (27814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.60/4.24 % (27814)Termination reason: Unknown
% 20.60/4.24 % (27814)Termination phase: Saturation
% 20.60/4.24
% 20.60/4.24 % (27814)Memory used [KB]: 7931
% 20.60/4.24 % (27814)Time elapsed: 0.137 s
% 20.60/4.24 % (27814)Instructions burned: 443 (million)
% 20.60/4.24 % (27814)------------------------------
% 20.60/4.24 % (27814)------------------------------
% 20.60/4.24 % (27810)Instruction limit reached!
% 20.60/4.24 % (27810)------------------------------
% 20.60/4.24 % (27810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.60/4.24 % (27810)Termination reason: Unknown
% 20.60/4.24 % (27810)Termination phase: Saturation
% 20.60/4.24
% 20.60/4.24 % (27810)Memory used [KB]: 12281
% 20.60/4.24 % (27810)Time elapsed: 0.311 s
% 20.60/4.24 % (27810)Instructions burned: 908 (million)
% 20.60/4.24 % (27810)------------------------------
% 20.60/4.24 % (27810)------------------------------
% 20.60/4.25 % (27815)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/594Mi)
% 20.60/4.25 % (27816)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/327Mi)
% 20.60/4.26 % (27813)Instruction limit reached!
% 20.60/4.26 % (27813)------------------------------
% 20.60/4.26 % (27813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.60/4.26 % (27813)Termination reason: Unknown
% 20.60/4.26 % (27813)Termination phase: Saturation
% 20.60/4.26
% 20.60/4.26 % (27813)Memory used [KB]: 9083
% 20.60/4.26 % (27813)Time elapsed: 0.176 s
% 20.60/4.26 % (27813)Instructions burned: 593 (million)
% 20.60/4.26 % (27813)------------------------------
% 20.60/4.26 % (27813)------------------------------
% 20.60/4.27 % (27817)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/5411Mi)
% 20.60/4.28 % (27781)Instruction limit reached!
% 20.60/4.28 % (27781)------------------------------
% 20.60/4.28 % (27781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.60/4.28 % (27781)Termination reason: Unknown
% 20.60/4.28 % (27781)Termination phase: Saturation
% 20.60/4.28
% 20.60/4.28 % (27781)Memory used [KB]: 16886
% 20.60/4.28 % (27781)Time elapsed: 1.686 s
% 20.60/4.28 % (27781)Instructions burned: 5205 (million)
% 20.60/4.28 % (27781)------------------------------
% 20.60/4.28 % (27781)------------------------------
% 20.60/4.29 % (27812)Instruction limit reached!
% 20.60/4.29 % (27812)------------------------------
% 20.60/4.29 % (27812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.60/4.29 % (27812)Termination reason: Unknown
% 20.60/4.29 % (27812)Termination phase: Saturation
% 20.60/4.29
% 20.60/4.29 % (27812)Memory used [KB]: 9466
% 20.60/4.29 % (27812)Time elapsed: 0.265 s
% 20.60/4.29 % (27812)Instructions burned: 865 (million)
% 20.60/4.29 % (27812)------------------------------
% 20.60/4.29 % (27812)------------------------------
% 20.60/4.29 % (27818)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/1762Mi)
% 20.60/4.30 % (27819)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/96Mi)
% 21.57/4.32 % (27819)Instruction limit reached!
% 21.57/4.32 % (27819)------------------------------
% 21.57/4.32 % (27819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/4.32 % (27819)Termination reason: Unknown
% 21.57/4.32 % (27819)Termination phase: Property scanning
% 21.57/4.32
% 21.57/4.32 % (27819)Memory used [KB]: 2430
% 21.57/4.32 % (27819)Time elapsed: 0.027 s
% 21.57/4.32 % (27819)Instructions burned: 96 (million)
% 21.57/4.32 % (27819)------------------------------
% 21.57/4.32 % (27819)------------------------------
% 21.57/4.33 % (27809)Instruction limit reached!
% 21.57/4.33 % (27809)------------------------------
% 21.57/4.33 % (27809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/4.33 % (27809)Termination reason: Unknown
% 21.57/4.33 % (27809)Termination phase: Saturation
% 21.57/4.33
% 21.57/4.33 % (27809)Memory used [KB]: 13304
% 21.57/4.33 % (27809)Time elapsed: 0.428 s
% 21.57/4.33 % (27809)Instructions burned: 1376 (million)
% 21.57/4.33 % (27809)------------------------------
% 21.57/4.33 % (27809)------------------------------
% 21.57/4.33 % (27820)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/503Mi)
% 21.57/4.34 % (27821)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_27230 for (2973ds/90Mi)
% 21.57/4.36 % (27816)Instruction limit reached!
% 21.57/4.36 % (27816)------------------------------
% 21.57/4.36 % (27816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/4.36 % (27816)Termination reason: Unknown
% 21.57/4.36 % (27816)Termination phase: Saturation
% 21.57/4.36
% 21.57/4.36 % (27816)Memory used [KB]: 7803
% 21.57/4.36 % (27816)Time elapsed: 0.104 s
% 21.57/4.36 % (27816)Instructions burned: 327 (million)
% 21.57/4.36 % (27816)------------------------------
% 21.57/4.36 % (27816)------------------------------
% 21.57/4.36 % (27821)Instruction limit reached!
% 21.57/4.36 % (27821)------------------------------
% 21.57/4.36 % (27821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/4.36 % (27821)Termination reason: Unknown
% 21.57/4.36 % (27821)Termination phase: Function definition elimination
% 21.57/4.36
% 21.57/4.36 % (27821)Memory used [KB]: 2174
% 21.57/4.36 % (27821)Time elapsed: 0.029 s
% 21.57/4.36 % (27821)Instructions burned: 91 (million)
% 21.57/4.36 % (27821)------------------------------
% 21.57/4.36 % (27821)------------------------------
% 21.57/4.37 % (27822)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_27230 for (2972ds/593Mi)
% 21.57/4.37 % (27823)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_27230 for (2972ds/1512Mi)
% 21.90/4.44 % (27815)Instruction limit reached!
% 21.90/4.44 % (27815)------------------------------
% 21.90/4.44 % (27815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/4.44 % (27815)Termination reason: Unknown
% 21.90/4.44 % (27815)Termination phase: Saturation
% 21.90/4.44
% 21.90/4.44 % (27815)Memory used [KB]: 7803
% 21.90/4.44 % (27815)Time elapsed: 0.186 s
% 21.90/4.44 % (27815)Instructions burned: 596 (million)
% 21.90/4.44 % (27815)------------------------------
% 21.90/4.44 % (27815)------------------------------
% 21.90/4.45 % (27824)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_27230 for (2971ds/569Mi)
% 21.90/4.49 % (27820)Instruction limit reached!
% 21.90/4.49 % (27820)------------------------------
% 21.90/4.49 % (27820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/4.49 % (27820)Termination reason: Unknown
% 21.90/4.49 % (27820)Termination phase: Saturation
% 21.90/4.49
% 21.90/4.49 % (27820)Memory used [KB]: 8443
% 21.90/4.49 % (27820)Time elapsed: 0.158 s
% 21.90/4.49 % (27820)Instructions burned: 504 (million)
% 21.90/4.49 % (27820)------------------------------
% 21.90/4.49 % (27820)------------------------------
% 22.36/4.50 % (27825)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_27230 for (2971ds/501Mi)
% 22.36/4.55 % (27822)Instruction limit reached!
% 22.36/4.55 % (27822)------------------------------
% 22.36/4.55 % (27822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.36/4.55 % (27822)Termination reason: Unknown
% 22.36/4.55 % (27822)Termination phase: Saturation
% 22.36/4.55
% 22.36/4.55 % (27822)Memory used [KB]: 8059
% 22.36/4.55 % (27822)Time elapsed: 0.180 s
% 22.36/4.55 % (27822)Instructions burned: 595 (million)
% 22.36/4.55 % (27822)------------------------------
% 22.36/4.55 % (27822)------------------------------
% 22.36/4.56 % (27826)lrs+1010_1:3_cnfonf=conj_eager:fsr=off:prag=on:sd=4:sgt=10:ss=axioms:i=562:si=on:rtra=on_0 on DTF2THF_27230 for (2970ds/562Mi)
% 22.65/4.62 % (27824)Instruction limit reached!
