%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT239^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n013.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 08:45:15 AM UTC 2026
% Result : Theorem 2.53s 1.51s
% Output : Refutation 2.53s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT239^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n013.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
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 06:11:53 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : DAT239^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.46 % Domain : Data Structures
% 0.29/0.46 % Problem : Red-black trees 1450
% 0.29/0.46 % Version : [Bla16] axioms : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.29/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Bla16]
% 0.29/0.46 % Names : rbt_impl__1450.p [Bla16]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% 0.29/0.46 % Syntax : Number of formulae : 346 ( 97 unt; 45 typ; 0 def)
% 0.29/0.46 % Number of atoms : 855 ( 257 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.29/0.46 % Number of connectives : 3434 ( 64 ~; 22 |; 37 &;2983 @)
% 0.29/0.46 % ( 0 <=>; 328 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 16 ( 6 avg)
% 0.29/0.46 % Number of types : 4 ( 3 usr)
% 0.29/0.46 % Number of type conns : 54 ( 54 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 43 ( 42 usr; 2 con; 0-3 aty)
% 0.29/0.46 % Number of variables : 652 ( 9 ^; 597 !; 8 ?; 652 :)
% 0.29/0.46 % ( 38 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.46 % SPC : TH1_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.46 % 2016-07-13 14:46:57.593
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 %----Could-be-implicit typings (4)
% 0.29/0.46 thf(ty_t_Num_Onum,type,
% 0.29/0.46 num: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Nat_Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Int_Oint,type,
% 0.29/0.46 int: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_itself,type,
% 0.29/0.46 itself: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 %----Explicit typings (41)
% 0.29/0.46 thf(sy_cl_Groups_Oone,type,
% 0.29/0.46 one:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.46 zero:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Num_Onumeral,type,
% 0.29/0.46 numeral:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.29/0.46 field:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Num_Oneg__numeral,type,
% 0.29/0.46 neg_numeral:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.46 group_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Omonoid__add,type,
% 0.29/0.46 monoid_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.46 preorder:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.29/0.46 zero_neq_one:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.29/0.46 semiring_char_0:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.29/0.46 zero_less_one:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Divides_Osemiring__div,type,
% 0.29/0.46 semiring_div:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.29/0.46 division_ring:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/0.46 semigroup_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.29/0.46 semidom_divide:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.46 linordered_idom:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.29/0.46 linordered_field:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/0.46 ab_semigroup_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.29/0.46 linordered_semidom:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Divides_Osemiring__div__parity,type,
% 0.29/0.46 semiring_div_parity:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/0.46 cancel_semigroup_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Divides_Osemiring__numeral__div,type,
% 0.29/0.46 semiring_numeral_div:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.29/0.46 cancel1352612707id_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/0.46 linord219039673up_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/0.46 ordere779506340up_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.29/0.46 linord1659791738miring:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.46 canoni770627133id_add:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/0.46 ordere236663937imp_le:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.29/0.46 ordere516151231imp_le:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.29/0.46 one_one:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/0.46 plus_plus:
% 0.29/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.46 zero_zero:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.29/0.46 neg_numeral_dbl:
% 0.29/0.46 !>[A: $tType] : ( A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Num_Onum_OBit0,type,
% 0.29/0.46 bit0: num > num ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Num_Onum_OOne,type,
% 0.29/0.46 one2: num ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.29/0.46 numeral_numeral:
% 0.29/0.46 !>[A: $tType] : ( num > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.46 ord_less:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.46 ord_less_eq:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pure_Otype,type,
% 0.29/0.46 type:
% 0.29/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.29/0.46 divide_divide:
% 0.29/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_na____,type,
% 0.29/0.46 na: nat ).
% 0.29/0.46
% 0.29/0.46 %----Relevant facts (251)
% 0.29/0.46 thf(fact_0_ns,axiom,
% 0.29/0.46 ~ ( ( na
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 | ( na
% 0.29/0.46 = ( one_one @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ns
% 0.29/0.46 thf(fact_1_False,axiom,
% 0.29/0.46 ~ ( ord_less_eq @ nat @ na @ ( one_one @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % False
% 0.29/0.46 thf(fact_2_Divides_Odiv__less,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( ( divide_divide @ nat @ M @ N )
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Divides.div_less
% 0.29/0.46 thf(fact_3_half__gt__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % half_gt_zero
% 0.29/0.46 thf(fact_4_half__gt__zero__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % half_gt_zero_iff
% 0.29/0.46 thf(fact_5_neq0__conv,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 % neq0_conv
% 0.29/0.46 thf(fact_6_div__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_0
% 0.29/0.46 thf(fact_7_divide__eq__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( ( A2
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 | ( B
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_eq_0_iff
% 0.29/0.46 thf(fact_8_div__by__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_by_0
% 0.29/0.46 thf(fact_9_divide__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [C: A,A2: A,B: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ C @ A2 )
% 0.29/0.46 = ( divide_divide @ A @ C @ B ) )
% 0.29/0.46 = ( ( C
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 | ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_cancel_left
% 0.29/0.46 thf(fact_10_divide__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ A2 @ C )
% 0.29/0.46 = ( divide_divide @ A @ B @ C ) )
% 0.29/0.46 = ( ( C
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 | ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_cancel_right
% 0.29/0.46 thf(fact_11_divide__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_zero
% 0.29/0.46 thf(fact_12_divide__zero__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_zero_left
% 0.29/0.46 thf(fact_13_zdiv__numeral__Bit0,axiom,
% 0.29/0.46 ! [V: num,W: num] :
% 0.29/0.46 ( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
% 0.29/0.46 = ( divide_divide @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_numeral_Bit0
% 0.29/0.46 thf(fact_14_num_Oinject_I1_J,axiom,
% 0.29/0.46 ! [X2: num,Y2: num] :
% 0.29/0.46 ( ( ( bit0 @ X2 )
% 0.29/0.46 = ( bit0 @ Y2 ) )
% 0.29/0.46 = ( X2 = Y2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % num.inject(1)
% 0.29/0.46 thf(fact_15_numeral__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/0.46 => ! [M: num,N: num] :
% 0.29/0.46 ( ( ( numeral_numeral @ A @ M )
% 0.29/0.46 = ( numeral_numeral @ A @ N ) )
% 0.29/0.46 = ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_eq_iff
% 0.29/0.46 thf(fact_16_numeral__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [M: num,N: num] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/0.46 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_iff
% 0.29/0.46 thf(fact_17_numeral__less__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [M: num,N: num] :
% 0.29/0.46 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/0.46 = ( ord_less @ num @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_less_iff
% 0.29/0.46 thf(fact_18_division__ring__divide__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % division_ring_divide_zero
% 0.29/0.46 thf(fact_19_divide__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_1
% 0.29/0.46 thf(fact_20_le0,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/0.46
% 0.29/0.46 % le0
% 0.29/0.46 thf(fact_21_one__eq__numeral__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ( ( ( one_one @ A )
% 0.29/0.46 = ( numeral_numeral @ A @ N ) )
% 0.29/0.46 = ( one2 = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_eq_numeral_iff
% 0.29/0.46 thf(fact_22_numeral__eq__one__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ( ( ( numeral_numeral @ A @ N )
% 0.29/0.46 = ( one_one @ A ) )
% 0.29/0.46 = ( N = one2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_eq_one_iff
% 0.29/0.46 thf(fact_23_zero__eq__1__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_eq_1_divide_iff
% 0.29/0.46 thf(fact_24_one__divide__eq__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_divide_eq_0_iff
% 0.29/0.46 thf(fact_25_eq__divide__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ( one_one @ A )
% 0.29/0.46 = ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 & ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_divide_eq_1
% 0.29/0.46 thf(fact_26_divide__eq__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ B @ A2 )
% 0.29/0.46 = ( one_one @ A ) )
% 0.29/0.46 = ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 & ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_eq_eq_1
% 0.29/0.46 thf(fact_27_divide__self__if,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( A2
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) ) )
% 0.29/0.46 & ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_self_if
% 0.29/0.46 thf(fact_28_divide__self,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/0.46 = ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_self
% 0.29/0.46 thf(fact_29_one__eq__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ( one_one @ A )
% 0.29/0.46 = ( divide_divide @ A @ A2 @ B ) )
% 0.29/0.46 = ( ( B
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 & ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_eq_divide_iff
% 0.29/0.46 thf(fact_30_div__self,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/0.46 = ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_self
% 0.29/0.46 thf(fact_31_divide__eq__1__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/0.46 = ( one_one @ A ) )
% 0.29/0.46 = ( ( B
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 & ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_eq_1_iff
% 0.29/0.46 thf(fact_32_less__one,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_one
% 0.29/0.46 thf(fact_33_numeral__le__one__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_le_one_iff
% 0.29/0.46 thf(fact_34_zero__le__divide__1__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_divide_1_iff
% 0.29/0.46 thf(fact_35_divide__le__0__1__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_0_1_iff
% 0.29/0.46 thf(fact_36_one__less__numeral__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.29/0.46 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_less_numeral_iff
% 0.29/0.46 thf(fact_37_zero__less__divide__1__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_divide_1_iff
% 0.29/0.46 thf(fact_38_less__divide__eq__1__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_divide_eq_1_pos
% 0.29/0.46 thf(fact_39_less__divide__eq__1__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_divide_eq_1_neg
% 0.29/0.46 thf(fact_40_divide__less__eq__1__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_eq_1_pos
% 0.29/0.46 thf(fact_41_divide__less__eq__1__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_eq_1_neg
% 0.29/0.46 thf(fact_42_divide__less__0__1__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_0_1_iff
% 0.29/0.46 thf(fact_43_le__divide__eq__1__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_divide_eq_1_pos
% 0.29/0.46 thf(fact_44_le__divide__eq__1__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_divide_eq_1_neg
% 0.29/0.46 thf(fact_45_divide__le__eq__1__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_eq_1_pos
% 0.29/0.46 thf(fact_46_divide__le__eq__1__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_eq_1_neg
% 0.29/0.46 thf(fact_47_one__div__two__eq__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_div_parity @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_div_two_eq_zero
% 0.29/0.46 thf(fact_48_le__numeral__extra_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(4)
% 0.29/0.46 thf(fact_49_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_50_le__trans,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_51_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_52_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_53_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_54_not__one__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_one_le_zero
% 0.29/0.46 thf(fact_55_zero__le__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_one
% 0.29/0.46 thf(fact_56_le__numeral__extra_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(1)
% 0.29/0.46 thf(fact_57_le__numeral__extra_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(2)
% 0.29/0.46 thf(fact_58_one__le__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_le_numeral
% 0.29/0.46 thf(fact_59_le__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(3)
% 0.29/0.46 thf(fact_60_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_61_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_62_less__mono__imp__le__mono,axiom,
% 0.29/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.46 ( ! [I2: nat,J2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/0.46 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_mono_imp_le_mono
% 0.29/0.46 thf(fact_63_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_64_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_65_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.46 | ( M2 = N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_66_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_67_nat__less__le,axiom,
% 0.29/0.46 ( ( ord_less @ nat )
% 0.29/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 & ( M2 != N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_68_div__le__dividend,axiom,
% 0.29/0.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.29/0.46
% 0.29/0.46 % div_le_dividend
% 0.29/0.46 thf(fact_69_div__le__mono,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_le_mono
% 0.29/0.46 thf(fact_70_not__numeral__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_numeral_le_zero
% 0.29/0.46 thf(fact_71_zero__le__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_numeral
% 0.29/0.46 thf(fact_72_divide__right__mono__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ 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 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_right_mono_neg
% 0.29/0.46 thf(fact_73_divide__nonpos__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonpos
% 0.29/0.46 thf(fact_74_divide__nonpos__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonneg
% 0.29/0.46 thf(fact_75_divide__nonneg__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonpos
% 0.29/0.46 thf(fact_76_divide__nonneg__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonneg
% 0.29/0.46 thf(fact_77_zero__le__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.29/0.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
% 0.29/0.46 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_divide_iff
% 0.29/0.46 thf(fact_78_divide__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ 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 @ ( zero_zero @ A ) @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_right_mono
% 0.29/0.46 thf(fact_79_divide__le__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
% 0.29/0.46 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_0_iff
% 0.29/0.46 thf(fact_80_ex__least__nat__le,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ( P @ N )
% 0.29/0.46 => ? [K2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.29/0.46 & ! [I3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ K2 )
% 0.29/0.46 => ~ ( P @ I3 ) )
% 0.29/0.46 & ( P @ K2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_least_nat_le
% 0.29/0.46 thf(fact_81_zero__neq__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( zero_zero @ A )
% 0.29/0.46 != ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_neq_one
% 0.29/0.46 thf(fact_82_less__numeral__extra_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(4)
% 0.29/0.46 thf(fact_83_div__by__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_by_1
% 0.29/0.46 thf(fact_84_le__divide__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_divide_eq_1
% 0.29/0.46 thf(fact_85_divide__le__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less_eq @ A @ B @ A2 ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/0.46 | ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_eq_1
% 0.29/0.46 thf(fact_86_divide__nonpos__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_pos
% 0.29/0.46 thf(fact_87_divide__nonpos__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_neg
% 0.29/0.46 thf(fact_88_divide__nonneg__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_pos
% 0.29/0.46 thf(fact_89_divide__nonneg__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_neg
% 0.29/0.46 thf(fact_90_semiring__numeral__div__class_Odiv__positive,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_numeral_div @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_numeral_div_class.div_positive
% 0.29/0.46 thf(fact_91_divide__le__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/0.46 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_cancel
% 0.29/0.46 thf(fact_92_semiring__numeral__div__class_Odiv__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_numeral_div @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ( divide_divide @ A @ A2 @ B )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_numeral_div_class.div_less
% 0.29/0.46 thf(fact_93_frac__less2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/0.46 => ( ( ord_less @ A @ W @ Z )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_less2
% 0.29/0.46 thf(fact_94_frac__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/0.46 => ( ( ord_less_eq @ A @ W @ Z )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_less
% 0.29/0.46 thf(fact_95_frac__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/0.46 => ( ( ord_less_eq @ A @ W @ Z )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_le
% 0.29/0.46 thf(fact_96_Divides_Odiv__positive,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_eq @ nat @ N @ M )
% 0.29/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Divides.div_positive
% 0.29/0.46 thf(fact_97_div__le__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_le_mono2
% 0.29/0.46 thf(fact_98_linorder__neqE__linordered__idom,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_linordered_idom
% 0.29/0.46 thf(fact_99_linordered__field__no__ub,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X3: A] :
% 0.29/0.46 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_ub
% 0.29/0.46 thf(fact_100_linordered__field__no__lb,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X3: A] :
% 0.29/0.46 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_lb
% 0.29/0.46 thf(fact_101_infinite__descent__measure,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,V2: A > nat,X: A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ~ ( P @ X4 )
% 0.29/0.46 => ? [Y4: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
% 0.29/0.46 & ~ ( P @ Y4 ) ) )
% 0.29/0.46 => ( P @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent_measure
% 0.29/0.46 thf(fact_102_measure__induct__rule,axiom,
% 0.29/0.46 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ! [Y4: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.29/0.46 => ( P @ Y4 ) )
% 0.29/0.46 => ( P @ X4 ) )
% 0.29/0.46 => ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct_rule
% 0.29/0.46 thf(fact_103_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_104_infinite__descent,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ~ ( P @ N3 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 & ~ ( P @ M3 ) ) )
% 0.29/0.46 => ( P @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent
% 0.29/0.46 thf(fact_105_nat__less__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ! [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 => ( P @ M3 ) )
% 0.29/0.46 => ( P @ N3 ) )
% 0.29/0.46 => ( P @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_106_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_107_measure__induct,axiom,
% 0.29/0.46 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ! [Y4: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.29/0.46 => ( P @ Y4 ) )
% 0.29/0.46 => ( P @ X4 ) )
