%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP131^2 : 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 : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:48 AM UTC 2026
% Result : ContradictoryAxioms 2.77s 1.74s
% Output : Refutation 2.77s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 5
% Syntax : Number of formulae : 39 ( 39 unt; 0 typ; 0 def)
% Number of atoms : 161 ( 70 equ; 0 cnn)
% Maximal formula atoms : 1 ( 4 avg)
% Number of connectives : 378 ( 11 ~; 0 |; 0 &; 353 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 35 usr; 6 con; 0-3 aty)
% ( 14 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 50 ( 19 ^; 31 !; 0 ?; 50 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_6,type,
nat: $tType ).
thf(func_def_0,type,
nat: $tType ).
thf(func_def_1,type,
zero:
!>[X0: $tType] : $o ).
thf(func_def_2,type,
group_add:
!>[X0: $tType] : $o ).
thf(func_def_3,type,
monoid_add:
!>[X0: $tType] : $o ).
thf(func_def_4,type,
ab_group_add:
!>[X0: $tType] : $o ).
thf(func_def_5,type,
wellorder:
!>[X0: $tType] : $o ).
thf(func_def_6,type,
semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_7,type,
semigroup_mult:
!>[X0: $tType] : $o ).
thf(func_def_8,type,
comm_monoid_add:
!>[X0: $tType] : $o ).
thf(func_def_9,type,
ab_semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_10,type,
comm_monoid_diff:
!>[X0: $tType] : $o ).
thf(func_def_11,type,
ab_semigroup_mult:
!>[X0: $tType] : $o ).
thf(func_def_12,type,
cancel_semigroup_add:
!>[X0: $tType] : $o ).
thf(func_def_13,type,
ordered_ab_group_add:
!>[X0: $tType] : $o ).
thf(func_def_14,type,
cancel1352612707id_add:
!>[X0: $tType] : $o ).
thf(func_def_15,type,
cancel146912293up_add:
!>[X0: $tType] : $o ).
thf(func_def_16,type,
linord219039673up_add:
!>[X0: $tType] : $o ).
thf(func_def_17,type,
ordere779506340up_add:
!>[X0: $tType] : $o ).
thf(func_def_18,type,
canoni770627133id_add:
!>[X0: $tType] : $o ).
thf(func_def_19,type,
ordere236663937imp_le:
!>[X0: $tType] : $o ).
thf(func_def_20,type,
ordere223160158up_add:
!>[X0: $tType] : $o ).
thf(func_def_21,type,
ordere623563068d_diff:
!>[X0: $tType] : $o ).
thf(func_def_22,type,
strict2144017051up_add:
!>[X0: $tType] : $o ).
thf(func_def_23,type,
ordere516151231imp_le:
!>[X0: $tType] : $o ).
thf(func_def_24,type,
minus_minus:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_25,type,
plus_plus:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_26,type,
times_times:
!>[X0: $tType] : ( X0 > X0 > X0 ) ).
thf(func_def_27,type,
zero_zero:
!>[X0: $tType] : X0 ).
thf(func_def_28,type,
number2016821345itions: ( nat > nat ) > nat > $o ).
thf(func_def_29,type,
ord_less:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_30,type,
ord_less_eq:
!>[X0: $tType] : ( X0 > X0 > $o ) ).
thf(func_def_31,type,
i: nat ).
thf(func_def_32,type,
k: nat ).
thf(func_def_33,type,
n: nat ).
thf(func_def_34,type,
p: nat > nat ).
thf(func_def_52,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f2850,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2849]) ).
thf(f2849,plain,
$true = $false,
inference(equality_proxy_clausification,[status(thm)],[f2815]) ).
thf(f2815,plain,
( $true
= ( $false = $true ) ),
inference(boolean_simplification,[status(thm)],[f2814]) ).
thf(f2814,plain,
( $true
= ( ( ~ $true ) = $true ) ),
inference(forward_demodulation,[status(thm)],[f2770,f1106]) ).
thf(f1106,plain,
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n )
= $true ),
inference(cnf_transformation,[status(thm)],[f336]) ).
thf(f336,plain,
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n )
= $true ),
inference(fool_elimination,[status(thm)],[f335]) ).
thf(f335,plain,
ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n,
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n,
file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_2_upper__bound) ).
thf(f2770,plain,
( $true
= ( ( ~ ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n ) )
= $true ) ),
inference(superposition,[status(thm)],[f2725,f1064]) ).
thf(f1064,plain,
( ( ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f382]) ).
thf(f382,plain,
( ( ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f381]) ).
thf(f381,plain,
ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ),
file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_3_lower__bound) ).
thf(f2725,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( ~ ( ord_less_eq @ nat @ X2 @ X1 ) )
= ( ord_less @ nat @ X1 @ X2 ) ) ),
inference(forward_demodulation,[status(thm)],[f2724,f2617]) ).
thf(f2617,plain,
! [X0: nat,X1: nat] :
( ( ~ ( ord_less_eq @ nat @ X0 @ X1 ) )
= ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ X0 @ X1 ) ) ),
inference(superposition,[status(thm)],[f1518,f2507]) ).
thf(f2507,plain,
! [X2: nat,X1: nat] :
( ( zero_zero
@ ( nat
= ( minus_minus @ nat @ X2 @ X1 ) ) )
= ( ord_less_eq @ nat @ X2 @ X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f2501]) ).
thf(f2501,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( ord_less_eq @ nat @ X2 @ X1 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ X2 @ X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2500]) ).
thf(f2500,plain,
! [X2: nat,X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less_eq @ nat @ Y0 @ X1 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ Y0 @ X1 ) ) ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f2465]) ).
thf(f2465,plain,
! [X1: nat] :
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq @ nat @ Y0 @ X1 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ Y0 @ X1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2464]) ).
thf(f2464,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq @ nat @ Y1 @ Y0 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ Y1 @ Y0 ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1109]) ).
thf(f1109,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq @ nat @ Y1 @ Y0 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ Y1 @ Y0 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f354]) ).
thf(f354,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq @ nat @ Y1 @ Y0 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ Y1 @ Y0 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f353]) ).
thf(f353,plain,
! [X0: nat,X1: nat] :
( ( ord_less_eq @ nat @ X0 @ X1 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f81]) ).
thf(f81,axiom,
! [X5: nat,X3: nat] :
( ( ord_less_eq @ nat @ X5 @ X3 )
= ( zero_zero
@ ( nat
= ( minus_minus @ nat @ X5 @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_80_diff__is__0__eq) ).
thf(f1518,plain,
! [X1: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ X1 )
= ( ~ ( zero_zero @ ( nat = X1 ) ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f1517]) ).
thf(f1517,plain,
! [X1: nat] :
( $true
= ( ( ord_less @ nat @ zero_zero @ nat @ X1 )
= ( ~ ( zero_zero @ ( nat = X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1516]) ).
thf(f1516,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ Y0 )
= ( ~ ( zero_zero @ ( nat = Y0 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1034]) ).
thf(f1034,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ Y0 )
= ( ~ ( zero_zero @ ( nat = Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f806]) ).
thf(f806,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ Y0 )
= ( ~ ( zero_zero @ ( nat = Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f805]) ).
thf(f805,plain,
! [X0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ X0 )
= ( zero_zero @ ( nat != X0 ) ) ),
inference(rectify,[status(thm)],[f52]) ).
thf(f52,axiom,
! [X8: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ X8 )
= ( zero_zero @ ( nat != X8 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_51_bot__nat__0_Onot__eq__extremum) ).
thf(f2724,plain,
! [X2: nat,X1: nat] :
( $true
= ( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ X2 @ X1 ) )
= ( ord_less @ nat @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2723]) ).
thf(f2723,plain,
! [X2: nat,X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ Y0 @ X1 ) )
= ( ord_less @ nat @ X1 @ Y0 ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f2665]) ).
thf(f2665,plain,
! [X1: nat] :
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ Y0 @ X1 ) )
= ( ord_less @ nat @ X1 @ Y0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2664]) ).
thf(f2664,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ Y1 @ Y0 ) )
= ( ord_less @ nat @ Y0 @ Y1 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1131]) ).
thf(f1131,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ Y1 @ Y0 ) )
= ( ord_less @ nat @ Y0 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f476]) ).
thf(f476,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ Y1 @ Y0 ) )
= ( ord_less @ nat @ Y0 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f475]) ).
thf(f475,plain,
! [X0: nat,X1: nat] :
( ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ X0 @ X1 ) )
= ( ord_less @ nat @ X1 @ X0 ) ),
inference(rectify,[status(thm)],[f79]) ).
thf(f79,axiom,
! [X3: nat,X5: nat] :
( ( ord_less @ nat @ X5 @ X3 )
= ( ord_less @ nat @ zero_zero @ nat @ ( minus_minus @ nat @ X3 @ X5 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_78_zero__less__diff) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP131^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n029.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:29 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP131^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Number_Partition problem prob_126__5325690_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Number_Partition/prob_126__5325690_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : ContradictoryAxioms
% 0.29/0.45 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 309 ( 90 unt; 35 typ; 0 def)
% 0.29/0.45 % Number of atoms : 733 ( 278 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 4 ( 2 avg)
% 0.29/0.45 % Number of connectives : 2936 ( 41 ~; 10 |; 22 &;2585 @)
% 0.29/0.45 % ( 0 <=>; 278 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 15 ( 6 avg)
% 0.29/0.45 % Number of types : 2 ( 1 usr)
% 0.29/0.45 % Number of type conns : 31 ( 31 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 36 ( 34 usr; 4 con; 0-3 aty)
% 0.29/0.45 % Number of variables : 794 ( 21 ^; 736 !; 8 ?; 794 :)
% 0.29/0.45 % ( 29 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.45 % SPC : TH1_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:27:44.817
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (1)
% 0.29/0.45 thf(ty_t_Nat_Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (34)
% 0.29/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.45 zero:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.45 group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Omonoid__add,type,
% 0.29/0.45 monoid_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oab__group__add,type,
% 0.29/0.45 ab_group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.45 wellorder:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/0.45 semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.29/0.45 semigroup_mult:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.29/0.45 comm_monoid_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/0.45 ab_semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.29/0.45 comm_monoid_diff:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.29/0.45 ab_semigroup_mult:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/0.45 cancel_semigroup_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.29/0.45 ordered_ab_group_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.29/0.45 cancel1352612707id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.29/0.45 cancel146912293up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/0.45 linord219039673up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/0.45 ordere779506340up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.45 canoni770627133id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/0.45 ordere236663937imp_le:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.29/0.45 ordere223160158up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 0.29/0.45 ordere623563068d_diff:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.29/0.45 strict2144017051up_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.29/0.45 ordere516151231imp_le:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.29/0.45 minus_minus:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/0.45 plus_plus:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.29/0.45 times_times:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.45 zero_zero:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 0.29/0.45 number2016821345itions: ( nat > nat ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.45 ord_less:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.45 ord_less_eq:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_i____,type,
% 0.29/0.45 i: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_k,type,
% 0.29/0.45 k: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_n,type,
% 0.29/0.45 n: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_p,type,
% 0.29/0.45 p: nat > nat ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (254)
% 0.29/0.45 thf(fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062,axiom,
% 0.29/0.45 ord_less_eq @ nat @ i @ n ).
% 0.29/0.45
% 0.29/0.45 % \<open>i \<le> n\<close>
% 0.29/0.45 thf(fact_1_assms_I1_J,axiom,
% 0.29/0.45 number2016821345itions @ p @ n ).
% 0.29/0.45
% 0.29/0.45 % assms(1)
% 0.29/0.45 thf(fact_2_upper__bound,axiom,
% 0.29/0.45 ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n ).
% 0.29/0.45
% 0.29/0.45 % upper_bound
% 0.29/0.45 thf(fact_3_lower__bound,axiom,
% 0.29/0.45 ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) ).
% 0.29/0.45
% 0.29/0.45 % lower_bound
% 0.29/0.45 thf(fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 0.29/0.45 ord_less_eq @ nat @ k @ n ).
% 0.29/0.45
% 0.29/0.45 % \<open>k \<le> n\<close>
% 0.29/0.45 thf(fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062,axiom,
% 0.29/0.45 ( ( p @ i )
% 0.29/0.45 != ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>p i \<noteq> 0\<close>
% 0.29/0.45 thf(fact_6_assms_I2_J,axiom,
% 0.29/0.45 ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) ).
% 0.29/0.45
% 0.29/0.45 % assms(2)
% 0.29/0.45 thf(fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062,axiom,
% 0.29/0.45 i != k ).
% 0.29/0.45
% 0.29/0.45 % \<open>i \<noteq> k\<close>
% 0.29/0.45 thf(fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062,axiom,
% 0.29/0.45 ord_less @ nat @ ( minus_minus @ nat @ n @ k ) @ i ).
% 0.29/0.45
% 0.29/0.45 % \<open>n - k < i\<close>
% 0.29/0.45 thf(fact_9_partitions__zero,axiom,
% 0.29/0.45 ! [P: nat > nat] :
% 0.29/0.45 ( ( number2016821345itions @ P @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( P
% 0.29/0.45 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % partitions_zero
% 0.29/0.45 thf(fact_10_partitions__bounds,axiom,
% 0.29/0.45 ! [P: nat > nat,N: nat,I2: nat] :
% 0.29/0.45 ( ( number2016821345itions @ P @ N )
% 0.29/0.45 => ( ord_less_eq @ nat @ ( P @ I2 ) @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % partitions_bounds
% 0.29/0.45 thf(fact_11_mult__le__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.29/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_le_cancel2
% 0.29/0.45 thf(fact_12_nat__mult__le__cancel__disj,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.29/0.45 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_mult_le_cancel_disj
% 0.29/0.45 thf(fact_13_mult__less__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.29/0.45 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_less_cancel2
% 0.29/0.45 thf(fact_14_nat__0__less__mult__iff,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.45 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_0_less_mult_iff
% 0.29/0.45 thf(fact_15_nat__mult__less__cancel__disj,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.29/0.45 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_mult_less_cancel_disj
% 0.29/0.45 thf(fact_16_add__gr__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.45 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_gr_0
% 0.29/0.45 thf(fact_17_add__less__same__cancel1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [B: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.45 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_same_cancel1
% 0.29/0.45 thf(fact_18_add__less__same__cancel2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.45 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_same_cancel2
% 0.29/0.45 thf(fact_19_less__add__same__cancel1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.45 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_add_same_cancel1
% 0.29/0.45 thf(fact_20_less__add__same__cancel2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.45 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_add_same_cancel2
% 0.29/0.45 thf(fact_21_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.45 thf(fact_22_add__right__cancel,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel_semigroup_add @ A )
% 0.29/0.45 => ! [B: A,A2: A,C: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.45 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.45 = ( B = C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_right_cancel
% 0.29/0.45 thf(fact_23_add__left__cancel,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel_semigroup_add @ A )
% 0.29/0.45 => ! [A2: A,B: A,C: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.45 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.45 = ( B = C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_left_cancel
% 0.29/0.45 thf(fact_24_le__zero__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_zero_eq
% 0.29/0.45 thf(fact_25_not__gr__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_gr_zero
% 0.29/0.45 thf(fact_26_add__le__cancel__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere236663937imp_le @ A )
% 0.29/0.45 => ! [A2: A,C: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_cancel_right
% 0.29/0.45 thf(fact_27_add__le__cancel__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere236663937imp_le @ A )
% 0.29/0.45 => ! [C: A,A2: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_cancel_left
% 0.29/0.45 thf(fact_28_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( plus_plus @ A @ X @ Y ) )
% 0.29/0.45 = ( ( X
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 & ( Y
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_eq_add_iff_both_eq_0
% 0.29/0.45 thf(fact_29_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ X @ Y )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( ( X
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 & ( Y
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_eq_0_iff_both_eq_0
% 0.29/0.45 thf(fact_30_add__cancel__right__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.45 = ( B
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_right_right
% 0.29/0.45 thf(fact_31_add__cancel__right__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.45 = ( B
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_right_left
% 0.29/0.45 thf(fact_32_add__cancel__left__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.45 = A2 )
% 0.29/0.45 = ( B
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_left_right
% 0.29/0.45 thf(fact_33_add__cancel__left__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [B: A,A2: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.45 = A2 )
% 0.29/0.45 = ( B
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_cancel_left_left
% 0.29/0.45 thf(fact_34_double__zero__sym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % double_zero_sym
% 0.29/0.45 thf(fact_35_double__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( A2
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % double_zero
% 0.29/0.45 thf(fact_36_add_Oright__neutral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( monoid_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % add.right_neutral
% 0.29/0.45 thf(fact_37_add_Oleft__neutral,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( monoid_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % add.left_neutral
% 0.29/0.45 thf(fact_38_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( minus_minus @ A @ A2 @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % cancel_comm_monoid_add_class.diff_cancel
% 0.29/0.45 thf(fact_39_diff__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel1352612707id_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_zero
% 0.29/0.45 thf(fact_40_zero__diff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( comm_monoid_diff @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_diff
% 0.29/0.45 thf(fact_41_diff__0__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( group_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_0_right
% 0.29/0.45 thf(fact_42_diff__self,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( group_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( minus_minus @ A @ A2 @ A2 )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_self
% 0.29/0.45 thf(fact_43_add__less__cancel__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere236663937imp_le @ A )
% 0.29/0.45 => ! [A2: A,C: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.45 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_cancel_right
% 0.29/0.45 thf(fact_44_add__less__cancel__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere236663937imp_le @ A )
% 0.29/0.45 => ! [C: A,A2: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.45 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_less_cancel_left
% 0.29/0.45 thf(fact_45_add__diff__cancel__right_H,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel146912293up_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_right'
% 0.29/0.45 thf(fact_46_add__diff__cancel__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel146912293up_add @ A )
% 0.29/0.45 => ! [A2: A,C: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.45 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_right
% 0.29/0.45 thf(fact_47_add__diff__cancel__left_H,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel146912293up_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ A2 )
% 0.29/0.45 = B ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_left'
% 0.29/0.45 thf(fact_48_add__diff__cancel__left,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( cancel146912293up_add @ A )
% 0.29/0.45 => ! [C: A,A2: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.45 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel_left
% 0.29/0.45 thf(fact_49_diff__add__cancel,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( group_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_add_cancel
% 0.29/0.45 thf(fact_50_add__diff__cancel,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( group_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_diff_cancel
% 0.29/0.45 thf(fact_51_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A2: nat] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.not_eq_extremum
% 0.29/0.45 thf(fact_52_less__nat__zero__code,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_nat_zero_code
% 0.29/0.45 thf(fact_53_neq0__conv,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( N
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq0_conv
% 0.29/0.45 thf(fact_54_bot__nat__0_Oextremum,axiom,
% 0.29/0.45 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.extremum
% 0.29/0.45 thf(fact_55_le0,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.45
% 0.29/0.45 % le0
% 0.29/0.45 thf(fact_56_Nat_Oadd__0__right,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.29/0.45 = M ) ).