% 22.65/4.62 % (27824)------------------------------
% 22.65/4.62 % (27824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.65/4.62 % (27824)Termination reason: Unknown
% 22.65/4.62 % (27824)Termination phase: Saturation
% 22.65/4.62
% 22.65/4.62 % (27824)Memory used [KB]: 8699
% 22.65/4.62 % (27824)Time elapsed: 0.172 s
% 22.65/4.62 % (27824)Instructions burned: 571 (million)
% 22.65/4.62 % (27824)------------------------------
% 22.65/4.62 % (27824)------------------------------
% 22.65/4.63 % (27827)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_27230 for (2970ds/406Mi)
% 22.65/4.65 % (27825)Instruction limit reached!
% 22.65/4.65 % (27825)------------------------------
% 22.65/4.65 % (27825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.65/4.65 % (27825)Termination reason: Unknown
% 22.65/4.65 % (27825)Termination phase: Saturation
% 22.65/4.65
% 22.65/4.65 % (27825)Memory used [KB]: 8443
% 22.65/4.65 % (27825)Time elapsed: 0.155 s
% 22.65/4.65 % (27825)Instructions burned: 502 (million)
% 22.65/4.65 % (27825)------------------------------
% 22.65/4.65 % (27825)------------------------------
% 22.65/4.66 % (27828)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_27230 for (2969ds/582Mi)
% 23.09/4.72 % (27826)Instruction limit reached!
% 23.09/4.72 % (27826)------------------------------
% 23.09/4.72 % (27826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.09/4.72 % (27826)Termination reason: Unknown
% 23.09/4.72 % (27826)Termination phase: Saturation
% 23.09/4.72
% 23.09/4.72 % (27826)Memory used [KB]: 8187
% 23.09/4.72 % (27826)Time elapsed: 0.167 s
% 23.09/4.72 % (27826)Instructions burned: 562 (million)
% 23.09/4.72 % (27826)------------------------------
% 23.09/4.72 % (27826)------------------------------
% 23.09/4.73 % (27829)lrs+2_1:1_au=on:bd=off:bsr=unit_only:sd=2:sgt=32:sos=all:ss=axioms:i=456:si=on:rtra=on_0 on DTF2THF_27230 for (2969ds/456Mi)
% 23.09/4.75 % (27827)Instruction limit reached!
% 23.09/4.75 % (27827)------------------------------
% 23.09/4.75 % (27827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.09/4.75 % (27827)Termination reason: Unknown
% 23.09/4.75 % (27827)Termination phase: Saturation
% 23.09/4.75
% 23.09/4.75 % (27827)Memory used [KB]: 8059
% 23.09/4.75 % (27827)Time elapsed: 0.130 s
% 23.09/4.75 % (27827)Instructions burned: 409 (million)
% 23.09/4.75 % (27827)------------------------------
% 23.09/4.75 % (27827)------------------------------
% 23.09/4.76 % (27830)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_27230 for (2968ds/393Mi)
% 23.09/4.78 % (27818)Instruction limit reached!
% 23.09/4.78 % (27818)------------------------------
% 23.09/4.78 % (27818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.09/4.78 % (27818)Termination reason: Unknown
% 23.09/4.78 % (27818)Termination phase: Saturation
% 23.09/4.78
% 23.09/4.78 % (27818)Memory used [KB]: 9594
% 23.09/4.78 % (27818)Time elapsed: 0.483 s
% 23.09/4.78 % (27818)Instructions burned: 1765 (million)
% 23.09/4.78 % (27818)------------------------------
% 23.09/4.78 % (27818)------------------------------
% 23.09/4.79 % (27831)lrs+1010_1:6_au=on:cnfonf=conj_eager:nwc=10.0:pe=on:prag=on:s2a=on:ss=axioms:st=1.5:i=1335:si=on:rtra=on_0 on DTF2THF_27230 for (2968ds/1335Mi)
% 24.00/4.85 % (27828)Instruction limit reached!
% 24.00/4.85 % (27828)------------------------------
% 24.00/4.85 % (27828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.00/4.85 % (27828)Termination reason: Unknown
% 24.00/4.85 % (27828)Termination phase: Saturation
% 24.00/4.85
% 24.00/4.85 % (27828)Memory used [KB]: 7675
% 24.00/4.85 % (27828)Time elapsed: 0.185 s
% 24.00/4.85 % (27828)Instructions burned: 584 (million)
% 24.00/4.85 % (27828)------------------------------
% 24.00/4.85 % (27828)------------------------------
% 24.00/4.85 % (27823)Instruction limit reached!
% 24.00/4.85 % (27823)------------------------------
% 24.00/4.85 % (27823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.00/4.85 % (27823)Termination reason: Unknown
% 24.00/4.85 % (27823)Termination phase: Saturation
% 24.00/4.85
% 24.00/4.85 % (27823)Memory used [KB]: 9083
% 24.00/4.85 % (27823)Time elapsed: 0.476 s
% 24.00/4.85 % (27823)Instructions burned: 1513 (million)
% 24.00/4.85 % (27823)------------------------------
% 24.00/4.85 % (27823)------------------------------
% 24.00/4.86 % (27832)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=3373:si=on:rtra=on_0 on DTF2THF_27230 for (2967ds/3373Mi)
% 24.00/4.86 % (27833)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_27230 for (2967ds/474Mi)
% 24.00/4.87 % (27829)Instruction limit reached!
% 24.00/4.87 % (27829)------------------------------
% 24.00/4.87 % (27829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.00/4.87 % (27829)Termination reason: Unknown
% 24.00/4.87 % (27829)Termination phase: Saturation
% 24.00/4.87
% 24.00/4.87 % (27829)Memory used [KB]: 8315
% 24.00/4.87 % (27829)Time elapsed: 0.144 s
% 24.00/4.87 % (27829)Instructions burned: 457 (million)
% 24.00/4.87 % (27829)------------------------------
% 24.00/4.87 % (27829)------------------------------
% 24.00/4.88 % (27830)Instruction limit reached!
% 24.00/4.88 % (27830)------------------------------
% 24.00/4.88 % (27830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.00/4.88 % (27830)Termination reason: Unknown
% 24.00/4.88 % (27830)Termination phase: Saturation
% 24.00/4.88
% 24.00/4.88 % (27830)Memory used [KB]: 7547
% 24.00/4.88 % (27830)Time elapsed: 0.113 s
% 24.00/4.88 % (27830)Instructions burned: 394 (million)
% 24.00/4.88 % (27830)------------------------------
% 24.00/4.88 % (27830)------------------------------
% 24.00/4.88 % (27834)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_27230 for (2967ds/157Mi)
% 24.00/4.89 % (27835)dis+1010_1:1_amm=off:cs=on:kws=precedence:nwc=10.0:piset=all:s2a=on:sac=on:sp=unary_frequency:spb=goal:ss=axioms:st=3.0:i=590:si=on:rtra=on_0 on DTF2THF_27230 for (2967ds/590Mi)
% 25.05/4.93 % (27834)Instruction limit reached!
% 25.05/4.93 % (27834)------------------------------
% 25.05/4.93 % (27834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.05/4.93 % (27834)Termination reason: Unknown
% 25.05/4.93 % (27834)Termination phase: Property scanning
% 25.05/4.93
% 25.05/4.93 % (27834)Memory used [KB]: 2430
% 25.05/4.93 % (27834)Time elapsed: 0.043 s
% 25.05/4.93 % (27834)Instructions burned: 159 (million)
% 25.05/4.93 % (27834)------------------------------
% 25.05/4.93 % (27834)------------------------------
% 25.05/4.93 % (27836)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_27230 for (2967ds/104Mi)
% 25.05/4.97 % (27836)Instruction limit reached!