% 0.29/0.46 => ( P @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct
% 0.29/0.46 thf(fact_108_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_109_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_110_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_111_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_112_not__one__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_one_less_zero
% 0.29/0.46 thf(fact_113_zero__less__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_less_one @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_one
% 0.29/0.46 thf(fact_114_less__numeral__extra_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(1)
% 0.29/0.46 thf(fact_115_less__numeral__extra_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(2)
% 0.29/0.46 thf(fact_116_not__numeral__less__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_numeral_less_one
% 0.29/0.46 thf(fact_117_numeral__One,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( numeral_numeral @ A @ one2 )
% 0.29/0.46 = ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % numeral_One
% 0.29/0.46 thf(fact_118_right__inverse__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( B
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/0.46 = ( one_one @ A ) )
% 0.29/0.46 = ( A2 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % right_inverse_eq
% 0.29/0.46 thf(fact_119_numerals_I1_J,axiom,
% 0.29/0.46 ( ( numeral_numeral @ nat @ one2 )
% 0.29/0.46 = ( one_one @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % numerals(1)
% 0.29/0.46 thf(fact_120_div__eq__dividend__iff,axiom,
% 0.29/0.46 ! [A2: nat,B: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( zero_zero @ nat ) )
% 0.29/0.46 => ( ( ( divide_divide @ nat @ A2 @ B )
% 0.29/0.46 = A2 )
% 0.29/0.46 = ( B
% 0.29/0.46 = ( one_one @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_eq_dividend_iff
% 0.29/0.46 thf(fact_121_less__divide__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less @ A @ A2 @ B ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_divide_eq_1
% 0.29/0.46 thf(fact_122_divide__less__eq__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [B: A,A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less @ A @ B @ A2 ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less @ A @ A2 @ B ) )
% 0.29/0.46 | ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_eq_1
% 0.29/0.46 thf(fact_123_div__less__dividend,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.29/0.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.46 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_less_dividend
% 0.29/0.46 thf(fact_124_less__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(3)
% 0.29/0.46 thf(fact_125_zero__neq__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ( ( zero_zero @ A )
% 0.29/0.46 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_neq_numeral
% 0.29/0.46 thf(fact_126_num_Odistinct_I1_J,axiom,
% 0.29/0.46 ! [X2: num] :
% 0.29/0.46 ( one2
% 0.29/0.46 != ( bit0 @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % num.distinct(1)
% 0.29/0.46 thf(fact_127_infinite__descent0__measure,axiom,
% 0.29/0.46 ! [A: $tType,V2: A > nat,P: A > $o,X: A] :
% 0.29/0.46 ( ! [X4: A] :
% 0.29/0.46 ( ( ( V2 @ X4 )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 => ( P @ X4 ) )
% 0.29/0.46 => ( ! [X4: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X4 ) )
% 0.29/0.46 => ( ~ ( P @ X4 )
% 0.29/0.46 => ? [Y4: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
% 0.29/0.46 & ~ ( P @ Y4 ) ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0_measure
% 0.29/0.46 thf(fact_128_less__nat__zero__code,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_nat_zero_code
% 0.29/0.46 thf(fact_129_infinite__descent0,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [N3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.29/0.46 => ( ~ ( P @ N3 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 & ~ ( P @ M3 ) ) ) )
% 0.29/0.46 => ( P @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0
% 0.29/0.46 thf(fact_130_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_131_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_132_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_133_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_134_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_135_not__numeral__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] :
% 0.29/0.46 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_numeral_less_zero
% 0.29/0.46 thf(fact_136_zero__less__numeral,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_numeral
% 0.29/0.46 thf(fact_137_divide__strict__right__mono__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ 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 @ C @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_strict_right_mono_neg
% 0.29/0.46 thf(fact_138_divide__strict__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ 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 @ ( zero_zero @ A ) @ C )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_strict_right_mono
% 0.29/0.46 thf(fact_139_zero__less__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_divide_iff
% 0.29/0.46 thf(fact_140_divide__less__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,C: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B ) )
% 0.29/0.46 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ B @ A2 ) )
% 0.29/0.46 & ( C
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_cancel
% 0.29/0.46 thf(fact_141_divide__less__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
% 0.29/0.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.46 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_less_0_iff
% 0.29/0.46 thf(fact_142_divide__pos__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_pos_pos
% 0.29/0.46 thf(fact_143_divide__pos__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_pos_neg
% 0.29/0.46 thf(fact_144_divide__neg__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_neg_pos
% 0.29/0.46 thf(fact_145_divide__neg__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_neg_neg
% 0.29/0.46 thf(fact_146_divide__numeral__1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A @ ( type @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_numeral_1
% 0.29/0.46 thf(fact_147_div__eq__0__iff,axiom,
% 0.29/0.46 ! [A2: nat,B: nat] :
% 0.29/0.46 ( ( ( divide_divide @ nat @ A2 @ B )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 = ( ( ord_less @ nat @ A2 @ B )
% 0.29/0.46 | ( B
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_eq_0_iff
% 0.29/0.46 thf(fact_148_div__2__gt__zero,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.29/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_2_gt_zero
% 0.29/0.46 thf(fact_149_zero__not__eq__two,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_div_parity @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( zero_zero @ A )
% 0.29/0.46 != ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_not_eq_two
% 0.29/0.46 thf(fact_150_semiring__norm_I76_J,axiom,
% 0.29/0.46 ! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(76)
% 0.29/0.46 thf(fact_151_semiring__norm_I83_J,axiom,
% 0.29/0.46 ! [N: num] :
% 0.29/0.46 ( one2
% 0.29/0.46 != ( bit0 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(83)
% 0.29/0.46 thf(fact_152_semiring__norm_I85_J,axiom,
% 0.29/0.46 ! [M: num] :
% 0.29/0.46 ( ( bit0 @ M )
% 0.29/0.46 != one2 ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(85)
% 0.29/0.46 thf(fact_153_not__gr__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type @ 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 % not_gr_zero
% 0.29/0.46 thf(fact_154_le__zero__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_zero_eq
% 0.29/0.46 thf(fact_155_dbl__simps_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 0.29/0.46 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dbl_simps(3)
% 0.29/0.46 thf(fact_156_semiring__norm_I75_J,axiom,
% 0.29/0.46 ! [M: num] :
% 0.29/0.46 ~ ( ord_less @ num @ M @ one2 ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(75)
% 0.29/0.46 thf(fact_157_semiring__norm_I78_J,axiom,
% 0.29/0.46 ! [M: num,N: num] :
% 0.29/0.46 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.46 = ( ord_less @ num @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(78)
% 0.29/0.46 thf(fact_158_semiring__norm_I87_J,axiom,
% 0.29/0.46 ! [M: num,N: num] :
% 0.29/0.46 ( ( ( bit0 @ M )
% 0.29/0.46 = ( bit0 @ N ) )
% 0.29/0.46 = ( M = N ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(87)
% 0.29/0.46 thf(fact_159_semiring__norm_I71_J,axiom,
% 0.29/0.46 ! [M: num,N: num] :
% 0.29/0.46 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/0.46 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(71)
% 0.29/0.46 thf(fact_160_semiring__norm_I68_J,axiom,
% 0.29/0.46 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(68)
% 0.29/0.46 thf(fact_161_dbl__simps_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/0.46 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dbl_simps(2)
% 0.29/0.46 thf(fact_162_semiring__norm_I69_J,axiom,
% 0.29/0.46 ! [M: num] :
% 0.29/0.46 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 0.29/0.46
% 0.29/0.46 % semiring_norm(69)
% 0.29/0.46 thf(fact_163_dbl__simps_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/0.46 => ! [K: num] :
% 0.29/0.46 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 0.29/0.46 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dbl_simps(5)
% 0.29/0.46 thf(fact_164_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
% 0.29/0.46 ! [X: int,Y: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/0.46 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ X @ Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Divides.transfer_nat_int_function_closures(1)
% 0.29/0.46 thf(fact_165_zdiv__mono1,axiom,
% 0.29/0.46 ! [A2: int,A3: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ A2 @ A3 )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A3 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_mono1
% 0.29/0.46 thf(fact_166_zdiv__mono2,axiom,
% 0.29/0.46 ! [A2: int,B2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ int @ B2 @ B )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_mono2
% 0.29/0.46 thf(fact_167_zdiv__eq__0__iff,axiom,
% 0.29/0.46 ! [I: int,K: int] :
% 0.29/0.46 ( ( ( divide_divide @ int @ I @ K )
% 0.29/0.46 = ( zero_zero @ int ) )
% 0.29/0.46 = ( ( K
% 0.29/0.46 = ( zero_zero @ int ) )
% 0.29/0.46 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 0.29/0.46 & ( ord_less @ int @ I @ K ) )
% 0.29/0.46 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 0.29/0.46 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_eq_0_iff
% 0.29/0.46 thf(fact_168_zdiv__mono1__neg,axiom,
% 0.29/0.46 ! [A2: int,A3: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ A2 @ A3 )
% 0.29/0.46 => ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_mono1_neg
% 0.29/0.46 thf(fact_169_zdiv__mono2__neg,axiom,
% 0.29/0.46 ! [A2: int,B2: int,B: int] :
% 0.29/0.46 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ int @ B2 @ B )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_mono2_neg
% 0.29/0.46 thf(fact_170_div__neg__pos__less0,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/0.46 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_neg_pos_less0
% 0.29/0.46 thf(fact_171_int__div__less__self,axiom,
% 0.29/0.46 ! [X: int,K: int] :
% 0.29/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ X )
% 0.29/0.46 => ( ( ord_less @ int @ ( one_one @ int ) @ K )
% 0.29/0.46 => ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % int_div_less_self
% 0.29/0.46 thf(fact_172_div__nonneg__neg__le0,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_nonneg_neg_le0
% 0.29/0.46 thf(fact_173_div__nonpos__pos__le0,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_nonpos_pos_le0
% 0.29/0.46 thf(fact_174_div__neg__neg__trivial,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ B @ A2 )
% 0.29/0.46 => ( ( divide_divide @ int @ A2 @ B )
% 0.29/0.46 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_neg_neg_trivial
% 0.29/0.46 thf(fact_175_div__pos__pos__trivial,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ int @ A2 @ B )
% 0.29/0.46 => ( ( divide_divide @ int @ A2 @ B )
% 0.29/0.46 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_pos_pos_trivial
% 0.29/0.46 thf(fact_176_neg__imp__zdiv__neg__iff,axiom,
% 0.29/0.46 ! [B: int,A2: int] :
% 0.29/0.46 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 0.29/0.46 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_imp_zdiv_neg_iff
% 0.29/0.46 thf(fact_177_pos__imp__zdiv__neg__iff,axiom,
% 0.29/0.46 ! [B: int,A2: int] :
% 0.29/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/0.46 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 0.29/0.46 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pos_imp_zdiv_neg_iff
% 0.29/0.46 thf(fact_178_pos__imp__zdiv__pos__iff,axiom,
% 0.29/0.46 ! [K: int,I: int] :
% 0.29/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
% 0.29/0.46 = ( ord_less_eq @ int @ K @ I ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pos_imp_zdiv_pos_iff
% 0.29/0.46 thf(fact_179_neg__imp__zdiv__nonneg__iff,axiom,
% 0.29/0.46 ! [B: int,A2: int] :
% 0.29/0.46 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/0.46 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neg_imp_zdiv_nonneg_iff
% 0.29/0.46 thf(fact_180_pos__imp__zdiv__nonneg__iff,axiom,
% 0.29/0.46 ! [B: int,A2: int] :
% 0.29/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/0.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/0.46 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pos_imp_zdiv_nonneg_iff
% 0.29/0.46 thf(fact_181_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.29/0.46 ! [A2: int,B: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/0.46 = ( ( ord_less_eq @ int @ B @ A2 )
% 0.29/0.46 & ( ord_less @ int @ ( zero_zero @ int ) @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonneg1_imp_zdiv_pos_iff
% 0.29/0.46 thf(fact_182_le__num__One__iff,axiom,
% 0.29/0.46 ! [X: num] :
% 0.29/0.46 ( ( ord_less_eq @ num @ X @ one2 )
% 0.29/0.46 = ( X = one2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_num_One_iff
% 0.29/0.46 thf(fact_183_zero__reorient,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero @ A @ ( type @ 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_184_one__reorient,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( one @ A @ ( type @ A ) )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ( one_one @ A )
% 0.29/0.46 = X )
% 0.29/0.46 = ( X
% 0.29/0.46 = ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_reorient
% 0.29/0.46 thf(fact_185_zero__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type @ 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_186_gr__zeroI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type @ 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_187_not__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_zero
% 0.29/0.47 thf(fact_188_gr__implies__not__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [M: A,N: A] :
% 0.29/0.47 ( ( ord_less @ A @ M @ N )
% 0.29/0.47 => ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % gr_implies_not_zero
% 0.29/0.47 thf(fact_189_zero__less__iff__neq__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [N: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.47 = ( N
% 0.29/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_iff_neq_zero
% 0.29/0.47 thf(fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
% 0.29/0.47 ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Nat_Transfer.transfer_nat_int_function_closures(7)
% 0.29/0.47 thf(fact_191_order__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A @ ( type @ A ) )
% 0.29/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_refl
% 0.29/0.47 thf(fact_192_Suc__n__div__2__gt__zero,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Suc_n_div_2_gt_zero
% 0.29/0.47 thf(fact_193_add__right__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_cancel
% 0.29/0.47 thf(fact_194_add__left__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_cancel
% 0.29/0.47 thf(fact_195_add__le__cancel__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_cancel_right
% 0.29/0.47 thf(fact_196_add__le__cancel__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_cancel_left
% 0.29/0.47 thf(fact_197_add_Oleft__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_neutral
% 0.29/0.47 thf(fact_198_add_Oright__neutral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( monoid_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.right_neutral
% 0.29/0.47 thf(fact_199_double__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.29/0.47 = ( zero_zero @ A ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_zero
% 0.29/0.47 thf(fact_200_double__zero__sym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ( zero_zero @ A )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( A2
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_zero_sym
% 0.29/0.47 thf(fact_201_add__cancel__left__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = A2 )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_left_left
% 0.29/0.47 thf(fact_202_add__cancel__left__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = A2 )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_left_right
% 0.29/0.47 thf(fact_203_add__cancel__right__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_right_left
% 0.29/0.47 thf(fact_204_add__cancel__right__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( B
% 0.29/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_cancel_right_right
% 0.29/0.47 thf(fact_205_add__less__cancel__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_cancel_left
% 0.29/0.47 thf(fact_206_add__less__cancel__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_cancel_right
% 0.29/0.47 thf(fact_207_numeral__plus__numeral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ! [M: num,N: num] :
% 0.29/0.47 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/0.47 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % numeral_plus_numeral
% 0.29/0.47 thf(fact_208_add__numeral__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ! [V: num,W: num,Z: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
% 0.29/0.47 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_numeral_left
% 0.29/0.47 thf(fact_209_add__is__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus @ nat @ M @ N )
% 0.29/0.47 = ( zero_zero @ nat ) )
% 0.29/0.47 = ( ( M
% 0.29/0.47 = ( zero_zero @ nat ) )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_is_0
% 0.29/0.47 thf(fact_210_Nat_Oadd__0__right,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 % Nat.add_0_right
% 0.29/0.47 thf(fact_211_nat__add__left__cancel__less,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_add_left_cancel_less
% 0.29/0.47 thf(fact_212_nat__add__left__cancel__le,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 % nat_add_left_cancel_le
% 0.29/0.47 thf(fact_213_add__le__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_same_cancel1
% 0.29/0.47 thf(fact_214_add__le__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_same_cancel2
% 0.29/0.47 thf(fact_215_le__add__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_add_same_cancel1
% 0.29/0.47 thf(fact_216_le__add__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_add_same_cancel2
% 0.29/0.47 thf(fact_217_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.47 thf(fact_218_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.47 thf(fact_219_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.47 thf(fact_220_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.47 thf(fact_221_less__add__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_add_same_cancel2
% 0.29/0.47 thf(fact_222_less__add__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_add_same_cancel1
% 0.29/0.47 thf(fact_223_add__less__same__cancel2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_same_cancel2
% 0.29/0.47 thf(fact_224_add__less__same__cancel1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_less_same_cancel1
% 0.29/0.47 thf(fact_225_add__gr__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 0.29/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/0.47 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_gr_0
% 0.29/0.47 thf(fact_226_numeral__plus__one,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ! [N: num] :
% 0.29/0.47 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.29/0.47 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % numeral_plus_one
% 0.29/0.47 thf(fact_227_one__plus__numeral,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ! [N: num] :
% 0.29/0.47 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.29/0.47 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % one_plus_numeral
% 0.29/0.47 thf(fact_228_div__add__self2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( B
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/0.47 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_add_self2
% 0.29/0.47 thf(fact_229_div__add__self1,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A] :
% 0.29/0.47 ( ( B
% 0.29/0.47 != ( zero_zero @ A ) )
% 0.29/0.47 => ( ( divide_divide @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/0.47 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % div_add_self1
% 0.29/0.47 thf(fact_230_one__add__one,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.29/0.47 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % one_add_one
% 0.29/0.47 thf(fact_231_add__self__div__2,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 % add_self_div_2
% 0.29/0.47 thf(fact_232_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
% 0.29/0.47 ord_less_eq @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ).