% 0.29/0.45
% 0.29/0.45 % Nat.add_0_right
% 0.29/0.45 thf(fact_57_add__is__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( plus_plus @ nat @ M @ N )
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 = ( ( M
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 & ( N
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_is_0
% 0.29/0.45 thf(fact_58_nat__add__left__cancel__less,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.45 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_add_left_cancel_less
% 0.29/0.45 thf(fact_59_diff__self__eq__0,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( minus_minus @ nat @ M @ M )
% 0.29/0.45 = ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_self_eq_0
% 0.29/0.45 thf(fact_60_diff__0__eq__0,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.45 = ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_0_eq_0
% 0.29/0.45 thf(fact_61_mult__cancel2,axiom,
% 0.29/0.45 ! [M: nat,K: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times @ nat @ M @ K )
% 0.29/0.45 = ( times_times @ nat @ N @ K ) )
% 0.29/0.45 = ( ( M = N )
% 0.29/0.45 | ( K
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_cancel2
% 0.29/0.45 thf(fact_62_mult__cancel1,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times @ nat @ K @ M )
% 0.29/0.45 = ( times_times @ nat @ K @ N ) )
% 0.29/0.45 = ( ( M = N )
% 0.29/0.45 | ( K
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_cancel1
% 0.29/0.45 thf(fact_63_mult__0__right,axiom,
% 0.29/0.45 ! [M: nat] :
% 0.29/0.45 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 0.29/0.45 = ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_0_right
% 0.29/0.45 thf(fact_64_mult__is__0,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ( times_times @ nat @ M @ N )
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 = ( ( M
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 | ( N
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % mult_is_0
% 0.29/0.45 thf(fact_65_nat__add__left__cancel__le,axiom,
% 0.29/0.45 ! [K: nat,M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.45 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_add_left_cancel_le
% 0.29/0.45 thf(fact_66_diff__diff__cancel,axiom,
% 0.29/0.45 ! [I2: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ I2 @ N )
% 0.29/0.45 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
% 0.29/0.45 = I2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_diff_cancel
% 0.29/0.45 thf(fact_67_diff__diff__left,axiom,
% 0.29/0.45 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.45 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 0.29/0.45 = ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_diff_left
% 0.29/0.45 thf(fact_68_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.45 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.45 thf(fact_69_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.45 thf(fact_70_le__add__same__cancel2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.45 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_add_same_cancel2
% 0.29/0.45 thf(fact_71_le__add__same__cancel1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.45 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_add_same_cancel1
% 0.29/0.45 thf(fact_72_add__le__same__cancel2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_same_cancel2
% 0.29/0.45 thf(fact_73_add__le__same__cancel1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordere516151231imp_le @ A )
% 0.29/0.45 => ! [B: A,A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % add_le_same_cancel1
% 0.29/0.45 thf(fact_74_diff__ge__0__iff__ge,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordered_ab_group_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.29/0.45 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_ge_0_iff_ge
% 0.29/0.45 thf(fact_75_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linord219039673up_add @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.45 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.45 thf(fact_76_diff__gt__0__iff__gt,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ordered_ab_group_add @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.29/0.45 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_gt_0_iff_gt
% 0.29/0.45 thf(fact_77_diff__add__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( comm_monoid_diff @ A )
% 0.29/0.45 => ! [A2: A,B: A] :
% 0.29/0.45 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_add_zero
% 0.29/0.45 thf(fact_78_zero__less__diff,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 0.29/0.45 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_diff
% 0.29/0.45 thf(fact_79_diff__is__0__eq_H,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.45 => ( ( minus_minus @ nat @ M @ N )
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % diff_is_0_eq'
% 0.29/0.45 thf(fact_80_diff__is__0__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( minus_minus @ nat @ M @ N )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_is_0_eq
% 0.29/0.46 thf(fact_81_Nat_Odiff__diff__right,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.29/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_diff_right
% 0.29/0.46 thf(fact_82_Nat_Oadd__diff__assoc2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.add_diff_assoc2
% 0.29/0.46 thf(fact_83_Nat_Oadd__diff__assoc,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.29/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.add_diff_assoc
% 0.29/0.46 thf(fact_84_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel146912293up_add @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.29/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.29/0.46 thf(fact_85_diff__eq__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.29/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.29/0.46 => ( ( A2 = B )
% 0.29/0.46 = ( C = D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_eq
% 0.29/0.46 thf(fact_86_diff__commute,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 0.29/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_commute
% 0.29/0.46 thf(fact_87_diff__eq__diff__less__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.29/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_less_eq
% 0.29/0.46 thf(fact_88_diff__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_right_mono
% 0.29/0.46 thf(fact_89_diff__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_left_mono
% 0.29/0.46 thf(fact_90_diff__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,D: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ D @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mono
% 0.29/0.46 thf(fact_91_eq__iff__diff__eq__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.29/0.46 = ( ^ [A3: A,B2: A] :
% 0.29/0.46 ( ( minus_minus @ A @ A3 @ B2 )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff_diff_eq_0
% 0.29/0.46 thf(fact_92_diff__strict__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_strict_right_mono
% 0.29/0.46 thf(fact_93_diff__strict__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.29/0.46 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_strict_left_mono
% 0.29/0.46 thf(fact_94_diff__eq__diff__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.29/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.29/0.46 => ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_diff_less
% 0.29/0.46 thf(fact_95_diff__strict__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,D: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less @ A @ D @ C )
% 0.29/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_strict_mono
% 0.29/0.46 thf(fact_96_add__implies__diff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel1352612707id_add @ A )
% 0.29/0.46 => ! [C: A,B: A,A2: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ C @ B )
% 0.29/0.46 = A2 )
% 0.29/0.46 => ( C
% 0.29/0.46 = ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_implies_diff
% 0.29/0.46 thf(fact_97_diff__diff__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel146912293up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_diff_add
% 0.29/0.46 thf(fact_98_diff__add__eq__diff__diff__swap,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_eq_diff_diff_swap
% 0.29/0.46 thf(fact_99_diff__add__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_eq
% 0.29/0.46 thf(fact_100_diff__diff__eq2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_diff_eq2
% 0.29/0.46 thf(fact_101_add__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_diff_eq
% 0.29/0.46 thf(fact_102_eq__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( minus_minus @ A @ C @ B ) )
% 0.29/0.46 = ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.46 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_diff_eq
% 0.29/0.46 thf(fact_103_diff__eq__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.29/0.46 = C )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_eq_eq
% 0.29/0.46 thf(fact_104_group__cancel_Osub1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_group_add @ A )
% 0.29/0.46 => ! [A4: A,K: A,A2: A,B: A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( plus_plus @ A @ K @ A2 ) )
% 0.29/0.46 => ( ( minus_minus @ A @ A4 @ B )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.sub1
% 0.29/0.46 thf(fact_105_minus__nat_Odiff__0,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % minus_nat.diff_0
% 0.29/0.46 thf(fact_106_diffs0__imp__equal,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( minus_minus @ nat @ M @ N )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 => ( ( ( minus_minus @ nat @ N @ M )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diffs0_imp_equal
% 0.29/0.46 thf(fact_107_less__imp__diff__less,axiom,
% 0.29/0.46 ! [J: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ J @ K )
% 0.29/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_diff_less
% 0.29/0.46 thf(fact_108_diff__less__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( ( ord_less @ nat @ M @ L )
% 0.29/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less_mono2
% 0.29/0.46 thf(fact_109_diff__le__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono2
% 0.29/0.46 thf(fact_110_le__diff__iff_H,axiom,
% 0.29/0.46 ! [A2: nat,C: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A2 @ C )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ B @ C )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff'
% 0.29/0.46 thf(fact_111_diff__le__self,axiom,
% 0.29/0.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_self
% 0.29/0.46 thf(fact_112_diff__le__mono,axiom,
% 0.29/0.46 ! [M: nat,N: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_mono
% 0.29/0.46 thf(fact_113_Nat_Odiff__diff__eq,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.29/0.46 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_diff_eq
% 0.29/0.46 thf(fact_114_le__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_iff
% 0.29/0.46 thf(fact_115_eq__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.29/0.46 => ( ( ( minus_minus @ nat @ M @ K )
% 0.29/0.46 = ( minus_minus @ nat @ N @ K ) )
% 0.29/0.46 = ( M = N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_diff_iff
% 0.29/0.46 thf(fact_116_diff__add__inverse2,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_inverse2
% 0.29/0.46 thf(fact_117_diff__add__inverse,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_inverse
% 0.29/0.46 thf(fact_118_diff__cancel2,axiom,
% 0.29/0.46 ! [M: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.29/0.46 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_cancel2
% 0.29/0.46 thf(fact_119_Nat_Odiff__cancel,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.46 = ( minus_minus @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_cancel
% 0.29/0.46 thf(fact_120_diff__mult__distrib2,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 0.29/0.46 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mult_distrib2
% 0.29/0.46 thf(fact_121_diff__mult__distrib,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K: nat] :
% 0.29/0.46 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 0.29/0.46 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_mult_distrib
% 0.29/0.46 thf(fact_122_le__iff__diff__le__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_diff_le_0
% 0.29/0.46 thf(fact_123_less__iff__diff__less__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [A3: A,B2: A] : ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_iff_diff_less_0
% 0.29/0.46 thf(fact_124_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ( minus_minus @ A @ B @ A2 )
% 0.29/0.46 = C )
% 0.29/0.46 = ( B
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.29/0.46 thf(fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ A2 ) )
% 0.29/0.46 = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.29/0.46 thf(fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.29/0.46 thf(fact_127_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.29/0.46 thf(fact_128_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.29/0.46 thf(fact_129_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.29/0.46 thf(fact_130_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.29/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.29/0.46 thf(fact_131_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.29/0.46 thf(fact_132_le__add__diff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add_diff
% 0.29/0.46 thf(fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere623563068d_diff @ A )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ A2 )
% 0.29/0.46 = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add
% 0.29/0.46 thf(fact_134_le__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_eq
% 0.29/0.46 thf(fact_135_diff__le__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_le_eq
% 0.29/0.46 thf(fact_136_less__diff__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 0.29/0.46 = ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_eq
% 0.29/0.46 thf(fact_137_diff__less__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordered_ab_group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less_eq
% 0.29/0.46 thf(fact_138_diff__less,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less
% 0.29/0.46 thf(fact_139_diff__add__0,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 0.29/0.46 = ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_add_0
% 0.29/0.46 thf(fact_140_diff__less__mono,axiom,
% 0.29/0.46 ! [A2: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ C @ A2 )
% 0.29/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_less_mono
% 0.29/0.46 thf(fact_141_less__diff__iff,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ N )
% 0.29/0.46 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 0.29/0.46 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_iff
% 0.29/0.46 thf(fact_142_add__diff__inverse__nat,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ~ ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 0.29/0.46 = M ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_diff_inverse_nat
% 0.29/0.46 thf(fact_143_less__diff__conv,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.29/0.46 = ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_conv
% 0.29/0.46 thf(fact_144_Nat_Ole__imp__diff__is__add,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ( minus_minus @ nat @ J @ I2 )
% 0.29/0.46 = K )
% 0.29/0.46 = ( J
% 0.29/0.46 = ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.le_imp_diff_is_add
% 0.29/0.46 thf(fact_145_Nat_Odiff__add__assoc2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K )
% 0.29/0.46 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_add_assoc2
% 0.29/0.46 thf(fact_146_Nat_Odiff__add__assoc,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 0.29/0.46 = ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.diff_add_assoc
% 0.29/0.46 thf(fact_147_Nat_Ole__diff__conv2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.le_diff_conv2
% 0.29/0.46 thf(fact_148_le__diff__conv,axiom,
% 0.29/0.46 ! [J: nat,K: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_diff_conv
% 0.29/0.46 thf(fact_149_nat__diff__split__asm,axiom,
% 0.29/0.46 ! [P2: nat > $o,A2: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ ( minus_minus @ nat @ A2 @ B ) )
% 0.29/0.46 = ( ~ ( ( ( ord_less @ nat @ A2 @ B )
% 0.29/0.46 & ~ ( P2 @ ( zero_zero @ nat ) ) )
% 0.29/0.46 | ? [D2: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus @ nat @ B @ D2 ) )
% 0.29/0.46 & ~ ( P2 @ D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_split_asm
% 0.29/0.46 thf(fact_150_nat__diff__split,axiom,
% 0.29/0.46 ! [P2: nat > $o,A2: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ ( minus_minus @ nat @ A2 @ B ) )
% 0.29/0.46 = ( ( ( ord_less @ nat @ A2 @ B )
% 0.29/0.46 => ( P2 @ ( zero_zero @ nat ) ) )
% 0.29/0.46 & ! [D2: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( plus_plus @ nat @ B @ D2 ) )
% 0.29/0.46 => ( P2 @ D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_split
% 0.29/0.46 thf(fact_151_less__diff__conv2,axiom,
% 0.29/0.46 ! [K: nat,J: nat,I2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ J )
% 0.29/0.46 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 0.29/0.46 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_diff_conv2
% 0.29/0.46 thf(fact_152_nat__diff__add__eq2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_add_eq2
% 0.29/0.46 thf(fact_153_nat__diff__add__eq1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ J @ I2 )
% 0.29/0.46 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_diff_add_eq1
% 0.29/0.46 thf(fact_154_nat__le__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_add_iff2
% 0.29/0.46 thf(fact_155_nat__le__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ J @ I2 )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_add_iff1
% 0.29/0.46 thf(fact_156_nat__eq__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
% 0.29/0.46 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( M
% 0.29/0.46 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_add_iff2
% 0.29/0.46 thf(fact_157_nat__eq__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ J @ I2 )
% 0.29/0.46 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
% 0.29/0.46 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M )
% 0.29/0.46 = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_eq_add_iff1
% 0.29/0.46 thf(fact_158_nat__less__add__iff2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_add_iff2
% 0.29/0.46 thf(fact_159_nat__less__add__iff1,axiom,
% 0.29/0.46 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ J @ I2 )
% 0.29/0.46 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.29/0.46 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_add_iff1
% 0.29/0.46 thf(fact_160_zero__reorient,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = X )
% 0.29/0.46 = ( X
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_reorient
% 0.29/0.46 thf(fact_161_measure__induct__rule,axiom,
% 0.29/0.46 ! [B3: $tType,A: $tType] :
% 0.29/0.46 ( ( wellorder @ B3 )
% 0.29/0.46 => ! [F: A > B3,P2: A > $o,A2: A] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ! [Y3: A] :
% 0.29/0.46 ( ( ord_less @ B3 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.29/0.46 => ( P2 @ Y3 ) )
% 0.29/0.46 => ( P2 @ X2 ) )
% 0.29/0.46 => ( P2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct_rule
% 0.29/0.46 thf(fact_162_measure__induct,axiom,
% 0.29/0.46 ! [B3: $tType,A: $tType] :
% 0.29/0.46 ( ( wellorder @ B3 )
% 0.29/0.46 => ! [F: A > B3,P2: A > $o,A2: A] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ! [Y3: A] :