% 25.05/4.97 % (27836)------------------------------
% 25.05/4.97 % (27836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.05/4.97 % (27836)Termination reason: Unknown
% 25.05/4.97 % (27836)Termination phase: Property scanning
% 25.05/4.97
% 25.05/4.97 % (27836)Memory used [KB]: 2174
% 25.05/4.97 % (27836)Time elapsed: 0.032 s
% 25.05/4.97 % (27836)Instructions burned: 106 (million)
% 25.05/4.97 % (27836)------------------------------
% 25.05/4.97 % (27836)------------------------------
% 25.05/4.98 % (27837)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_27230 for (2966ds/414Mi)
% 25.05/5.00 % (27833)Instruction limit reached!
% 25.05/5.00 % (27833)------------------------------
% 25.05/5.00 % (27833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.05/5.00 % (27833)Termination reason: Unknown
% 25.05/5.00 % (27833)Termination phase: Saturation
% 25.05/5.00
% 25.05/5.00 % (27833)Memory used [KB]: 8059
% 25.05/5.00 % (27833)Time elapsed: 0.142 s
% 25.05/5.00 % (27833)Instructions burned: 474 (million)
% 25.05/5.00 % (27833)------------------------------
% 25.05/5.00 % (27833)------------------------------
% 26.72/5.02 % (27838)dis+10_1:1_av=off:bsd=on:bsr=unit_only:fd=off:hud=8:nm=16:prag=on:i=715:si=on:rtra=on_0 on DTF2THF_27230 for (2966ds/715Mi)
% 26.72/5.06 % (27835)Instruction limit reached!
% 26.72/5.06 % (27835)------------------------------
% 26.72/5.06 % (27835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.72/5.06 % (27835)Termination reason: Unknown
% 26.72/5.06 % (27835)Termination phase: Saturation
% 26.72/5.06
% 26.72/5.06 % (27835)Memory used [KB]: 8443
% 26.72/5.06 % (27835)Time elapsed: 0.179 s
% 26.72/5.06 % (27835)Instructions burned: 591 (million)
% 26.72/5.06 % (27835)------------------------------
% 26.72/5.06 % (27835)------------------------------
% 26.72/5.07 % (27839)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=961:si=on:rtra=on_0 on DTF2THF_27230 for (2965ds/961Mi)
% 26.72/5.09 % (27837)Instruction limit reached!
% 26.72/5.09 % (27837)------------------------------
% 26.72/5.09 % (27837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.72/5.09 % (27837)Termination reason: Unknown
% 26.72/5.09 % (27837)Termination phase: Saturation
% 26.72/5.09
% 26.72/5.09 % (27837)Memory used [KB]: 7675
% 26.72/5.09 % (27837)Time elapsed: 0.120 s
% 26.72/5.09 % (27837)Instructions burned: 414 (million)
% 26.72/5.09 % (27837)------------------------------
% 26.72/5.09 % (27837)------------------------------
% 26.72/5.10 % (27840)lrs-1002_1:128_abs=on:bd=preordered:cbe=off:e2e=on:fde=unused:fe=off:tgt=full:i=1213:si=on:rtra=on_0 on DTF2THF_27230 for (2965ds/1213Mi)
% 27.27/5.16 % (27831)Instruction limit reached!
% 27.27/5.16 % (27831)------------------------------
% 27.27/5.16 % (27831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.27/5.16 % (27831)Termination reason: Unknown
% 27.27/5.16 % (27831)Termination phase: Saturation
% 27.27/5.16
% 27.27/5.16 % (27831)Memory used [KB]: 9466
% 27.27/5.16 % (27831)Time elapsed: 0.377 s
% 27.27/5.16 % (27831)Instructions burned: 1335 (million)
% 27.27/5.16 % (27831)------------------------------
% 27.27/5.16 % (27831)------------------------------
% 27.27/5.17 % (27841)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_27230 for (2964ds/1979Mi)
% 27.89/5.30 % (27838)Instruction limit reached!
% 27.89/5.30 % (27838)------------------------------
% 27.89/5.30 % (27838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.89/5.30 % (27838)Termination reason: Unknown
% 27.89/5.30 % (27838)Termination phase: Saturation
% 27.89/5.30
% 27.89/5.30 % (27838)Memory used [KB]: 4093
% 27.89/5.30 % (27838)Time elapsed: 0.309 s
% 27.89/5.30 % (27838)Instructions burned: 717 (million)
% 27.89/5.30 % (27838)------------------------------
% 27.89/5.30 % (27838)------------------------------
% 27.89/5.31 % (27842)dis+1002_1:1_amm=off:au=on:bsd=on:cbe=off:cnfonf=lazy_gen:sp=unary_frequency:i=481:si=on:rtra=on_0 on DTF2THF_27230 for (2963ds/481Mi)
% 27.89/5.38 % (27839)Instruction limit reached!
% 27.89/5.38 % (27839)------------------------------
% 27.89/5.38 % (27839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.89/5.38 % (27839)Termination reason: Unknown
% 27.89/5.38 % (27839)Termination phase: Saturation
% 27.89/5.38
% 27.89/5.38 % (27839)Memory used [KB]: 7547
% 27.89/5.38 % (27839)Time elapsed: 0.327 s
% 27.89/5.38 % (27839)Instructions burned: 962 (million)
% 27.89/5.38 % (27839)------------------------------
% 27.89/5.38 % (27839)------------------------------
% 27.89/5.39 % (27843)dis+10_8:1_apa=on:cnfonf=off:e2e=on:plsq=on:plsqc=1:plsqr=32,1:ss=axioms:i=783:si=on:rtra=on_0 on DTF2THF_27230 for (2962ds/783Mi)
% 28.80/5.46 % (27842)Instruction limit reached!
% 28.80/5.46 % (27842)------------------------------
% 28.80/5.46 % (27842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.80/5.46 % (27842)Termination reason: Unknown
% 28.80/5.46 % (27842)Termination phase: Saturation
% 28.80/5.46
% 28.80/5.46 % (27842)Memory used [KB]: 7931
% 28.80/5.46 % (27842)Time elapsed: 0.172 s
% 28.80/5.46 % (27842)Instructions burned: 483 (million)
% 28.80/5.46 % (27842)------------------------------
% 28.80/5.46 % (27842)------------------------------
% 28.80/5.47 % (27844)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=432:si=on:rtra=on_0 on DTF2THF_27230 for (2961ds/432Mi)
% 29.26/5.50 % (27840)Instruction limit reached!
% 29.26/5.50 % (27840)------------------------------
% 29.26/5.50 % (27840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.26/5.50 % (27840)Termination reason: Unknown
% 29.26/5.50 % (27840)Termination phase: Saturation
% 29.26/5.50
% 29.26/5.50 % (27840)Memory used [KB]: 8315
% 29.26/5.50 % (27840)Time elapsed: 0.416 s
% 29.26/5.50 % (27840)Instructions burned: 1215 (million)
% 29.26/5.50 % (27840)------------------------------
% 29.26/5.50 % (27840)------------------------------
% 29.26/5.51 % (27845)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=1670:si=on:rtra=on_0 on DTF2THF_27230 for (2961ds/1670Mi)
% 29.26/5.53 % (27817)Instruction limit reached!
% 29.26/5.53 % (27817)------------------------------
% 29.26/5.53 % (27817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.26/5.53 % (27817)Termination reason: Unknown
% 29.26/5.53 % (27817)Termination phase: Saturation
% 29.26/5.53
% 29.26/5.53 % (27817)Memory used [KB]: 8571
% 29.26/5.53 % (27817)Time elapsed: 1.267 s
% 29.26/5.53 % (27817)Instructions burned: 5413 (million)
% 29.26/5.53 % (27817)------------------------------
% 29.26/5.53 % (27817)------------------------------
% 29.26/5.54 % (27846)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_27230 for (2960ds/90Mi)
% 29.26/5.57 % (27846)Instruction limit reached!
% 29.26/5.57 % (27846)------------------------------
% 29.26/5.57 % (27846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.26/5.57 % (27846)Termination reason: Unknown
% 29.26/5.57 % (27846)Termination phase: Function definition elimination
% 29.26/5.57
% 29.26/5.57 % (27846)Memory used [KB]: 2302
% 29.26/5.57 % (27846)Time elapsed: 0.030 s
% 29.26/5.57 % (27846)Instructions burned: 92 (million)
% 29.26/5.57 % (27846)------------------------------
% 29.26/5.57 % (27846)------------------------------
% 29.26/5.58 % (27847)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_27230 for (2960ds/346Mi)
% 29.95/5.61 % (27844)Instruction limit reached!