% 0.29/0.47
% 0.29/0.47 % Nat_Transfer.transfer_nat_int_function_closures(6)
% 0.29/0.47 thf(fact_233_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.29/0.47 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/0.47
% 0.29/0.47 % Nat_Transfer.transfer_nat_int_function_closures(5)
% 0.29/0.47 thf(fact_234_dbl__def,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/0.47 => ( ( neg_numeral_dbl @ A )
% 0.29/0.47 = ( ^ [X5: A] : ( plus_plus @ A @ X5 @ X5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dbl_def
% 0.29/0.47 thf(fact_235_add__right__imp__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 => ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_imp_eq
% 0.29/0.47 thf(fact_236_add__left__imp__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 => ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_imp_eq
% 0.29/0.47 thf(fact_237_add_Oleft__commute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_commute
% 0.29/0.47 thf(fact_238_add_Ocommute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ( ( plus_plus @ A )
% 0.29/0.47 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.commute
% 0.29/0.47 thf(fact_239_add_Oright__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [B: A,A2: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/0.47 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.right_cancel
% 0.29/0.47 thf(fact_240_add_Oleft__cancel,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( group_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.47 = ( B = C ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.left_cancel
% 0.29/0.47 thf(fact_241_add_Oassoc,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add.assoc
% 0.29/0.47 thf(fact_242_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( I = J )
% 0.29/0.47 & ( K = L ) )
% 0.29/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.29/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(4)
% 0.29/0.47 thf(fact_243_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/0.47 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.47 thf(fact_244_add__le__imp__le__right,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,C: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_imp_le_right
% 0.29/0.47 thf(fact_245_add__le__imp__le__left,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/0.47 => ! [C: A,A2: A,B: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_le_imp_le_left
% 0.29/0.47 thf(fact_246_le__iff__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/0.47 => ( ( ord_less_eq @ A )
% 0.29/0.47 = ( ^ [A4: A,B3: A] :
% 0.29/0.47 ? [C2: A] :
% 0.29/0.47 ( B3
% 0.29/0.47 = ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_iff_add
% 0.29/0.47 thf(fact_247_add__right__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_right_mono
% 0.29/0.47 thf(fact_248_add__left__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_left_mono
% 0.29/0.47 thf(fact_249_add__mono,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.47 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono
% 0.29/0.47 thf(fact_250_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.47 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.47 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.47 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % add_mono_thms_linordered_semiring(1)
% 0.29/0.47
% 0.29/0.47 %----Type constructors (49)
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/0.47 ordere516151231imp_le @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.47 ordere236663937imp_le @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.47 linord1659791738miring @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.47 ordere779506340up_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.29/0.47 linord219039673up_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/0.47 cancel1352612707id_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Divides_Osemiring__numeral__div,axiom,
% 0.29/0.47 semiring_numeral_div @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.47 cancel_semigroup_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Divides_Osemiring__div__parity,axiom,
% 0.29/0.47 semiring_div_parity @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.29/0.47 linordered_semidom @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.29/0.47 ab_semigroup_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.29/0.47 linordered_idom @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 0.29/0.47 semidom_divide @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.29/0.47 semigroup_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Divides_Osemiring__div,axiom,
% 0.29/0.47 semiring_div @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
% 0.29/0.47 zero_less_one @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.29/0.47 semiring_char_0 @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 0.29/0.47 zero_neq_one @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Orderings_Opreorder,axiom,
% 0.29/0.47 preorder @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 0.29/0.47 monoid_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.29/0.47 group_add @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.29/0.47 neg_numeral @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 0.29/0.47 numeral @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.29/0.47 zero @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.29/0.47 one @ int @ ( type @ int ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1,axiom,
% 0.29/0.47 ordere516151231imp_le @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_2,axiom,
% 0.29/0.47 ordere236663937imp_le @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.47 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_3,axiom,
% 0.29/0.47 linord1659791738miring @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_4,axiom,
% 0.29/0.47 ordere779506340up_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_5,axiom,
% 0.29/0.47 cancel1352612707id_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Divides_Osemiring__numeral__div_6,axiom,
% 0.29/0.47 semiring_numeral_div @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_7,axiom,
% 0.29/0.47 cancel_semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Divides_Osemiring__div__parity_8,axiom,
% 0.29/0.47 semiring_div_parity @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 0.29/0.47 linordered_semidom @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_10,axiom,
% 0.29/0.47 ab_semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Osemidom__divide_11,axiom,
% 0.29/0.47 semidom_divide @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add_12,axiom,
% 0.29/0.47 semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Divides_Osemiring__div_13,axiom,
% 0.29/0.47 semiring_div @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Ozero__less__one_14,axiom,
% 0.29/0.47 zero_less_one @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_15,axiom,
% 0.29/0.47 semiring_char_0 @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_16,axiom,
% 0.29/0.47 zero_neq_one @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
% 0.29/0.47 preorder @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__add_18,axiom,
% 0.29/0.47 monoid_add @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Num_Onumeral_19,axiom,
% 0.29/0.47 numeral @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ozero_20,axiom,
% 0.29/0.47 zero @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oone_21,axiom,
% 0.29/0.47 one @ nat @ ( type @ nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Num_Onum___Orderings_Opreorder_22,axiom,
% 0.29/0.47 preorder @ num @ ( type @ num ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.29/0.47 preorder @ $o @ ( type @ $o ) ).
% 0.29/0.47
% 0.29/0.47 %----Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 0.29/1.40 ---- Embedded in THF ---
% 0.29/1.40 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.29/1.40 %%% Generated on Tue Feb 24 06:11:54 EST 2026
% 0.29/1.40 %%% using '$$dhol' embedding, version 1.3.0.
% 0.29/1.40 %%% Logic specification used:
% 0.29/1.40 %%% thf(spec, logic, $$dhol).
% 0.29/1.40
% 0.29/1.40 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2DTF21282.p
% 0.29/1.40 thf(ty_t_Num_Onum, type, num: $tType).
% 0.29/1.40 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.29/1.40 thf(ty_t_Int_Oint, type, int: $tType).
% 0.29/1.40 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 0.29/1.40 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Divides_Osemiring__div, type, semiring_div: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Divides_Osemiring__div__parity, type, semiring_div_parity: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Divides_Osemiring__numeral__div, type, semiring_numeral_div: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere516151231imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.29/1.40 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 0.29/1.40 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 0.29/1.40 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.29/1.40 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 0.29/1.40 thf(sy_c_Num_Onum_OBit0, type, bit0: (num > num)).
% 0.29/1.40 thf(sy_c_Num_Onum_OOne, type, one2: num).
% 0.29/1.40 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 0.29/1.40 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.29/1.40 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.29/1.40 thf(sy_c_Pure_Otype, type, type: (!> [A:$tType]: ((itself @ A)))).
% 0.29/1.40 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 0.29/1.40 thf(sy_v_na____, type, na: nat).
% 0.29/1.40 thf(fact_0_ns, axiom, (~ (((na = (zero_zero @ nat)) | (na = (one_one @ nat)))))).
% 0.29/1.40 thf(fact_1_False, axiom, (~ ((((ord_less_eq @ nat) @ na) @ (one_one @ nat))))).
% 0.29/1.40 thf(fact_2_Divides_Odiv__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 0.29/1.40 thf(fact_3_half__gt__zero, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 0.29/1.40 thf(fact_4_half__gt__zero__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.29/1.40 thf(fact_5_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.29/1.40 thf(fact_6_div__0, axiom, (! [A:$tType]: ((((semiring_div @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_7_divide__eq__0__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A,B:A]: ((((((divide_divide @ A) @ A2) @ B) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B = (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_8_div__by__0, axiom, (! [A:$tType]: ((((semiring_div @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_9_divide__cancel__left, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [C:A,A2:A,B:A]: ((((((divide_divide @ A) @ C) @ A2) = (((divide_divide @ A) @ C) @ B)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 0.29/1.40 thf(fact_10_divide__cancel__right, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: ((((((divide_divide @ A) @ A2) @ C) = (((divide_divide @ A) @ B) @ C)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 0.29/1.40 thf(fact_11_divide__zero, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_12_divide__zero__left, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_13_zdiv__numeral__Bit0, axiom, (! [V:num,W:num]: (((((divide_divide @ int) @ ((numeral_numeral @ int) @ (bit0 @ V))) @ ((numeral_numeral @ int) @ (bit0 @ W))) = (((divide_divide @ int) @ ((numeral_numeral @ int) @ V)) @ ((numeral_numeral @ int) @ W)))))).
% 0.29/1.40 thf(fact_14_num_Oinject_I1_J, axiom, (! [X2:num,Y2:num]: ((((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))).
% 0.29/1.40 thf(fact_15_numeral__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [M:num,N:num]: (((((numeral_numeral @ A) @ M) = ((numeral_numeral @ A) @ N)) = (M = N)))))))).
% 0.29/1.40 thf(fact_16_numeral__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:num,N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less_eq @ num) @ M) @ N)))))))).
% 0.29/1.40 thf(fact_17_numeral__less__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:num,N:num]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ num) @ M) @ N)))))))).
% 0.29/1.40 thf(fact_18_division__ring__divide__zero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_19_divide__1, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 0.29/1.40 thf(fact_20_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.29/1.40 thf(fact_21_one__eq__numeral__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [N:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ N)) = (one2 = N)))))))).
% 0.29/1.40 thf(fact_22_numeral__eq__one__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [N:num]: (((((numeral_numeral @ A) @ N) = (one_one @ A)) = (N = one2)))))))).
% 0.29/1.40 thf(fact_23_zero__eq__1__divide__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_24_one__divide__eq__0__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_25_eq__divide__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: ((((one_one @ A) = (((divide_divide @ A) @ B) @ A2)) = ((A2 != (zero_zero @ A)) & (A2 = B))))))))).
% 0.29/1.40 thf(fact_26_divide__eq__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: ((((((divide_divide @ A) @ B) @ A2) = (one_one @ A)) = ((A2 != (zero_zero @ A)) & (A2 = B))))))))).
% 0.29/1.40 thf(fact_27_divide__self__if, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (! [A2:A]: ((((A2 = (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (zero_zero @ A))) & ((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A)))))))))).
% 0.29/1.40 thf(fact_28_divide__self, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))).
% 0.29/1.40 thf(fact_29_one__eq__divide__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A,B:A]: ((((one_one @ A) = (((divide_divide @ A) @ A2) @ B)) = ((B != (zero_zero @ A)) & (A2 = B))))))))).
% 0.29/1.40 thf(fact_30_div__self, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))).
% 0.29/1.40 thf(fact_31_divide__eq__1__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A,B:A]: ((((((divide_divide @ A) @ A2) @ B) = (one_one @ A)) = ((B != (zero_zero @ A)) & (A2 = B))))))))).
% 0.29/1.40 thf(fact_32_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 0.29/1.40 thf(fact_33_numeral__le__one__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = (((ord_less_eq @ num) @ N) @ one2)))))))).
% 0.29/1.40 thf(fact_34_zero__le__divide__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.29/1.40 thf(fact_35_divide__le__0__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_36_one__less__numeral__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: (((((ord_less @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ num) @ one2) @ N)))))))).
% 0.29/1.40 thf(fact_37_zero__less__divide__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.29/1.40 thf(fact_38_less__divide__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less @ A) @ A2) @ B))))))))).
% 0.29/1.40 thf(fact_39_less__divide__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less @ A) @ B) @ A2))))))))).
% 0.29/1.40 thf(fact_40_divide__less__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ B) @ A2))))))))).
% 0.29/1.40 thf(fact_41_divide__less__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ A2) @ B))))))))).
% 0.29/1.40 thf(fact_42_divide__less__0__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_43_le__divide__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ A2) @ B))))))))).
% 0.29/1.40 thf(fact_44_le__divide__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ B) @ A2))))))))).
% 0.29/1.40 thf(fact_45_divide__le__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ B) @ A2))))))))).
% 0.29/1.40 thf(fact_46_divide__le__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A2) @ B))))))))).
% 0.29/1.40 thf(fact_47_one__div__two__eq__zero, axiom, (! [A:$tType]: ((((semiring_div_parity @ A) @ (type @ A)) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 0.29/1.40 thf(fact_48_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 0.29/1.40 thf(fact_49_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 0.29/1.40 thf(fact_50_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 0.29/1.40 thf(fact_51_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.29/1.40 thf(fact_52_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 0.29/1.40 thf(fact_53_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 0.29/1.40 thf(fact_54_not__one__le__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_55_zero__le__one, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.29/1.40 thf(fact_56_le__numeral__extra_I1_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.29/1.40 thf(fact_57_le__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_58_one__le__numeral, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.29/1.40 thf(fact_59_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 0.29/1.40 thf(fact_60_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 0.29/1.40 thf(fact_61_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 0.29/1.40 thf(fact_62_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 0.29/1.40 thf(fact_63_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 0.29/1.40 thf(fact_64_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.29/1.40 thf(fact_65_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 0.29/1.40 thf(fact_66_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 0.29/1.40 thf(fact_67_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 0.29/1.40 thf(fact_68_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 0.29/1.40 thf(fact_69_div__le__mono, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N) @ K)))))).
% 0.29/1.40 thf(fact_70_not__numeral__le__zero, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_71_zero__le__numeral, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.29/1.40 thf(fact_72_divide__right__mono__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ C)) @ (((divide_divide @ A) @ A2) @ C)))))))))).
% 0.29/1.40 thf(fact_73_divide__nonpos__nonpos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_74_divide__nonpos__nonneg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_75_divide__nonneg__nonpos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_76_divide__nonneg__nonneg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_77_zero__le__divide__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))))))))))).
% 0.29/1.40 thf(fact_78_divide__right__mono, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 0.29/1.40 thf(fact_79_divide__le__0__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))))).
% 0.29/1.40 thf(fact_80_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((~ ((P @ (zero_zero @ nat)))) => ((P @ N) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ K2)))))))))).
% 0.29/1.40 thf(fact_81_zero__neq__one, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type @ A)) => ((zero_zero @ A) != (one_one @ A)))))).
% 0.29/1.40 thf(fact_82_less__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 0.29/1.40 thf(fact_83_div__by__1, axiom, (! [A:$tType]: ((((semiring_div @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 0.29/1.40 thf(fact_84_le__divide__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ A2) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B) @ A2)))))))))).
% 0.29/1.40 thf(fact_85_divide__le__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ A2) @ B)) | (A2 = (zero_zero @ A))))))))))).