% 0.29/0.46 ( ( ord_less @ B3 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 0.29/0.46 => ( P2 @ Y3 ) )
% 0.29/0.46 => ( P2 @ X2 ) )
% 0.29/0.46 => ( P2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct
% 0.29/0.46 thf(fact_163_add__right__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 => ( B = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_imp_eq
% 0.29/0.46 thf(fact_164_add__left__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 => ( B = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_imp_eq
% 0.29/0.46 thf(fact_165_add_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_commute
% 0.29/0.46 thf(fact_166_add_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ( ( plus_plus @ A )
% 0.29/0.46 = ( ^ [A3: A,B2: A] : ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.commute
% 0.29/0.46 thf(fact_167_add_Oright__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 = ( B = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_cancel
% 0.29/0.46 thf(fact_168_add_Oleft__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( B = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_cancel
% 0.29/0.46 thf(fact_169_add_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.assoc
% 0.29/0.46 thf(fact_170_group__cancel_Oadd2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [B4: A,K: A,B: A,A2: A] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( plus_plus @ A @ K @ B ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A2 @ B4 )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add2
% 0.29/0.46 thf(fact_171_group__cancel_Oadd1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A4: A,K: A,A2: A,B: A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( plus_plus @ A @ K @ A2 ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A4 @ B )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add1
% 0.29/0.46 thf(fact_172_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ( plus_plus @ A @ I2 @ K )
% 0.29/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(4)
% 0.29/0.46 thf(fact_173_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.46 thf(fact_174_mult_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_mult @ A )
% 0.29/0.46 => ! [B: A,A2: A,C: A] :
% 0.29/0.46 ( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
% 0.29/0.46 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.left_commute
% 0.29/0.46 thf(fact_175_mult_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_mult @ A )
% 0.29/0.46 => ( ( times_times @ A )
% 0.29/0.46 = ( ^ [A3: A,B2: A] : ( times_times @ A @ B2 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.commute
% 0.29/0.46 thf(fact_176_mult_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semigroup_mult @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult.assoc
% 0.29/0.46 thf(fact_177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_mult @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 0.29/0.46 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.46 thf(fact_178_infinite__descent__measure,axiom,
% 0.29/0.46 ! [A: $tType,P2: A > $o,V: A > nat,X: A] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ~ ( P2 @ X2 )
% 0.29/0.46 => ? [Y3: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.29/0.46 & ~ ( P2 @ Y3 ) ) )
% 0.29/0.46 => ( P2 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent_measure
% 0.29/0.46 thf(fact_179_linorder__neqE__nat,axiom,
% 0.29/0.46 ! [X: nat,Y: nat] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.29/0.46 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_nat
% 0.29/0.46 thf(fact_180_infinite__descent,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ~ ( P2 @ N2 )
% 0.29/0.46 => ? [M2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.46 & ~ ( P2 @ M2 ) ) )
% 0.29/0.46 => ( P2 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent
% 0.29/0.46 thf(fact_181_nat__less__induct,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ! [M2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.46 => ( P2 @ M2 ) )
% 0.29/0.46 => ( P2 @ N2 ) )
% 0.29/0.46 => ( P2 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_182_less__irrefl__nat,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl_nat
% 0.29/0.46 thf(fact_183_less__not__refl3,axiom,
% 0.29/0.46 ! [S: nat,T: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ S @ T )
% 0.29/0.46 => ( S != T ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl3
% 0.29/0.46 thf(fact_184_less__not__refl2,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ N @ M )
% 0.29/0.46 => ( M != N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl2
% 0.29/0.46 thf(fact_185_less__not__refl,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl
% 0.29/0.46 thf(fact_186_nat__neq__iff,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M != N )
% 0.29/0.46 = ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_neq_iff
% 0.29/0.46 thf(fact_187_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [P2: nat > $o,K: nat,B: nat] :
% 0.29/0.46 ( ( P2 @ K )
% 0.29/0.46 => ( ! [Y4: nat] :
% 0.29/0.46 ( ( P2 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 0.29/0.46 => ? [X2: nat] :
% 0.29/0.46 ( ( P2 @ X2 )
% 0.29/0.46 & ! [Y3: nat] :
% 0.29/0.46 ( ( P2 @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.ex_has_greatest_nat
% 0.29/0.46 thf(fact_188_nat__le__linear,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_linear
% 0.29/0.46 thf(fact_189_le__antisym,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_antisym
% 0.29/0.46 thf(fact_190_eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M = N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_imp_le
% 0.29/0.46 thf(fact_191_le__trans,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_192_le__refl,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % le_refl
% 0.29/0.46 thf(fact_193_zero__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le
% 0.29/0.46 thf(fact_194_zero__less__iff__neq__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.46 = ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_iff_neq_zero
% 0.29/0.46 thf(fact_195_gr__implies__not__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [M: A,N: A] :
% 0.29/0.46 ( ( ord_less @ A @ M @ N )
% 0.29/0.46 => ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not_zero
% 0.29/0.46 thf(fact_196_not__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_zero
% 0.29/0.46 thf(fact_197_gr__zeroI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( N
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_zeroI
% 0.29/0.46 thf(fact_198_add__le__imp__le__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_right
% 0.29/0.46 thf(fact_199_add__le__imp__le__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_left
% 0.29/0.46 thf(fact_200_le__iff__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A3: A,B2: A] :
% 0.29/0.46 ? [C2: A] :
% 0.29/0.46 ( B2
% 0.29/0.46 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_add
% 0.29/0.46 thf(fact_201_add__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_mono
% 0.29/0.46 thf(fact_202_less__eqE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ~ ! [C3: A] :
% 0.29/0.46 ( B
% 0.29/0.46 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eqE
% 0.29/0.46 thf(fact_203_add__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_mono
% 0.29/0.46 thf(fact_204_add__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono
% 0.29/0.46 thf(fact_205_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(1)
% 0.29/0.46 thf(fact_206_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(2)
% 0.29/0.46 thf(fact_207_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I2 @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(3)
% 0.29/0.46 thf(fact_208_add_Ogroup__left__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.group_left_neutral
% 0.29/0.46 thf(fact_209_add_Ocomm__neutral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.comm_neutral
% 0.29/0.46 thf(fact_210_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % comm_monoid_add_class.add_0
% 0.29/0.46 thf(fact_211_add__less__imp__less__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_right
% 0.29/0.46 thf(fact_212_add__less__imp__less__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_left
% 0.29/0.46 thf(fact_213_add__strict__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_right_mono
% 0.29/0.46 thf(fact_214_add__strict__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_left_mono
% 0.29/0.46 thf(fact_215_add__strict__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( strict2144017051up_add @ A )
% 0.29/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less @ A @ C @ D )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_mono
% 0.29/0.46 thf(fact_216_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I2 @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(1)
% 0.29/0.46 thf(fact_217_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I2 = J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(2)
% 0.29/0.46 thf(fact_218_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I2: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I2 @ J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(5)
% 0.29/0.46 thf(fact_219_infinite__descent0__measure,axiom,
% 0.29/0.46 ! [A: $tType,V: A > nat,P2: A > $o,X: A] :
% 0.29/0.46 ( ! [X2: A] :
% 0.29/0.46 ( ( ( V @ X2 )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 => ( P2 @ X2 ) )
% 0.29/0.46 => ( ! [X2: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.29/0.46 => ( ~ ( P2 @ X2 )
% 0.29/0.46 => ? [Y3: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 0.29/0.46 & ~ ( P2 @ Y3 ) ) ) )
% 0.29/0.46 => ( P2 @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0_measure
% 0.29/0.46 thf(fact_220_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_strict
% 0.29/0.46 thf(fact_221_infinite__descent0,axiom,
% 0.29/0.46 ! [P2: nat > $o,N: nat] :
% 0.29/0.46 ( ( P2 @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.29/0.46 => ( ~ ( P2 @ N2 )
% 0.29/0.46 => ? [M2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.46 & ~ ( P2 @ M2 ) ) ) )
% 0.29/0.46 => ( P2 @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0
% 0.29/0.46 thf(fact_222_gr__implies__not0,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( N
% 0.29/0.46 != ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not0
% 0.29/0.46 thf(fact_223_less__zeroE,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_zeroE
% 0.29/0.46 thf(fact_224_not__less0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less0
% 0.29/0.46 thf(fact_225_not__gr0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr0
% 0.29/0.46 thf(fact_226_gr0I,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( N
% 0.29/0.46 != ( zero_zero @ nat ) )
% 0.29/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr0I
% 0.29/0.46 thf(fact_227_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_uniqueI
% 0.29/0.46 thf(fact_228_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_unique
% 0.29/0.46 thf(fact_229_le__0__eq,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_0_eq
% 0.29/0.46 thf(fact_230_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_nat.simps(1)
% 0.29/0.46 thf(fact_231_add__eq__self__zero,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( plus_plus @ nat @ M @ N )
% 0.29/0.46 = M )
% 0.29/0.46 => ( N
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_eq_self_zero
% 0.29/0.46 thf(fact_232_plus__nat_Oadd__0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 = N ) ).
% 0.29/0.46
% 0.29/0.46 % plus_nat.add_0
% 0.29/0.46 thf(fact_233_less__mono__imp__le__mono,axiom,
% 0.29/0.46 ! [F: nat > nat,I2: nat,J: nat] :
% 0.29/0.46 ( ! [I3: nat,J2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ J2 )
% 0.29/0.46 => ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_mono_imp_le_mono
% 0.29/0.46 thf(fact_234_le__neq__implies__less,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ( M != N )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_implies_less
% 0.29/0.46 thf(fact_235_less__or__eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 | ( M = N ) )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_or_eq_imp_le
% 0.29/0.46 thf(fact_236_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 | ( M3 = N3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_237_less__imp__le__nat,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le_nat
% 0.29/0.46 thf(fact_238_nat__less__le,axiom,
% 0.29/0.46 ( ( ord_less @ nat )
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.29/0.46 & ( M3 != N3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_239_less__add__eq__less,axiom,
% 0.29/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ K @ L )
% 0.29/0.46 => ( ( ( plus_plus @ nat @ M @ L )
% 0.29/0.46 = ( plus_plus @ nat @ K @ N ) )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_eq_less
% 0.29/0.46 thf(fact_240_trans__less__add2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add2
% 0.29/0.46 thf(fact_241_trans__less__add1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add1
% 0.29/0.46 thf(fact_242_add__less__mono1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono1
% 0.29/0.46 thf(fact_243_not__add__less2,axiom,
% 0.29/0.46 ! [J: nat,I2: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less2
% 0.29/0.46 thf(fact_244_not__add__less1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less1
% 0.29/0.46 thf(fact_245_add__less__mono,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less @ nat @ K @ L )
% 0.29/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono
% 0.29/0.46 thf(fact_246_add__lessD1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 0.29/0.46 => ( ord_less @ nat @ I2 @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_lessD1
% 0.29/0.46 thf(fact_247_nat__mult__eq__cancel__disj,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ( times_times @ nat @ K @ M )
% 0.29/0.46 = ( times_times @ nat @ K @ N ) )
% 0.29/0.46 = ( ( K
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 | ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_mult_eq_cancel_disj
% 0.29/0.46 thf(fact_248_mult__0,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 = ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_0
% 0.29/0.46 thf(fact_249_nat__le__iff__add,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M3: nat,N3: nat] :
% 0.29/0.46 ? [K2: nat] :
% 0.29/0.46 ( N3
% 0.29/0.46 = ( plus_plus @ nat @ M3 @ K2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_iff_add
% 0.29/0.46 thf(fact_250_trans__le__add2,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add2
% 0.29/0.46 thf(fact_251_trans__le__add1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add1
% 0.29/0.46 thf(fact_252_add__le__mono1,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono1
% 0.29/0.46 thf(fact_253_add__le__mono,axiom,
% 0.29/0.46 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ L )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono
% 0.29/0.46
% 0.29/0.46 % Type constructors (19)
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.46 ordere516151231imp_le @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.29/0.46 strict2144017051up_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 0.29/0.46 ordere623563068d_diff @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.46 ordere223160158up_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.46 ordere236663937imp_le @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.46 canoni770627133id_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.46 ordere779506340up_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.29/0.46 cancel146912293up_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.46 cancel1352612707id_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.46 cancel_semigroup_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.29/0.46 ab_semigroup_mult @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.29/0.46 comm_monoid_diff @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.29/0.46 ab_semigroup_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.46 comm_monoid_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.29/0.46 semigroup_mult @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.29/0.46 semigroup_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.29/0.46 wellorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.29/0.46 monoid_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.29/0.46 zero @ nat ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 $false ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.61/1.44 ---- Embedded in THF ---
% 2.61/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.44 %%% Generated on Tue Feb 24 08:27:30 EST 2026
% 2.61/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.44 %%% Logic specification used:
% 2.61/1.44 %%% thf(spec, logic, $$dhol).
% 2.61/1.44
% 2.61/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2DTF14594.p
% See solution above
% 2.61/1.45 ------------------------
% 2.61/1.47 ---- Cleaned THF ---
% 2.61/1.47 thf(ty_t_Nat_Onat,type,
% 2.61/1.47 nat: $tType ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ozero,type,
% 2.61/1.47 zero:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ogroup__add,type,
% 2.61/1.47 group_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Omonoid__add,type,
% 2.61/1.47 monoid_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oab__group__add,type,
% 2.61/1.47 ab_group_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Orderings_Owellorder,type,
% 2.61/1.47 wellorder:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.61/1.47 semigroup_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Osemigroup__mult,type,
% 2.61/1.47 semigroup_mult:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.61/1.47 comm_monoid_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.61/1.47 ab_semigroup_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.61/1.47 comm_monoid_diff:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 2.61/1.47 ab_semigroup_mult:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.61/1.47 cancel_semigroup_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.61/1.47 ordered_ab_group_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.61/1.47 cancel1352612707id_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.61/1.47 cancel146912293up_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.61/1.47 linord219039673up_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.61/1.47 ordere779506340up_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.61/1.47 canoni770627133id_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.61/1.47 ordere236663937imp_le:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.61/1.47 ordere223160158up_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 2.61/1.47 ordere623563068d_diff:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.61/1.47 strict2144017051up_add:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 2.61/1.47 ordere516151231imp_le:
% 2.61/1.47 !>[A: $tType] : $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.61/1.47 minus_minus:
% 2.61/1.47 !>[A: $tType] : ( A > A > A ) ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.61/1.47 plus_plus:
% 2.61/1.47 !>[A: $tType] : ( A > A > A ) ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 2.61/1.47 times_times:
% 2.61/1.47 !>[A: $tType] : ( A > A > A ) ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.61/1.47 zero_zero:
% 2.61/1.47 !>[A: $tType] : A ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
% 2.61/1.47 number2016821345itions: ( nat > nat ) > nat > $o ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.61/1.47 ord_less:
% 2.61/1.47 !>[A: $tType] : ( A > A > $o ) ).
