%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP130^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:47 AM UTC 2026
% Result : Theorem 2.55s 1.58s
% Output : Refutation 2.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP130^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n017.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:25:56 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP130^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer NthRoot_Impl problem prob_360__3277026_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : NthRoot_Impl/prob_360__3277026_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.00 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 328 ( 120 unt; 36 typ; 0 def)
% 0.29/0.45 % Number of atoms : 768 ( 209 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 7 ( 2 avg)
% 0.29/0.45 % Number of connectives : 2926 ( 75 ~; 22 |; 38 &;2425 @)
% 0.29/0.45 % ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 16 ( 6 avg)
% 0.29/0.45 % Number of types : 4 ( 3 usr)
% 0.29/0.45 % Number of type conns : 50 ( 50 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 34 ( 33 usr; 7 con; 0-3 aty)
% 0.29/0.45 % Number of variables : 648 ( 7 ^; 590 !; 27 ?; 648 :)
% 0.29/0.45 % ( 24 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.45 % SPC : TH1_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:26:40.961
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (3)
% 0.29/0.45 thf(ty_t_Real_Oreal,type,
% 0.29/0.45 real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Nat_Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_t_Int_Oint,type,
% 0.29/0.45 int: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (33)
% 0.29/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.29/0.45 zero:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Fields_Ofield,type,
% 0.29/0.45 field:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Oring__1,type,
% 0.29/0.45 ring_1:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Nat_Oring__char__0,type,
% 0.29/0.45 ring_char_0:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.45 order:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Osemiring__1,type,
% 0.29/0.45 semiring_1:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.29/0.45 monoid_mult:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.45 wellorder:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Fields_Odivision__ring,type,
% 0.29/0.45 division_ring:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Parity_Osemiring__bits,type,
% 0.29/0.45 semiring_bits:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.29/0.45 semidom_divide:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.45 linordered_idom:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Fields_Olinordered__field,type,
% 0.29/0.45 linordered_field:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.29/0.45 linordered_semidom:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.29/0.45 semiri134348788visors:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.29/0.45 archim1804426504_field:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.45 canoni770627133id_add:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.29/0.45 unique1598680935umeral:
% 0.29/0.45 !>[A: $tType] : $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.29/0.45 zero_zero:
% 0.29/0.45 !>[A: $tType] : A ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.29/0.45 ring_1_of_int:
% 0.29/0.45 !>[A: $tType] : ( int > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_OSuc,type,
% 0.29/0.45 suc: nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_NthRoot__Impl__Mirabelle__yrjxwmcnbt_Ofixed__root,type,
% 0.29/0.45 nthRoo220209705d_root: nat > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.45 ord_less:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.45 ord_less_eq:
% 0.29/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Power_Opower__class_Opower,type,
% 0.29/0.45 power_power:
% 0.29/0.45 !>[A: $tType] : ( A > nat > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.29/0.45 divide_divide:
% 0.29/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_NY____,type,
% 0.29/0.45 ny: real ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_na____,type,
% 0.29/0.45 na: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_p,type,
% 0.29/0.45 p: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_pm,type,
% 0.29/0.45 pm: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_x,type,
% 0.29/0.45 x: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_y_H____,type,
% 0.29/0.45 y: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_ya____,type,
% 0.29/0.45 ya: int ).
% 0.29/0.45
% 0.29/0.45 % Relevant facts (253)
% 0.29/0.45 thf(fact_0_n00,axiom,
% 0.29/0.45 ( na
% 0.29/0.45 != ( zero_zero @ int ) ) ).
% 0.29/0.45
% 0.29/0.45 % n00
% 0.29/0.45 thf(fact_1_of__int__less__of__int__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [B: int,W: nat,X: int] :
% 0.29/0.45 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 0.29/0.45 = ( ord_less @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_less_of_int_power_cancel_iff
% 0.29/0.45 thf(fact_2_of__int__power__less__of__int__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [X: int,B: int,W: nat] :
% 0.29/0.45 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.29/0.45 = ( ord_less @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_power_less_of_int_cancel_iff
% 0.29/0.45 thf(fact_3_of__int__power,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_1 @ A )
% 0.29/0.45 => ! [Z: int,N: nat] :
% 0.29/0.45 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 0.29/0.45 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_power
% 0.29/0.45 thf(fact_4_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [B: int,W: nat,X: int] :
% 0.29/0.45 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 0.29/0.45 = ( ring_1_of_int @ A @ X ) )
% 0.29/0.45 = ( ( power_power @ int @ B @ W )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_eq_of_int_power_cancel_iff
% 0.29/0.45 thf(fact_5_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [X: int,B: int,W: nat] :
% 0.29/0.45 ( ( ( ring_1_of_int @ A @ X )
% 0.29/0.45 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.29/0.45 = ( X
% 0.29/0.45 = ( power_power @ int @ B @ W ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_power_eq_of_int_cancel_iff
% 0.29/0.45 thf(fact_6_of__int__less__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [W: int,Z: int] :
% 0.29/0.45 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( ord_less @ int @ W @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_less_iff
% 0.29/0.45 thf(fact_7_n0,axiom,
% 0.29/0.45 ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ na ) ).
% 0.29/0.45
% 0.29/0.45 % n0
% 0.29/0.45 thf(fact_8_y0,axiom,
% 0.29/0.45 ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ ya ) ).