% 0.29/1.40 thf(fact_86_divide__nonpos__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_87_divide__nonpos__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_88_divide__nonneg__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_89_divide__nonneg__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_90_semiring__numeral__div__class_Odiv__positive, axiom, (! [A:$tType]: ((((semiring_numeral_div @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)))))))))).
% 0.29/1.40 thf(fact_91_divide__le__cancel, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ A2) @ B)) & ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B) @ A2)))))))))).
% 0.29/1.40 thf(fact_92_semiring__numeral__div__class_Odiv__less, axiom, (! [A:$tType]: ((((semiring_numeral_div @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ B) => ((((divide_divide @ A) @ A2) @ B) = (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_93_frac__less2, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A,W:A,Z:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 0.29/1.40 thf(fact_94_frac__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 0.29/1.40 thf(fact_95_frac__le, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 0.29/1.40 thf(fact_96_Divides_Odiv__positive, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ M) @ N))))))).
% 0.29/1.40 thf(fact_97_div__le__mono2, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ K) @ N)) @ (((divide_divide @ nat) @ K) @ M))))))).
% 0.29/1.40 thf(fact_98_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.29/1.40 thf(fact_99_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 0.29/1.40 thf(fact_100_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X3:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X3))))))))).
% 0.29/1.40 thf(fact_101_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V2:(A > nat),X:A]: (((! [X4:A]: (((~ ((P @ X4))) => (? [Y4:A]: (((((ord_less @ nat) @ (V2 @ Y4)) @ (V2 @ X4)) & (~ ((P @ Y4))))))))) => (P @ X))))).
% 0.29/1.40 thf(fact_102_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y4:A]: (((((ord_less @ nat) @ (F @ Y4)) @ (F @ X4)) => (P @ Y4)))) => (P @ X4)))) => (P @ A2))))).
% 0.29/1.40 thf(fact_103_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 0.29/1.40 thf(fact_104_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P @ M3))))))))) => (P @ N))))).
% 0.29/1.40 thf(fact_105_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P @ M3)))) => (P @ N3)))) => (P @ N))))).
% 0.29/1.40 thf(fact_106_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.29/1.40 thf(fact_107_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X4:A]: (((! [Y4:A]: (((((ord_less @ nat) @ (F @ Y4)) @ (F @ X4)) => (P @ Y4)))) => (P @ X4)))) => (P @ A2))))).
% 0.29/1.40 thf(fact_108_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 0.29/1.40 thf(fact_109_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 0.29/1.40 thf(fact_110_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 0.29/1.40 thf(fact_111_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 0.29/1.40 thf(fact_112_not__one__less__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_113_zero__less__one, axiom, (! [A:$tType]: ((((zero_less_one @ A) @ (type @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.29/1.40 thf(fact_114_less__numeral__extra_I1_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 0.29/1.40 thf(fact_115_less__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_116_not__numeral__less__one, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)))))))))).
% 0.29/1.40 thf(fact_117_numeral__One, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => (((numeral_numeral @ A) @ one2) = (one_one @ A)))))).
% 0.29/1.40 thf(fact_118_right__inverse__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (! [B:A,A2:A]: (((B != (zero_zero @ A)) => (((((divide_divide @ A) @ A2) @ B) = (one_one @ A)) = (A2 = B))))))))).
% 0.29/1.40 thf(fact_119_numerals_I1_J, axiom, (((numeral_numeral @ nat) @ one2) = (one_one @ nat))).
% 0.29/1.40 thf(fact_120_div__eq__dividend__iff, axiom, (! [A2:nat,B:nat]: (((A2 != (zero_zero @ nat)) => (((((divide_divide @ nat) @ A2) @ B) = A2) = (B = (one_one @ nat))))))).
% 0.29/1.40 thf(fact_121_less__divide__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ A2) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B) @ A2)))))))))).
% 0.29/1.40 thf(fact_122_divide__less__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ A2) @ B)) | (A2 = (zero_zero @ A))))))))))).
% 0.29/1.40 thf(fact_123_div__less__dividend, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))))).
% 0.29/1.40 thf(fact_124_less__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 0.29/1.40 thf(fact_125_zero__neq__numeral, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [N:num]: (((zero_zero @ A) != ((numeral_numeral @ A) @ N)))))))).
% 0.29/1.40 thf(fact_126_num_Odistinct_I1_J, axiom, (! [X2:num]: ((one2 != (bit0 @ X2))))).
% 0.29/1.40 thf(fact_127_infinite__descent0__measure, axiom, (! [A:$tType,V2:(A > nat),P:(A > $o),X:A]: (((! [X4:A]: ((((V2 @ X4) = (zero_zero @ nat)) => (P @ X4)))) => ((! [X4:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V2 @ X4)) => ((~ ((P @ X4))) => (? [Y4:A]: (((((ord_less @ nat) @ (V2 @ Y4)) @ (V2 @ X4)) & (~ ((P @ Y4)))))))))) => (P @ X)))))).
% 0.29/1.40 thf(fact_128_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.29/1.40 thf(fact_129_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((~ ((P @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P @ M3)))))))))) => (P @ N)))))).
% 0.29/1.40 thf(fact_130_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 0.29/1.40 thf(fact_131_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.29/1.40 thf(fact_132_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 0.29/1.40 thf(fact_133_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 0.29/1.40 thf(fact_134_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 0.29/1.40 thf(fact_135_not__numeral__less__zero, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_136_zero__less__numeral, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:num]: ((((ord_less @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 0.29/1.40 thf(fact_137_divide__strict__right__mono__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 0.29/1.40 thf(fact_138_divide__strict__right__mono, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 0.29/1.40 thf(fact_139_zero__less__divide__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B) @ (zero_zero @ A))))))))))).
% 0.29/1.40 thf(fact_140_divide__less__cancel, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ A2) @ B)) & (((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ B) @ A2)) & (C != (zero_zero @ A))))))))))).
% 0.29/1.40 thf(fact_141_divide__less__0__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))))).
% 0.29/1.40 thf(fact_142_divide__pos__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_143_divide__pos__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_144_divide__neg__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_145_divide__neg__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 0.29/1.40 thf(fact_146_divide__numeral__1, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ one2)) = A2))))))).
% 0.29/1.40 thf(fact_147_div__eq__0__iff, axiom, (! [A2:nat,B:nat]: ((((((divide_divide @ nat) @ A2) @ B) = (zero_zero @ nat)) = ((((ord_less @ nat) @ A2) @ B) | (B = (zero_zero @ nat))))))).
% 0.29/1.40 thf(fact_148_div__2__gt__zero, axiom, (! [N:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.29/1.40 thf(fact_149_zero__not__eq__two, axiom, (! [A:$tType]: ((((semiring_div_parity @ A) @ (type @ A)) => ((zero_zero @ A) != ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.29/1.40 thf(fact_150_semiring__norm_I76_J, axiom, (! [N:num]: ((((ord_less @ num) @ one2) @ (bit0 @ N))))).
% 0.29/1.40 thf(fact_151_semiring__norm_I83_J, axiom, (! [N:num]: ((one2 != (bit0 @ N))))).
% 0.29/1.40 thf(fact_152_semiring__norm_I85_J, axiom, (! [M:num]: (((bit0 @ M) != one2)))).
% 0.29/1.40 thf(fact_153_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_154_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_155_dbl__simps_I3_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type @ A)) => (((neg_numeral_dbl @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.29/1.40 thf(fact_156_semiring__norm_I75_J, axiom, (! [M:num]: ((~ ((((ord_less @ num) @ M) @ one2)))))).
% 0.29/1.40 thf(fact_157_semiring__norm_I78_J, axiom, (! [M:num,N:num]: (((((ord_less @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (((ord_less @ num) @ M) @ N))))).
% 0.29/1.40 thf(fact_158_semiring__norm_I87_J, axiom, (! [M:num,N:num]: ((((bit0 @ M) = (bit0 @ N)) = (M = N))))).
% 0.29/1.40 thf(fact_159_semiring__norm_I71_J, axiom, (! [M:num,N:num]: (((((ord_less_eq @ num) @ (bit0 @ M)) @ (bit0 @ N)) = (((ord_less_eq @ num) @ M) @ N))))).
% 0.29/1.40 thf(fact_160_semiring__norm_I68_J, axiom, (! [N:num]: ((((ord_less_eq @ num) @ one2) @ N)))).
% 0.29/1.40 thf(fact_161_dbl__simps_I2_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type @ A)) => (((neg_numeral_dbl @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 0.29/1.40 thf(fact_162_semiring__norm_I69_J, axiom, (! [M:num]: ((~ ((((ord_less_eq @ num) @ (bit0 @ M)) @ one2)))))).
% 0.29/1.40 thf(fact_163_dbl__simps_I5_J, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type @ A)) => (! [K:num]: ((((neg_numeral_dbl @ A) @ ((numeral_numeral @ A) @ K)) = ((numeral_numeral @ A) @ (bit0 @ K))))))))).
% 0.29/1.40 thf(fact_164_Divides_Otransfer__nat__int__function__closures_I1_J, axiom, (! [X:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ X) @ Y))))))).
% 0.29/1.40 thf(fact_165_zdiv__mono1, axiom, (! [A2:int,A3:int,B:int]: (((((ord_less_eq @ int) @ A2) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (((divide_divide @ int) @ A3) @ B))))))).
% 0.29/1.40 thf(fact_166_zdiv__mono2, axiom, (! [A2:int,B2:int,B:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ B2) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (((divide_divide @ int) @ A2) @ B2)))))))).
% 0.29/1.40 thf(fact_167_zdiv__eq__0__iff, axiom, (! [I:int,K:int]: ((((((divide_divide @ int) @ I) @ K) = (zero_zero @ int)) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) & (((ord_less @ int) @ I) @ K)) | ((((ord_less_eq @ int) @ I) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I)))))))).
% 0.29/1.40 thf(fact_168_zdiv__mono1__neg, axiom, (! [A2:int,A3:int,B:int]: (((((ord_less_eq @ int) @ A2) @ A3) => ((((ord_less @ int) @ B) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A3) @ B)) @ (((divide_divide @ int) @ A2) @ B))))))).
% 0.29/1.40 thf(fact_169_zdiv__mono2__neg, axiom, (! [A2:int,B2:int,B:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ B2) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (((divide_divide @ int) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_170_div__neg__pos__less0, axiom, (! [A2:int,B:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_171_int__div__less__self, axiom, (! [X:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ X) => ((((ord_less @ int) @ (one_one @ int)) @ K) => (((ord_less @ int) @ (((divide_divide @ int) @ X) @ K)) @ X)))))).
% 0.29/1.40 thf(fact_172_div__nonneg__neg__le0, axiom, (! [A2:int,B:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ B) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_173_div__nonpos__pos__le0, axiom, (! [A2:int,B:int]: (((((ord_less_eq @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_174_div__neg__neg__trivial, axiom, (! [A2:int,B:int]: (((((ord_less_eq @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ B) @ A2) => ((((divide_divide @ int) @ A2) @ B) = (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_175_div__pos__pos__trivial, axiom, (! [A2:int,B:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ A2) @ B) => ((((divide_divide @ int) @ A2) @ B) = (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_176_neg__imp__zdiv__neg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ B) @ (zero_zero @ int)) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int)) = (((ord_less @ int) @ (zero_zero @ int)) @ A2)))))).
% 0.29/1.40 thf(fact_177_pos__imp__zdiv__neg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int)) = (((ord_less @ int) @ A2) @ (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_178_pos__imp__zdiv__pos__iff, axiom, (! [K:int,I:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ I) @ K)) = (((ord_less_eq @ int) @ K) @ I)))))).
% 0.29/1.40 thf(fact_179_neg__imp__zdiv__nonneg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ B) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B)) = (((ord_less_eq @ int) @ A2) @ (zero_zero @ int))))))).
% 0.29/1.40 thf(fact_180_pos__imp__zdiv__nonneg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A2)))))).
% 0.29/1.40 thf(fact_181_nonneg1__imp__zdiv__pos__iff, axiom, (! [A2:int,B:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B)) = ((((ord_less_eq @ int) @ B) @ A2) & (((ord_less @ int) @ (zero_zero @ int)) @ B))))))).
% 0.29/1.40 thf(fact_182_le__num__One__iff, axiom, (! [X:num]: (((((ord_less_eq @ num) @ X) @ one2) = (X = one2))))).
% 0.29/1.40 thf(fact_183_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type @ A)) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_184_one__reorient, axiom, (! [A:$tType]: ((((one @ A) @ (type @ A)) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 0.29/1.40 thf(fact_185_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 0.29/1.40 thf(fact_186_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 0.29/1.40 thf(fact_187_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 0.29/1.40 thf(fact_188_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_189_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ ((numeral_numeral @ int) @ (bit0 @ one2)))).
% 0.29/1.40 thf(fact_191_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.29/1.40 thf(fact_192_Suc__n__div__2__gt__zero, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))).
% 0.29/1.40 thf(fact_193_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 0.29/1.40 thf(fact_194_add__left__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 0.29/1.40 thf(fact_195_add__le__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_196_add__le__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_197_add_Oleft__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))).
% 0.29/1.40 thf(fact_198_add_Oright__neutral, axiom, (! [A:$tType]: ((((monoid_add @ A) @ (type @ A)) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 0.29/1.40 thf(fact_199_double__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_200_double__zero__sym, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_201_add__cancel__left__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [B:A,A2:A]: ((((((plus_plus @ A) @ B) @ A2) = A2) = (B = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_202_add__cancel__left__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A2:A,B:A]: ((((((plus_plus @ A) @ A2) @ B) = A2) = (B = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_203_add__cancel__right__left, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ B) @ A2)) = (B = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_204_add__cancel__right__right, axiom, (! [A:$tType]: ((((cancel1352612707id_add @ A) @ (type @ A)) => (! [A2:A,B:A]: (((A2 = (((plus_plus @ A) @ A2) @ B)) = (B = (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_205_add__less__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_206_add__less__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_207_numeral__plus__numeral, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => (! [M:num,N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ N))))))))).
% 0.29/1.40 thf(fact_208_add__numeral__left, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => (! [V:num,W:num,Z:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W)) @ Z)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V) @ W))) @ Z)))))))).
% 0.29/1.40 thf(fact_209_add__is__0, axiom, (! [M:nat,N:nat]: ((((((plus_plus @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 0.29/1.40 thf(fact_210_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 0.29/1.40 thf(fact_211_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 0.29/1.40 thf(fact_212_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 0.29/1.40 thf(fact_213_add__le__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_214_add__le__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_215_le__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 0.29/1.40 thf(fact_216_le__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))).
% 0.29/1.40 thf(fact_217_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_218_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.29/1.40 thf(fact_219_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 0.29/1.40 thf(fact_220_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_221_less__add__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 0.29/1.40 thf(fact_222_less__add__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B)) = (((ord_less @ A) @ (zero_zero @ A)) @ B)))))))).
% 0.29/1.40 thf(fact_223_add__less__same__cancel2, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_224_add__less__same__cancel1, axiom, (! [A:$tType]: ((((ordere516151231imp_le @ A) @ (type @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 0.29/1.40 thf(fact_225_add__gr__0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 0.29/1.40 thf(fact_226_numeral__plus__one, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => (! [N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ N) @ one2))))))))).
% 0.29/1.40 thf(fact_227_one__plus__numeral, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => (! [N:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ N))))))))).
% 0.29/1.40 thf(fact_228_div__add__self2, axiom, (! [A:$tType]: ((((semiring_div @ A) @ (type @ A)) => (! [B:A,A2:A]: (((B != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B)) @ B) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (one_one @ A)))))))))).
% 0.29/1.40 thf(fact_229_div__add__self1, axiom, (! [A:$tType]: ((((semiring_div @ A) @ (type @ A)) => (! [B:A,A2:A]: (((B != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B) @ A2)) @ B) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (one_one @ A)))))))))).