% 2.61/1.47
% 2.61/1.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.61/1.47 ord_less_eq:
% 2.61/1.47 !>[A: $tType] : ( A > A > $o ) ).
% 2.61/1.47
% 2.61/1.47 thf(sy_v_i____,type,
% 2.61/1.47 i: nat ).
% 2.61/1.47
% 2.61/1.47 thf(sy_v_k,type,
% 2.61/1.47 k: nat ).
% 2.61/1.47
% 2.61/1.47 thf(sy_v_n,type,
% 2.61/1.47 n: nat ).
% 2.61/1.47
% 2.61/1.47 thf(sy_v_p,type,
% 2.61/1.47 p: nat > nat ).
% 2.61/1.47
% 2.61/1.47 thf(fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062,axiom,
% 2.61/1.47 ord_less_eq @ nat @ i @ n ).
% 2.61/1.47
% 2.61/1.47 thf(fact_1_assms_I1_J,axiom,
% 2.61/1.47 number2016821345itions @ p @ n ).
% 2.61/1.47
% 2.61/1.47 thf(fact_2_upper__bound,axiom,
% 2.61/1.47 ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n ).
% 2.61/1.47
% 2.61/1.47 thf(fact_3_lower__bound,axiom,
% 2.61/1.47 ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 2.61/1.47 ord_less_eq @ nat @ k @ n ).
% 2.61/1.47
% 2.61/1.47 thf(fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062,axiom,
% 2.61/1.47 ( ( p @ i )
% 2.61/1.47 != ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_6_assms_I2_J,axiom,
% 2.61/1.47 ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062,axiom,
% 2.61/1.47 i != k ).
% 2.61/1.47
% 2.61/1.47 thf(fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062,axiom,
% 2.61/1.47 ord_less @ nat @ ( minus_minus @ nat @ n @ k ) @ i ).
% 2.61/1.47
% 2.61/1.47 thf(fact_9_partitions__zero,axiom,
% 2.61/1.47 ! [P: nat > nat] :
% 2.61/1.47 ( ( number2016821345itions @ P @ ( zero_zero @ nat ) )
% 2.61/1.47 = ( P
% 2.61/1.47 = ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_10_partitions__bounds,axiom,
% 2.61/1.47 ! [P: nat > nat,N: nat,I2: nat] :
% 2.61/1.47 ( ( number2016821345itions @ P @ N )
% 2.61/1.47 => ( ord_less_eq @ nat @ ( P @ I2 ) @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_11_mult__le__cancel2,axiom,
% 2.61/1.47 ! [M: nat,K: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.61/1.47 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_12_nat__mult__le__cancel__disj,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.61/1.47 => ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_13_mult__less__cancel2,axiom,
% 2.61/1.47 ! [M: nat,K: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.61/1.47 & ( ord_less @ nat @ M @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_14_nat__0__less__mult__iff,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.61/1.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_15_nat__mult__less__cancel__disj,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.61/1.47 & ( ord_less @ nat @ M @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_16_add__gr__0,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 2.61/1.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.61/1.47 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_17_add__less__same__cancel1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [B: A,A2: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.61/1.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_18_add__less__same__cancel2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.61/1.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_19_less__add__same__cancel1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.61/1.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_20_less__add__same__cancel2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.61/1.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_21_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.61/1.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_22_add__right__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel_semigroup_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ B @ A2 )
% 2.61/1.47 = ( plus_plus @ A @ C @ A2 ) )
% 2.61/1.47 = ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_23_add__left__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel_semigroup_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ A2 @ B )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ C ) )
% 2.61/1.47 = ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_24_le__zero__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [N: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.61/1.47 = ( N
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_25_not__gr__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [N: A] :
% 2.61/1.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.61/1.47 = ( N
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_26_add__le__cancel__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_27_add__le__cancel__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [C: A,A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_28_zero__eq__add__iff__both__eq__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [X: A,Y: A] :
% 2.61/1.47 ( ( ( zero_zero @ A )
% 2.61/1.47 = ( plus_plus @ A @ X @ Y ) )
% 2.61/1.47 = ( ( X
% 2.61/1.47 = ( zero_zero @ A ) )
% 2.61/1.47 & ( Y
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_29_add__eq__0__iff__both__eq__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [X: A,Y: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ X @ Y )
% 2.61/1.47 = ( zero_zero @ A ) )
% 2.61/1.47 = ( ( X
% 2.61/1.47 = ( zero_zero @ A ) )
% 2.61/1.47 & ( Y
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_30_add__cancel__right__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 = ( plus_plus @ A @ A2 @ B ) )
% 2.61/1.47 = ( B
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_31_add__cancel__right__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 = ( plus_plus @ A @ B @ A2 ) )
% 2.61/1.47 = ( B
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_32_add__cancel__left__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ A2 @ B )
% 2.61/1.47 = A2 )
% 2.61/1.47 = ( B
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_33_add__cancel__left__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [B: A,A2: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ B @ A2 )
% 2.61/1.47 = A2 )
% 2.61/1.47 = ( B
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_34_double__zero__sym,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ( zero_zero @ A )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ A2 ) )
% 2.61/1.47 = ( A2
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_35_double__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ A2 @ A2 )
% 2.61/1.47 = ( zero_zero @ A ) )
% 2.61/1.47 = ( A2
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_36_add_Oright__neutral,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( monoid_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_37_add_Oleft__neutral,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( monoid_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_38_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ A2 )
% 2.61/1.47 = ( zero_zero @ A ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_39_diff__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_40_zero__diff,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_diff @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.61/1.47 = ( zero_zero @ A ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_41_diff__0__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_42_diff__self,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ A2 )
% 2.61/1.47 = ( zero_zero @ A ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_43_add__less__cancel__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_44_add__less__cancel__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [C: A,A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.61/1.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_45_add__diff__cancel__right_H,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_46_add__diff__cancel__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_47_add__diff__cancel__left_H,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ A2 )
% 2.61/1.47 = B ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_48_add__diff__cancel__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [C: A,A2: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.61/1.47 = ( minus_minus @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_49_diff__add__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_50_add__diff__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_51_bot__nat__0_Onot__eq__extremum,axiom,
% 2.61/1.47 ! [A2: nat] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 != ( zero_zero @ nat ) )
% 2.61/1.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_52_less__nat__zero__code,axiom,
% 2.61/1.47 ! [N: nat] :
% 2.61/1.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_53_neq0__conv,axiom,
% 2.61/1.47 ! [N: nat] :
% 2.61/1.47 ( ( N
% 2.61/1.47 != ( zero_zero @ nat ) )
% 2.61/1.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_54_bot__nat__0_Oextremum,axiom,
% 2.61/1.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_55_le0,axiom,
% 2.61/1.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_56_Nat_Oadd__0__right,axiom,
% 2.61/1.47 ! [M: nat] :
% 2.61/1.47 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 2.61/1.47 = M ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_57_add__is__0,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ( plus_plus @ nat @ M @ N )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 = ( ( M
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 & ( N
% 2.61/1.47 = ( zero_zero @ nat ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_58_nat__add__left__cancel__less,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.61/1.47 = ( ord_less @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_59_diff__self__eq__0,axiom,
% 2.61/1.47 ! [M: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ M @ M )
% 2.61/1.47 = ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_60_diff__0__eq__0,axiom,
% 2.61/1.47 ! [N: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.61/1.47 = ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_61_mult__cancel2,axiom,
% 2.61/1.47 ! [M: nat,K: nat,N: nat] :
% 2.61/1.47 ( ( ( times_times @ nat @ M @ K )
% 2.61/1.47 = ( times_times @ nat @ N @ K ) )
% 2.61/1.47 = ( ( M = N )
% 2.61/1.47 | ( K
% 2.61/1.47 = ( zero_zero @ nat ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_62_mult__cancel1,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ( times_times @ nat @ K @ M )
% 2.61/1.47 = ( times_times @ nat @ K @ N ) )
% 2.61/1.47 = ( ( M = N )
% 2.61/1.47 | ( K
% 2.61/1.47 = ( zero_zero @ nat ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_63_mult__0__right,axiom,
% 2.61/1.47 ! [M: nat] :
% 2.61/1.47 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 2.61/1.47 = ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_64_mult__is__0,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ( times_times @ nat @ M @ N )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 = ( ( M
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 | ( N
% 2.61/1.47 = ( zero_zero @ nat ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_65_nat__add__left__cancel__le,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_66_diff__diff__cancel,axiom,
% 2.61/1.47 ! [I2: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ N )
% 2.61/1.47 => ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
% 2.61/1.47 = I2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_67_diff__diff__left,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 2.61/1.47 = ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_68_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.61/1.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_69_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_70_le__add__same__cancel2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 2.61/1.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_71_le__add__same__cancel1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.61/1.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_72_add__le__same__cancel2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_73_add__le__same__cancel1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere516151231imp_le @ A )
% 2.61/1.47 => ! [B: A,A2: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_74_diff__ge__0__iff__ge,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.61/1.47 = ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_75_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( linord219039673up_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 2.61/1.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_76_diff__gt__0__iff__gt,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.61/1.47 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_77_diff__add__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 2.61/1.47 = ( zero_zero @ A ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_78_zero__less__diff,axiom,
% 2.61/1.47 ! [N: nat,M: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 2.61/1.47 = ( ord_less @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_79_diff__is__0__eq_H,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.47 => ( ( minus_minus @ nat @ M @ N )
% 2.61/1.47 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_80_diff__is__0__eq,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ( minus_minus @ nat @ M @ N )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_81_Nat_Odiff__diff__right,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.61/1.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_82_Nat_Oadd__diff__assoc2,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 2.61/1.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_83_Nat_Oadd__diff__assoc,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.61/1.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_84_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 2.61/1.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_85_diff__eq__diff__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A,D: A] :
% 2.61/1.47 ( ( ( minus_minus @ A @ A2 @ B )
% 2.61/1.47 = ( minus_minus @ A @ C @ D ) )
% 2.61/1.47 => ( ( A2 = B )
% 2.61/1.47 = ( C = D ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_86_diff__commute,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
% 2.61/1.47 = ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_87_diff__eq__diff__less__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A,D: A] :
% 2.61/1.47 ( ( ( minus_minus @ A @ A2 @ B )
% 2.61/1.47 = ( minus_minus @ A @ C @ D ) )
% 2.61/1.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 = ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_88_diff__right__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_89_diff__left__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ B @ A2 )
% 2.61/1.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_90_diff__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,D: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less_eq @ A @ D @ C )
% 2.61/1.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_91_eq__iff__diff__eq__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.61/1.47 = ( ^ [A3: A,B2: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A3 @ B2 )
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_92_diff__strict__right__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_93_diff__strict__left__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ B @ A2 )
% 2.61/1.47 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_94_diff__eq__diff__less,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A,D: A] :
% 2.61/1.47 ( ( ( minus_minus @ A @ A2 @ B )
% 2.61/1.47 = ( minus_minus @ A @ C @ D ) )
% 2.61/1.47 => ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_95_diff__strict__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,D: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less @ A @ D @ C )
% 2.61/1.47 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_96_add__implies__diff,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel1352612707id_add @ A )
% 2.61/1.47 => ! [C: A,B: A,A2: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ C @ B )
% 2.61/1.47 = A2 )
% 2.61/1.47 => ( C
% 2.61/1.47 = ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_97_diff__diff__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel146912293up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_98_diff__add__eq__diff__diff__swap,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_99_diff__add__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_100_diff__diff__eq2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_101_add__diff__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ C ) )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_102_eq__diff__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 = ( minus_minus @ A @ C @ B ) )
% 2.61/1.47 = ( ( plus_plus @ A @ A2 @ B )
% 2.61/1.47 = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_103_diff__eq__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ( minus_minus @ A @ A2 @ B )
% 2.61/1.47 = C )
% 2.61/1.47 = ( A2
% 2.61/1.47 = ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_104_group__cancel_Osub1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_group_add @ A )
% 2.61/1.47 => ! [A4: A,K: A,A2: A,B: A] :
% 2.61/1.47 ( ( A4
% 2.61/1.47 = ( plus_plus @ A @ K @ A2 ) )
% 2.61/1.47 => ( ( minus_minus @ A @ A4 @ B )
% 2.61/1.47 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_105_minus__nat_Odiff__0,axiom,
% 2.61/1.47 ! [M: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 2.61/1.47 = M ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_106_diffs0__imp__equal,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ( minus_minus @ nat @ M @ N )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 => ( ( ( minus_minus @ nat @ N @ M )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 => ( M = N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_107_less__imp__diff__less,axiom,
% 2.61/1.47 ! [J: nat,K: nat,N: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ J @ K )
% 2.61/1.47 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_108_diff__less__mono2,axiom,
% 2.61/1.47 ! [M: nat,N: nat,L: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ M @ N )
% 2.61/1.47 => ( ( ord_less @ nat @ M @ L )
% 2.61/1.47 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_109_diff__le__mono2,axiom,
% 2.61/1.47 ! [M: nat,N: nat,L: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_110_le__diff__iff_H,axiom,
% 2.61/1.47 ! [A2: nat,C: nat,B: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ A2 @ C )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ B @ C )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_111_diff__le__self,axiom,
% 2.61/1.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_112_diff__le__mono,axiom,
% 2.61/1.47 ! [M: nat,N: nat,L: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.47 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_113_Nat_Odiff__diff__eq,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ M )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ K @ N )
% 2.61/1.47 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.61/1.47 = ( minus_minus @ nat @ M @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_114_le__diff__iff,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ M )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ K @ N )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_115_eq__diff__iff,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ M )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ K @ N )
% 2.61/1.47 => ( ( ( minus_minus @ nat @ M @ K )
% 2.61/1.47 = ( minus_minus @ nat @ N @ K ) )
% 2.61/1.47 = ( M = N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_116_diff__add__inverse2,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
% 2.61/1.47 = M ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_117_diff__add__inverse,axiom,
% 2.61/1.47 ! [N: nat,M: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
% 2.61/1.47 = M ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_118_diff__cancel2,axiom,
% 2.61/1.47 ! [M: nat,K: nat,N: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 2.61/1.47 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_119_Nat_Odiff__cancel,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.61/1.47 = ( minus_minus @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_120_diff__mult__distrib2,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
% 2.61/1.47 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_121_diff__mult__distrib,axiom,
% 2.61/1.47 ! [M: nat,N: nat,K: nat] :
% 2.61/1.47 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
% 2.61/1.47 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_122_le__iff__diff__le__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ( ( ord_less_eq @ A )
% 2.61/1.47 = ( ^ [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_123_less__iff__diff__less__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ( ( ord_less @ A )
% 2.61/1.47 = ( ^ [A3: A,B2: A] : ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_124_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( ( minus_minus @ A @ B @ A2 )
% 2.61/1.47 = C )
% 2.61/1.47 = ( B
% 2.61/1.47 = ( plus_plus @ A @ C @ A2 ) ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ A2 ) )
% 2.61/1.47 = B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_127_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 )
% 2.61/1.47 = ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_128_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_129_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 )
% 2.61/1.47 = ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_130_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.61/1.47 = ( minus_minus @ A @ ( plus_plus @ A @ C @ B ) @ A2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_131_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B @ A2 ) )
% 2.61/1.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_132_le__add__diff,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B @ C ) @ A2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere623563068d_diff @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ A2 )
% 2.61/1.47 = B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_134_le__diff__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 2.61/1.47 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_135_diff__le__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_136_less__diff__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C @ B ) )
% 2.61/1.47 = ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_137_diff__less__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordered_ab_group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( ord_less @ A @ A2 @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_138_diff__less,axiom,
% 2.61/1.47 ! [N: nat,M: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.61/1.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.61/1.47 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_139_diff__add__0,axiom,
% 2.61/1.47 ! [N: nat,M: nat] :
% 2.61/1.47 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
% 2.61/1.47 = ( zero_zero @ nat ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_140_diff__less__mono,axiom,
% 2.61/1.47 ! [A2: nat,B: nat,C: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ A2 @ B )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ C @ A2 )
% 2.61/1.47 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_141_less__diff__iff,axiom,
% 2.61/1.47 ! [K: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ M )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ K @ N )
% 2.61/1.47 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