% 0.29/0.45
% 0.29/0.45 % y0
% 0.29/0.45 thf(fact_9_of__int__eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [W: int,Z: int] :
% 0.29/0.45 ( ( ( ring_1_of_int @ A @ W )
% 0.29/0.45 = ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( W = Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_eq_iff
% 0.29/0.45 thf(fact_10_nat_Oinject,axiom,
% 0.29/0.45 ! [X2: nat,Y2: nat] :
% 0.29/0.45 ( ( ( suc @ X2 )
% 0.29/0.45 = ( suc @ Y2 ) )
% 0.29/0.45 = ( X2 = Y2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat.inject
% 0.29/0.45 thf(fact_11_old_Onat_Oinject,axiom,
% 0.29/0.45 ! [Nat: nat,Nat2: nat] :
% 0.29/0.45 ( ( ( suc @ Nat )
% 0.29/0.45 = ( suc @ Nat2 ) )
% 0.29/0.45 = ( Nat = Nat2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.nat.inject
% 0.29/0.45 thf(fact_12_y00,axiom,
% 0.29/0.45 ( ya
% 0.29/0.45 != ( zero_zero @ int ) ) ).
% 0.29/0.45
% 0.29/0.45 % y00
% 0.29/0.45 thf(fact_13__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062,axiom,
% 0.29/0.45 ( ( ord_less @ int @ ( zero_zero @ int ) @ ya )
% 0.29/0.45 | ( ya
% 0.29/0.45 = ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>0 < y \<or> y = 0\<close>
% 0.29/0.45 thf(fact_14_Suc__less__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.45 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_less_eq
% 0.29/0.45 thf(fact_15_Suc__mono,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ M @ N )
% 0.29/0.45 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_mono
% 0.29/0.45 thf(fact_16_lessI,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % lessI
% 0.29/0.45 thf(fact_17_not_I1_J,axiom,
% 0.29/0.45 ~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
% 0.29/0.45 | ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not(1)
% 0.29/0.45 thf(fact_18_not_I2_J,axiom,
% 0.29/0.45 ~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
% 0.29/0.45 | ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not(2)
% 0.29/0.45 thf(fact_19__C1_Oprems_C_I2_J,axiom,
% 0.29/0.45 ord_less_eq @ int @ ( zero_zero @ int ) @ ya ).
% 0.29/0.45
% 0.29/0.45 % "1.prems"(2)
% 0.29/0.45 thf(fact_20__C1_Oprems_C_I3_J,axiom,
% 0.29/0.45 ord_less_eq @ int @ ( zero_zero @ int ) @ na ).
% 0.29/0.45
% 0.29/0.45 % "1.prems"(3)
% 0.29/0.45 thf(fact_21_of__int__0,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_1 @ A )
% 0.29/0.45 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0
% 0.29/0.45 thf(fact_22_of__int__0__eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ( zero_zero @ A )
% 0.29/0.45 = ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( Z
% 0.29/0.45 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0_eq_iff
% 0.29/0.45 thf(fact_23_of__int__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ring_char_0 @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ( ring_1_of_int @ A @ Z )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( Z
% 0.29/0.45 = ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_eq_0_iff
% 0.29/0.45 thf(fact_24_of__int__less__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_less_0_iff
% 0.29/0.45 thf(fact_25_of__int__0__less__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_0_less_iff
% 0.29/0.45 thf(fact_26_less__int__code_I1_J,axiom,
% 0.29/0.45 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_int_code(1)
% 0.29/0.45 thf(fact_27_not__less__less__Suc__eq,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ~ ( ord_less @ nat @ N @ M )
% 0.29/0.45 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.29/0.45 = ( N = M ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_less_Suc_eq
% 0.29/0.45 thf(fact_28_strict__inc__induct,axiom,
% 0.29/0.45 ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ J )
% 0.29/0.45 => ( ! [I2: nat] :
% 0.29/0.45 ( ( J
% 0.29/0.45 = ( suc @ I2 ) )
% 0.29/0.45 => ( P @ I2 ) )
% 0.29/0.45 => ( ! [I2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I2 @ J )
% 0.29/0.45 => ( ( P @ ( suc @ I2 ) )
% 0.29/0.45 => ( P @ I2 ) ) )
% 0.29/0.45 => ( P @ I ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % strict_inc_induct
% 0.29/0.45 thf(fact_29_less__Suc__induct,axiom,
% 0.29/0.45 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ J )
% 0.29/0.45 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.29/0.45 => ( ! [I2: nat,J2: nat,K: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/0.45 => ( ( ord_less @ nat @ J2 @ K )
% 0.29/0.45 => ( ( P @ I2 @ J2 )
% 0.29/0.45 => ( ( P @ J2 @ K )
% 0.29/0.45 => ( P @ I2 @ K ) ) ) ) )
% 0.29/0.45 => ( P @ I @ J ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_Suc_induct
% 0.29/0.45 thf(fact_30_less__trans__Suc,axiom,
% 0.29/0.45 ! [I: nat,J: nat,K2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ J )
% 0.29/0.45 => ( ( ord_less @ nat @ J @ K2 )
% 0.29/0.45 => ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_trans_Suc
% 0.29/0.45 thf(fact_31_Suc__less__SucD,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.45 => ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_less_SucD
% 0.29/0.45 thf(fact_32_less__antisym,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ~ ( ord_less @ nat @ N @ M )
% 0.29/0.45 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.29/0.45 => ( M = N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_antisym
% 0.29/0.45 thf(fact_33_Suc__less__eq2,axiom,