% 0.29/1.40 thf(fact_230_one__add__one, axiom, (! [A:$tType]: ((((numeral @ A) @ (type @ A)) => ((((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 0.29/1.40 thf(fact_231_add__self__div__2, axiom, (! [M:nat]: (((((divide_divide @ nat) @ (((plus_plus @ nat) @ M) @ M)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = M)))).
% 0.29/1.40 thf(fact_232_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (one_one @ int))).
% 0.29/1.40 thf(fact_233_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 0.29/1.40 thf(fact_234_dbl__def, axiom, (! [A:$tType]: ((((neg_numeral @ A) @ (type @ A)) => ((neg_numeral_dbl @ A) = (^ [X5:A]: ((((plus_plus @ A) @ X5) @ X5)))))))).
% 0.29/1.40 thf(fact_235_add__right__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B = C)))))))).
% 0.29/1.40 thf(fact_236_add__left__imp__eq, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) => (B = C)))))))).
% 0.29/1.40 thf(fact_237_add_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => (! [B:A,A2:A,C:A]: (((((plus_plus @ A) @ B) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 0.29/1.40 thf(fact_238_add_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => ((plus_plus @ A) = (^ [A4:A,B3:A]: ((((plus_plus @ A) @ B3) @ A4)))))))).
% 0.29/1.40 thf(fact_239_add_Oright__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 0.29/1.40 thf(fact_240_add_Oleft__cancel, axiom, (! [A:$tType]: ((((group_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 0.29/1.40 thf(fact_241_add_Oassoc, axiom, (! [A:$tType]: ((((semigroup_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 0.29/1.40 thf(fact_242_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 0.29/1.40 thf(fact_243_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B) @ C))))))))).
% 0.29/1.40 thf(fact_244_add__le__imp__le__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_245_add__le__imp__le__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 0.29/1.40 thf(fact_246_le__iff__add, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B3:A]: ((? [C2:A]: ((B3 = (((plus_plus @ A) @ A4) @ C2))))))))))).
% 0.29/1.40 thf(fact_247_add__right__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C))))))))).
% 0.29/1.40 thf(fact_248_add__left__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B))))))))).
% 0.29/1.40 thf(fact_249_add__mono, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [A2:A,B:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ D)))))))))).
% 0.29/1.40 thf(fact_250_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: ((((ordere779506340up_add @ A) @ (type @ A)) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, ((ordere516151231imp_le @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, ((linord1659791738miring @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, ((ordere779506340up_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, ((linord219039673up_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, ((cancel1352612707id_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Divides_Osemiring__numeral__div, axiom, ((semiring_numeral_div @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Divides_Osemiring__div__parity, axiom, ((semiring_div_parity @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, ((ab_semigroup_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, ((linordered_idom @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, ((semidom_divide @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, ((semigroup_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Divides_Osemiring__div, axiom, ((semiring_div @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Ozero__less__one, axiom, ((zero_less_one @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, ((semiring_char_0 @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Rings_Ozero__neq__one, axiom, ((zero_neq_one @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Orderings_Opreorder, axiom, ((preorder @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, ((monoid_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, ((group_add @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, ((neg_numeral @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Num_Onumeral, axiom, ((numeral @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Ozero, axiom, ((zero @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Int_Oint___Groups_Oone, axiom, ((one @ int) @ (type @ int))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1, axiom, ((ordere516151231imp_le @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_2, axiom, ((ordere236663937imp_le @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_3, axiom, ((linord1659791738miring @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_4, axiom, ((ordere779506340up_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_5, axiom, ((cancel1352612707id_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Divides_Osemiring__numeral__div_6, axiom, ((semiring_numeral_div @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_7, axiom, ((cancel_semigroup_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Divides_Osemiring__div__parity_8, axiom, ((semiring_div_parity @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9, axiom, ((linordered_semidom @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_10, axiom, ((ab_semigroup_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Rings_Osemidom__divide_11, axiom, ((semidom_divide @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Osemigroup__add_12, axiom, ((semigroup_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Divides_Osemiring__div_13, axiom, ((semiring_div @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Rings_Ozero__less__one_14, axiom, ((zero_less_one @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_15, axiom, ((semiring_char_0 @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_16, axiom, ((zero_neq_one @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Orderings_Opreorder_17, axiom, ((preorder @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Omonoid__add_18, axiom, ((monoid_add @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Num_Onumeral_19, axiom, ((numeral @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Ozero_20, axiom, ((zero @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Nat_Onat___Groups_Oone_21, axiom, ((one @ nat) @ (type @ nat))).
% 0.29/1.40 thf(tcon_Num_Onum___Orderings_Opreorder_22, axiom, ((preorder @ num) @ (type @ num))).
% 0.29/1.40 thf(tcon_HOL_Obool___Orderings_Opreorder_23, axiom, ((preorder @ $o) @ (type @ $o))).
% 0.29/1.40 thf(conj_0, conjecture, (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ na) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))).
% 0.29/1.40 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2DTF21282.p
% 0.29/1.40 ------------------------
% 0.29/1.42 ---- Cleaned THF ---
% 0.29/1.43 thf(ty_t_Num_Onum,type,
% 0.29/1.43 num: $tType ).
% 0.29/1.43
% 0.29/1.43 thf(ty_t_Nat_Onat,type,
% 0.29/1.43 nat: $tType ).
% 0.29/1.43
% 0.29/1.43 thf(ty_t_Int_Oint,type,
% 0.29/1.43 int: $tType ).
% 0.29/1.43
% 0.29/1.43 thf(ty_t_itself,type,
% 0.29/1.43 itself: $tType > $tType ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Oone,type,
% 0.29/1.43 one:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Ozero,type,
% 0.29/1.43 zero:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Num_Onumeral,type,
% 0.29/1.43 numeral:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Fields_Ofield,type,
% 0.29/1.43 field:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Num_Oneg__numeral,type,
% 0.29/1.43 neg_numeral:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/1.43 group_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Omonoid__add,type,
% 0.29/1.43 monoid_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/1.43 preorder:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.29/1.43 zero_neq_one:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.29/1.43 semiring_char_0:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.29/1.43 zero_less_one:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Divides_Osemiring__div,type,
% 0.29/1.43 semiring_div:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Fields_Odivision__ring,type,
% 0.29/1.43 division_ring:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/1.43 semigroup_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.29/1.43 semidom_divide:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/1.43 linordered_idom:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Fields_Olinordered__field,type,
% 0.29/1.43 linordered_field:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/1.43 ab_semigroup_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.29/1.43 linordered_semidom:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Divides_Osemiring__div__parity,type,
% 0.29/1.43 semiring_div_parity:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/1.43 cancel_semigroup_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Divides_Osemiring__numeral__div,type,
% 0.29/1.43 semiring_numeral_div:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.29/1.43 cancel1352612707id_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.29/1.43 linord219039673up_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/1.43 ordere779506340up_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.29/1.43 linord1659791738miring:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/1.43 canoni770627133id_add:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/1.43 ordere236663937imp_le:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 0.29/1.43 ordere516151231imp_le:
% 0.29/1.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.29/1.43 one_one:
% 0.29/1.43 !>[A: $tType] : A ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/1.43 plus_plus:
% 0.29/1.43 !>[A: $tType] : ( A > A > A ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/1.43 zero_zero:
% 0.29/1.43 !>[A: $tType] : A ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.29/1.43 neg_numeral_dbl:
% 0.29/1.43 !>[A: $tType] : ( A > A ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Num_Onum_OBit0,type,
% 0.29/1.43 bit0: num > num ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Num_Onum_OOne,type,
% 0.29/1.43 one2: num ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.29/1.43 numeral_numeral:
% 0.29/1.43 !>[A: $tType] : ( num > A ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/1.43 ord_less:
% 0.29/1.43 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/1.43 ord_less_eq:
% 0.29/1.43 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Pure_Otype,type,
% 0.29/1.43 type:
% 0.29/1.43 !>[A: $tType] : ( itself @ A ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.29/1.43 divide_divide:
% 0.29/1.43 !>[A: $tType] : ( A > A > A ) ).
% 0.29/1.43
% 0.29/1.43 thf(sy_v_na____,type,
% 0.29/1.43 na: nat ).
% 0.29/1.43
% 0.29/1.43 thf(fact_0_ns,axiom,
% 0.29/1.43 ~ ( ( na
% 0.29/1.43 = ( zero_zero @ nat ) )
% 0.29/1.43 | ( na
% 0.29/1.43 = ( one_one @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_1_False,axiom,
% 0.29/1.43 ~ ( ord_less_eq @ nat @ na @ ( one_one @ nat ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_2_Divides_Odiv__less,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M @ N )
% 0.29/1.43 => ( ( divide_divide @ nat @ M @ N )
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_3_half__gt__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_4_half__gt__zero__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.29/1.43 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_5_neq0__conv,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( N
% 0.29/1.43 != ( zero_zero @ nat ) )
% 0.29/1.43 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_6_div__0,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_7_divide__eq__0__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 = ( ( A2
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 | ( B
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_8_div__by__0,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_9_divide__cancel__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [C: A,A2: A,B: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ C @ A2 )
% 0.29/1.43 = ( divide_divide @ A @ C @ B ) )
% 0.29/1.43 = ( ( C
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 | ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_10_divide__cancel__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ A2 @ C )
% 0.29/1.43 = ( divide_divide @ A @ B @ C ) )
% 0.29/1.43 = ( ( C
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 | ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_11_divide__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_12_divide__zero__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_13_zdiv__numeral__Bit0,axiom,
% 0.29/1.43 ! [V: num,W: num] :
% 0.29/1.43 ( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
% 0.29/1.43 = ( divide_divide @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_14_num_Oinject_I1_J,axiom,
% 0.29/1.43 ! [X2: num,Y2: num] :
% 0.29/1.43 ( ( ( bit0 @ X2 )
% 0.29/1.43 = ( bit0 @ Y2 ) )
% 0.29/1.43 = ( X2 = Y2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_15_numeral__eq__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/1.43 => ! [M: num,N: num] :
% 0.29/1.43 ( ( ( numeral_numeral @ A @ M )
% 0.29/1.43 = ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( M = N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_16_numeral__le__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [M: num,N: num] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( ord_less_eq @ num @ M @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_17_numeral__less__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [M: num,N: num] :
% 0.29/1.43 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( ord_less @ num @ M @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_18_division__ring__divide__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_19_divide__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.29/1.43 = A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_20_le0,axiom,
% 0.29/1.43 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_21_one__eq__numeral__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( ( one_one @ A )
% 0.29/1.43 = ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( one2 = N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_22_numeral__eq__one__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( ( numeral_numeral @ A @ N )
% 0.29/1.43 = ( one_one @ A ) )
% 0.29/1.43 = ( N = one2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_23_zero__eq__1__divide__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ( zero_zero @ A )
% 0.29/1.43 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/1.43 = ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_24_one__divide__eq__0__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 = ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_25_eq__divide__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ( one_one @ A )
% 0.29/1.43 = ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ( A2
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 & ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_26_divide__eq__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ B @ A2 )
% 0.29/1.43 = ( one_one @ A ) )
% 0.29/1.43 = ( ( A2
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 & ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_27_divide__self__if,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ( A2
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/1.43 = ( zero_zero @ A ) ) )
% 0.29/1.43 & ( ( A2
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/1.43 = ( one_one @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_28_divide__self,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( A2
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/1.43 = ( one_one @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_29_one__eq__divide__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ( one_one @ A )
% 0.29/1.43 = ( divide_divide @ A @ A2 @ B ) )
% 0.29/1.43 = ( ( B
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 & ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_30_div__self,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semidom_divide @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( A2
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.29/1.43 = ( one_one @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_31_divide__eq__1__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/1.43 = ( one_one @ A ) )
% 0.29/1.43 = ( ( B
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 & ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_32_less__one,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.29/1.43 = ( N
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_33_numeral__le__one__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_34_zero__le__divide__1__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/1.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_35_divide__le__0__1__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_36_one__less__numeral__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_37_zero__less__divide__1__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.29/1.43 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_38_less__divide__eq__1__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_39_less__divide__eq__1__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_40_divide__less__eq__1__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ord_less @ A @ B @ A2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_41_divide__less__eq__1__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_42_divide__less__0__1__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_43_le__divide__eq__1__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_44_le__divide__eq__1__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_45_divide__le__eq__1__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ord_less_eq @ A @ B @ A2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_46_divide__le__eq__1__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_47_one__div__two__eq__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div_parity @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_48_le__numeral__extra_I4_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_49_le__refl,axiom,
% 0.29/1.43 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_50_le__trans,axiom,
% 0.29/1.43 ! [I: nat,J: nat,K: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/1.43 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/1.43 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_51_eq__imp__le,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( M = N )
% 0.29/1.43 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_52_le__antisym,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/1.43 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/1.43 => ( M = N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_53_nat__le__linear,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/1.43 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_54_not__one__le__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_55_zero__le__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_56_le__numeral__extra_I1_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_57_le__numeral__extra_I2_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_58_one__le__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_59_le__numeral__extra_I3_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_60_le__0__eq,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.29/1.43 = ( N
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_61_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/1.43 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_62_less__mono__imp__le__mono,axiom,
% 0.29/1.43 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/1.43 ( ! [I2: nat,J2: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/1.43 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/1.43 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/1.43 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_63_le__neq__implies__less,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/1.43 => ( ( M != N )
% 0.29/1.43 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_64_less__or__eq__imp__le,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ( ord_less @ nat @ M @ N )
% 0.29/1.43 | ( M = N ) )
% 0.29/1.43 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_65_le__eq__less__or__eq,axiom,
% 0.29/1.43 ( ( ord_less_eq @ nat )
% 0.29/1.43 = ( ^ [M2: nat,N2: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/1.43 | ( M2 = N2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_66_less__imp__le__nat,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M @ N )
% 0.29/1.43 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_67_nat__less__le,axiom,
% 0.29/1.43 ( ( ord_less @ nat )
% 0.29/1.43 = ( ^ [M2: nat,N2: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/1.43 & ( M2 != N2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_68_div__le__dividend,axiom,
% 0.29/1.43 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_69_div__le__mono,axiom,
% 0.29/1.43 ! [M: nat,N: nat,K: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/1.43 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_70_not__numeral__le__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_71_zero__le__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_72_divide__right__mono__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/1.43 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_73_divide__nonpos__nonpos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_74_divide__nonpos__nonneg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_75_divide__nonneg__nonpos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_76_divide__nonneg__nonneg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_77_zero__le__divide__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.29/1.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