% 2.61/1.47 = ( ord_less @ nat @ M @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_142_add__diff__inverse__nat,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ~ ( ord_less @ nat @ M @ N )
% 2.61/1.47 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
% 2.61/1.47 = M ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_143_less__diff__conv,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.61/1.47 = ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_144_Nat_Ole__imp__diff__is__add,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( ( minus_minus @ nat @ J @ I2 )
% 2.61/1.47 = K )
% 2.61/1.47 = ( J
% 2.61/1.47 = ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_145_Nat_Odiff__add__assoc2,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K )
% 2.61/1.47 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_146_Nat_Odiff__add__assoc,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 2.61/1.47 = ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_147_Nat_Ole__diff__conv2,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_148_le__diff__conv,axiom,
% 2.61/1.47 ! [J: nat,K: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 2.61/1.47 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_149_nat__diff__split__asm,axiom,
% 2.61/1.47 ! [P2: nat > $o,A2: nat,B: nat] :
% 2.61/1.47 ( ( P2 @ ( minus_minus @ nat @ A2 @ B ) )
% 2.61/1.47 = ( ~ ( ( ( ord_less @ nat @ A2 @ B )
% 2.61/1.47 & ~ ( P2 @ ( zero_zero @ nat ) ) )
% 2.61/1.47 | ? [D2: nat] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 = ( plus_plus @ nat @ B @ D2 ) )
% 2.61/1.47 & ~ ( P2 @ D2 ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_150_nat__diff__split,axiom,
% 2.61/1.47 ! [P2: nat > $o,A2: nat,B: nat] :
% 2.61/1.47 ( ( P2 @ ( minus_minus @ nat @ A2 @ B ) )
% 2.61/1.47 = ( ( ( ord_less @ nat @ A2 @ B )
% 2.61/1.47 => ( P2 @ ( zero_zero @ nat ) ) )
% 2.61/1.47 & ! [D2: nat] :
% 2.61/1.47 ( ( A2
% 2.61/1.47 = ( plus_plus @ nat @ B @ D2 ) )
% 2.61/1.47 => ( P2 @ D2 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_151_less__diff__conv2,axiom,
% 2.61/1.47 ! [K: nat,J: nat,I2: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ K @ J )
% 2.61/1.47 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
% 2.61/1.47 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_152_nat__diff__add__eq2,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_153_nat__diff__add__eq1,axiom,
% 2.61/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ J @ I2 )
% 2.61/1.47 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_154_nat__le__add__iff2,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_155_nat__le__add__iff1,axiom,
% 2.61/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ J @ I2 )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_156_nat__eq__add__iff2,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
% 2.61/1.47 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( M
% 2.61/1.47 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_157_nat__eq__add__iff1,axiom,
% 2.61/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ J @ I2 )
% 2.61/1.47 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
% 2.61/1.47 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M )
% 2.61/1.47 = N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_158_nat__less__add__iff2,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_159_nat__less__add__iff1,axiom,
% 2.61/1.47 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ J @ I2 )
% 2.61/1.47 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 2.61/1.47 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_160_zero__reorient,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( zero @ A )
% 2.61/1.47 => ! [X: A] :
% 2.61/1.47 ( ( ( zero_zero @ A )
% 2.61/1.47 = X )
% 2.61/1.47 = ( X
% 2.61/1.47 = ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_161_measure__induct__rule,axiom,
% 2.61/1.47 ! [B3: $tType,A: $tType] :
% 2.61/1.47 ( ( wellorder @ B3 )
% 2.61/1.47 => ! [F: A > B3,P2: A > $o,A2: A] :
% 2.61/1.47 ( ! [X2: A] :
% 2.61/1.47 ( ! [Y3: A] :
% 2.61/1.47 ( ( ord_less @ B3 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 2.61/1.47 => ( P2 @ Y3 ) )
% 2.61/1.47 => ( P2 @ X2 ) )
% 2.61/1.47 => ( P2 @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_162_measure__induct,axiom,
% 2.61/1.47 ! [B3: $tType,A: $tType] :
% 2.61/1.47 ( ( wellorder @ B3 )
% 2.61/1.47 => ! [F: A > B3,P2: A > $o,A2: A] :
% 2.61/1.47 ( ! [X2: A] :
% 2.61/1.47 ( ! [Y3: A] :
% 2.61/1.47 ( ( ord_less @ B3 @ ( F @ Y3 ) @ ( F @ X2 ) )
% 2.61/1.47 => ( P2 @ Y3 ) )
% 2.61/1.47 => ( P2 @ X2 ) )
% 2.61/1.47 => ( P2 @ A2 ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_163_add__right__imp__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel_semigroup_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ B @ A2 )
% 2.61/1.47 = ( plus_plus @ A @ C @ A2 ) )
% 2.61/1.47 => ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_164_add__left__imp__eq,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( cancel_semigroup_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ A2 @ B )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ C ) )
% 2.61/1.47 => ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_165_add_Oleft__commute,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_166_add_Ocommute,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_add @ A )
% 2.61/1.47 => ( ( plus_plus @ A )
% 2.61/1.47 = ( ^ [A3: A,B2: A] : ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_167_add_Oright__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ B @ A2 )
% 2.61/1.47 = ( plus_plus @ A @ C @ A2 ) )
% 2.61/1.47 = ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_168_add_Oleft__cancel,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ( plus_plus @ A @ A2 @ B )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ C ) )
% 2.61/1.47 = ( B = C ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_169_add_Oassoc,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( semigroup_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_170_group__cancel_Oadd2,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_add @ A )
% 2.61/1.47 => ! [B4: A,K: A,B: A,A2: A] :
% 2.61/1.47 ( ( B4
% 2.61/1.47 = ( plus_plus @ A @ K @ B ) )
% 2.61/1.47 => ( ( plus_plus @ A @ A2 @ B4 )
% 2.61/1.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_171_group__cancel_Oadd1,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_add @ A )
% 2.61/1.47 => ! [A4: A,K: A,A2: A,B: A] :
% 2.61/1.47 ( ( A4
% 2.61/1.47 = ( plus_plus @ A @ K @ A2 ) )
% 2.61/1.47 => ( ( plus_plus @ A @ A4 @ B )
% 2.61/1.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_172_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( I2 = J )
% 2.61/1.47 & ( K = L ) )
% 2.61/1.47 => ( ( plus_plus @ A @ I2 @ K )
% 2.61/1.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_173_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_174_mult_Oleft__commute,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_mult @ A )
% 2.61/1.47 => ! [B: A,A2: A,C: A] :
% 2.61/1.47 ( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
% 2.61/1.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_175_mult_Ocommute,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_mult @ A )
% 2.61/1.47 => ( ( times_times @ A )
% 2.61/1.47 = ( ^ [A3: A,B2: A] : ( times_times @ A @ B2 @ A3 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_176_mult_Oassoc,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( semigroup_mult @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ab_semigroup_mult @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
% 2.61/1.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_178_infinite__descent__measure,axiom,
% 2.61/1.47 ! [A: $tType,P2: A > $o,V: A > nat,X: A] :
% 2.61/1.47 ( ! [X2: A] :
% 2.61/1.47 ( ~ ( P2 @ X2 )
% 2.61/1.47 => ? [Y3: A] :
% 2.61/1.47 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 2.61/1.47 & ~ ( P2 @ Y3 ) ) )
% 2.61/1.47 => ( P2 @ X ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_179_linorder__neqE__nat,axiom,
% 2.61/1.47 ! [X: nat,Y: nat] :
% 2.61/1.47 ( ( X != Y )
% 2.61/1.47 => ( ~ ( ord_less @ nat @ X @ Y )
% 2.61/1.47 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_180_infinite__descent,axiom,
% 2.61/1.47 ! [P2: nat > $o,N: nat] :
% 2.61/1.47 ( ! [N2: nat] :
% 2.61/1.47 ( ~ ( P2 @ N2 )
% 2.61/1.47 => ? [M2: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ M2 @ N2 )
% 2.61/1.47 & ~ ( P2 @ M2 ) ) )
% 2.61/1.47 => ( P2 @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_181_nat__less__induct,axiom,
% 2.61/1.47 ! [P2: nat > $o,N: nat] :
% 2.61/1.47 ( ! [N2: nat] :
% 2.61/1.47 ( ! [M2: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ M2 @ N2 )
% 2.61/1.47 => ( P2 @ M2 ) )
% 2.61/1.47 => ( P2 @ N2 ) )
% 2.61/1.47 => ( P2 @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_182_less__irrefl__nat,axiom,
% 2.61/1.47 ! [N: nat] :
% 2.61/1.47 ~ ( ord_less @ nat @ N @ N ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_183_less__not__refl3,axiom,
% 2.61/1.47 ! [S: nat,T: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ S @ T )
% 2.61/1.47 => ( S != T ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_184_less__not__refl2,axiom,
% 2.61/1.47 ! [N: nat,M: nat] :
% 2.61/1.47 ( ( ord_less @ nat @ N @ M )
% 2.61/1.47 => ( M != N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_185_less__not__refl,axiom,
% 2.61/1.47 ! [N: nat] :
% 2.61/1.47 ~ ( ord_less @ nat @ N @ N ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_186_nat__neq__iff,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( M != N )
% 2.61/1.47 = ( ( ord_less @ nat @ M @ N )
% 2.61/1.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_187_Nat_Oex__has__greatest__nat,axiom,
% 2.61/1.47 ! [P2: nat > $o,K: nat,B: nat] :
% 2.61/1.47 ( ( P2 @ K )
% 2.61/1.47 => ( ! [Y4: nat] :
% 2.61/1.47 ( ( P2 @ Y4 )
% 2.61/1.47 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 2.61/1.47 => ? [X2: nat] :
% 2.61/1.47 ( ( P2 @ X2 )
% 2.61/1.47 & ! [Y3: nat] :
% 2.61/1.47 ( ( P2 @ Y3 )
% 2.61/1.47 => ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_188_nat__le__linear,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_189_le__antisym,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ N @ M )
% 2.61/1.47 => ( M = N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_190_eq__imp__le,axiom,
% 2.61/1.47 ! [M: nat,N: nat] :
% 2.61/1.47 ( ( M = N )
% 2.61/1.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_191_le__trans,axiom,
% 2.61/1.47 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.47 => ( ( ord_less_eq @ nat @ J @ K )
% 2.61/1.47 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_192_le__refl,axiom,
% 2.61/1.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_193_zero__le,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_194_zero__less__iff__neq__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [N: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.61/1.47 = ( N
% 2.61/1.47 != ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_195_gr__implies__not__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [M: A,N: A] :
% 2.61/1.47 ( ( ord_less @ A @ M @ N )
% 2.61/1.47 => ( N
% 2.61/1.47 != ( zero_zero @ A ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_196_not__less__zero,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [N: A] :
% 2.61/1.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_197_gr__zeroI,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [N: A] :
% 2.61/1.47 ( ( N
% 2.61/1.47 != ( zero_zero @ A ) )
% 2.61/1.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_198_add__le__imp__le__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_199_add__le__imp__le__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [C: A,A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.61/1.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_200_le__iff__add,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ( ( ord_less_eq @ A )
% 2.61/1.47 = ( ^ [A3: A,B2: A] :
% 2.61/1.47 ? [C2: A] :
% 2.61/1.47 ( B2
% 2.61/1.47 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_201_add__right__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_202_less__eqE,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( canoni770627133id_add @ A )
% 2.61/1.47 => ! [A2: A,B: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ~ ! [C3: A] :
% 2.61/1.47 ( B
% 2.61/1.47 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_203_add__left__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_204_add__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A,D: A] :
% 2.61/1.47 ( ( ord_less_eq @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less_eq @ A @ C @ D )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_205_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.61/1.47 & ( ord_less_eq @ A @ K @ L ) )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_206_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( I2 = J )
% 2.61/1.47 & ( ord_less_eq @ A @ K @ L ) )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_207_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere779506340up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( ord_less_eq @ A @ I2 @ J )
% 2.61/1.47 & ( K = L ) )
% 2.61/1.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_208_add_Ogroup__left__neutral,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( group_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_209_add_Ocomm__neutral,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_210_comm__monoid__add__class_Oadd__0,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( comm_monoid_add @ A )
% 2.61/1.47 => ! [A2: A] :
% 2.61/1.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 2.61/1.47 = A2 ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_211_add__less__imp__less__right,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [A2: A,C: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 2.61/1.47 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_212_add__less__imp__less__left,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere236663937imp_le @ A )
% 2.61/1.47 => ! [C: A,A2: A,B: A] :
% 2.61/1.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 2.61/1.47 => ( ord_less @ A @ A2 @ B ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_213_add__strict__right__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere223160158up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_214_add__strict__left__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere223160158up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_215_add__strict__mono,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( strict2144017051up_add @ A )
% 2.61/1.47 => ! [A2: A,B: A,C: A,D: A] :
% 2.61/1.47 ( ( ord_less @ A @ A2 @ B )
% 2.61/1.47 => ( ( ord_less @ A @ C @ D )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_216_add__mono__thms__linordered__field_I1_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere223160158up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( ord_less @ A @ I2 @ J )
% 2.61/1.47 & ( K = L ) )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_217_add__mono__thms__linordered__field_I2_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere223160158up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( I2 = J )
% 2.61/1.47 & ( ord_less @ A @ K @ L ) )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_218_add__mono__thms__linordered__field_I5_J,axiom,
% 2.61/1.47 ! [A: $tType] :
% 2.61/1.47 ( ( ordere223160158up_add @ A )
% 2.61/1.47 => ! [I2: A,J: A,K: A,L: A] :
% 2.61/1.47 ( ( ( ord_less @ A @ I2 @ J )
% 2.61/1.47 & ( ord_less @ A @ K @ L ) )
% 2.61/1.47 => ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.61/1.47
% 2.61/1.47 thf(fact_219_infinite__descent0__measure,axiom,
% 2.61/1.47 ! [A: $tType,V: A > nat,P2: A > $o,X: A] :
% 2.61/1.47 ( ! [X2: A] :
% 2.61/1.47 ( ( ( V @ X2 )
% 2.61/1.47 = ( zero_zero @ nat ) )
% 2.61/1.47 => ( P2 @ X2 ) )
% 2.61/1.47 => ( ! [X2: A] :
% 2.61/1.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 2.61/1.47 => ( ~ ( P2 @ X2 )
% 2.61/1.48 => ? [Y3: A] :
% 2.61/1.48 ( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
% 2.61/1.48 & ~ ( P2 @ Y3 ) ) ) )
% 2.61/1.48 => ( P2 @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_220_bot__nat__0_Oextremum__strict,axiom,
% 2.61/1.48 ! [A2: nat] :
% 2.61/1.48 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_221_infinite__descent0,axiom,
% 2.61/1.48 ! [P2: nat > $o,N: nat] :
% 2.61/1.48 ( ( P2 @ ( zero_zero @ nat ) )
% 2.61/1.48 => ( ! [N2: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.61/1.48 => ( ~ ( P2 @ N2 )
% 2.61/1.48 => ? [M2: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ M2 @ N2 )
% 2.61/1.48 & ~ ( P2 @ M2 ) ) ) )
% 2.61/1.48 => ( P2 @ N ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_222_gr__implies__not0,axiom,
% 2.61/1.48 ! [M: nat,N: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ M @ N )
% 2.61/1.48 => ( N
% 2.61/1.48 != ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_223_less__zeroE,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_224_not__less0,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_225_not__gr0,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.61/1.48 = ( N
% 2.61/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_226_gr0I,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ( ( N
% 2.61/1.48 != ( zero_zero @ nat ) )
% 2.61/1.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_227_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.61/1.48 ! [A2: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.61/1.48 => ( A2
% 2.61/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_228_bot__nat__0_Oextremum__unique,axiom,
% 2.61/1.48 ! [A2: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.61/1.48 = ( A2
% 2.61/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_229_le__0__eq,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.61/1.48 = ( N
% 2.61/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_230_less__eq__nat_Osimps_I1_J,axiom,
% 2.61/1.48 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_231_add__eq__self__zero,axiom,
% 2.61/1.48 ! [M: nat,N: nat] :
% 2.61/1.48 ( ( ( plus_plus @ nat @ M @ N )
% 2.61/1.48 = M )
% 2.61/1.48 => ( N
% 2.61/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_232_plus__nat_Oadd__0,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 2.61/1.48 = N ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_233_less__mono__imp__le__mono,axiom,
% 2.61/1.48 ! [F: nat > nat,I2: nat,J: nat] :
% 2.61/1.48 ( ! [I3: nat,J2: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ I3 @ J2 )
% 2.61/1.48 => ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 2.61/1.48 => ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_234_le__neq__implies__less,axiom,
% 2.61/1.48 ! [M: nat,N: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ M @ N )
% 2.61/1.48 => ( ( M != N )
% 2.61/1.48 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_235_less__or__eq__imp__le,axiom,
% 2.61/1.48 ! [M: nat,N: nat] :
% 2.61/1.48 ( ( ( ord_less @ nat @ M @ N )
% 2.61/1.48 | ( M = N ) )
% 2.61/1.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_236_le__eq__less__or__eq,axiom,
% 2.61/1.48 ( ( ord_less_eq @ nat )
% 2.61/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ M3 @ N3 )
% 2.61/1.48 | ( M3 = N3 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_237_less__imp__le__nat,axiom,
% 2.61/1.48 ! [M: nat,N: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ M @ N )
% 2.61/1.48 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_238_nat__less__le,axiom,
% 2.61/1.48 ( ( ord_less @ nat )
% 2.61/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.61/1.48 & ( M3 != N3 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_239_less__add__eq__less,axiom,
% 2.61/1.48 ! [K: nat,L: nat,M: nat,N: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ K @ L )
% 2.61/1.48 => ( ( ( plus_plus @ nat @ M @ L )
% 2.61/1.48 = ( plus_plus @ nat @ K @ N ) )
% 2.61/1.48 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_240_trans__less__add2,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_241_trans__less__add1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_242_add__less__mono1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_243_not__add__less2,axiom,
% 2.61/1.48 ! [J: nat,I2: nat] :
% 2.61/1.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_244_not__add__less1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat] :
% 2.61/1.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_245_add__less__mono,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ I2 @ J )
% 2.61/1.48 => ( ( ord_less @ nat @ K @ L )
% 2.61/1.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_246_add__lessD1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.48 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
% 2.61/1.48 => ( ord_less @ nat @ I2 @ K ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_247_nat__mult__eq__cancel__disj,axiom,
% 2.61/1.48 ! [K: nat,M: nat,N: nat] :
% 2.61/1.48 ( ( ( times_times @ nat @ K @ M )
% 2.61/1.48 = ( times_times @ nat @ K @ N ) )
% 2.61/1.48 = ( ( K
% 2.61/1.48 = ( zero_zero @ nat ) )
% 2.61/1.48 | ( M = N ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_248_mult__0,axiom,
% 2.61/1.48 ! [N: nat] :
% 2.61/1.48 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 2.61/1.48 = ( zero_zero @ nat ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_249_nat__le__iff__add,axiom,
% 2.61/1.48 ( ( ord_less_eq @ nat )
% 2.61/1.48 = ( ^ [M3: nat,N3: nat] :
% 2.61/1.48 ? [K2: nat] :
% 2.61/1.48 ( N3
% 2.61/1.48 = ( plus_plus @ nat @ M3 @ K2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_250_trans__le__add2,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_251_trans__le__add1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,M: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_252_add__le__mono1,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,K: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_253_add__le__mono,axiom,
% 2.61/1.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 2.61/1.48 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.61/1.48 => ( ( ord_less_eq @ nat @ K @ L )
% 2.61/1.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 2.61/1.48 ordere516151231imp_le @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.61/1.48 strict2144017051up_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 2.61/1.48 ordere623563068d_diff @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.61/1.48 ordere223160158up_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.61/1.48 ordere236663937imp_le @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.61/1.48 canoni770627133id_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.61/1.48 ordere779506340up_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.61/1.48 cancel146912293up_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 2.61/1.48 cancel1352612707id_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.61/1.48 cancel_semigroup_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 2.61/1.48 ab_semigroup_mult @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.61/1.48 comm_monoid_diff @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.61/1.48 ab_semigroup_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.61/1.48 comm_monoid_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 2.61/1.48 semigroup_mult @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.61/1.48 semigroup_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.61/1.48 wellorder @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 2.61/1.48 monoid_add @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.61/1.48 zero @ nat ).