% 29.95/5.61 % (27844)------------------------------
% 29.95/5.61 % (27844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.95/5.61 % (27844)Termination reason: Unknown
% 29.95/5.61 % (27844)Termination phase: Saturation
% 29.95/5.61
% 29.95/5.61 % (27844)Memory used [KB]: 7803
% 29.95/5.61 % (27844)Time elapsed: 0.138 s
% 29.95/5.61 % (27844)Instructions burned: 432 (million)
% 29.95/5.61 % (27844)------------------------------
% 29.95/5.61 % (27844)------------------------------
% 29.95/5.62 % (27848)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=401:si=on:rtra=on_0 on DTF2THF_27230 for (2960ds/401Mi)
% 29.95/5.63 % (27843)Instruction limit reached!
% 29.95/5.63 % (27843)------------------------------
% 29.95/5.63 % (27843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.95/5.63 % (27843)Termination reason: Unknown
% 29.95/5.63 % (27843)Termination phase: Saturation
% 29.95/5.63
% 29.95/5.63 % (27843)Memory used [KB]: 9210
% 29.95/5.63 % (27843)Time elapsed: 0.247 s
% 29.95/5.63 % (27843)Instructions burned: 783 (million)
% 29.95/5.63 % (27843)------------------------------
% 29.95/5.63 % (27843)------------------------------
% 29.95/5.64 % (27849)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_27230 for (2959ds/6454Mi)
% 29.95/5.68 % (27847)Instruction limit reached!
% 29.95/5.68 % (27847)------------------------------
% 29.95/5.68 % (27847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.95/5.68 % (27847)Termination reason: Unknown
% 29.95/5.68 % (27847)Termination phase: Saturation
% 29.95/5.68
% 29.95/5.68 % (27847)Memory used [KB]: 7675
% 29.95/5.68 % (27847)Time elapsed: 0.104 s
% 29.95/5.68 % (27847)Instructions burned: 348 (million)
% 29.95/5.68 % (27847)------------------------------
% 29.95/5.68 % (27847)------------------------------
% 29.95/5.69 % (27850)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:kws=arity:prag=on:i=283:si=on:rtra=on_0 on DTF2THF_27230 for (2959ds/283Mi)
% 30.66/5.73 % (27848)Instruction limit reached!
% 30.66/5.73 % (27848)------------------------------
% 30.66/5.73 % (27848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.66/5.73 % (27848)Termination reason: Unknown
% 30.66/5.73 % (27848)Termination phase: Saturation
% 30.66/5.73
% 30.66/5.73 % (27848)Memory used [KB]: 8059
% 30.66/5.73 % (27848)Time elapsed: 0.118 s
% 30.66/5.73 % (27848)Instructions burned: 403 (million)
% 30.66/5.73 % (27848)------------------------------
% 30.66/5.73 % (27848)------------------------------
% 30.66/5.74 % (27851)ott+10_1:1_aac=none:apa=on:au=on:cnfonf=off:ntd=on:nwc=12.0:sos=on:i=865:si=on:rtra=on_0 on DTF2THF_27230 for (2958ds/865Mi)
% 30.66/5.78 % (27850)Instruction limit reached!
% 30.66/5.78 % (27850)------------------------------
% 30.66/5.78 % (27850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.66/5.78 % (27850)Termination reason: Unknown
% 30.66/5.78 % (27850)Termination phase: Saturation
% 30.66/5.78
% 30.66/5.78 % (27850)Memory used [KB]: 7419
% 30.66/5.78 % (27850)Time elapsed: 0.091 s
% 30.66/5.78 % (27850)Instructions burned: 283 (million)
% 30.66/5.78 % (27850)------------------------------
% 30.66/5.78 % (27850)------------------------------
% 30.66/5.79 % (27852)lrs+1002_3:4_au=on:av=off:cnfonf=off:fd=off:fsr=off:ins=2:sd=1:sos=on:ss=axioms:i=4528:si=on:rtra=on_0 on DTF2THF_27230 for (2958ds/4528Mi)
% 31.10/5.80 % (27841)Instruction limit reached!
% 31.10/5.80 % (27841)------------------------------
% 31.10/5.80 % (27841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.10/5.80 % (27841)Termination reason: Unknown
% 31.10/5.80 % (27841)Termination phase: Saturation
% 31.10/5.80
% 31.10/5.80 % (27841)Memory used [KB]: 7931
% 31.10/5.80 % (27841)Time elapsed: 0.653 s
% 31.10/5.80 % (27841)Instructions burned: 1981 (million)
% 31.10/5.80 % (27841)------------------------------
% 31.10/5.80 % (27841)------------------------------
% 31.10/5.81 % (27853)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=1001:si=on:rtra=on_0 on DTF2THF_27230 for (2958ds/1001Mi)
% 31.94/5.91 % (27832)Instruction limit reached!
% 31.94/5.91 % (27832)------------------------------
% 31.94/5.91 % (27832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.94/5.91 % (27832)Termination reason: Unknown
% 31.94/5.91 % (27832)Termination phase: Saturation
% 31.94/5.91
% 31.94/5.91 % (27832)Memory used [KB]: 12409
% 31.94/5.91 % (27832)Time elapsed: 1.074 s
% 31.94/5.91 % (27832)Instructions burned: 3375 (million)
% 31.94/5.91 % (27832)------------------------------
% 31.94/5.91 % (27832)------------------------------
% 31.94/5.92 % (27854)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_27230 for (2957ds/102Mi)
% 31.94/5.95 % (27854)Instruction limit reached!
% 31.94/5.95 % (27854)------------------------------
% 31.94/5.95 % (27854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.94/5.95 % (27854)Termination reason: Unknown
% 31.94/5.95 % (27854)Termination phase: Property scanning
% 31.94/5.95
% 31.94/5.95 % (27854)Memory used [KB]: 2302
% 31.94/5.95 % (27854)Time elapsed: 0.029 s
% 31.94/5.95 % (27854)Instructions burned: 104 (million)
% 31.94/5.95 % (27854)------------------------------
% 31.94/5.95 % (27854)------------------------------
% 31.94/5.96 % (27855)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_27230 for (2956ds/952Mi)
% 31.94/6.02 % (27845)Instruction limit reached!
% 31.94/6.02 % (27845)------------------------------
% 31.94/6.02 % (27845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.94/6.02 % (27845)Termination reason: Unknown
% 31.94/6.02 % (27845)Termination phase: Saturation
% 31.94/6.02
% 31.94/6.02 % (27845)Memory used [KB]: 10490
% 31.94/6.02 % (27845)Time elapsed: 0.513 s
% 31.94/6.02 % (27845)Instructions burned: 1670 (million)
% 31.94/6.02 % (27845)------------------------------
% 31.94/6.02 % (27845)------------------------------
% 31.94/6.02 % (27851)Instruction limit reached!
% 31.94/6.02 % (27851)------------------------------
% 31.94/6.02 % (27851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.94/6.02 % (27851)Termination reason: Unknown
% 31.94/6.02 % (27851)Termination phase: Saturation
% 31.94/6.02
% 31.94/6.02 % (27851)Memory used [KB]: 8955
% 31.94/6.02 % (27851)Time elapsed: 0.278 s
% 31.94/6.02 % (27851)Instructions burned: 866 (million)
% 31.94/6.02 % (27851)------------------------------
% 31.94/6.02 % (27851)------------------------------
% 33.81/6.03 % (27856)lrs+1010_1:1024_add=large:c=on:cbe=off:cnfonf=lazy_gen:fde=none:nm=30:piset=and:sac=on:uhcvi=on:i=578:si=on:rtra=on_0 on DTF2THF_27230 for (2956ds/578Mi)
% 33.81/6.03 % (27857)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_27230 for (2956ds/643Mi)
% 33.81/6.05 % (27853)Instruction limit reached!