% 0.29/1.43 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_78_divide__right__mono,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/1.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_79_divide__le__0__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
% 0.29/1.43 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_80_ex__least__nat__le,axiom,
% 0.29/1.43 ! [P: nat > $o,N: nat] :
% 0.29/1.43 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/1.43 => ( ( P @ N )
% 0.29/1.43 => ? [K2: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.29/1.43 & ! [I3: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ I3 @ K2 )
% 0.29/1.43 => ~ ( P @ I3 ) )
% 0.29/1.43 & ( P @ K2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_81_zero__neq__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( zero_neq_one @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( zero_zero @ A )
% 0.29/1.43 != ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_82_less__numeral__extra_I4_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_83_div__by__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.29/1.43 = A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_84_le__divide__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_85_divide__le__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less_eq @ A @ B @ A2 ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/1.43 | ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_86_divide__nonpos__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_87_divide__nonpos__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_88_divide__nonneg__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_89_divide__nonneg__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_90_semiring__numeral__div__class_Odiv__positive,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_numeral_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 0.29/1.43 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_91_divide__le__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.29/1.43 => ( ord_less_eq @ A @ A2 @ B ) )
% 0.29/1.43 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_92_semiring__numeral__div__class_Odiv__less,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_numeral_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ A @ A2 @ B )
% 0.29/1.43 => ( ( divide_divide @ A @ A2 @ B )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_93_frac__less2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/1.43 => ( ( ord_less @ A @ W @ Z )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_94_frac__less,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less @ A @ X @ Y )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/1.43 => ( ( ord_less_eq @ A @ W @ Z )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_95_frac__le,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A,W: A,Z: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.29/1.43 => ( ( ord_less_eq @ A @ W @ Z )
% 0.29/1.43 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_96_Divides_Odiv__positive,axiom,
% 0.29/1.43 ! [N: nat,M: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/1.43 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/1.43 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_97_div__le__mono2,axiom,
% 0.29/1.43 ! [M: nat,N: nat,K: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/1.43 => ( ( ord_less_eq @ nat @ M @ N )
% 0.29/1.43 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_98_linorder__neqE__linordered__idom,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_idom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( X != Y )
% 0.29/1.43 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/1.43 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_99_linordered__field__no__ub,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X3: A] :
% 0.29/1.43 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_100_linordered__field__no__lb,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X3: A] :
% 0.29/1.43 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_101_infinite__descent__measure,axiom,
% 0.29/1.43 ! [A: $tType,P: A > $o,V2: A > nat,X: A] :
% 0.29/1.43 ( ! [X4: A] :
% 0.29/1.43 ( ~ ( P @ X4 )
% 0.29/1.43 => ? [Y4: A] :
% 0.29/1.43 ( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
% 0.29/1.43 & ~ ( P @ Y4 ) ) )
% 0.29/1.43 => ( P @ X ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_102_measure__induct__rule,axiom,
% 0.29/1.43 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.29/1.43 ( ! [X4: A] :
% 0.29/1.43 ( ! [Y4: A] :
% 0.29/1.43 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.29/1.43 => ( P @ Y4 ) )
% 0.29/1.43 => ( P @ X4 ) )
% 0.29/1.43 => ( P @ A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_103_linorder__neqE__nat,axiom,
% 0.29/1.43 ! [X: nat,Y: nat] :
% 0.29/1.43 ( ( X != Y )
% 0.29/1.43 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.29/1.43 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_104_infinite__descent,axiom,
% 0.29/1.43 ! [P: nat > $o,N: nat] :
% 0.29/1.43 ( ! [N3: nat] :
% 0.29/1.43 ( ~ ( P @ N3 )
% 0.29/1.43 => ? [M3: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/1.43 & ~ ( P @ M3 ) ) )
% 0.29/1.43 => ( P @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_105_nat__less__induct,axiom,
% 0.29/1.43 ! [P: nat > $o,N: nat] :
% 0.29/1.43 ( ! [N3: nat] :
% 0.29/1.43 ( ! [M3: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/1.43 => ( P @ M3 ) )
% 0.29/1.43 => ( P @ N3 ) )
% 0.29/1.43 => ( P @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_106_less__irrefl__nat,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_107_measure__induct,axiom,
% 0.29/1.43 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.29/1.43 ( ! [X4: A] :
% 0.29/1.43 ( ! [Y4: A] :
% 0.29/1.43 ( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
% 0.29/1.43 => ( P @ Y4 ) )
% 0.29/1.43 => ( P @ X4 ) )
% 0.29/1.43 => ( P @ A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_108_less__not__refl3,axiom,
% 0.29/1.43 ! [S: nat,T: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ S @ T )
% 0.29/1.43 => ( S != T ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_109_less__not__refl2,axiom,
% 0.29/1.43 ! [N: nat,M: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ N @ M )
% 0.29/1.43 => ( M != N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_110_less__not__refl,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_111_nat__neq__iff,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( M != N )
% 0.29/1.43 = ( ( ord_less @ nat @ M @ N )
% 0.29/1.43 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_112_not__one__less__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord1659791738miring @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_113_zero__less__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( zero_less_one @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_114_less__numeral__extra_I1_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_115_less__numeral__extra_I2_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_116_not__numeral__less__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_117_numeral__One,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( numeral_numeral @ A @ one2 )
% 0.29/1.43 = ( one_one @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_118_right__inverse__eq,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( division_ring @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( B
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ( divide_divide @ A @ A2 @ B )
% 0.29/1.43 = ( one_one @ A ) )
% 0.29/1.43 = ( A2 = B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_119_numerals_I1_J,axiom,
% 0.29/1.43 ( ( numeral_numeral @ nat @ one2 )
% 0.29/1.43 = ( one_one @ nat ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_120_div__eq__dividend__iff,axiom,
% 0.29/1.43 ! [A2: nat,B: nat] :
% 0.29/1.43 ( ( A2
% 0.29/1.43 != ( zero_zero @ nat ) )
% 0.29/1.43 => ( ( ( divide_divide @ nat @ A2 @ B )
% 0.29/1.43 = A2 )
% 0.29/1.43 = ( B
% 0.29/1.43 = ( one_one @ nat ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_121_less__divide__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B @ A2 ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less @ A @ A2 @ B ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_122_divide__less__eq__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( divide_divide @ A @ B @ A2 ) @ ( one_one @ A ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less @ A @ B @ A2 ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less @ A @ A2 @ B ) )
% 0.29/1.43 | ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_123_div__less__dividend,axiom,
% 0.29/1.43 ! [N: nat,M: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.29/1.43 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/1.43 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_124_less__numeral__extra_I3_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_125_zero__neq__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( zero_zero @ A )
% 0.29/1.43 != ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_126_num_Odistinct_I1_J,axiom,
% 0.29/1.43 ! [X2: num] :
% 0.29/1.43 ( one2
% 0.29/1.43 != ( bit0 @ X2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_127_infinite__descent0__measure,axiom,
% 0.29/1.43 ! [A: $tType,V2: A > nat,P: A > $o,X: A] :
% 0.29/1.43 ( ! [X4: A] :
% 0.29/1.43 ( ( ( V2 @ X4 )
% 0.29/1.43 = ( zero_zero @ nat ) )
% 0.29/1.43 => ( P @ X4 ) )
% 0.29/1.43 => ( ! [X4: A] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X4 ) )
% 0.29/1.43 => ( ~ ( P @ X4 )
% 0.29/1.43 => ? [Y4: A] :
% 0.29/1.43 ( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
% 0.29/1.43 & ~ ( P @ Y4 ) ) ) )
% 0.29/1.43 => ( P @ X ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_128_less__nat__zero__code,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_129_infinite__descent0,axiom,
% 0.29/1.43 ! [P: nat > $o,N: nat] :
% 0.29/1.43 ( ( P @ ( zero_zero @ nat ) )
% 0.29/1.43 => ( ! [N3: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.29/1.43 => ( ~ ( P @ N3 )
% 0.29/1.43 => ? [M3: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/1.43 & ~ ( P @ M3 ) ) ) )
% 0.29/1.43 => ( P @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_130_gr__implies__not0,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ M @ N )
% 0.29/1.43 => ( N
% 0.29/1.43 != ( zero_zero @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_131_less__zeroE,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_132_not__less0,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_133_not__gr0,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.29/1.43 = ( N
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_134_gr0I,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( N
% 0.29/1.43 != ( zero_zero @ nat ) )
% 0.29/1.43 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_135_not__numeral__less__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_136_zero__less__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_137_divide__strict__right__mono__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A,C: A] :
% 0.29/1.43 ( ( ord_less @ A @ B @ A2 )
% 0.29/1.43 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_138_divide__strict__right__mono,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ B )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_139_zero__less__divide__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_140_divide__less__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.29/1.43 => ( ord_less @ A @ A2 @ B ) )
% 0.29/1.43 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less @ A @ B @ A2 ) )
% 0.29/1.43 & ( C
% 0.29/1.43 != ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_141_divide__less__0__iff,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
% 0.29/1.43 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_142_divide__pos__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_143_divide__pos__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/1.43 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_144_divide__neg__pos,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.29/1.43 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_145_divide__neg__neg,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linordered_field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A,Y: A] :
% 0.29/1.43 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_146_divide__numeral__1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( field @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
% 0.29/1.43 = A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_147_div__eq__0__iff,axiom,
% 0.29/1.43 ! [A2: nat,B: nat] :
% 0.29/1.43 ( ( ( divide_divide @ nat @ A2 @ B )
% 0.29/1.43 = ( zero_zero @ nat ) )
% 0.29/1.43 = ( ( ord_less @ nat @ A2 @ B )
% 0.29/1.43 | ( B
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_148_div__2__gt__zero,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.29/1.43 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_149_zero__not__eq__two,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div_parity @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( zero_zero @ A )
% 0.29/1.43 != ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_150_semiring__norm_I76_J,axiom,
% 0.29/1.43 ! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_151_semiring__norm_I83_J,axiom,
% 0.29/1.43 ! [N: num] :
% 0.29/1.43 ( one2
% 0.29/1.43 != ( bit0 @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_152_semiring__norm_I85_J,axiom,
% 0.29/1.43 ! [M: num] :
% 0.29/1.43 ( ( bit0 @ M )
% 0.29/1.43 != one2 ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_153_not__gr__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: A] :
% 0.29/1.43 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/1.43 = ( N
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_154_le__zero__eq,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/1.43 = ( N
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_155_dbl__simps_I3_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_156_semiring__norm_I75_J,axiom,
% 0.29/1.43 ! [M: num] :
% 0.29/1.43 ~ ( ord_less @ num @ M @ one2 ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_157_semiring__norm_I78_J,axiom,
% 0.29/1.43 ! [M: num,N: num] :
% 0.29/1.43 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/1.43 = ( ord_less @ num @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_158_semiring__norm_I87_J,axiom,
% 0.29/1.43 ! [M: num,N: num] :
% 0.29/1.43 ( ( ( bit0 @ M )
% 0.29/1.43 = ( bit0 @ N ) )
% 0.29/1.43 = ( M = N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_159_semiring__norm_I71_J,axiom,
% 0.29/1.43 ! [M: num,N: num] :
% 0.29/1.43 ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.29/1.43 = ( ord_less_eq @ num @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_160_semiring__norm_I68_J,axiom,
% 0.29/1.43 ! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_161_dbl__simps_I2_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
% 0.29/1.43 = ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_162_semiring__norm_I69_J,axiom,
% 0.29/1.43 ! [M: num] :
% 0.29/1.43 ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_163_dbl__simps_I5_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ! [K: num] :
% 0.29/1.43 ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_164_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
% 0.29/1.43 ! [X: int,Y: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.29/1.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
% 0.29/1.43 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ X @ Y ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_165_zdiv__mono1,axiom,
% 0.29/1.43 ! [A2: int,A3: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ A2 @ A3 )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A3 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_166_zdiv__mono2,axiom,
% 0.29/1.43 ! [A2: int,B2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.29/1.43 => ( ( ord_less_eq @ int @ B2 @ B )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_167_zdiv__eq__0__iff,axiom,
% 0.29/1.43 ! [I: int,K: int] :
% 0.29/1.43 ( ( ( divide_divide @ int @ I @ K )
% 0.29/1.43 = ( zero_zero @ int ) )
% 0.29/1.43 = ( ( K
% 0.29/1.43 = ( zero_zero @ int ) )
% 0.29/1.43 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
% 0.29/1.43 & ( ord_less @ int @ I @ K ) )
% 0.29/1.43 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
% 0.29/1.43 & ( ord_less @ int @ K @ I ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_168_zdiv__mono1__neg,axiom,
% 0.29/1.43 ! [A2: int,A3: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ A2 @ A3 )
% 0.29/1.43 => ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_169_zdiv__mono2__neg,axiom,
% 0.29/1.43 ! [A2: int,B2: int,B: int] :
% 0.29/1.43 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.29/1.43 => ( ( ord_less_eq @ int @ B2 @ B )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_170_div__neg__pos__less0,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/1.43 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_171_int__div__less__self,axiom,
% 0.29/1.43 ! [X: int,K: int] :
% 0.29/1.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ X )
% 0.29/1.43 => ( ( ord_less @ int @ ( one_one @ int ) @ K )
% 0.29/1.43 => ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_172_div__nonneg__neg__le0,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_173_div__nonpos__pos__le0,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/1.43 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_174_div__neg__neg__trivial,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less @ int @ B @ A2 )
% 0.29/1.43 => ( ( divide_divide @ int @ A2 @ B )
% 0.29/1.43 = ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_175_div__pos__pos__trivial,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ int @ A2 @ B )
% 0.29/1.43 => ( ( divide_divide @ int @ A2 @ B )
% 0.29/1.43 = ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_176_neg__imp__zdiv__neg__iff,axiom,
% 0.29/1.43 ! [B: int,A2: int] :
% 0.29/1.43 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 0.29/1.43 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_177_pos__imp__zdiv__neg__iff,axiom,
% 0.29/1.43 ! [B: int,A2: int] :
% 0.29/1.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/1.43 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 0.29/1.43 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_178_pos__imp__zdiv__pos__iff,axiom,
% 0.29/1.43 ! [K: int,I: int] :
% 0.29/1.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
% 0.29/1.43 = ( ord_less_eq @ int @ K @ I ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_179_neg__imp__zdiv__nonneg__iff,axiom,
% 0.29/1.43 ! [B: int,A2: int] :
% 0.29/1.43 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 0.29/1.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/1.43 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_180_pos__imp__zdiv__nonneg__iff,axiom,
% 0.29/1.43 ! [B: int,A2: int] :
% 0.29/1.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 0.29/1.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/1.43 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_181_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.29/1.43 ! [A2: int,B: int] :
% 0.29/1.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.29/1.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 0.29/1.43 = ( ( ord_less_eq @ int @ B @ A2 )
% 0.29/1.43 & ( ord_less @ int @ ( zero_zero @ int ) @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_182_le__num__One__iff,axiom,
% 0.29/1.43 ! [X: num] :
% 0.29/1.43 ( ( ord_less_eq @ num @ X @ one2 )
% 0.29/1.43 = ( X = one2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_183_zero__reorient,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( zero @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A] :
% 0.29/1.43 ( ( ( zero_zero @ A )
% 0.29/1.43 = X )
% 0.29/1.43 = ( X
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_184_one__reorient,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( one @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A] :
% 0.29/1.43 ( ( ( one_one @ A )
% 0.29/1.43 = X )
% 0.29/1.43 = ( X
% 0.29/1.43 = ( one_one @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_185_zero__le,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_186_gr__zeroI,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: A] :
% 0.29/1.43 ( ( N
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_187_not__less__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: A] :
% 0.29/1.43 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_188_gr__implies__not__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [M: A,N: A] :
% 0.29/1.43 ( ( ord_less @ A @ M @ N )
% 0.29/1.43 => ( N
% 0.29/1.43 != ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_189_zero__less__iff__neq__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/1.43 = ( N
% 0.29/1.43 != ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_190_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
% 0.29/1.43 ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_191_order__refl,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( preorder @ A @ ( type @ A ) )
% 0.29/1.43 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_192_Suc__n__div__2__gt__zero,axiom,
% 0.29/1.43 ! [N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/1.43 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_193_add__right__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/1.43 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/1.43 = ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_194_add__left__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/1.43 = ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_195_add__le__cancel__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_196_add__le__cancel__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [C: A,A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_197_add_Oleft__neutral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( monoid_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/1.43 = A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_198_add_Oright__neutral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( monoid_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