% 2.61/1.48
% 2.61/1.48 thf(conj_0,conjecture,
% 2.61/1.48 $false ).
% 2.61/1.48 ------------------------
% 2.77/1.50 % (14870)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_14594 for (2999ds/27Mi)
% 2.77/1.50 % (14868)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/183Mi)
% 2.77/1.50 % (14872)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_14594 for (2999ds/2Mi)
% 2.77/1.50 % (14874)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/18Mi)
% 2.77/1.50 % (14872)Instruction limit reached!
% 2.77/1.50 % (14872)------------------------------
% 2.77/1.50 % (14872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.50 % (14872)Termination reason: Unknown
% 2.77/1.50 % (14872)Termination phase: shuffling
% 2.77/1.50
% 2.77/1.50 % (14872)Memory used [KB]: 1279
% 2.77/1.50 % (14872)Time elapsed: 0.002 s
% 2.77/1.50 % (14872)Instructions burned: 3 (million)
% 2.77/1.50 % (14872)------------------------------
% 2.77/1.50 % (14872)------------------------------
% 2.77/1.51 % (14869)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_14594 for (2999ds/4Mi)
% 2.77/1.51 % (14874)Instruction limit reached!
% 2.77/1.51 % (14874)------------------------------
% 2.77/1.51 % (14874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.51 % (14874)Termination reason: Unknown
% 2.77/1.51 % (14874)Termination phase: Property scanning
% 2.77/1.51
% 2.77/1.51 % (14874)Memory used [KB]: 1535
% 2.77/1.51 % (14874)Time elapsed: 0.008 s
% 2.77/1.51 % (14874)Instructions burned: 20 (million)
% 2.77/1.51 % (14874)------------------------------
% 2.77/1.51 % (14874)------------------------------
% 2.77/1.51 % (14871)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/2Mi)
% 2.77/1.51 % (14870)Instruction limit reached!
% 2.77/1.51 % (14870)------------------------------
% 2.77/1.51 % (14870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.51 % (14870)Termination reason: Unknown
% 2.77/1.51 % (14870)Termination phase: Property scanning
% 2.77/1.51
% 2.77/1.51 % (14870)Memory used [KB]: 1663
% 2.77/1.51 % (14870)Time elapsed: 0.010 s
% 2.77/1.51 % (14870)Instructions burned: 27 (million)
% 2.77/1.51 % (14870)------------------------------
% 2.77/1.51 % (14870)------------------------------
% 2.77/1.51 % (14873)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_14594 for (2999ds/275Mi)
% 2.77/1.51 % (14871)Instruction limit reached!
% 2.77/1.51 % (14871)------------------------------
% 2.77/1.51 % (14871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.51 % (14871)Termination reason: Unknown
% 2.77/1.51 % (14871)Termination phase: shuffling
% 2.77/1.51
% 2.77/1.51 % (14871)Memory used [KB]: 1279
% 2.77/1.51 % (14871)Time elapsed: 0.004 s
% 2.77/1.51 % (14871)Instructions burned: 3 (million)
% 2.77/1.51 % (14871)------------------------------
% 2.77/1.51 % (14871)------------------------------
% 2.77/1.51 % (14869)Instruction limit reached!
% 2.77/1.51 % (14869)------------------------------
% 2.77/1.51 % (14869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.51 % (14869)Termination reason: Unknown
% 2.77/1.51 % (14869)Termination phase: shuffling
% 2.77/1.51
% 2.77/1.51 % (14869)Memory used [KB]: 1279
% 2.77/1.51 % (14869)Time elapsed: 0.005 s
% 2.77/1.51 % (14869)Instructions burned: 5 (million)
% 2.77/1.51 % (14869)------------------------------
% 2.77/1.51 % (14869)------------------------------
% 2.77/1.51 % (14875)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/3Mi)
% 2.77/1.51 % (14875)Instruction limit reached!
% 2.77/1.51 % (14875)------------------------------
% 2.77/1.51 % (14875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.51 % (14875)Termination reason: Unknown
% 2.77/1.51 % (14875)Termination phase: shuffling
% 2.77/1.51
% 2.77/1.51 % (14875)Memory used [KB]: 1279
% 2.77/1.51 % (14875)Time elapsed: 0.003 s
% 2.77/1.51 % (14875)Instructions burned: 4 (million)
% 2.77/1.51 % (14875)------------------------------
% 2.77/1.51 % (14875)------------------------------
% 2.77/1.52 % (14877)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_14594 for (2999ds/15Mi)
% 2.77/1.52 % (14876)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_14594 for (2999ds/37Mi)
% 2.77/1.52 % (14877)Instruction limit reached!
% 2.77/1.52 % (14877)------------------------------
% 2.77/1.52 % (14877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.52 % (14877)Termination reason: Unknown
% 2.77/1.52 % (14877)Termination phase: shuffling
% 2.77/1.52
% 2.77/1.52 % (14877)Memory used [KB]: 1535
% 2.77/1.52 % (14877)Time elapsed: 0.006 s
% 2.77/1.52 % (14877)Instructions burned: 15 (million)
% 2.77/1.52 % (14877)------------------------------
% 2.77/1.52 % (14877)------------------------------
% 2.77/1.52 % (14880)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/7Mi)
% 2.77/1.53 % (14873)Refutation not found, incomplete strategy
% 2.77/1.53 % (14873)------------------------------
% 2.77/1.53 % (14873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (14873)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.53
% 2.77/1.53
% 2.77/1.53 % (14873)Memory used [KB]: 6012
% 2.77/1.53 % (14873)Time elapsed: 0.018 s
% 2.77/1.53 % (14873)Instructions burned: 29 (million)
% 2.77/1.53 % (14873)------------------------------
% 2.77/1.53 % (14873)------------------------------
% 2.77/1.53 % (14880)Instruction limit reached!
% 2.77/1.53 % (14880)------------------------------
% 2.77/1.53 % (14880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (14880)Termination reason: Unknown
% 2.77/1.53 % (14880)Termination phase: shuffling
% 2.77/1.53
% 2.77/1.53 % (14880)Memory used [KB]: 1407
% 2.77/1.53 % (14880)Time elapsed: 0.004 s
% 2.77/1.53 % (14880)Instructions burned: 7 (million)
% 2.77/1.53 % (14880)------------------------------
% 2.77/1.53 % (14880)------------------------------
% 2.77/1.53 % (14878)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_14594 for (2999ds/3Mi)
% 2.77/1.53 % (14879)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_14594 for (2999ds/1041Mi)
% 2.77/1.53 % (14878)Instruction limit reached!
% 2.77/1.53 % (14878)------------------------------
% 2.77/1.53 % (14878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (14878)Termination reason: Unknown
% 2.77/1.53 % (14878)Termination phase: shuffling
% 2.77/1.53
% 2.77/1.53 % (14878)Memory used [KB]: 1279
% 2.77/1.53 % (14878)Time elapsed: 0.004 s
% 2.77/1.53 % (14878)Instructions burned: 4 (million)
% 2.77/1.53 % (14878)------------------------------
% 2.77/1.53 % (14878)------------------------------
% 2.77/1.53 % (14876)Instruction limit reached!
% 2.77/1.53 % (14876)------------------------------
% 2.77/1.53 % (14876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (14876)Termination reason: Unknown
% 2.77/1.53 % (14876)Termination phase: Property scanning
% 2.77/1.53
% 2.77/1.53 % (14876)Memory used [KB]: 2046
% 2.77/1.53 % (14876)Time elapsed: 0.013 s
% 2.77/1.53 % (14876)Instructions burned: 39 (million)
% 2.77/1.53 % (14876)------------------------------
% 2.77/1.53 % (14876)------------------------------
% 2.77/1.53 % (14881)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/16Mi)
% 2.77/1.54 % (14882)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_14594 for (2999ds/3Mi)
% 2.77/1.54 % (14882)Instruction limit reached!
% 2.77/1.54 % (14882)------------------------------
% 2.77/1.54 % (14882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.54 % (14882)Termination reason: Unknown
% 2.77/1.54 % (14882)Termination phase: shuffling
% 2.77/1.54
% 2.77/1.54 % (14882)Memory used [KB]: 1279
% 2.77/1.54 % (14882)Time elapsed: 0.003 s
% 2.77/1.54 % (14882)Instructions burned: 4 (million)
% 2.77/1.54 % (14882)------------------------------
% 2.77/1.54 % (14882)------------------------------
% 2.77/1.54 % (14881)Instruction limit reached!
% 2.77/1.54 % (14881)------------------------------
% 2.77/1.54 % (14881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.54 % (14881)Termination reason: Unknown
% 2.77/1.54 % (14881)Termination phase: shuffling
% 2.77/1.54
% 2.77/1.54 % (14881)Memory used [KB]: 1535
% 2.77/1.54 % (14881)Time elapsed: 0.006 s
% 2.77/1.54 % (14881)Instructions burned: 16 (million)
% 2.77/1.54 % (14881)------------------------------
% 2.77/1.54 % (14881)------------------------------
% 2.77/1.54 % (14885)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/3Mi)
% 2.77/1.54 % (14884)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/7Mi)
% 2.77/1.54 % (14885)Instruction limit reached!
% 2.77/1.54 % (14885)------------------------------
% 2.77/1.54 % (14885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.54 % (14885)Termination reason: Unknown
% 2.77/1.54 % (14885)Termination phase: shuffling
% 2.77/1.54
% 2.77/1.54 % (14885)Memory used [KB]: 1279
% 2.77/1.54 % (14885)Time elapsed: 0.003 s
% 2.77/1.54 % (14885)Instructions burned: 5 (million)
% 2.77/1.54 % (14885)------------------------------
% 2.77/1.54 % (14885)------------------------------
% 2.77/1.54 % (14884)Instruction limit reached!
% 2.77/1.54 % (14884)------------------------------
% 2.77/1.54 % (14884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.54 % (14884)Termination reason: Unknown
% 2.77/1.54 % (14884)Termination phase: shuffling
% 2.77/1.54
% 2.77/1.54 % (14884)Memory used [KB]: 1407
% 2.77/1.54 % (14884)Time elapsed: 0.004 s
% 2.77/1.54 % (14884)Instructions burned: 8 (million)
% 2.77/1.54 % (14884)------------------------------
% 2.77/1.54 % (14884)------------------------------
% 2.77/1.54 % (14883)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_14594 for (2999ds/3Mi)
% 2.77/1.55 % (14883)Instruction limit reached!
% 2.77/1.55 % (14883)------------------------------
% 2.77/1.55 % (14883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (14883)Termination reason: Unknown
% 2.77/1.55 % (14883)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (14883)Memory used [KB]: 1279
% 2.77/1.55 % (14883)Time elapsed: 0.004 s
% 2.77/1.55 % (14883)Instructions burned: 4 (million)
% 2.77/1.55 % (14883)------------------------------
% 2.77/1.55 % (14883)------------------------------
% 2.77/1.55 % (14886)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/4Mi)
% 2.77/1.55 % (14887)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/18Mi)
% 2.77/1.55 % (14886)Instruction limit reached!
% 2.77/1.55 % (14886)------------------------------
% 2.77/1.55 % (14886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (14886)Termination reason: Unknown
% 2.77/1.55 % (14886)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (14886)Memory used [KB]: 1279
% 2.77/1.55 % (14886)Time elapsed: 0.002 s
% 2.77/1.55 % (14886)Instructions burned: 4 (million)
% 2.77/1.55 % (14886)------------------------------
% 2.77/1.55 % (14886)------------------------------
% 2.77/1.55 % (14888)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/710Mi)
% 2.77/1.55 % (14889)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/6Mi)
% 2.77/1.55 % (14887)Instruction limit reached!
% 2.77/1.55 % (14887)------------------------------
% 2.77/1.55 % (14887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (14887)Termination reason: Unknown
% 2.77/1.55 % (14887)Termination phase: Property scanning
% 2.77/1.55
% 2.77/1.55 % (14887)Memory used [KB]: 1535
% 2.77/1.55 % (14887)Time elapsed: 0.007 s
% 2.77/1.55 % (14887)Instructions burned: 19 (million)
% 2.77/1.55 % (14887)------------------------------
% 2.77/1.55 % (14887)------------------------------
% 2.77/1.56 % (14889)Instruction limit reached!