% 33.81/6.05 % (27853)------------------------------
% 33.81/6.05 % (27853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.81/6.05 % (27853)Termination reason: Unknown
% 33.81/6.05 % (27853)Termination phase: Saturation
% 33.81/6.05
% 33.81/6.05 % (27853)Memory used [KB]: 7419
% 33.81/6.05 % (27853)Time elapsed: 0.237 s
% 33.81/6.05 % (27853)Instructions burned: 1004 (million)
% 33.81/6.05 % (27853)------------------------------
% 33.81/6.05 % (27853)------------------------------
% 33.81/6.06 % (27858)lrs+1010_1:1_cnfonf=off:ntd=on:nwc=3.0:s2a=on:sp=const_frequency:i=865:si=on:rtra=on_0 on DTF2THF_27230 for (2955ds/865Mi)
% 33.81/6.08 % (27805)First to succeed.
% 33.81/6.09 % (27805)Refutation found. Thanks to Tanya!
% 33.81/6.09 % SZS status Theorem for DTF2THF_27230
% 33.81/6.09 % SZS output start Proof for DTF2THF_27230
% 33.81/6.09 thf(type_def_5, type, carrier_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(type_def_6, type, set: ($tType) > $tType).
% 33.81/6.09 thf(type_def_8, type, ring_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(type_def_9, type, aGroup_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(type_def_10, type, module_ext: ($tType * $tType * $tType) > $tType).
% 33.81/6.09 thf(type_def_11, type, product_unit: $tType).
% 33.81/6.09 thf(type_def_12, type, nat: $tType).
% 33.81/6.09 thf(type_def_13, type, a: $tType).
% 33.81/6.09 thf(type_def_14, type, b: $tType).
% 33.81/6.09 thf(type_def_15, type, c: $tType).
% 33.81/6.09 thf(type_def_16, type, e: $tType).
% 33.81/6.09 thf(type_def_17, type, f: $tType).
% 33.81/6.09 thf(type_def_18, type, d: $tType).
% 33.81/6.09 thf(func_def_0, type, carrier_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(func_def_1, type, module_ext: ($tType * $tType * $tType) > $tType).
% 33.81/6.09 thf(func_def_2, type, aGroup_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(func_def_3, type, ring_ext: ($tType * $tType) > $tType).
% 33.81/6.09 thf(func_def_4, type, product_unit: $tType).
% 33.81/6.09 thf(func_def_5, type, set: ($tType) > $tType).
% 33.81/6.09 thf(func_def_6, type, nat: $tType).
% 33.81/6.09 thf(func_def_7, type, f: $tType).
% 33.81/6.09 thf(func_def_8, type, e: $tType).
% 33.81/6.09 thf(func_def_9, type, d: $tType).
% 33.81/6.09 thf(func_def_10, type, c: $tType).
% 33.81/6.09 thf(func_def_11, type, b: $tType).
% 33.81/6.09 thf(func_def_12, type, a: $tType).
% 33.81/6.09 thf(func_def_13, type, type: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_14, type, zero: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_15, type, order: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_16, type, group_add: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_17, type, wellorder: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_18, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_19, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_20, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_21, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 33.81/6.09 thf(func_def_22, type, carrier: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ X1 > set @ X0)).
% 33.81/6.09 thf(func_def_23, type, ring: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > $o)).
% 33.81/6.09 thf(func_def_24, type, zero2: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ X1 > X0)).
% 33.81/6.09 thf(func_def_25, type, ideal: !>[X0: $tType, X1: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > set @ X0 > $o)).
% 33.81/6.09 thf(func_def_26, type, annihilator: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X0)).
% 33.81/6.09 thf(func_def_27, type, hom: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > carrier_ext @ X4 @ aGroup_ext @ X4 @ module_ext @ X4 @ X0 @ X5 > carrier_ext @ (X2 > X4) @ aGroup_ext @ (X2 > X4) @ module_ext @ (X2 > X4) @ X0 @ product_unit)).
% 33.81/6.09 thf(func_def_28, type, module: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ module_ext @ X0 @ X1 @ X2 > carrier_ext @ X1 @ aGroup_ext @ X1 @ ring_ext @ X1 @ X3 > $o)).
% 33.81/6.09 thf(func_def_29, type, free_generator: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X2 > $o)).
% 33.81/6.09 thf(func_def_30, type, generator: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X2 > $o)).
% 33.81/6.09 thf(func_def_31, type, l_comb: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > nat > (nat > X0) > (nat > X2) > X2)).
% 33.81/6.09 thf(func_def_32, type, linear_span: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X0 > set @ X2 > set @ X2)).
% 33.81/6.09 thf(func_def_33, type, mdl: !>[X0: $tType, X1: $tType, X2: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ module_ext @ X0 @ X1 @ X2 > set @ X0 > carrier_ext @ X0 @ aGroup_ext @ X0 @ module_ext @ X0 @ X1 @ product_unit)).
% 33.81/6.09 thf(func_def_34, type, misomorphic: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > carrier_ext @ X4 @ aGroup_ext @ X4 @ module_ext @ X4 @ X0 @ X5 > $o)).
% 33.81/6.09 thf(func_def_35, type, quotie135653060odules: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X2 > set @ X2 > set @ X0)).
% 33.81/6.09 thf(func_def_36, type, smodule_ideal_coeff: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X0 > set @ X2)).
% 33.81/6.09 thf(func_def_37, type, submodule: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X2 > $o)).
% 33.81/6.09 thf(func_def_38, type, algebr833503410le_fgs: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X1 > carrier_ext @ X2 @ aGroup_ext @ X2 @ module_ext @ X2 @ X0 @ X3 > set @ X2 > set @ X2)).
% 33.81/6.09 thf(func_def_39, type, pi: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ (X0 > X1))).
% 33.81/6.09 thf(func_def_40, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 33.81/6.09 thf(func_def_41, type, zero_zero: !>[X0: $tType]:(X0)).
% 33.81/6.09 thf(func_def_42, type, suc: nat > nat).
% 33.81/6.09 thf(func_def_43, type, bot_bot: !>[X0: $tType]:(X0)).
% 33.81/6.09 thf(func_def_44, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 33.81/6.09 thf(func_def_45, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 33.81/6.09 thf(func_def_46, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 33.81/6.09 thf(func_def_47, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 33.81/6.09 thf(func_def_48, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 33.81/6.09 thf(func_def_49, type, h: set @ a).
% 33.81/6.09 thf(func_def_50, type, h1: set @ a).
% 33.81/6.09 thf(func_def_51, type, m: carrier_ext @ a @ aGroup_ext @ a @ module_ext @ a @ b @ c).
% 33.81/6.09 thf(func_def_52, type, n: carrier_ext @ e @ aGroup_ext @ e @ module_ext @ e @ b @ f).
% 33.81/6.09 thf(func_def_53, type, r: carrier_ext @ b @ aGroup_ext @ b @ ring_ext @ b @ d).
% 33.81/6.09 thf(func_def_54, type, g: nat > a).
% 33.81/6.09 thf(func_def_55, type, h2: a).
% 33.81/6.09 thf(func_def_56, type, m2: nat).
% 33.81/6.09 thf(func_def_57, type, t: nat > b).
% 33.81/6.09 thf(func_def_58, type, x: a).
% 33.81/6.09 thf(func_def_59, type, xa: nat).
% 33.81/6.09 thf(func_def_70, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 33.81/6.09 thf(func_def_71, type, sK1: nat > nat).
% 33.81/6.09 thf(func_def_72, type, sK2: nat > nat).
% 33.81/6.09 thf(func_def_73, type, sK3: a > set @ b > nat).
% 33.81/6.09 thf(func_def_74, type, sK4: a > set @ b > nat > b).
% 33.81/6.09 thf(func_def_75, type, sK5: a > set @ b > nat > a).
% 33.81/6.09 thf(func_def_76, type, sK6: nat > (nat > $o) > nat).
% 33.81/6.09 thf(func_def_77, type, sK7: nat > (nat > $o) > nat).
% 33.81/6.09 thf(func_def_78, type, sK8: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_79, type, sK9: nat > nat > nat).
% 33.81/6.09 thf(func_def_80, type, sK10: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 33.81/6.09 thf(func_def_81, type, sK11: !>[X0: $tType]:((X0 > $o) > X0)).