% 0.29/0.45 ! [N: nat,M: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.29/0.45 = ( ? [M2: nat] :
% 0.29/0.45 ( ( M
% 0.29/0.45 = ( suc @ M2 ) )
% 0.29/0.45 & ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_less_eq2
% 0.29/0.45 thf(fact_34_All__less__Suc,axiom,
% 0.29/0.45 ! [N: nat,P: nat > $o] :
% 0.29/0.45 ( ( ! [I3: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.29/0.45 => ( P @ I3 ) ) )
% 0.29/0.45 = ( ( P @ N )
% 0.29/0.45 & ! [I3: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I3 @ N )
% 0.29/0.45 => ( P @ I3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % All_less_Suc
% 0.29/0.45 thf(fact_35_not__less__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.29/0.45 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_eq
% 0.29/0.45 thf(fact_36_less__Suc__eq,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ M @ N )
% 0.29/0.45 | ( M = N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_Suc_eq
% 0.29/0.45 thf(fact_37_Ex__less__Suc,axiom,
% 0.29/0.45 ! [N: nat,P: nat > $o] :
% 0.29/0.45 ( ( ? [I3: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.29/0.45 & ( P @ I3 ) ) )
% 0.29/0.45 = ( ( P @ N )
% 0.29/0.45 | ? [I3: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I3 @ N )
% 0.29/0.45 & ( P @ I3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Ex_less_Suc
% 0.29/0.45 thf(fact_38_less__SucI,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ M @ N )
% 0.29/0.45 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_SucI
% 0.29/0.45 thf(fact_39_less__SucE,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.29/0.45 => ( ~ ( ord_less @ nat @ M @ N )
% 0.29/0.45 => ( M = N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_SucE
% 0.29/0.45 thf(fact_40_Suc__lessI,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ M @ N )
% 0.29/0.45 => ( ( ( suc @ M )
% 0.29/0.45 != N )
% 0.29/0.45 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_lessI
% 0.29/0.45 thf(fact_41_Suc__lessE,axiom,
% 0.29/0.45 ! [I: nat,K2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( suc @ I ) @ K2 )
% 0.29/0.45 => ~ ! [J2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ J2 )
% 0.29/0.45 => ( K2
% 0.29/0.45 != ( suc @ J2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_lessE
% 0.29/0.45 thf(fact_42_Suc__lessD,axiom,
% 0.29/0.45 ! [M: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.29/0.45 => ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_lessD
% 0.29/0.45 thf(fact_43_Nat_OlessE,axiom,
% 0.29/0.45 ! [I: nat,K2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ K2 )
% 0.29/0.45 => ( ( K2
% 0.29/0.45 != ( suc @ I ) )
% 0.29/0.45 => ~ ! [J2: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ I @ J2 )
% 0.29/0.45 => ( K2
% 0.29/0.45 != ( suc @ J2 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Nat.lessE
% 0.29/0.45 thf(fact_44_of__int__pos,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.45 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_pos
% 0.29/0.45 thf(fact_45_measure__induct__rule,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( wellorder @ B2 )
% 0.29/0.45 => ! [F: A > B2,P: A > $o,A2: A] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ! [Y: A] :
% 0.29/0.45 ( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
% 0.29/0.45 => ( P @ Y ) )
% 0.29/0.45 => ( P @ X3 ) )
% 0.29/0.45 => ( P @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % measure_induct_rule
% 0.29/0.45 thf(fact_46_measure__induct,axiom,
% 0.29/0.45 ! [B2: $tType,A: $tType] :
% 0.29/0.45 ( ( wellorder @ B2 )
% 0.29/0.45 => ! [F: A > B2,P: A > $o,A2: A] :
% 0.29/0.45 ( ! [X3: A] :
% 0.29/0.45 ( ! [Y: A] :
% 0.29/0.45 ( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
% 0.29/0.45 => ( P @ Y ) )
% 0.29/0.45 => ( P @ X3 ) )
% 0.29/0.45 => ( P @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % measure_induct
% 0.29/0.45 thf(fact_47_n__not__Suc__n,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( N
% 0.29/0.45 != ( suc @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % n_not_Suc_n
% 0.29/0.45 thf(fact_48_Suc__inject,axiom,
% 0.29/0.45 ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ( suc @ X )
% 0.29/0.45 = ( suc @ Y3 ) )
% 0.29/0.45 => ( X = Y3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Suc_inject
% 0.29/0.45 thf(fact_49_lift__Suc__mono__less__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [F: nat > A,N: nat,M: nat] :
% 0.29/0.45 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.45 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.29/0.45 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % lift_Suc_mono_less_iff
% 0.29/0.45 thf(fact_50_lift__Suc__mono__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A )
% 0.29/0.45 => ! [F: nat > A,N: nat,N3: nat] :
% 0.29/0.45 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.45 => ( ( ord_less @ nat @ N @ N3 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % lift_Suc_mono_less
% 0.29/0.45 thf(fact_51_power__0__Suc,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiring_1 @ A )
% 0.29/0.45 => ! [N: nat] :
% 0.29/0.45 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 0.29/0.45 = ( zero_zero @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_0_Suc
% 0.29/0.45 thf(fact_52_NY0,axiom,
% 0.29/0.45 ord_less @ real @ ( zero_zero @ real ) @ ny ).