% 0.29/1.43 = A2 ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_199_double__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ A2 @ A2 )
% 0.29/1.43 = ( zero_zero @ A ) )
% 0.29/1.43 = ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_200_double__zero__sym,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ( zero_zero @ A )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/1.43 = ( A2
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_201_add__cancel__left__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/1.43 = A2 )
% 0.29/1.43 = ( B
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_202_add__cancel__left__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/1.43 = A2 )
% 0.29/1.43 = ( B
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_203_add__cancel__right__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( A2
% 0.29/1.43 = ( plus_plus @ A @ B @ A2 ) )
% 0.29/1.43 = ( B
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_204_add__cancel__right__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel1352612707id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( A2
% 0.29/1.43 = ( plus_plus @ A @ A2 @ B ) )
% 0.29/1.43 = ( B
% 0.29/1.43 = ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_205_add__less__cancel__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [C: A,A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_206_add__less__cancel__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_207_numeral__plus__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ! [M: num,N: num] :
% 0.29/1.43 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_208_add__numeral__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ! [V: num,W: num,Z: A] :
% 0.29/1.43 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
% 0.29/1.43 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_209_add__is__0,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ( plus_plus @ nat @ M @ N )
% 0.29/1.43 = ( zero_zero @ nat ) )
% 0.29/1.43 = ( ( M
% 0.29/1.43 = ( zero_zero @ nat ) )
% 0.29/1.43 & ( N
% 0.29/1.43 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_210_Nat_Oadd__0__right,axiom,
% 0.29/1.43 ! [M: nat] :
% 0.29/1.43 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
% 0.29/1.43 = M ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_211_nat__add__left__cancel__less,axiom,
% 0.29/1.43 ! [K: nat,M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/1.43 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_212_nat__add__left__cancel__le,axiom,
% 0.29/1.43 ! [K: nat,M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/1.43 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_213_add__le__same__cancel1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_214_add__le__same__cancel2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_215_le__add__same__cancel1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/1.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_216_le__add__same__cancel2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_217_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_218_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/1.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_219_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
% 0.29/1.43 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_220_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( linord219039673up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
% 0.29/1.43 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_221_less__add__same__cancel2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
% 0.29/1.43 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_222_less__add__same__cancel1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
% 0.29/1.43 = ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_223_add__less__same__cancel2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/1.43 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_224_add__less__same__cancel1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere516151231imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/1.43 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_225_add__gr__0,axiom,
% 0.29/1.43 ! [M: nat,N: nat] :
% 0.29/1.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
% 0.29/1.43 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.29/1.43 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_226_numeral__plus__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_227_one__plus__numeral,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ! [N: num] :
% 0.29/1.43 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_228_div__add__self2,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( B
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
% 0.29/1.43 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_229_div__add__self1,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semiring_div @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A] :
% 0.29/1.43 ( ( B
% 0.29/1.43 != ( zero_zero @ A ) )
% 0.29/1.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
% 0.29/1.43 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_230_one__add__one,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.29/1.43 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_231_add__self__div__2,axiom,
% 0.29/1.43 ! [M: nat] :
% 0.29/1.43 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.29/1.43 = M ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_232_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
% 0.29/1.43 ord_less_eq @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_233_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.29/1.43 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_234_dbl__def,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( neg_numeral @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( neg_numeral_dbl @ A )
% 0.29/1.43 = ( ^ [X5: A] : ( plus_plus @ A @ X5 @ X5 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_235_add__right__imp__eq,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/1.43 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/1.43 => ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_236_add__left__imp__eq,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( cancel_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/1.43 => ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_237_add_Oleft__commute,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A,C: A] :
% 0.29/1.43 ( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_238_add_Ocommute,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( plus_plus @ A )
% 0.29/1.43 = ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_239_add_Oright__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( group_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [B: A,A2: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ B @ A2 )
% 0.29/1.43 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/1.43 = ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_240_add_Oleft__cancel,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( group_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ( plus_plus @ A @ A2 @ B )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/1.43 = ( B = C ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_241_add_Oassoc,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_242_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [I: A,J: A,K: A,L: A] :
% 0.29/1.43 ( ( ( I = J )
% 0.29/1.43 & ( K = L ) )
% 0.29/1.43 => ( ( plus_plus @ A @ I @ K )
% 0.29/1.43 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_243_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ab_semigroup_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
% 0.29/1.43 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_244_add__le__imp__le__right,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,C: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.29/1.43 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_245_add__le__imp__le__left,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere236663937imp_le @ A @ ( type @ A ) )
% 0.29/1.43 => ! [C: A,A2: A,B: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.29/1.43 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_246_le__iff__add,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.29/1.43 => ( ( ord_less_eq @ A )
% 0.29/1.43 = ( ^ [A4: A,B3: A] :
% 0.29/1.43 ? [C2: A] :
% 0.29/1.43 ( B3
% 0.29/1.43 = ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_247_add__right__mono,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/1.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_248_add__left__mono,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/1.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_249_add__mono,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [A2: A,B: A,C: A,D: A] :
% 0.29/1.43 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/1.43 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/1.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(fact_250_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/1.43 ! [A: $tType] :
% 0.29/1.43 ( ( ordere779506340up_add @ A @ ( type @ A ) )
% 0.29/1.43 => ! [I: A,J: A,K: A,L: A] :
% 0.29/1.43 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/1.43 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/1.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.29/1.43 ordere516151231imp_le @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/1.43 ordere236663937imp_le @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/1.43 linord1659791738miring @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/1.43 ordere779506340up_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.29/1.43 linord219039673up_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.29/1.43 cancel1352612707id_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Divides_Osemiring__numeral__div,axiom,
% 0.29/1.43 semiring_numeral_div @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 0.29/1.43 cancel_semigroup_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Divides_Osemiring__div__parity,axiom,
% 0.29/1.43 semiring_div_parity @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.29/1.43 linordered_semidom @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 0.29/1.43 ab_semigroup_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.29/1.43 linordered_idom @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 0.29/1.43 semidom_divide @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 0.29/1.43 semigroup_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Divides_Osemiring__div,axiom,
% 0.29/1.43 semiring_div @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
% 0.29/1.43 zero_less_one @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.29/1.43 semiring_char_0 @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 0.29/1.43 zero_neq_one @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Orderings_Opreorder,axiom,
% 0.29/1.43 preorder @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 0.29/1.43 monoid_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.29/1.43 group_add @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 0.29/1.43 neg_numeral @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 0.29/1.43 numeral @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.29/1.43 zero @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 0.29/1.43 one @ int @ ( type @ int ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1,axiom,
% 0.29/1.43 ordere516151231imp_le @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_2,axiom,
% 0.29/1.43 ordere236663937imp_le @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/1.43 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_3,axiom,
% 0.29/1.43 linord1659791738miring @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_4,axiom,
% 0.29/1.43 ordere779506340up_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_5,axiom,
% 0.29/1.43 cancel1352612707id_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Divides_Osemiring__numeral__div_6,axiom,
% 0.29/1.43 semiring_numeral_div @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_7,axiom,
% 0.29/1.43 cancel_semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Divides_Osemiring__div__parity_8,axiom,
% 0.29/1.43 semiring_div_parity @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_9,axiom,
% 0.29/1.43 linordered_semidom @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_10,axiom,
% 0.29/1.43 ab_semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Rings_Osemidom__divide_11,axiom,
% 0.29/1.43 semidom_divide @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Osemigroup__add_12,axiom,
% 0.29/1.43 semigroup_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Divides_Osemiring__div_13,axiom,
% 0.29/1.43 semiring_div @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Rings_Ozero__less__one_14,axiom,
% 0.29/1.43 zero_less_one @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_15,axiom,
% 0.29/1.43 semiring_char_0 @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_16,axiom,
% 0.29/1.43 zero_neq_one @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
% 0.29/1.43 preorder @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Omonoid__add_18,axiom,
% 0.29/1.43 monoid_add @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Num_Onumeral_19,axiom,
% 0.29/1.43 numeral @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Ozero_20,axiom,
% 0.29/1.43 zero @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Nat_Onat___Groups_Oone_21,axiom,
% 0.29/1.43 one @ nat @ ( type @ nat ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_Num_Onum___Orderings_Opreorder_22,axiom,
% 0.29/1.43 preorder @ num @ ( type @ num ) ).
% 0.29/1.43
% 0.29/1.43 thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
% 0.29/1.43 preorder @ $o @ ( type @ $o ) ).
% 0.29/1.43
% 0.29/1.43 thf(conj_0,conjecture,
% 0.29/1.43 ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.29/1.43 ------------------------
% 0.29/1.45 % (21443)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_21282 for (2999ds/4Mi)
% 0.29/1.45 % (21445)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/2Mi)
% 0.29/1.45 % (21445)Instruction limit reached!
% 0.29/1.45 % (21445)------------------------------
% 0.29/1.45 % (21445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.45 % (21445)Termination reason: Unknown
% 0.29/1.45 % (21445)Termination phase: shuffling
% 0.29/1.45
% 0.29/1.45 % (21445)Memory used [KB]: 1279
% 0.29/1.45 % (21445)Time elapsed: 0.002 s
% 0.29/1.45 % (21445)Instructions burned: 3 (million)
% 0.29/1.45 % (21445)------------------------------
% 0.29/1.45 % (21445)------------------------------
% 0.29/1.45 % (21443)Instruction limit reached!
% 0.29/1.45 % (21443)------------------------------
% 0.29/1.45 % (21443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.45 % (21443)Termination reason: Unknown
% 0.29/1.45 % (21443)Termination phase: shuffling
% 0.29/1.45
% 0.29/1.45 % (21443)Memory used [KB]: 1407
% 0.29/1.45 % (21443)Time elapsed: 0.003 s
% 0.29/1.45 % (21443)Instructions burned: 5 (million)
% 0.29/1.45 % (21443)------------------------------
% 0.29/1.45 % (21443)------------------------------
% 0.29/1.46 % (21442)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/183Mi)
% 0.29/1.46 % (21446)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_21282 for (2999ds/2Mi)
% 0.29/1.46 % (21447)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_21282 for (2999ds/275Mi)
% 0.29/1.46 % (21446)Instruction limit reached!
% 0.29/1.46 % (21446)------------------------------
% 0.29/1.46 % (21446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.46 % (21446)Termination reason: Unknown
% 0.29/1.46 % (21446)Termination phase: shuffling
% 0.29/1.46
% 0.29/1.46 % (21446)Memory used [KB]: 1279
% 0.29/1.46 % (21446)Time elapsed: 0.004 s
% 0.29/1.46 % (21446)Instructions burned: 3 (million)
% 0.29/1.46 % (21444)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_21282 for (2999ds/27Mi)
% 0.29/1.46 % (21446)------------------------------
% 0.29/1.46 % (21446)------------------------------
% 0.29/1.46 % (21449)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/3Mi)
% 0.29/1.46 % (21449)Instruction limit reached!
% 0.29/1.46 % (21449)------------------------------
% 0.29/1.46 % (21449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.46 % (21449)Termination reason: Unknown
% 0.29/1.46 % (21449)Termination phase: shuffling
% 0.29/1.46
% 0.29/1.46 % (21449)Memory used [KB]: 1279
% 0.29/1.46 % (21449)Time elapsed: 0.002 s
% 0.29/1.46 % (21449)Instructions burned: 3 (million)
% 0.29/1.46 % (21449)------------------------------
% 0.29/1.46 % (21449)------------------------------
% 0.29/1.46 % (21448)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/18Mi)
% 0.29/1.47 % (21450)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_21282 for (2999ds/37Mi)
% 0.29/1.47 % (21444)Instruction limit reached!
% 0.29/1.47 % (21444)------------------------------
% 0.29/1.47 % (21444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.47 % (21444)Termination reason: Unknown
% 0.29/1.47 % (21444)Termination phase: Property scanning
% 0.29/1.47
% 0.29/1.47 % (21444)Memory used [KB]: 1663
% 0.29/1.47 % (21444)Time elapsed: 0.010 s
% 0.29/1.47 % (21444)Instructions burned: 28 (million)
% 0.29/1.47 % (21444)------------------------------
% 0.29/1.47 % (21444)------------------------------
% 0.29/1.47 % (21448)Instruction limit reached!
% 0.29/1.47 % (21448)------------------------------
% 0.29/1.47 % (21448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.47 % (21448)Termination reason: Unknown
% 0.29/1.47 % (21448)Termination phase: Property scanning
% 0.29/1.47
% 0.29/1.47 % (21448)Memory used [KB]: 1663
% 0.29/1.47 % (21448)Time elapsed: 0.008 s
% 0.29/1.47 % (21448)Instructions burned: 18 (million)
% 0.29/1.47 % (21448)------------------------------
% 0.29/1.47 % (21448)------------------------------
% 0.29/1.47 % (21452)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_21282 for (2999ds/3Mi)
% 0.29/1.47 % (21452)Instruction limit reached!
% 0.29/1.47 % (21452)------------------------------
% 0.29/1.47 % (21452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.47 % (21452)Termination reason: Unknown
% 0.29/1.47 % (21452)Termination phase: shuffling
% 0.29/1.47
% 0.29/1.47 % (21452)Memory used [KB]: 1407
% 0.29/1.47 % (21452)Time elapsed: 0.003 s
% 0.29/1.47 % (21452)Instructions burned: 5 (million)
% 0.29/1.47 % (21452)------------------------------
% 0.29/1.47 % (21452)------------------------------
% 0.29/1.48 % (21451)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_21282 for (2999ds/15Mi)
% 0.29/1.48 % (21453)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_21282 for (2999ds/1041Mi)
% 0.29/1.48 % (21451)Instruction limit reached!
% 0.29/1.48 % (21451)------------------------------
% 0.29/1.48 % (21451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.48 % (21451)Termination reason: Unknown
% 0.29/1.48 % (21451)Termination phase: shuffling
% 0.29/1.48
% 0.29/1.48 % (21451)Memory used [KB]: 1663
% 0.29/1.48 % (21451)Time elapsed: 0.007 s
% 0.29/1.48 % (21451)Instructions burned: 17 (million)
% 0.29/1.48 % (21451)------------------------------
% 0.29/1.48 % (21451)------------------------------
% 0.29/1.48 % (21454)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/7Mi)
% 0.29/1.48 % (21455)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/16Mi)
% 0.29/1.48 % (21450)Instruction limit reached!
% 0.29/1.48 % (21450)------------------------------
% 0.29/1.48 % (21450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.48 % (21450)Termination reason: Unknown
% 0.29/1.48 % (21450)Termination phase: Property scanning
% 0.29/1.48
% 0.29/1.48 % (21450)Memory used [KB]: 1791
% 0.29/1.48 % (21450)Time elapsed: 0.018 s
% 0.29/1.48 % (21450)Instructions burned: 37 (million)
% 0.29/1.48 % (21450)------------------------------
% 0.29/1.48 % (21450)------------------------------
% 0.29/1.48 % (21454)Instruction limit reached!
% 0.29/1.48 % (21454)------------------------------
% 0.29/1.48 % (21454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.48 % (21454)Termination reason: Unknown
% 0.29/1.48 % (21454)Termination phase: shuffling
% 0.29/1.48
% 0.29/1.48 % (21454)Memory used [KB]: 1407
% 0.29/1.48 % (21454)Time elapsed: 0.004 s
% 0.29/1.48 % (21454)Instructions burned: 8 (million)
% 0.29/1.48 % (21454)------------------------------
% 0.29/1.48 % (21454)------------------------------
% 0.29/1.49 % (21455)Instruction limit reached!
% 0.29/1.49 % (21455)------------------------------
% 0.29/1.49 % (21455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.49 % (21455)Termination reason: Unknown
% 0.29/1.49 % (21455)Termination phase: Property scanning
% 0.29/1.49
% 0.29/1.49 % (21455)Memory used [KB]: 1663
% 0.29/1.49 % (21455)Time elapsed: 0.007 s
% 0.29/1.49 % (21455)Instructions burned: 18 (million)
% 0.29/1.49 % (21455)------------------------------
% 0.29/1.49 % (21455)------------------------------
% 0.29/1.49 % (21456)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_21282 for (2999ds/3Mi)
% 0.29/1.49 % (21456)Instruction limit reached!
% 0.29/1.49 % (21456)------------------------------
% 0.29/1.49 % (21456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.29/1.49 % (21456)Termination reason: Unknown
% 0.29/1.49 % (21456)Termination phase: shuffling
% 0.29/1.49
% 0.29/1.49 % (21456)Memory used [KB]: 1279
% 0.29/1.49 % (21456)Time elapsed: 0.002 s
% 0.29/1.49 % (21456)Instructions burned: 3 (million)
% 0.29/1.49 % (21456)------------------------------
% 0.29/1.49 % (21456)------------------------------
% 0.29/1.49 % (21458)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/7Mi)
% 2.53/1.50 % (21458)Instruction limit reached!