% 33.81/6.09 thf(func_def_82, type, sK12: !>[X0: $tType]:((nat > X0) > nat)).
% 33.81/6.09 thf(func_def_83, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 33.81/6.09 thf(func_def_84, type, sK14: nat > nat > nat).
% 33.81/6.09 thf(func_def_85, type, sK15: !>[X0: $tType]:((nat > X0) > (nat > X0) > nat > nat)).
% 33.81/6.09 thf(func_def_86, type, sK16: !>[X0: $tType]:((nat > X0) > nat)).
% 33.81/6.09 thf(func_def_87, type, sK17: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_88, type, sK18: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_89, type, sK19: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_90, type, sK20: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_91, type, sK21: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_92, type, sK22: (nat > $o) > nat > nat > nat).
% 33.81/6.09 thf(func_def_93, type, sK23: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 33.81/6.09 thf(func_def_94, type, sK24: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_95, type, sK25: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_96, type, sK26: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 33.81/6.09 thf(func_def_97, type, sK27: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_98, type, sK28: nat > nat > nat).
% 33.81/6.09 thf(func_def_99, type, sK29: (nat > a) > (nat > a) > nat > nat).
% 33.81/6.09 thf(func_def_100, type, sK30: nat > nat > nat).
% 33.81/6.09 thf(func_def_101, type, sK31: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_102, type, sK32: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_103, type, sK33: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_104, type, sK34: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_105, type, sK35: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_106, type, sK36: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_107, type, sK37: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_108, type, sK38: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_109, type, sK39: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_110, type, sK40: nat > (nat > $o) > nat).
% 33.81/6.09 thf(func_def_111, type, sK41: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 33.81/6.09 thf(func_def_112, type, sK42: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 33.81/6.09 thf(func_def_113, type, sK43: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 33.81/6.09 thf(func_def_114, type, sK44: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_115, type, sK45: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_116, type, sK46: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_117, type, sK47: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_118, type, sK48: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_119, type, sK49: (nat > nat) > nat).
% 33.81/6.09 thf(func_def_120, type, sK50: (nat > nat) > nat).
% 33.81/6.09 thf(func_def_121, type, sK51: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_122, type, sK52: nat > nat).
% 33.81/6.09 thf(func_def_123, type, sK53: !>[X0: $tType]:((nat > X0) > nat)).
% 33.81/6.09 thf(func_def_124, type, sK54: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_125, type, sK55: !>[X0: $tType]:((nat > X0) > (nat > X0) > nat > nat)).
% 33.81/6.09 thf(func_def_126, type, sK56: (nat > $o) > nat > nat > nat).
% 33.81/6.09 thf(func_def_127, type, sK57: (nat > a) > (nat > a) > nat > nat).
% 33.81/6.09 thf(func_def_128, type, sK58: a > set @ a > nat).
% 33.81/6.09 thf(func_def_129, type, sK59: a > set @ a > nat > a).
% 33.81/6.09 thf(func_def_130, type, sK60: a > set @ a > nat > b).
% 33.81/6.09 thf(func_def_131, type, sK61: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_132, type, sK62: nat > nat).
% 33.81/6.09 thf(func_def_133, type, sK63: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_134, type, sK64: (nat > $o) > nat > nat).
% 33.81/6.09 thf(func_def_135, type, sK65: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_136, type, sK66: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_137, type, sK67: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_138, type, sK68: (nat > nat > $o) > nat).
% 33.81/6.09 thf(func_def_139, type, sK69: (nat > $o) > nat).
% 33.81/6.09 thf(func_def_140, type, sK70: !>[X0: $tType]:((nat > X0) > nat)).
% 33.81/6.09 thf(func_def_141, type, sK71: !>[X0: $tType]:(set @ X0 > X0)).
% 33.81/6.09 thf(func_def_142, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X0 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat)).
% 33.81/6.09 thf(func_def_143, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X0 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat > X0)).
% 33.81/6.09 thf(func_def_144, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X0 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat > X3)).
% 33.81/6.09 thf(func_def_145, type, sK75: nat > nat).
% 33.81/6.09 thf(func_def_146, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X3 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat)).
% 33.81/6.09 thf(func_def_147, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X3 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat > X3)).
% 33.81/6.09 thf(func_def_148, type, sK78: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(X3 > set @ X3 > carrier_ext @ X0 @ aGroup_ext @ X0 @ ring_ext @ X0 @ X2 > carrier_ext @ X3 @ aGroup_ext @ X3 @ module_ext @ X3 @ X0 @ X1 > nat > X0)).
% 33.81/6.09 thf(func_def_150, type, ph80: !>[X0: $tType]:(X0)).
% 33.81/6.09 thf(f23289,plain,(
% 33.81/6.09 $false),
% 33.81/6.09 inference(avatar_sat_refutation,[status(thm)],[f1783,f1784,f19861,f20079,f23284])).
% 33.81/6.09 thf(f23284,plain,(
% 33.81/6.09 ~spl79_1 | ~spl79_275 | spl79_276),
% 33.81/6.09 inference(avatar_contradiction_clause,[status(thm)],[f23283])).
% 33.81/6.09 thf(f23283,plain,(
% 33.81/6.09 $false | (~spl79_1 | ~spl79_275 | spl79_276)),
% 33.81/6.09 inference(subsumption_resolution,[status(thm)],[f23282,f18442])).
% 33.81/6.09 thf(f18442,plain,(
% 33.81/6.09 (m2 != xa) | spl79_276),
% 33.81/6.09 inference(avatar_component_clause,[status(thm)],[f18441])).
% 33.81/6.09 thf(f18441,definition,(
% 33.81/6.09 spl79_276 <=> (m2 = xa)),
% 33.81/6.09 introduced(definition,[new_symbols(naming,[spl79_276])],[avatar_definition])).
% 33.81/6.09 thf(f23282,plain,(
% 33.81/6.09 (m2 = xa) | (~spl79_1 | ~spl79_275)),
% 33.81/6.09 inference(subsumption_resolution,[status(thm)],[f23244,f1778])).
% 33.81/6.09 thf(f1778,plain,(
% 33.81/6.09 (h2 = (g @ xa)) | ~spl79_1),
% 33.81/6.09 inference(avatar_component_clause,[status(thm)],[f1776])).
% 33.81/6.09 thf(f1776,definition,(
% 33.81/6.09 spl79_1 <=> (h2 = (g @ xa))),
% 33.81/6.09 introduced(definition,[new_symbols(naming,[spl79_1])],[avatar_definition])).
% 33.81/6.09 thf(f23244,plain,(
% 33.81/6.09 (h2 != (g @ xa)) | (m2 = xa) | ~spl79_275),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f23205])).
% 33.81/6.09 thf(f23205,plain,(
% 33.81/6.09 (m2 = xa) | ($true != $true) | (h2 != (g @ xa)) | ~spl79_275),
% 33.81/6.09 inference(superposition,[status(thm)],[f1822,f18438])).
% 33.81/6.09 thf(f18438,plain,(
% 33.81/6.09 ((ord_less_eq @ nat @ xa @ m2) = $true) | ~spl79_275),
% 33.81/6.09 inference(avatar_component_clause,[status(thm)],[f18437])).
% 33.81/6.09 thf(f18437,definition,(
% 33.81/6.09 spl79_275 <=> ((ord_less_eq @ nat @ xa @ m2) = $true)),
% 33.81/6.09 introduced(definition,[new_symbols(naming,[spl79_275])],[avatar_definition])).
% 33.81/6.09 thf(f1822,plain,(
% 33.81/6.09 ( ! [X0 : nat] : (($true != (ord_less_eq @ nat @ X0 @ m2)) | (h2 != (g @ X0)) | (m2 = X0)) )),
% 33.81/6.09 inference(forward_demodulation,[status(thm)],[f1821,f1638])).
% 33.81/6.09 thf(f1638,plain,(
% 33.81/6.09 (h2 = (g @ m2))),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f268])).
% 33.81/6.09 thf(f268,axiom,(
% 33.81/6.09 (h2 = (g @ m2))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_3)).