% 2.77/1.56 % (14889)------------------------------
% 2.77/1.56 % (14889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (14889)Termination reason: Unknown
% 2.77/1.56 % (14889)Termination phase: shuffling
% 2.77/1.56
% 2.77/1.56 % (14889)Memory used [KB]: 1407
% 2.77/1.56 % (14889)Time elapsed: 0.003 s
% 2.77/1.56 % (14889)Instructions burned: 6 (million)
% 2.77/1.56 % (14889)------------------------------
% 2.77/1.56 % (14889)------------------------------
% 2.77/1.56 % (14891)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_14594 for (2999ds/21Mi)
% 2.77/1.56 % (14888)Refutation not found, incomplete strategy
% 2.77/1.56 % (14888)------------------------------
% 2.77/1.56 % (14888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (14888)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.56
% 2.77/1.56
% 2.77/1.56 % (14888)Memory used [KB]: 6012
% 2.77/1.56 % (14888)Time elapsed: 0.011 s
% 2.77/1.56 % (14888)Instructions burned: 29 (million)
% 2.77/1.56 % (14888)------------------------------
% 2.77/1.56 % (14888)------------------------------
% 2.77/1.56 % (14868)Instruction limit reached!
% 2.77/1.56 % (14868)------------------------------
% 2.77/1.56 % (14868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (14868)Termination reason: Unknown
% 2.77/1.56 % (14868)Termination phase: Saturation
% 2.77/1.56
% 2.77/1.56 % (14868)Memory used [KB]: 6652
% 2.77/1.56 % (14868)Time elapsed: 0.062 s
% 2.77/1.56 % (14868)Instructions burned: 184 (million)
% 2.77/1.56 % (14868)------------------------------
% 2.77/1.56 % (14868)------------------------------
% 2.77/1.56 % (14892)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/5Mi)
% 2.77/1.56 % (14890)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_14594 for (2999ds/902Mi)
% 2.77/1.56 % (14893)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_14594 for (2999ds/6Mi)
% 2.77/1.56 % (14891)Instruction limit reached!
% 2.77/1.56 % (14891)------------------------------
% 2.77/1.56 % (14891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (14891)Termination reason: Unknown
% 2.77/1.56 % (14891)Termination phase: Property scanning
% 2.77/1.56
% 2.77/1.56 % (14891)Memory used [KB]: 1663
% 2.77/1.56 % (14891)Time elapsed: 0.008 s
% 2.77/1.56 % (14891)Instructions burned: 23 (million)
% 2.77/1.56 % (14891)------------------------------
% 2.77/1.56 % (14891)------------------------------
% 2.77/1.57 % (14892)Instruction limit reached!
% 2.77/1.57 % (14892)------------------------------
% 2.77/1.57 % (14892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.57 % (14892)Termination reason: Unknown
% 2.77/1.57 % (14892)Termination phase: shuffling
% 2.77/1.57
% 2.77/1.57 % (14892)Memory used [KB]: 1407
% 2.77/1.57 % (14892)Time elapsed: 0.003 s
% 2.77/1.57 % (14892)Instructions burned: 7 (million)
% 2.77/1.57 % (14892)------------------------------
% 2.77/1.57 % (14892)------------------------------
% 2.77/1.57 % (14893)Instruction limit reached!
% 2.77/1.57 % (14893)------------------------------
% 2.77/1.57 % (14893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.57 % (14893)Termination reason: Unknown
% 2.77/1.57 % (14893)Termination phase: shuffling
% 2.77/1.57
% 2.77/1.57 % (14893)Memory used [KB]: 1407
% 2.77/1.57 % (14893)Time elapsed: 0.003 s
% 2.77/1.57 % (14893)Instructions burned: 7 (million)
% 2.77/1.57 % (14893)------------------------------
% 2.77/1.57 % (14893)------------------------------
% 2.77/1.57 % (14894)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/377Mi)
% 2.77/1.57 % (14895)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_14594 for (2999ds/779Mi)
% 2.77/1.57 % (14896)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_14594 for (2999ds/19Mi)
% 2.77/1.57 % (14897)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/879Mi)
% 2.77/1.58 % (14898)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_14594 for (2999ds/17Mi)
% 2.77/1.58 % (14896)Instruction limit reached!
% 2.77/1.58 % (14896)------------------------------
% 2.77/1.58 % (14896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (14896)Termination reason: Unknown
% 2.77/1.58 % (14896)Termination phase: shuffling
% 2.77/1.58
% 2.77/1.58 % (14896)Memory used [KB]: 1663
% 2.77/1.58 % (14896)Time elapsed: 0.006 s
% 2.77/1.58 % (14896)Instructions burned: 19 (million)
% 2.77/1.58 % (14896)------------------------------
% 2.77/1.58 % (14896)------------------------------
% 2.77/1.58 % (14898)Instruction limit reached!
% 2.77/1.58 % (14898)------------------------------
% 2.77/1.58 % (14898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (14898)Termination reason: Unknown
% 2.77/1.58 % (14898)Termination phase: Property scanning
% 2.77/1.58
% 2.77/1.58 % (14898)Memory used [KB]: 1663
% 2.77/1.58 % (14898)Time elapsed: 0.007 s
% 2.77/1.58 % (14898)Instructions burned: 19 (million)
% 2.77/1.58 % (14898)------------------------------
% 2.77/1.58 % (14898)------------------------------
% 2.77/1.58 % (14897)Refutation not found, incomplete strategy
% 2.77/1.58 % (14897)------------------------------
% 2.77/1.58 % (14897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (14897)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.58
% 2.77/1.58
% 2.77/1.58 % (14897)Memory used [KB]: 6140
% 2.77/1.58 % (14897)Time elapsed: 0.010 s
% 2.77/1.58 % (14897)Instructions burned: 31 (million)
% 2.77/1.58 % (14897)------------------------------
% 2.77/1.58 % (14897)------------------------------
% 2.77/1.59 % (14899)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_14594 for (2998ds/3Mi)
% 2.77/1.59 % (14899)Instruction limit reached!
% 2.77/1.59 % (14899)------------------------------
% 2.77/1.59 % (14899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (14899)Termination reason: Unknown
% 2.77/1.59 % (14899)Termination phase: shuffling
% 2.77/1.59
% 2.77/1.59 % (14899)Memory used [KB]: 1279
% 2.77/1.59 % (14899)Time elapsed: 0.002 s
% 2.77/1.59 % (14899)Instructions burned: 4 (million)
% 2.77/1.59 % (14899)------------------------------
% 2.77/1.59 % (14899)------------------------------
% 2.77/1.59 % (14900)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_14594 for (2998ds/30Mi)
% 2.77/1.59 % (14901)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_14594 for (2998ds/127Mi)
% 2.77/1.59 % (14890)Refutation not found, incomplete strategy
% 2.77/1.59 % (14890)------------------------------
% 2.77/1.59 % (14890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (14890)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.59
% 2.77/1.59
% 2.77/1.59 % (14890)Memory used [KB]: 6524
% 2.77/1.59 % (14890)Time elapsed: 0.032 s
% 2.77/1.59 % (14890)Instructions burned: 77 (million)
% 2.77/1.59 % (14890)------------------------------
% 2.77/1.59 % (14890)------------------------------
% 2.77/1.60 % (14902)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_14594 for (2998ds/100Mi)
% 2.77/1.60 % (14900)Instruction limit reached!
% 2.77/1.60 % (14900)------------------------------
% 2.77/1.60 % (14900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.60 % (14900)Termination reason: Unknown
% 2.77/1.60 % (14900)Termination phase: Preprocessing 1
% 2.77/1.60
% 2.77/1.60 % (14900)Memory used [KB]: 1663
% 2.77/1.60 % (14900)Time elapsed: 0.010 s
% 2.77/1.60 % (14900)Instructions burned: 30 (million)
% 2.77/1.60 % (14900)------------------------------
% 2.77/1.60 % (14900)------------------------------
% 2.77/1.61 % (14903)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_14594 for (2998ds/3Mi)
% 2.77/1.61 % (14903)Instruction limit reached!
% 2.77/1.61 % (14903)------------------------------
% 2.77/1.61 % (14903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.61 % (14903)Termination reason: Unknown
% 2.77/1.61 % (14903)Termination phase: shuffling
% 2.77/1.61
% 2.77/1.61 % (14903)Memory used [KB]: 1279
% 2.77/1.61 % (14903)Time elapsed: 0.002 s
% 2.77/1.61 % (14903)Instructions burned: 3 (million)
% 2.77/1.61 % (14903)------------------------------
% 2.77/1.61 % (14903)------------------------------
% 2.77/1.61 % (14904)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_14594 for (2998ds/20Mi)
% 2.77/1.62 % (14904)Instruction limit reached!
% 2.77/1.62 % (14904)------------------------------
% 2.77/1.62 % (14904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.62 % (14904)Termination reason: Unknown
% 2.77/1.62 % (14904)Termination phase: Property scanning
% 2.77/1.62
% 2.77/1.62 % (14904)Memory used [KB]: 1535
% 2.77/1.62 % (14904)Time elapsed: 0.008 s
% 2.77/1.62 % (14904)Instructions burned: 22 (million)
% 2.77/1.62 % (14904)------------------------------
% 2.77/1.62 % (14904)------------------------------
% 2.77/1.62 % (14905)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_14594 for (2998ds/86Mi)
% 2.77/1.62 % (14902)Instruction limit reached!
% 2.77/1.62 % (14902)------------------------------
% 2.77/1.62 % (14902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.62 % (14902)Termination reason: Unknown
% 2.77/1.62 % (14902)Termination phase: Saturation
% 2.77/1.62
% 2.77/1.62 % (14902)Memory used [KB]: 6652
% 2.77/1.62 % (14902)Time elapsed: 0.027 s
% 2.77/1.62 % (14902)Instructions burned: 100 (million)
% 2.77/1.62 % (14902)------------------------------
% 2.77/1.62 % (14902)------------------------------
% 2.77/1.63 % (14906)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_14594 for (2998ds/107Mi)
% 2.77/1.63 % (14901)Instruction limit reached!
% 2.77/1.63 % (14901)------------------------------
% 2.77/1.63 % (14901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.63 % (14901)Termination reason: Unknown
% 2.77/1.63 % (14901)Termination phase: Saturation
% 2.77/1.63
% 2.77/1.63 % (14901)Memory used [KB]: 6780
% 2.77/1.63 % (14901)Time elapsed: 0.040 s
% 2.77/1.63 % (14901)Instructions burned: 127 (million)
% 2.77/1.63 % (14901)------------------------------
% 2.77/1.63 % (14901)------------------------------
% 2.77/1.63 % (14907)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_14594 for (2998ds/39Mi)
% 2.77/1.64 % (14906)Refutation not found, incomplete strategy
% 2.77/1.64 % (14906)------------------------------
% 2.77/1.64 % (14906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.64 % (14906)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.64
% 2.77/1.64
% 2.77/1.64 % (14906)Memory used [KB]: 6140
% 2.77/1.64 % (14906)Time elapsed: 0.010 s
% 2.77/1.64 % (14906)Instructions burned: 31 (million)
% 2.77/1.64 % (14906)------------------------------
% 2.77/1.64 % (14906)------------------------------
% 2.77/1.64 % (14908)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_14594 for (2998ds/448Mi)
% 2.77/1.64 % (14905)Instruction limit reached!
% 2.77/1.64 % (14905)------------------------------
% 2.77/1.64 % (14905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.64 % (14905)Termination reason: Unknown
% 2.77/1.64 % (14905)Termination phase: Saturation
% 2.77/1.64
% 2.77/1.64 % (14905)Memory used [KB]: 6524
% 2.77/1.64 % (14905)Time elapsed: 0.029 s
% 2.77/1.64 % (14905)Instructions burned: 86 (million)
% 2.77/1.64 % (14907)Instruction limit reached!
% 2.77/1.64 % (14907)------------------------------
% 2.77/1.64 % (14907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.64 % (14905)------------------------------
% 2.77/1.64 % (14905)------------------------------
% 2.77/1.64 % (14907)Termination reason: Unknown
% 2.77/1.64 % (14907)Termination phase: Property scanning
% 2.77/1.64
% 2.77/1.64 % (14907)Memory used [KB]: 2046
% 2.77/1.64 % (14907)Time elapsed: 0.011 s
% 2.77/1.64 % (14907)Instructions burned: 39 (million)
% 2.77/1.64 % (14907)------------------------------
% 2.77/1.64 % (14907)------------------------------
% 2.77/1.64 % (14909)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_14594 for (2998ds/46Mi)
% 2.77/1.65 % (14910)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_14594 for (2998ds/98Mi)
% 2.77/1.65 % (14911)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_14594 for (2998ds/507Mi)
% 2.77/1.65 % (14909)Refutation not found, incomplete strategy
% 2.77/1.65 % (14909)------------------------------
% 2.77/1.65 % (14909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.65 % (14909)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.65
% 2.77/1.65
% 2.77/1.65 % (14909)Memory used [KB]: 6140
% 2.77/1.65 % (14909)Time elapsed: 0.010 s
% 2.77/1.65 % (14909)Instructions burned: 30 (million)
% 2.77/1.65 % (14909)------------------------------
% 2.77/1.65 % (14909)------------------------------
% 2.77/1.66 % (14912)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_14594 for (2998ds/149Mi)
% 2.77/1.68 % (14910)Instruction limit reached!
% 2.77/1.68 % (14910)------------------------------
% 2.77/1.68 % (14910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (14910)Termination reason: Unknown
% 2.77/1.68 % (14910)Termination phase: Saturation
% 2.77/1.68
% 2.77/1.68 % (14910)Memory used [KB]: 2046
% 2.77/1.68 % (14910)Time elapsed: 0.027 s
% 2.77/1.68 % (14910)Instructions burned: 100 (million)
% 2.77/1.68 % (14910)------------------------------
% 2.77/1.68 % (14910)------------------------------
% 2.77/1.68 % (14911)Refutation not found, incomplete strategy
% 2.77/1.68 % (14911)------------------------------
% 2.77/1.68 % (14911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (14911)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.68
% 2.77/1.68
% 2.77/1.68 % (14911)Memory used [KB]: 6652
% 2.77/1.68 % (14911)Time elapsed: 0.027 s
% 2.77/1.68 % (14911)Instructions burned: 80 (million)
% 2.77/1.68 % (14911)------------------------------
% 2.77/1.68 % (14911)------------------------------
% 2.77/1.69 % (14913)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_14594 for (2997ds/39Mi)
% 2.77/1.69 % (14914)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_14594 for (2997ds/6Mi)
% 2.77/1.69 % (14914)Instruction limit reached!
% 2.77/1.69 % (14914)------------------------------
% 2.77/1.69 % (14914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (14914)Termination reason: Unknown
% 2.77/1.69 % (14914)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (14914)Memory used [KB]: 1407
% 2.77/1.69 % (14914)Time elapsed: 0.004 s
% 2.77/1.69 % (14914)Instructions burned: 8 (million)
% 2.77/1.69 % (14914)------------------------------
% 2.77/1.69 % (14914)------------------------------
% 2.77/1.69 % (14894)Instruction limit reached!
% 2.77/1.69 % (14894)------------------------------
% 2.77/1.69 % (14894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (14894)Termination reason: Unknown
% 2.77/1.69 % (14894)Termination phase: Saturation
% 2.77/1.69
% 2.77/1.69 % (14894)Memory used [KB]: 7164
% 2.77/1.69 % (14894)Time elapsed: 0.124 s
% 2.77/1.69 % (14894)Instructions burned: 378 (million)
% 2.77/1.69 % (14894)------------------------------
% 2.77/1.69 % (14894)------------------------------
% 2.77/1.70 % (14913)Instruction limit reached!