% 0.29/0.45
% 0.29/0.45 % NY0
% 0.29/0.45 thf(fact_53_power__eq__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( semiri134348788visors @ A )
% 0.29/0.45 => ! [A2: A,N: nat] :
% 0.29/0.45 ( ( ( power_power @ A @ A2 @ N )
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 = ( ( A2
% 0.29/0.45 = ( zero_zero @ A ) )
% 0.29/0.45 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_eq_0_iff
% 0.29/0.45 thf(fact_54_not__gr__zero,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_gr_zero
% 0.29/0.45 thf(fact_55_realpow__pos__nth2,axiom,
% 0.29/0.45 ! [A2: real,N: nat] :
% 0.29/0.45 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.29/0.45 => ? [R: real] :
% 0.29/0.45 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.29/0.45 & ( ( power_power @ real @ R @ ( suc @ N ) )
% 0.29/0.45 = A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % realpow_pos_nth2
% 0.29/0.45 thf(fact_56_zero__less__power,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [A2: A,N: nat] :
% 0.29/0.45 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_power
% 0.29/0.45 thf(fact_57_NY__def,axiom,
% 0.29/0.45 ( ny
% 0.29/0.45 = ( divide_divide @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ pm ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % NY_def
% 0.29/0.45 thf(fact_58_ex__of__int__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1804426504_field @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [Z2: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_of_int_less
% 0.29/0.45 thf(fact_59_ex__less__of__int,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( archim1804426504_field @ A )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [Z2: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_less_of_int
% 0.29/0.45 thf(fact_60_yyn,axiom,
% 0.29/0.45 ord_less @ int @ na @ ( power_power @ int @ ya @ p ) ).
% 0.29/0.45
% 0.29/0.45 % yyn
% 0.29/0.45 thf(fact_61__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 0.29/0.45 ( pm
% 0.29/0.45 != ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % \<open>pm = 0 \<Longrightarrow> False\<close>
% 0.29/0.45 thf(fact_62_p0,axiom,
% 0.29/0.45 ( p
% 0.29/0.45 != ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % p0
% 0.29/0.45 thf(fact_63_pm0,axiom,
% 0.29/0.45 ord_less @ nat @ ( zero_zero @ nat ) @ pm ).
% 0.29/0.45
% 0.29/0.45 % pm0
% 0.29/0.45 thf(fact_64__C1_Oprems_C_I1_J,axiom,
% 0.29/0.45 ord_less_eq @ int @ na @ ( power_power @ int @ ya @ p ) ).
% 0.29/0.45
% 0.29/0.45 % "1.prems"(1)
% 0.29/0.45 thf(fact_65_p,axiom,
% 0.29/0.45 ( p
% 0.29/0.45 = ( suc @ pm ) ) ).
% 0.29/0.45
% 0.29/0.45 % p
% 0.29/0.45 thf(fact_66_fixed__root__axioms,axiom,
% 0.29/0.45 nthRoo220209705d_root @ p @ pm ).
% 0.29/0.45
% 0.29/0.45 % fixed_root_axioms
% 0.29/0.45 thf(fact_67_le__zero__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( canoni770627133id_add @ A )
% 0.29/0.45 => ! [N: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_zero_eq
% 0.29/0.45 thf(fact_68_nat__power__eq__Suc__0__iff,axiom,
% 0.29/0.45 ! [X: nat,M: nat] :
% 0.29/0.45 ( ( ( power_power @ nat @ X @ M )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.45 = ( ( M
% 0.29/0.45 = ( zero_zero @ nat ) )
% 0.29/0.45 | ( X
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_power_eq_Suc_0_iff
% 0.29/0.45 thf(fact_69_power__Suc__0,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.29/0.45 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_Suc_0
% 0.29/0.45 thf(fact_70_neq0__conv,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( N
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq0_conv
% 0.29/0.45 thf(fact_71_less__nat__zero__code,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_nat_zero_code
% 0.29/0.45 thf(fact_72_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A2: nat] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 != ( zero_zero @ nat ) )
% 0.29/0.45 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_nat_0.not_eq_extremum
% 0.29/0.45 thf(fact_73_nat__zero__less__power__iff,axiom,
% 0.29/0.45 ! [X: nat,N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.29/0.45 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.29/0.45 | ( N
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % nat_zero_less_power_iff
% 0.29/0.45 thf(fact_74_power__Suc0__right,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( monoid_mult @ A )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.45 = A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_Suc0_right
% 0.29/0.45 thf(fact_75_zero__less__Suc,axiom,
% 0.29/0.45 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_Suc
% 0.29/0.45 thf(fact_76_less__Suc0,axiom,
% 0.29/0.45 ! [N: nat] :
% 0.29/0.45 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.45 = ( N
% 0.29/0.45 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_Suc0
% 0.29/0.45 thf(fact_77_of__int__le__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [W: int,Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 0.29/0.45 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_le_iff
% 0.29/0.45 thf(fact_78_power__mono__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_semidom @ A )
% 0.29/0.45 => ! [A2: A,B: A,N: nat] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.45 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.45 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 0.29/0.45 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % power_mono_iff
% 0.29/0.45 thf(fact_79_of__int__le__0__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linordered_idom @ A )
% 0.29/0.45 => ! [Z: int] :
% 0.29/0.45 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 0.29/0.45 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % of_int_le_0_iff
% 0.29/0.45 thf(fact_80_of__int__0__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [Z: int] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 0.29/0.46 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_int_0_le_iff
% 0.29/0.46 thf(fact_81_of__int__le__of__int__power__cancel__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [B: int,W: nat,X: int] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 0.29/0.46 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_int_le_of_int_power_cancel_iff
% 0.29/0.46 thf(fact_82_of__int__power__le__of__int__cancel__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [X: int,B: int,W: nat] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 0.29/0.46 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_int_power_le_of_int_cancel_iff