% 33.81/6.09 thf(f1821,plain,(
% 33.81/6.09 ( ! [X0 : nat] : ((m2 = X0) | ((g @ m2) != (g @ X0)) | ($true != (ord_less_eq @ nat @ X0 @ m2))) )),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f1820])).
% 33.81/6.09 thf(f1820,plain,(
% 33.81/6.09 ( ! [X0 : nat] : ((m2 = X0) | ((g @ m2) != (g @ X0)) | ($true != (ord_less_eq @ nat @ X0 @ m2)) | ($true != $true)) )),
% 33.81/6.09 inference(superposition,[status(thm)],[f1590,f1743])).
% 33.81/6.09 thf(f1743,plain,(
% 33.81/6.09 ( ! [X1 : nat] : (($true = (ord_less_eq @ nat @ X1 @ X1))) )),
% 33.81/6.09 inference(equality_resolution,[status(thm)],[f1647])).
% 33.81/6.09 thf(f1647,plain,(
% 33.81/6.09 ( ! [X0 : nat,X1 : nat] : ((X0 != X1) | ($true = (ord_less_eq @ nat @ X0 @ X1))) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f957])).
% 33.81/6.09 thf(f957,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : (((X0 != X1) & ($true != (ord_less @ nat @ X0 @ X1))) | ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 33.81/6.09 inference(ennf_transformation,[status(thm)],[f582])).
% 33.81/6.09 thf(f582,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : ((($true = (ord_less @ nat @ X0 @ X1)) | (X0 = X1)) => ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f581])).
% 33.81/6.09 thf(f581,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : (((X0 = X1) | (ord_less @ nat @ X0 @ X1)) => (ord_less_eq @ nat @ X0 @ X1))),
% 33.81/6.09 inference(rectify,[status(thm)],[f115])).
% 33.81/6.09 thf(f115,axiom,(
% 33.81/6.09 ! [X26 : nat,X19 : nat] : (((X19 = X26) | (ord_less @ nat @ X26 @ X19)) => (ord_less_eq @ nat @ X26 @ X19))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',fact_114_less__or__eq__imp__le)).
% 33.81/6.09 thf(f1590,plain,(
% 33.81/6.09 ( ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ m2)) | (X0 = X1) | ((g @ X1) != (g @ X0)) | ((ord_less_eq @ nat @ X1 @ m2) != $true)) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f1120])).
% 33.81/6.09 thf(f1120,plain,(
% 33.81/6.09 ! [X0 : nat] : (($true != (ord_less_eq @ nat @ X0 @ m2)) | ! [X1 : nat] : (((g @ X1) != (g @ X0)) | (X0 = X1) | ((ord_less_eq @ nat @ X1 @ m2) != $true)))),
% 33.81/6.09 inference(flattening,[status(thm)],[f1119])).
% 33.81/6.09 thf(f1119,plain,(
% 33.81/6.09 ! [X0 : nat] : (! [X1 : nat] : (((X0 = X1) | ((g @ X1) != (g @ X0))) | ((ord_less_eq @ nat @ X1 @ m2) != $true)) | ($true != (ord_less_eq @ nat @ X0 @ m2)))),
% 33.81/6.09 inference(ennf_transformation,[status(thm)],[f606])).
% 33.81/6.09 thf(f606,plain,(
% 33.81/6.09 ! [X0 : nat] : (($true = (ord_less_eq @ nat @ X0 @ m2)) => ! [X1 : nat] : (((ord_less_eq @ nat @ X1 @ m2) = $true) => (((g @ X1) = (g @ X0)) => (X0 = X1))))),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f605])).
% 33.81/6.09 thf(f605,plain,(
% 33.81/6.09 ! [X0 : nat] : ((ord_less_eq @ nat @ X0 @ m2) => ! [X1 : nat] : ((ord_less_eq @ nat @ X1 @ m2) => (((g @ X1) = (g @ X0)) => (X0 = X1))))),
% 33.81/6.09 inference(rectify,[status(thm)],[f266])).
% 33.81/6.09 thf(f266,axiom,(
% 33.81/6.09 ! [X79 : nat] : ((ord_less_eq @ nat @ X79 @ m2) => ! [X48 : nat] : ((ord_less_eq @ nat @ X48 @ m2) => (((g @ X79) = (g @ X48)) => (X48 = X79))))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_1)).
% 33.81/6.09 thf(f20079,plain,(
% 33.81/6.09 spl79_275),
% 33.81/6.09 inference(avatar_split_clause,[status(thm)],[f20046,f18437])).
% 33.81/6.09 thf(f20046,plain,(
% 33.81/6.09 ((ord_less_eq @ nat @ xa @ m2) = $true)),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f20023])).
% 33.81/6.09 thf(f20023,plain,(
% 33.81/6.09 ((ord_less_eq @ nat @ xa @ m2) = $true) | ($true != $true)),
% 33.81/6.09 inference(superposition,[status(thm)],[f1646,f19939])).
% 33.81/6.09 thf(f19939,plain,(
% 33.81/6.09 ($true = (ord_less @ nat @ xa @ m2))),
% 33.81/6.09 inference(forward_demodulation,[status(thm)],[f19900,f6549])).
% 33.81/6.09 thf(f6549,plain,(
% 33.81/6.09 (m2 = (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))))),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f6540])).
% 33.81/6.09 thf(f6540,plain,(
% 33.81/6.09 (m2 = (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat)))) | ($true != $true)),
% 33.81/6.09 inference(superposition,[status(thm)],[f1419,f1458])).
% 33.81/6.09 thf(f1458,plain,(
% 33.81/6.09 ((ord_less @ nat @ zero_zero @ nat @ m2) = $true)),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f530])).
% 33.81/6.09 thf(f530,plain,(
% 33.81/6.09 ((ord_less @ nat @ zero_zero @ nat @ m2) = $true)),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f529])).
% 33.81/6.09 thf(f529,plain,(
% 33.81/6.09 (ord_less @ nat @ zero_zero @ nat @ m2)),
% 33.81/6.09 inference(rectify,[status(thm)],[f281])).
% 33.81/6.09 thf(f281,axiom,(
% 33.81/6.09 (ord_less @ nat @ zero_zero @ nat @ m2)),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_16)).
% 33.81/6.09 thf(f1419,plain,(
% 33.81/6.09 ( ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X0) != $true) | ((suc @ (minus_minus @ nat @ X0 @ (suc @ zero_zero @ nat))) = X0)) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f902])).
% 33.81/6.09 thf(f902,plain,(
% 33.81/6.09 ! [X0 : nat] : (((suc @ (minus_minus @ nat @ X0 @ (suc @ zero_zero @ nat))) = X0) | ((ord_less @ nat @ zero_zero @ nat @ X0) != $true))),
% 33.81/6.09 inference(ennf_transformation,[status(thm)],[f542])).
% 33.81/6.09 thf(f542,plain,(
% 33.81/6.09 ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X0) = $true) => ((suc @ (minus_minus @ nat @ X0 @ (suc @ zero_zero @ nat))) = X0))),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f541])).
% 33.81/6.09 thf(f541,plain,(
% 33.81/6.09 ! [X0 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X0) => ((suc @ (minus_minus @ nat @ X0 @ (suc @ zero_zero @ nat))) = X0))),
% 33.81/6.09 inference(rectify,[status(thm)],[f27])).
% 33.81/6.09 thf(f27,axiom,(
% 33.81/6.09 ! [X19 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X19) => ((suc @ (minus_minus @ nat @ X19 @ (suc @ zero_zero @ nat))) = X19))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',fact_26_Suc__pred)).
% 33.81/6.09 thf(f19900,plain,(
% 33.81/6.09 ($true = (ord_less @ nat @ xa @ (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat)))))),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f19875])).
% 33.81/6.09 thf(f19875,plain,(
% 33.81/6.09 ($true = (ord_less @ nat @ xa @ (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))))) | ($true != $true)),
% 33.81/6.09 inference(superposition,[status(thm)],[f1488,f1401])).
% 33.81/6.09 thf(f1401,plain,(
% 33.81/6.09 ((ord_less_eq @ nat @ xa @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))) = $true)),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f642])).