% 2.53/1.50 % (21458)------------------------------
% 2.53/1.50 % (21458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.50 % (21458)Termination reason: Unknown
% 2.53/1.50 % (21458)Termination phase: shuffling
% 2.53/1.50
% 2.53/1.50 % (21458)Memory used [KB]: 1407
% 2.53/1.50 % (21458)Time elapsed: 0.004 s
% 2.53/1.50 % (21458)Instructions burned: 8 (million)
% 2.53/1.50 % (21458)------------------------------
% 2.53/1.50 % (21458)------------------------------
% 2.53/1.50 % (21459)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/3Mi)
% 2.53/1.50 % (21457)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_21282 for (2999ds/3Mi)
% 2.53/1.50 % (21459)Instruction limit reached!
% 2.53/1.50 % (21459)------------------------------
% 2.53/1.50 % (21459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.50 % (21459)Termination reason: Unknown
% 2.53/1.50 % (21459)Termination phase: shuffling
% 2.53/1.50
% 2.53/1.50 % (21459)Memory used [KB]: 1279
% 2.53/1.50 % (21459)Time elapsed: 0.002 s
% 2.53/1.50 % (21459)Instructions burned: 3 (million)
% 2.53/1.50 % (21459)------------------------------
% 2.53/1.50 % (21459)------------------------------
% 2.53/1.50 % (21457)Instruction limit reached!
% 2.53/1.50 % (21457)------------------------------
% 2.53/1.50 % (21457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.50 % (21457)Termination reason: Unknown
% 2.53/1.50 % (21457)Termination phase: shuffling
% 2.53/1.50
% 2.53/1.50 % (21457)Memory used [KB]: 1407
% 2.53/1.50 % (21457)Time elapsed: 0.004 s
% 2.53/1.50 % (21457)Instructions burned: 5 (million)
% 2.53/1.50 % (21457)------------------------------
% 2.53/1.50 % (21457)------------------------------
% 2.53/1.50 % (21460)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/4Mi)
% 2.53/1.50 % (21460)Instruction limit reached!
% 2.53/1.50 % (21460)------------------------------
% 2.53/1.50 % (21460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.50 % (21460)Termination reason: Unknown
% 2.53/1.50 % (21460)Termination phase: shuffling
% 2.53/1.50
% 2.53/1.50 % (21460)Memory used [KB]: 1407
% 2.53/1.50 % (21460)Time elapsed: 0.003 s
% 2.53/1.50 % (21460)Instructions burned: 6 (million)
% 2.53/1.50 % (21460)------------------------------
% 2.53/1.50 % (21460)------------------------------
% 2.53/1.51 % (21461)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/18Mi)
% 2.53/1.51 % (21447)First to succeed.
% 2.53/1.51 % (21462)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21282 for (2999ds/710Mi)
% 2.53/1.51 % (21447)Refutation found. Thanks to Tanya!
% 2.53/1.51 % SZS status Theorem for DTF2THF_21282
% 2.53/1.51 % SZS output start Proof for DTF2THF_21282
% 2.53/1.51 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.53/1.51 thf(type_def_7, type, num: $tType).
% 2.53/1.51 thf(type_def_8, type, nat: $tType).
% 2.53/1.51 thf(type_def_9, type, int: $tType).
% 2.53/1.51 thf(func_def_0, type, num: $tType).
% 2.53/1.51 thf(func_def_1, type, nat: $tType).
% 2.53/1.51 thf(func_def_2, type, int: $tType).
% 2.53/1.51 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.53/1.51 thf(func_def_4, type, one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_5, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_6, type, numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_7, type, field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_8, type, neg_numeral: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_9, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_10, type, monoid_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_11, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_12, type, zero_neq_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_13, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_14, type, zero_less_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_15, type, semiring_div: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_16, type, division_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_17, type, semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_18, type, semidom_divide: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_20, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_21, type, ab_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_22, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_23, type, semiring_div_parity: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_24, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_25, type, semiring_numeral_div: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_26, type, cancel1352612707id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_27, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_28, type, ordere779506340up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_29, type, linord1659791738miring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_30, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_31, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_32, type, ordere516151231imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.53/1.51 thf(func_def_33, type, one_one: !>[X0: $tType]:(X0)).
% 2.53/1.51 thf(func_def_34, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.53/1.51 thf(func_def_35, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.53/1.51 thf(func_def_36, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 2.53/1.51 thf(func_def_37, type, bit0: num > num).
% 2.53/1.51 thf(func_def_38, type, one2: num).
% 2.53/1.51 thf(func_def_39, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 2.53/1.51 thf(func_def_40, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.53/1.51 thf(func_def_41, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.53/1.51 thf(func_def_42, type, type: !>[X0: $tType]:(itself @ X0)).
% 2.53/1.51 thf(func_def_43, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.53/1.51 thf(func_def_44, type, na: nat).
% 2.53/1.51 thf(func_def_57, type, sK0: !>[X0: $tType]:(X0 > X0)).
% 2.53/1.51 thf(func_def_58, type, sK1: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.53/1.51 thf(func_def_59, type, sK2: (nat > $o) > nat).
% 2.53/1.51 thf(func_def_60, type, sK3: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.53/1.51 thf(func_def_61, type, sK4: nat > (nat > $o) > nat).
% 2.53/1.51 thf(func_def_62, type, sK5: !>[X0: $tType]:(X0 > X0)).
% 2.53/1.51 thf(func_def_63, type, sK6: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.53/1.51 thf(func_def_64, type, sK7: (nat > $o) > nat).
% 2.53/1.51 thf(func_def_65, type, sK8: (nat > nat) > nat).
% 2.53/1.51 thf(func_def_66, type, sK9: (nat > nat) > nat).
% 2.53/1.51 thf(func_def_67, type, sK10: (nat > $o) > nat).
% 2.53/1.51 thf(func_def_68, type, sK11: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.53/1.51 thf(func_def_69, type, sK12: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.53/1.51 thf(func_def_71, type, ph14: !>[X0: $tType]:(X0)).
% 2.53/1.51 thf(f1631,plain,(
% 2.53/1.51 $false),
% 2.53/1.51 inference(subsumption_resolution,[status(thm)],[f1629,f1200])).
% 2.53/1.51 thf(f1200,plain,(
% 2.53/1.51 ((ord_less_eq @ nat @ na @ one_one @ nat) != $true)),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f904])).
% 2.53/1.51 thf(f904,plain,(
% 2.53/1.51 ((ord_less_eq @ nat @ na @ one_one @ nat) != $true)),
% 2.53/1.51 inference(flattening,[status(thm)],[f595])).
% 2.53/1.51 thf(f595,plain,(
% 2.53/1.51 ~((ord_less_eq @ nat @ na @ one_one @ nat) = $true)),
% 2.53/1.51 inference(fool_elimination,[status(thm)],[f594])).
% 2.53/1.51 thf(f594,plain,(
% 2.53/1.51 ~(ord_less_eq @ nat @ na @ one_one @ nat)),
% 2.53/1.51 inference(rectify,[status(thm)],[f2])).
% 2.53/1.51 thf(f2,axiom,(
% 2.53/1.51 ~(ord_less_eq @ nat @ na @ one_one @ nat)),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',fact_1_False)).
% 2.53/1.51 thf(f1629,plain,(
% 2.53/1.51 ((ord_less_eq @ nat @ na @ one_one @ nat) = $true)),
% 2.53/1.51 inference(trivial_inequality_removal,[status(thm)],[f1625])).
% 2.53/1.51 thf(f1625,plain,(
% 2.53/1.51 ($true != $true) | ((ord_less_eq @ nat @ na @ one_one @ nat) = $true)),
% 2.53/1.51 inference(superposition,[status(thm)],[f1434,f1292])).
% 2.53/1.51 thf(f1292,plain,(
% 2.53/1.51 ( ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less_eq @ nat @ X1 @ X0) = $true)) )),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f341])).
% 2.53/1.51 thf(f341,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : (((ord_less_eq @ nat @ X1 @ X0) = $true) | ($true = (ord_less_eq @ nat @ X0 @ X1)))),
% 2.53/1.51 inference(fool_elimination,[status(thm)],[f340])).
% 2.53/1.51 thf(f340,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) | (ord_less_eq @ nat @ X1 @ X0))),
% 2.53/1.51 inference(rectify,[status(thm)],[f54])).
% 2.53/1.51 thf(f54,axiom,(
% 2.53/1.51 ! [X1 : nat,X2 : nat] : ((ord_less_eq @ nat @ X1 @ X2) | (ord_less_eq @ nat @ X2 @ X1))),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',fact_53_nat__le__linear)).
% 2.53/1.51 thf(f1434,plain,(
% 2.53/1.51 ((ord_less_eq @ nat @ one_one @ nat @ na) != $true)),
% 2.53/1.51 inference(subsumption_resolution,[status(thm)],[f1433,f1195])).
% 2.53/1.51 thf(f1195,plain,(
% 2.53/1.51 (na != one_one @ nat)),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f965])).
% 2.53/1.51 thf(f965,plain,(
% 2.53/1.51 (na != one_one @ nat) & (na != zero_zero @ nat)),
% 2.53/1.51 inference(ennf_transformation,[status(thm)],[f1])).
% 2.53/1.51 thf(f1,axiom,(
% 2.53/1.51 ~((na = zero_zero @ nat) | (na = one_one @ nat))),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',fact_0_ns)).
% 2.53/1.51 thf(f1433,plain,(
% 2.53/1.51 (na = one_one @ nat) | ((ord_less_eq @ nat @ one_one @ nat @ na) != $true)),
% 2.53/1.51 inference(trivial_inequality_removal,[status(thm)],[f1428])).
% 2.53/1.51 thf(f1428,plain,(
% 2.53/1.51 ((ord_less_eq @ nat @ one_one @ nat @ na) != $true) | ($true != $true) | (na = one_one @ nat)),
% 2.53/1.51 inference(superposition,[status(thm)],[f1394,f1210])).
% 2.53/1.51 thf(f1210,plain,(
% 2.53/1.51 ( ! [X0 : nat,X1 : nat] : (((ord_less @ nat @ X0 @ X1) = $true) | ($true != (ord_less_eq @ nat @ X0 @ X1)) | (X0 = X1)) )),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f1003])).
% 2.53/1.51 thf(f1003,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : (($true != (ord_less_eq @ nat @ X0 @ X1)) | ((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1))),
% 2.53/1.51 inference(flattening,[status(thm)],[f1002])).
% 2.53/1.51 thf(f1002,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : ((((ord_less @ nat @ X0 @ X1) = $true) | (X0 = X1)) | ($true != (ord_less_eq @ nat @ X0 @ X1)))),
% 2.53/1.51 inference(ennf_transformation,[status(thm)],[f795])).
% 2.53/1.51 thf(f795,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : (($true = (ord_less_eq @ nat @ X0 @ X1)) => ((X0 != X1) => ((ord_less @ nat @ X0 @ X1) = $true)))),
% 2.53/1.51 inference(fool_elimination,[status(thm)],[f794])).
% 2.53/1.51 thf(f794,plain,(
% 2.53/1.51 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((X0 != X1) => (ord_less @ nat @ X0 @ X1)))),
% 2.53/1.51 inference(rectify,[status(thm)],[f64])).
% 2.53/1.51 thf(f64,axiom,(
% 2.53/1.51 ! [X1 : nat,X2 : nat] : ((ord_less_eq @ nat @ X1 @ X2) => ((X1 != X2) => (ord_less @ nat @ X1 @ X2)))),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',fact_63_le__neq__implies__less)).
% 2.53/1.51 thf(f1394,plain,(
% 2.53/1.51 ($true != (ord_less @ nat @ one_one @ nat @ na))),
% 2.53/1.51 inference(trivial_inequality_removal,[status(thm)],[f1380])).
% 2.53/1.51 thf(f1380,plain,(
% 2.53/1.51 ($true != $true) | ($true != (ord_less @ nat @ one_one @ nat @ na))),
% 2.53/1.51 inference(superposition,[status(thm)],[f1326,f1199])).
% 2.53/1.51 thf(f1199,plain,(
% 2.53/1.51 ( ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ X0 @ (numeral_numeral @ nat @ (bit0 @ one2)))) = $true) | ((ord_less @ nat @ one_one @ nat @ X0) != $true)) )),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f958])).
% 2.53/1.51 thf(f958,plain,(
% 2.53/1.51 ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ X0 @ (numeral_numeral @ nat @ (bit0 @ one2)))) = $true) | ((ord_less @ nat @ one_one @ nat @ X0) != $true))),
% 2.53/1.51 inference(ennf_transformation,[status(thm)],[f507])).
% 2.53/1.51 thf(f507,plain,(
% 2.53/1.51 ! [X0 : nat] : (((ord_less @ nat @ one_one @ nat @ X0) = $true) => ((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ X0 @ (numeral_numeral @ nat @ (bit0 @ one2)))) = $true))),
% 2.53/1.51 inference(fool_elimination,[status(thm)],[f506])).
% 2.53/1.51 thf(f506,plain,(
% 2.53/1.51 ! [X0 : nat] : ((ord_less @ nat @ one_one @ nat @ X0) => (ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ X0 @ (numeral_numeral @ nat @ (bit0 @ one2)))))),
% 2.53/1.51 inference(rectify,[status(thm)],[f149])).
% 2.53/1.51 thf(f149,axiom,(
% 2.53/1.51 ! [X2 : nat] : ((ord_less @ nat @ one_one @ nat @ X2) => (ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ X2 @ (numeral_numeral @ nat @ (bit0 @ one2)))))),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',fact_148_div__2__gt__zero)).
% 2.53/1.51 thf(f1326,plain,(
% 2.53/1.51 ((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2)))) != $true)),
% 2.53/1.51 inference(cnf_transformation,[status(thm)],[f882])).
% 2.53/1.51 thf(f882,plain,(
% 2.53/1.51 ((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2)))) != $true)),
% 2.53/1.51 inference(flattening,[status(thm)],[f879])).
% 2.53/1.51 thf(f879,plain,(
% 2.53/1.51 ~((ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2)))) = $true)),
% 2.53/1.51 inference(fool_elimination,[status(thm)],[f878])).
% 2.53/1.51 thf(f878,plain,(
% 2.53/1.51 ~(ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2))))),
% 2.53/1.51 inference(rectify,[status(thm)],[f302])).
% 2.53/1.51 thf(f302,negated_conjecture,(
% 2.53/1.51 ~(ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2))))),
% 2.53/1.51 inference(negated_conjecture,[status(cth)],[f301])).
% 2.53/1.51 thf(f301,conjecture,(
% 2.53/1.51 (ord_less @ nat @ zero_zero @ nat @ (divide_divide @ nat @ na @ (numeral_numeral @ nat @ (bit0 @ one2))))),
% 2.53/1.51 file('/export/starexec/sandbox2/tmp/tmp.lYVY7RQ28L/DTF2THF_21282.p',conj_0)).
% 2.53/1.51 % SZS output end Proof for DTF2THF_21282
% 2.53/1.51 % (21447)------------------------------
% 2.53/1.51 % (21447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.51 % (21447)Termination reason: Refutation
% 2.53/1.51
% 2.53/1.51 % (21447)Memory used [KB]: 6780
% 2.53/1.51 % (21447)Time elapsed: 0.050 s
% 2.53/1.51 % (21447)Instructions burned: 119 (million)
% 2.53/1.51 % (21447)------------------------------
% 2.53/1.51 % (21447)------------------------------
% 2.53/1.51 % (21441)Success in time 0.072 s
%------------------------------------------------------------------------------