% 0.29/0.46 thf(fact_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_nat__less__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ! [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N2 )
% 0.29/0.46 => ( P @ M3 ) )
% 0.29/0.46 => ( P @ N2 ) )
% 0.29/0.46 => ( P @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_94_infinite__descent,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ~ ( P @ N2 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N2 )
% 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_95_infinite__descent0,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.29/0.46 => ( ~ ( P @ N2 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N2 )
% 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_96_linorder__neqE__nat,axiom,
% 0.29/0.46 ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( X != Y3 )
% 0.29/0.46 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_nat
% 0.29/0.46 thf(fact_97_infinite__descent__measure,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ~ ( P @ X3 )
% 0.29/0.46 => ? [Y: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
% 0.29/0.46 & ~ ( P @ Y ) ) )
% 0.29/0.46 => ( P @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent_measure
% 0.29/0.46 thf(fact_98_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_strict
% 0.29/0.46 thf(fact_99_infinite__descent0__measure,axiom,
% 0.29/0.46 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( ( V @ X3 )
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 => ( P @ X3 ) )
% 0.29/0.46 => ( ! [X3: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 0.29/0.46 => ( ~ ( P @ X3 )
% 0.29/0.46 => ? [Y: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
% 0.29/0.46 & ~ ( P @ Y ) ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0_measure
% 0.29/0.46 thf(fact_100_nat__power__less__imp__less,axiom,
% 0.29/0.46 ! [I: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 0.29/0.46 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_power_less_imp_less
% 0.29/0.46 thf(fact_101_lift__Suc__antimono__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [F: nat > A,N: nat,N3: nat] :
% 0.29/0.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lift_Suc_antimono_le
% 0.29/0.46 thf(fact_102_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [F: nat > A,N: nat,N3: nat] :
% 0.29/0.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lift_Suc_mono_le
% 0.29/0.46 thf(fact_103_zero__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le
% 0.29/0.46 thf(fact_104_power__divide,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ A )
% 0.29/0.46 => ! [A2: A,B: A,N: nat] :
% 0.29/0.46 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B ) @ N )
% 0.29/0.46 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_divide
% 0.29/0.46 thf(fact_105_ex__le__of__int,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( archim1804426504_field @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_le_of_int
% 0.29/0.46 thf(fact_106_exists__least__lemma,axiom,
% 0.29/0.46 ! [P: nat > $o] :
% 0.29/0.46 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.29/0.46 => ? [N2: nat] :
% 0.29/0.46 ( ~ ( P @ N2 )
% 0.29/0.46 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % exists_least_lemma
% 0.29/0.46 thf(fact_107_power__gt__expt,axiom,
% 0.29/0.46 ! [N: nat,K2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.29/0.46 => ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_gt_expt
% 0.29/0.46 thf(fact_108_less__eq__int__code_I1_J,axiom,
% 0.29/0.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_int_code(1)
% 0.29/0.46 thf(fact_109_nat_Odistinct_I1_J,axiom,
% 0.29/0.46 ! [X2: nat] :
% 0.29/0.46 ( ( zero_zero @ nat )
% 0.29/0.46 != ( suc @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat.distinct(1)
% 0.29/0.46 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 0.29/0.46 ! [Nat3: nat] :
% 0.29/0.46 ( ( suc @ Nat3 )
% 0.29/0.46 != ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.nat.distinct(2)
% 0.29/0.46 thf(fact_111_old_Onat_Odistinct_I1_J,axiom,
% 0.29/0.46 ! [Nat2: nat] :
% 0.29/0.46 ( ( zero_zero @ nat )
% 0.29/0.46 != ( suc @ Nat2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.nat.distinct(1)
% 0.29/0.46 thf(fact_112_nat_OdiscI,axiom,
% 0.29/0.46 ! [Nat: nat,X2: nat] :
% 0.29/0.46 ( ( Nat
% 0.29/0.46 = ( suc @ X2 ) )
% 0.29/0.46 => ( Nat
% 0.29/0.46 != ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat.discI
% 0.29/0.46 thf(fact_113_nat__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( P @ N2 )
% 0.29/0.46 => ( P @ ( suc @ N2 ) ) )
% 0.29/0.46 => ( P @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_induct
% 0.29/0.46 thf(fact_114_diff__induct,axiom,
% 0.29/0.46 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.46 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
% 0.29/0.46 => ( ! [X3: nat,Y4: nat] :
% 0.29/0.46 ( ( P @ X3 @ Y4 )
% 0.29/0.46 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.29/0.46 => ( P @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diff_induct
% 0.29/0.46 thf(fact_115_zero__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,K2: nat] :
% 0.29/0.46 ( ( P @ K2 )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( P @ ( suc @ N2 ) )
% 0.29/0.46 => ( P @ N2 ) )
% 0.29/0.46 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_induct
% 0.29/0.46 thf(fact_116_Suc__neq__Zero,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( suc @ M )
% 0.29/0.46 != ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_neq_Zero
% 0.29/0.46 thf(fact_117_Zero__neq__Suc,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( zero_zero @ nat )
% 0.29/0.46 != ( suc @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % Zero_neq_Suc
% 0.29/0.46 thf(fact_118_Zero__not__Suc,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( zero_zero @ nat )
% 0.29/0.46 != ( suc @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % Zero_not_Suc
% 0.29/0.46 thf(fact_119_old_Onat_Oexhaust,axiom,
% 0.29/0.46 ! [Y3: nat] :
% 0.29/0.46 ( ( Y3
% 0.29/0.46 != ( zero_zero @ nat ) )
% 0.29/0.46 => ~ ! [Nat4: nat] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( suc @ Nat4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.nat.exhaust
% 0.29/0.46 thf(fact_120_old_Onat_Oinducts,axiom,
% 0.29/0.46 ! [P: nat > $o,Nat: nat] :
% 0.29/0.46 ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( ! [Nat4: nat] :
% 0.29/0.46 ( ( P @ Nat4 )
% 0.29/0.46 => ( P @ ( suc @ Nat4 ) ) )