% 33.81/6.09 thf(f642,plain,(
% 33.81/6.09 ((ord_less_eq @ nat @ xa @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))) = $true)),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f641])).
% 33.81/6.09 thf(f641,plain,(
% 33.81/6.09 (ord_less_eq @ nat @ xa @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat)))),
% 33.81/6.09 inference(rectify,[status(thm)],[f284])).
% 33.81/6.09 thf(f284,axiom,(
% 33.81/6.09 (ord_less_eq @ nat @ xa @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat)))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_19)).
% 33.81/6.09 thf(f1488,plain,(
% 33.81/6.09 ( ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less @ nat @ X0 @ (suc @ X1)) = $true)) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f1025])).
% 33.81/6.09 thf(f1025,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less @ nat @ X0 @ (suc @ X1)) = $true))),
% 33.81/6.09 inference(ennf_transformation,[status(thm)],[f670])).
% 33.81/6.09 thf(f670,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) => ((ord_less @ nat @ X0 @ (suc @ X1)) = $true))),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f669])).
% 33.81/6.09 thf(f669,plain,(
% 33.81/6.09 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => (ord_less @ nat @ X0 @ (suc @ X1)))),
% 33.81/6.09 inference(rectify,[status(thm)],[f162])).
% 33.81/6.09 thf(f162,axiom,(
% 33.81/6.09 ! [X26 : nat,X19 : nat] : ((ord_less_eq @ nat @ X26 @ X19) => (ord_less @ nat @ X26 @ (suc @ X19)))),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',fact_161_le__imp__less__Suc)).
% 33.81/6.09 thf(f1646,plain,(
% 33.81/6.09 ( ! [X0 : nat,X1 : nat] : (($true != (ord_less @ nat @ X0 @ X1)) | ($true = (ord_less_eq @ nat @ X0 @ X1))) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f957])).
% 33.81/6.09 thf(f19861,plain,(
% 33.81/6.09 ~spl79_276),
% 33.81/6.09 inference(avatar_contradiction_clause,[status(thm)],[f19860])).
% 33.81/6.09 thf(f19860,plain,(
% 33.81/6.09 $false | ~spl79_276),
% 33.81/6.09 inference(subsumption_resolution,[status(thm)],[f19859,f1605])).
% 33.81/6.09 thf(f1605,plain,(
% 33.81/6.09 ( ! [X0 : nat] : (((ord_less @ nat @ X0 @ X0) != $true)) )),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f837])).
% 33.81/6.09 thf(f837,plain,(
% 33.81/6.09 ! [X0 : nat] : ((ord_less @ nat @ X0 @ X0) != $true)),
% 33.81/6.09 inference(flattening,[status(thm)],[f608])).
% 33.81/6.09 thf(f608,plain,(
% 33.81/6.09 ! [X0 : nat] : ~((ord_less @ nat @ X0 @ X0) = $true)),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f607])).
% 33.81/6.09 thf(f607,plain,(
% 33.81/6.09 ! [X0 : nat] : ~(ord_less @ nat @ X0 @ X0)),
% 33.81/6.09 inference(rectify,[status(thm)],[f78])).
% 33.81/6.09 thf(f78,axiom,(
% 33.81/6.09 ! [X19 : nat] : ~(ord_less @ nat @ X19 @ X19)),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',fact_77_less__irrefl__nat)).
% 33.81/6.09 thf(f19859,plain,(
% 33.81/6.09 ($true = (ord_less @ nat @ m2 @ m2)) | ~spl79_276),
% 33.81/6.09 inference(forward_demodulation,[status(thm)],[f19778,f6549])).
% 33.81/6.09 thf(f19778,plain,(
% 33.81/6.09 ($true = (ord_less @ nat @ m2 @ (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))))) | ~spl79_276),
% 33.81/6.09 inference(trivial_inequality_removal,[status(thm)],[f19757])).
% 33.81/6.09 thf(f19757,plain,(
% 33.81/6.09 ($true != $true) | ($true = (ord_less @ nat @ m2 @ (suc @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat))))) | ~spl79_276),
% 33.81/6.09 inference(superposition,[status(thm)],[f1488,f19735])).
% 33.81/6.09 thf(f19735,plain,(
% 33.81/6.09 ($true = (ord_less_eq @ nat @ m2 @ (minus_minus @ nat @ m2 @ (suc @ zero_zero @ nat)))) | ~spl79_276),
% 33.81/6.09 inference(backward_demodulation,[status(thm)],[f1401,f18443])).
% 33.81/6.09 thf(f18443,plain,(
% 33.81/6.09 (m2 = xa) | ~spl79_276),
% 33.81/6.09 inference(avatar_component_clause,[status(thm)],[f18441])).
% 33.81/6.09 thf(f1784,plain,(
% 33.81/6.09 ~spl79_2),
% 33.81/6.09 inference(avatar_split_clause,[status(thm)],[f1356,f1780])).
% 33.81/6.09 thf(f1780,definition,(
% 33.81/6.09 spl79_2 <=> ((member @ a @ (g @ xa) @ h1) = $true)),
% 33.81/6.09 introduced(definition,[new_symbols(naming,[spl79_2])],[avatar_definition])).
% 33.81/6.09 thf(f1356,plain,(
% 33.81/6.09 ((member @ a @ (g @ xa) @ h1) != $true)),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f812])).
% 33.81/6.09 thf(f812,plain,(
% 33.81/6.09 ((member @ a @ (g @ xa) @ h1) != $true)),
% 33.81/6.09 inference(flattening,[status(thm)],[f454])).
% 33.81/6.09 thf(f454,plain,(
% 33.81/6.09 ~((member @ a @ (g @ xa) @ h1) = $true)),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f453])).
% 33.81/6.09 thf(f453,plain,(
% 33.81/6.09 ~(member @ a @ (g @ xa) @ h1)),
% 33.81/6.09 inference(rectify,[status(thm)],[f287])).
% 33.81/6.09 thf(f287,negated_conjecture,(
% 33.81/6.09 ~(member @ a @ (g @ xa) @ h1)),
% 33.81/6.09 inference(negated_conjecture,[status(cth)],[f286])).
% 33.81/6.09 thf(f286,conjecture,(
% 33.81/6.09 (member @ a @ (g @ xa) @ h1)),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_21)).
% 33.81/6.09 thf(f1783,plain,(
% 33.81/6.09 spl79_1 | spl79_2),
% 33.81/6.09 inference(avatar_split_clause,[status(thm)],[f1688,f1780,f1776])).
% 33.81/6.09 thf(f1688,plain,(
% 33.81/6.09 ((member @ a @ (g @ xa) @ h1) = $true) | (h2 = (g @ xa))),
% 33.81/6.09 inference(cnf_transformation,[status(thm)],[f678])).
% 33.81/6.09 thf(f678,plain,(
% 33.81/6.09 ((member @ a @ (g @ xa) @ h1) = $true) | (h2 = (g @ xa))),
% 33.81/6.09 inference(fool_elimination,[status(thm)],[f677])).
% 33.81/6.09 thf(f677,plain,(
% 33.81/6.09 (h2 = (g @ xa)) | (member @ a @ (g @ xa) @ h1)),
% 33.81/6.09 inference(rectify,[status(thm)],[f285])).
% 33.81/6.09 thf(f285,axiom,(
% 33.81/6.09 (h2 = (g @ xa)) | (member @ a @ (g @ xa) @ h1)),
% 33.81/6.09 file('/export/starexec/sandbox/tmp/tmp.cvBoRdIAaj/DTF2THF_27230.p',conj_20)).
% 33.81/6.09 % SZS output end Proof for DTF2THF_27230
% 33.81/6.09 % (27805)------------------------------
% 33.81/6.09 % (27805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.81/6.09 % (27805)Termination reason: Refutation
% 33.81/6.09
% 33.81/6.09 % (27805)Memory used [KB]: 15479
% 33.81/6.09 % (27805)Time elapsed: 2.601 s
% 33.81/6.09 % (27805)Instructions burned: 7762 (million)
% 33.81/6.09 % (27805)------------------------------
% 33.81/6.09 % (27805)------------------------------
% 33.81/6.09 % (27421)Success in time 4.452 s
%------------------------------------------------------------------------------