% 2.77/1.70 % (14913)------------------------------
% 2.77/1.70 % (14913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.70 % (14913)Termination reason: Unknown
% 2.77/1.70 % (14913)Termination phase: Property scanning
% 2.77/1.70
% 2.77/1.70 % (14913)Memory used [KB]: 1791
% 2.77/1.70 % (14913)Time elapsed: 0.013 s
% 2.77/1.70 % (14913)Instructions burned: 41 (million)
% 2.77/1.70 % (14913)------------------------------
% 2.77/1.70 % (14913)------------------------------
% 2.77/1.70 % (14915)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_14594 for (2997ds/1087Mi)
% 2.77/1.70 % (14916)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_14594 for (2997ds/47Mi)
% 2.77/1.71 % (14917)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_14594 for (2997ds/44Mi)
% 2.77/1.71 % (14912)Instruction limit reached!
% 2.77/1.71 % (14912)------------------------------
% 2.77/1.71 % (14912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (14912)Termination reason: Unknown
% 2.77/1.71 % (14912)Termination phase: Saturation
% 2.77/1.71
% 2.77/1.71 % (14912)Memory used [KB]: 6780
% 2.77/1.71 % (14912)Time elapsed: 0.048 s
% 2.77/1.71 % (14912)Instructions burned: 152 (million)
% 2.77/1.71 % (14912)------------------------------
% 2.77/1.71 % (14912)------------------------------
% 2.77/1.71 % (14915)Refutation not found, incomplete strategy
% 2.77/1.71 % (14915)------------------------------
% 2.77/1.71 % (14915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.71 % (14915)Termination reason: Refutation not found, incomplete strategy
% 2.77/1.71
% 2.77/1.71
% 2.77/1.71 % (14915)Memory used [KB]: 6140
% 2.77/1.71 % (14915)Time elapsed: 0.011 s
% 2.77/1.71 % (14915)Instructions burned: 31 (million)
% 2.77/1.71 % (14915)------------------------------
% 2.77/1.71 % (14915)------------------------------
% 2.77/1.72 % (14916)Instruction limit reached!
% 2.77/1.72 % (14916)------------------------------
% 2.77/1.72 % (14916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (14916)Termination reason: Unknown
% 2.77/1.72 % (14916)Termination phase: Property scanning
% 2.77/1.72
% 2.77/1.72 % (14916)Memory used [KB]: 1791
% 2.77/1.72 % (14916)Time elapsed: 0.015 s
% 2.77/1.72 % (14916)Instructions burned: 49 (million)
% 2.77/1.72 % (14916)------------------------------
% 2.77/1.72 % (14916)------------------------------
% 2.77/1.72 % (14918)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_14594 for (2997ds/432Mi)
% 2.77/1.72 % (14917)Instruction limit reached!
% 2.77/1.72 % (14917)------------------------------
% 2.77/1.72 % (14917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.72 % (14917)Termination reason: Unknown
% 2.77/1.72 % (14917)Termination phase: Property scanning
% 2.77/1.72
% 2.77/1.72 % (14917)Memory used [KB]: 2046
% 2.77/1.72 % (14917)Time elapsed: 0.013 s
% 2.77/1.72 % (14917)Instructions burned: 46 (million)
% 2.77/1.72 % (14917)------------------------------
% 2.77/1.72 % (14917)------------------------------
% 2.77/1.72 % (14919)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_14594 for (2997ds/62Mi)
% 2.77/1.73 % (14920)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_14594 for (2997ds/31Mi)
% 2.77/1.73 % (14921)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_14594 for (2997ds/96Mi)
% 2.77/1.74 % (14920)Instruction limit reached!
% 2.77/1.74 % (14920)------------------------------
% 2.77/1.74 % (14920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (14920)Termination reason: Unknown
% 2.77/1.74 % (14920)Termination phase: Preprocessing 3
% 2.77/1.74
% 2.77/1.74 % (14920)Memory used [KB]: 1663
% 2.77/1.74 % (14920)Time elapsed: 0.011 s
% 2.77/1.74 % (14920)Instructions burned: 32 (million)
% 2.77/1.74 % (14920)------------------------------
% 2.77/1.74 % (14920)------------------------------
% 2.77/1.74 % (14879)First to succeed.
% 2.77/1.74 % (14919)Instruction limit reached!
% 2.77/1.74 % (14919)------------------------------
% 2.77/1.74 % (14919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (14919)Termination reason: Unknown
% 2.77/1.74 % (14919)Termination phase: Property scanning
% 2.77/1.74
% 2.77/1.74 % (14919)Memory used [KB]: 2046
% 2.77/1.74 % (14919)Time elapsed: 0.020 s
% 2.77/1.74 % (14919)Instructions burned: 65 (million)
% 2.77/1.74 % (14919)------------------------------
% 2.77/1.74 % (14919)------------------------------
% 2.77/1.74 % (14879)Refutation found. Thanks to Tanya!
% 2.77/1.74 % SZS status ContradictoryAxioms for DTF2THF_14594
% 2.77/1.74 % SZS output start Proof for DTF2THF_14594
% 2.77/1.74 thf(type_def_6, type, nat: $tType).
% 2.77/1.74 thf(func_def_0, type, nat: $tType).
% 2.77/1.74 thf(func_def_1, type, zero: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_2, type, group_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_3, type, monoid_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_4, type, ab_group_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_5, type, wellorder: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_6, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_7, type, semigroup_mult: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_8, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_9, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_10, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_11, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_12, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_13, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_14, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_15, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_16, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_17, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_18, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_19, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_20, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_21, type, ordere623563068d_diff: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_22, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_23, type, ordere516151231imp_le: !>[X0: $tType]:($o)).
% 2.77/1.74 thf(func_def_24, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.77/1.74 thf(func_def_25, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.77/1.74 thf(func_def_26, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.77/1.74 thf(func_def_27, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.77/1.74 thf(func_def_28, type, number2016821345itions: (nat > nat) > nat > $o).
% 2.77/1.74 thf(func_def_29, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.74 thf(func_def_30, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.74 thf(func_def_31, type, i: nat).
% 2.77/1.74 thf(func_def_32, type, k: nat).
% 2.77/1.74 thf(func_def_33, type, n: nat).
% 2.77/1.74 thf(func_def_34, type, p: nat > nat).
% 2.77/1.74 thf(func_def_52, type, ph1: !>[X0: $tType]:(X0)).
% 2.77/1.74 thf(f2850,plain,(
% 2.77/1.74 $false),
% 2.77/1.74 inference(trivial_inequality_removal,[status(thm)],[f2849])).
% 2.77/1.74 thf(f2849,plain,(
% 2.77/1.74 ($true = $false)),
% 2.77/1.74 inference(equality_proxy_clausification,[status(thm)],[f2815])).
% 2.77/1.74 thf(f2815,plain,(
% 2.77/1.74 ($true = ($false = $true))),
% 2.77/1.74 inference(boolean_simplification,[status(thm)],[f2814])).
% 2.77/1.74 thf(f2814,plain,(
% 2.77/1.74 ($true = ((~ $true) = $true))),
% 2.77/1.74 inference(forward_demodulation,[status(thm)],[f2770,f1106])).
% 2.77/1.74 thf(f1106,plain,(
% 2.77/1.74 ((ord_less_eq @ nat @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)) @ n) = $true)),
% 2.77/1.74 inference(cnf_transformation,[status(thm)],[f336])).
% 2.77/1.74 thf(f336,plain,(
% 2.77/1.74 ((ord_less_eq @ nat @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)) @ n) = $true)),
% 2.77/1.74 inference(fool_elimination,[status(thm)],[f335])).
% 2.77/1.74 thf(f335,plain,(
% 2.77/1.74 (ord_less_eq @ nat @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)) @ n)),
% 2.77/1.74 inference(rectify,[status(thm)],[f3])).
% 2.77/1.74 thf(f3,axiom,(
% 2.77/1.74 (ord_less_eq @ nat @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)) @ n)),
% 2.77/1.74 file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_2_upper__bound)).
% 2.77/1.74 thf(f2770,plain,(
% 2.77/1.74 ($true = ((~ (ord_less_eq @ nat @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)) @ n)) = $true))),
% 2.77/1.74 inference(superposition,[status(thm)],[f2725,f1064])).
% 2.77/1.74 thf(f1064,plain,(
% 2.77/1.74 ((ord_less @ nat @ n @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k))) = $true)),
% 2.77/1.74 inference(cnf_transformation,[status(thm)],[f382])).
% 2.77/1.74 thf(f382,plain,(
% 2.77/1.74 ((ord_less @ nat @ n @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k))) = $true)),
% 2.77/1.74 inference(fool_elimination,[status(thm)],[f381])).
% 2.77/1.74 thf(f381,plain,(
% 2.77/1.74 (ord_less @ nat @ n @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)))),
% 2.77/1.74 inference(rectify,[status(thm)],[f4])).
% 2.77/1.74 thf(f4,axiom,(
% 2.77/1.74 (ord_less @ nat @ n @ (plus_plus @ nat @ (times_times @ nat @ (p @ i) @ i) @ (times_times @ nat @ (p @ k) @ k)))),
% 2.77/1.74 file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_3_lower__bound)).
% 2.77/1.74 thf(f2725,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (($true = ((~ (ord_less_eq @ nat @ X2 @ X1)) = (ord_less @ nat @ X1 @ X2)))) )),
% 2.77/1.74 inference(forward_demodulation,[status(thm)],[f2724,f2617])).
% 2.77/1.74 thf(f2617,plain,(
% 2.77/1.74 ( ! [X0 : nat,X1 : nat] : (((~ (ord_less_eq @ nat @ X0 @ X1)) = (ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ X0 @ X1)))) )),
% 2.77/1.74 inference(superposition,[status(thm)],[f1518,f2507])).
% 2.77/1.74 thf(f2507,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (((zero_zero @ nat = (minus_minus @ nat @ X2 @ X1)) = (ord_less_eq @ nat @ X2 @ X1))) )),
% 2.77/1.74 inference(equality_proxy_clausification,[status(thm)],[f2501])).
% 2.77/1.74 thf(f2501,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less_eq @ nat @ X2 @ X1) = (zero_zero @ nat = (minus_minus @ nat @ X2 @ X1))))) )),
% 2.77/1.74 inference(beta_eta_normalization,[status(thm)],[f2500])).
% 2.77/1.74 thf(f2500,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less_eq @ nat @ Y0 @ X1) = (zero_zero @ nat = (minus_minus @ nat @ Y0 @ X1)))) @ X2))) )),
% 2.77/1.74 inference(pi_clausification,[status(thm)],[f2465])).
% 2.77/1.74 thf(f2465,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq @ nat @ Y0 @ X1) = (zero_zero @ nat = (minus_minus @ nat @ Y0 @ X1))))))) )),
% 2.77/1.74 inference(beta_eta_normalization,[status(thm)],[f2464])).
% 2.77/1.74 thf(f2464,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) = (zero_zero @ nat = (minus_minus @ nat @ Y1 @ Y0)))))) @ X1))) )),
% 2.77/1.74 inference(pi_clausification,[status(thm)],[f1109])).
% 2.77/1.74 thf(f1109,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) = (zero_zero @ nat = (minus_minus @ nat @ Y1 @ Y0))))))))),
% 2.77/1.74 inference(cnf_transformation,[status(thm)],[f354])).
% 2.77/1.74 thf(f354,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) = (zero_zero @ nat = (minus_minus @ nat @ Y1 @ Y0))))))))),
% 2.77/1.74 inference(fool_elimination,[status(thm)],[f353])).
% 2.77/1.74 thf(f353,plain,(
% 2.77/1.74 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) = (zero_zero @ nat = (minus_minus @ nat @ X0 @ X1)))),
% 2.77/1.74 inference(rectify,[status(thm)],[f81])).
% 2.77/1.74 thf(f81,axiom,(
% 2.77/1.74 ! [X5 : nat,X3 : nat] : ((ord_less_eq @ nat @ X5 @ X3) = (zero_zero @ nat = (minus_minus @ nat @ X5 @ X3)))),
% 2.77/1.74 file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_80_diff__is__0__eq)).
% 2.77/1.74 thf(f1518,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X1) = (~ (zero_zero @ nat = X1)))) )),
% 2.77/1.74 inference(equality_proxy_clausification,[status(thm)],[f1517])).
% 2.77/1.74 thf(f1517,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = ((ord_less @ nat @ zero_zero @ nat @ X1) = (~ (zero_zero @ nat = X1))))) )),
% 2.77/1.74 inference(beta_eta_normalization,[status(thm)],[f1516])).
% 2.77/1.74 thf(f1516,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less @ nat @ zero_zero @ nat @ Y0) = (~ (zero_zero @ nat = Y0)))) @ X1))) )),
% 2.77/1.74 inference(pi_clausification,[status(thm)],[f1034])).
% 2.77/1.74 thf(f1034,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less @ nat @ zero_zero @ nat @ Y0) = (~ (zero_zero @ nat = Y0))))))),
% 2.77/1.74 inference(cnf_transformation,[status(thm)],[f806])).
% 2.77/1.74 thf(f806,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less @ nat @ zero_zero @ nat @ Y0) = (~ (zero_zero @ nat = Y0))))))),
% 2.77/1.74 inference(fool_elimination,[status(thm)],[f805])).
% 2.77/1.74 thf(f805,plain,(
% 2.77/1.74 ! [X0 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X0) = (zero_zero @ nat != X0))),
% 2.77/1.74 inference(rectify,[status(thm)],[f52])).
% 2.77/1.74 thf(f52,axiom,(
% 2.77/1.74 ! [X8 : nat] : ((ord_less @ nat @ zero_zero @ nat @ X8) = (zero_zero @ nat != X8))),
% 2.77/1.74 file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_51_bot__nat__0_Onot__eq__extremum)).
% 2.77/1.74 thf(f2724,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ X2 @ X1)) = (ord_less @ nat @ X1 @ X2)))) )),
% 2.77/1.74 inference(beta_eta_normalization,[status(thm)],[f2723])).
% 2.77/1.74 thf(f2723,plain,(
% 2.77/1.74 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ Y0 @ X1)) = (ord_less @ nat @ X1 @ Y0))) @ X2))) )),
% 2.77/1.74 inference(pi_clausification,[status(thm)],[f2665])).
% 2.77/1.74 thf(f2665,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ Y0 @ X1)) = (ord_less @ nat @ X1 @ Y0)))))) )),
% 2.77/1.74 inference(beta_eta_normalization,[status(thm)],[f2664])).
% 2.77/1.74 thf(f2664,plain,(
% 2.77/1.74 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ Y1 @ Y0)) = (ord_less @ nat @ Y0 @ Y1))))) @ X1))) )),
% 2.77/1.74 inference(pi_clausification,[status(thm)],[f1131])).
% 2.77/1.74 thf(f1131,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ Y1 @ Y0)) = (ord_less @ nat @ Y0 @ Y1)))))))),
% 2.77/1.74 inference(cnf_transformation,[status(thm)],[f476])).
% 2.77/1.74 thf(f476,plain,(
% 2.77/1.74 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ Y1 @ Y0)) = (ord_less @ nat @ Y0 @ Y1)))))))),
% 2.77/1.74 inference(fool_elimination,[status(thm)],[f475])).
% 2.77/1.74 thf(f475,plain,(
% 2.77/1.74 ! [X0 : nat,X1 : nat] : ((ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ X0 @ X1)) = (ord_less @ nat @ X1 @ X0))),
% 2.77/1.74 inference(rectify,[status(thm)],[f79])).
% 2.77/1.74 thf(f79,axiom,(
% 2.77/1.74 ! [X3 : nat,X5 : nat] : ((ord_less @ nat @ X5 @ X3) = (ord_less @ nat @ zero_zero @ nat @ (minus_minus @ nat @ X3 @ X5)))),
% 2.77/1.74 file('/export/starexec/sandbox/tmp/tmp.5WYWvp2lJI/DTF2THF_14594.p',fact_78_zero__less__diff)).
% 2.77/1.74 % SZS output end Proof for DTF2THF_14594
% 2.77/1.74 % (14879)------------------------------
% 2.77/1.74 % (14879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.74 % (14879)Termination reason: Refutation
% 2.77/1.74
% 2.77/1.74 % (14879)Memory used [KB]: 7931
% 2.77/1.74 % (14879)Time elapsed: 0.214 s
% 2.77/1.74 % (14879)Instructions burned: 553 (million)
% 2.77/1.74 % (14879)------------------------------
% 2.77/1.74 % (14879)------------------------------
% 2.77/1.74 % (14867)Success in time 0.26 s
%------------------------------------------------------------------------------