% 0.29/0.46 => ( P @ Nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.nat.inducts
% 0.29/0.46 thf(fact_121_not0__implies__Suc,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( N
% 0.29/0.46 != ( zero_zero @ nat ) )
% 0.29/0.46 => ? [M4: nat] :
% 0.29/0.46 ( N
% 0.29/0.46 = ( suc @ M4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not0_implies_Suc
% 0.29/0.46 thf(fact_122_less__Suc__eq__0__disj,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.29/0.46 = ( ( M
% 0.29/0.46 = ( zero_zero @ nat ) )
% 0.29/0.46 | ? [J3: nat] :
% 0.29/0.46 ( ( M
% 0.29/0.46 = ( suc @ J3 ) )
% 0.29/0.46 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_Suc_eq_0_disj
% 0.29/0.46 thf(fact_123_gr0__implies__Suc,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ? [M4: nat] :
% 0.29/0.46 ( N
% 0.29/0.46 = ( suc @ M4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr0_implies_Suc
% 0.29/0.46 thf(fact_124_All__less__Suc2,axiom,
% 0.29/0.46 ! [N: nat,P: nat > $o] :
% 0.29/0.46 ( ( ! [I3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.29/0.46 => ( P @ I3 ) ) )
% 0.29/0.46 = ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 & ! [I3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ N )
% 0.29/0.46 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % All_less_Suc2
% 0.29/0.46 thf(fact_125_gr0__conv__Suc,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 = ( ? [M5: nat] :
% 0.29/0.46 ( N
% 0.29/0.46 = ( suc @ M5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr0_conv_Suc
% 0.29/0.46 thf(fact_126_Ex__less__Suc2,axiom,
% 0.29/0.46 ! [N: nat,P: nat > $o] :
% 0.29/0.46 ( ( ? [I3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 0.29/0.46 & ( P @ I3 ) ) )
% 0.29/0.46 = ( ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 | ? [I3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I3 @ N )
% 0.29/0.46 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Ex_less_Suc2
% 0.29/0.46 thf(fact_127_power__eq__imp__eq__base,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,N: nat,B: A] :
% 0.29/0.46 ( ( ( power_power @ A @ A2 @ N )
% 0.29/0.46 = ( power_power @ A @ B @ N ) )
% 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 @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( A2 = B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_eq_imp_eq_base
% 0.29/0.46 thf(fact_128_power__eq__iff__eq__base,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [N: nat,A2: A,B: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 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 => ( ( ( power_power @ A @ A2 @ N )
% 0.29/0.46 = ( power_power @ A @ B @ N ) )
% 0.29/0.46 = ( A2 = B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_eq_iff_eq_base
% 0.29/0.46 thf(fact_129_zero__le__power,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_power
% 0.29/0.46 thf(fact_130_power__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,B: A,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_mono
% 0.29/0.46 thf(fact_131_of__int__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [Z: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_int_nonneg
% 0.29/0.46 thf(fact_132_power__less__imp__less__base,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,N: nat,B: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_less_imp_less_base
% 0.29/0.46 thf(fact_133_power__strict__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,B: A,N: nat] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_strict_mono
% 0.29/0.46 thf(fact_134_power__inject__base,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,N: nat,B: A] :
% 0.29/0.46 ( ( ( power_power @ A @ A2 @ ( suc @ N ) )
% 0.29/0.46 = ( power_power @ A @ B @ ( suc @ N ) ) )
% 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 => ( A2 = B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_inject_base
% 0.29/0.46 thf(fact_135_power__le__imp__le__base,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A,N: nat,B: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B @ ( suc @ N ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_le_imp_le_base
% 0.29/0.46 thf(fact_136_realpow__pos__nth,axiom,
% 0.29/0.46 ! [N: nat,A2: real] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.29/0.46 => ? [R: real] :
% 0.29/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.29/0.46 & ( ( power_power @ real @ R @ N )
% 0.29/0.46 = A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % realpow_pos_nth
% 0.29/0.46 thf(fact_137_realpow__pos__nth__unique,axiom,
% 0.29/0.46 ! [N: nat,A2: real] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.29/0.46 => ? [X3: real] :
% 0.29/0.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
% 0.29/0.46 & ( ( power_power @ real @ X3 @ N )
% 0.29/0.46 = A2 )
% 0.29/0.46 & ! [Y: real] :
% 0.29/0.46 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
% 0.29/0.46 & ( ( power_power @ real @ Y @ N )
% 0.29/0.46 = A2 ) )
% 0.29/0.46 => ( Y = X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % realpow_pos_nth_unique
% 0.29/0.46 thf(fact_138_zero__reorient,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ( zero_zero @ A )
% 0.29/0.46 = X )
% 0.29/0.46 = ( X
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_reorient
% 0.29/0.46 thf(fact_139_gr__zeroI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( N
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_zeroI
% 0.29/0.46 thf(fact_140_not__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_zero
% 0.29/0.46 thf(fact_141_gr__implies__not__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [M: A,N: A] :
% 0.29/0.46 ( ( ord_less @ A @ M @ N )
% 0.29/0.46 => ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not_zero
% 0.29/0.46 thf(fact_142_zero__less__iff__neq__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.46 = ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_iff_neq_zero
% 0.29/0.46 thf(fact_143_power__not__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiri134348788visors @ A )
% 0.29/0.46 => ! [A2: A,N: nat] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ( power_power @ A @ A2 @ N )
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % power_not_zero
% 0.29/0.46 thf(fact_144_zero__power,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_1 @ A )
% 0.29/0.46 => ! [N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.29/0.46 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_power
% 0.29/0.46 thf(fact_145_yyn_H,axiom,
% 0.29/0.46 ord_less @ int @ ( power_power @ int @ y @ p ) @ na ).
% 0.29/0.46
% 0.29/0.46 % yyn'
% 0.29/0.46 thf(fact_146_y_H0,axiom,
% 0.29/0.46 ord_less_eq @ int @ ( zero_zero @ int ) @ y ).
% 0.29/0.46
% 0.29/0.46 % y'0
% 0.29/0.46 thf(fact_147_compare__pow__less__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [P2: nat,X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ( ord_less @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
% 0.29/0.46 = ( ord_less @ A @ X @ Y3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compare_pow_less_iff
% 0.29/0.46 thf(fact_148_compare__pow__le__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [P2: nat,X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % compare_pow_le_iff
% 0.29/0.46 thf(fact_149_division__ring__divide__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( division_ring @ 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_150_bits__div__by__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_bits @ 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 % bits_div_by_0
% 0.29/0.46 thf(fact_151_divide__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ 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_152_divide__eq__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ 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_153_divide__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( field @ 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_154_bits__div__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semiring_bits @ 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 % bits_div_0
% 0.29/0.46 thf(fact_155_bot__nat__0_Oextremum,axiom,
% 0.29/0.46 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum
% 0.29/0.46 thf(fact_156_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_157_Suc__le__mono,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_le_mono
% 0.29/0.46 thf(fact_158_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_159_le__trans,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ J @ K2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_160_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_161_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_162_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_163_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [P: nat > $o,K2: nat,B: nat] :
% 0.29/0.46 ( ( P @ K2 )
% 0.29/0.46 => ( ! [Y4: nat] :
% 0.29/0.46 ( ( P @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 0.29/0.46 => ? [X3: nat] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 & ! [Y: nat] :
% 0.29/0.46 ( ( P @ Y )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.ex_has_greatest_nat
% 0.29/0.46 thf(fact_164_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_uniqueI
% 0.29/0.46 thf(fact_165_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.46 ! [A2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_unique
% 0.29/0.46 thf(fact_166_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_167_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_168_transitive__stepwise__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 0.29/0.46 => ( ! [X3: nat,Y4: nat,Z2: nat] :
% 0.29/0.46 ( ( R2 @ X3 @ Y4 )
% 0.29/0.46 => ( ( R2 @ Y4 @ Z2 )
% 0.29/0.46 => ( R2 @ X3 @ Z2 ) ) )
% 0.29/0.46 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 0.29/0.46 => ( R2 @ M @ N ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % transitive_stepwise_le
% 0.29/0.46 thf(fact_169_nat__induct__at__least,axiom,
% 0.29/0.46 ! [M: nat,N: nat,P: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ( P @ M )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.29/0.46 => ( ( P @ N2 )
% 0.29/0.46 => ( P @ ( suc @ N2 ) ) ) )
% 0.29/0.46 => ( P @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_induct_at_least
% 0.29/0.46 thf(fact_170_full__nat__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N2: nat] :
% 0.29/0.46 ( ! [M3: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
% 0.29/0.46 => ( P @ M3 ) )
% 0.29/0.46 => ( P @ N2 ) )
% 0.29/0.46 => ( P @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % full_nat_induct
% 0.29/0.46 thf(fact_171_not__less__eq__eq,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 @ ( suc @ N ) @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_eq_eq
% 0.29/0.46 thf(fact_172_Suc__n__not__le__n,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_n_not_le_n
% 0.29/0.46 thf(fact_173_le__Suc__eq,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.29/0.46 = ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 | ( M
% 0.29/0.46 = ( suc @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_Suc_eq
% 0.29/0.46 thf(fact_174_Suc__le__D,axiom,
% 0.29/0.46 ! [N: nat,M6: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
% 0.29/0.46 => ? [M4: nat] :
% 0.29/0.46 ( M6
% 0.29/0.46 = ( suc @ M4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_le_D
% 0.29/0.46 thf(fact_175_le__SucI,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 @ M @ ( suc @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_SucI
% 0.29/0.46 thf(fact_176_le__SucE,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.29/0.46 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( M
% 0.29/0.46 = ( suc @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_SucE
% 0.29/0.46 thf(fact_177_Suc__leD,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_leD
% 0.29/0.46 thf(fact_178_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_179_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_180_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_181_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M5: nat,N4: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M5 @ N4 )
% 0.29/0.46 | ( M5 = N4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_182_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_183_nat__less__le,axiom,
% 0.29/0.46 ( ( ord_less @ nat )
% 0.29/0.46 = ( ^ [M5: nat,N4: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M5 @ N4 )
% 0.29/0.46 & ( M5 != N4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_184_fixed__root_Op0,axiom,
% 0.29/0.46 ! [P2: nat,Pm: nat] :
% 0.29/0.46 ( ( nthRoo220209705d_root @ P2 @ Pm )
% 0.29/0.46 => ( P2
% 0.29/0.46 != ( zero_zero @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixed_root.p0
% 0.29/0.46 thf(fact_185_fixed__root_Ointro,axiom,
% 0.29/0.46 ! [P2: nat,Pm: nat] :
% 0.29/0.46 ( ( P2
% 0.29/0.46 = ( suc @ Pm ) )
% 0.29/0.46 => ( nthRoo220209705d_root @ P2 @ Pm ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixed_root.intro
% 0.29/0.46 thf(fact_186_fixed__root__def,axiom,
% 0.29/0.46 ( nthRoo220209705d_root
% 0.29/0.46 = ( ^ [P3: nat,Pm2: nat] :
% 0.29/0.46 ( P3
% 0.29/0.46 = ( suc @ Pm2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixed_root_def
% 0.29/0.46 thf(fact_187_fixed__root_Op,axiom,
% 0.29/0.46 ! [P2: nat,Pm: nat] :
% 0.29/0.46 ( ( nthRoo220209705d_root @ P2 @ Pm )
% 0.29/0.46 => ( P2
% 0.29/0.46 = ( suc @ Pm ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fixed_root.p
% 0.29/0.46 thf(fact_188_nat__one__le__power,axiom,
% 0.29/0.46 ! [I: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_one_le_power
% 0.29/0.46 thf(fact_189_ex__least__nat__le,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ( P @ N )
% 0.29/0.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ? [K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ N )
% 0.29/0.46 & ! [I4: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I4 @ K )
% 0.29/0.46 => ~ ( P @ I4 ) )
% 0.29/0.46 & ( P @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_least_nat_le
% 0.29/0.46 thf(fact_190_Suc__leI,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 @ ( suc @ M ) @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_leI
% 0.29/0.46 thf(fact_191_Suc__le__eq,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.29/0.46 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_le_eq
% 0.29/0.46 thf(fact_192_dec__induct,axiom,
% 0.29/0.46 ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( P @ I )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 0.29/0.46 => ( ( ord_less @ nat @ N2 @ J )
% 0.29/0.46 => ( ( P @ N2 )
% 0.29/0.46 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.29/0.46 => ( P @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dec_induct
% 0.29/0.46 thf(fact_193_inc__induct,axiom,
% 0.29/0.46 ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( P @ J )
% 0.29/0.46 => ( ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 0.29/0.46 => ( ( ord_less @ nat @ N2 @ J )
% 0.29/0.46 => ( ( P @ ( suc @ N2 ) )
% 0.29/0.46 => ( P @ N2 ) ) ) )
% 0.29/0.46 => ( P @ I ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % inc_induct
% 0.29/0.46 thf(fact_194_Suc__le__lessD,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_le_lessD
% 0.29/0.46 thf(fact_195_le__less__Suc__eq,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 @ nat @ N @ ( suc @ M ) )
% 0.29/0.46 = ( N = M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_Suc_eq
% 0.29/0.46 thf(fact_196_less__Suc__eq__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_Suc_eq_le
% 0.29/0.46 thf(fact_197_less__eq__Suc__le,axiom,
% 0.29/0.46 ( ( ord_less @ nat )
% 0.29/0.46 = ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_Suc_le
% 0.29/0.46 thf(fact_198_le__imp__less__Suc,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 @ nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_less_Suc
% 0.29/0.46 thf(fact_199_linordered__field__no__ub,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X4: A] :
% 0.29/0.46 ? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_ub
% 0.29/0.46 thf(fact_200_linordered__field__no__lb,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X4: A] :
% 0.29/0.46 ? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_lb
% 0.29/0.46 thf(fact_201_ex__least__nat__less,axiom,
% 0.29/0.46 ! [P: nat > $o,N: nat] :
% 0.29/0.46 ( ( P @ N )
% 0.29/0.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.29/0.46 => ? [K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ K @ N )
% 0.29/0.46 & ! [I4: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I4 @ K )
% 0.29/0.46 => ~ ( P @ I4 ) )
% 0.29/0.46 & ( P @ ( suc @ K ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_least_nat_less
% 0.29/0.46 thf(fact_202_divide__right__mono__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_203_divide__nonpos__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonpos
% 0.29/0.46 thf(fact_204_divide__nonpos__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonneg
% 0.29/0.46 thf(fact_205_divide__nonneg__nonpos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonpos
% 0.29/0.46 thf(fact_206_divide__nonneg__nonneg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonneg
% 0.29/0.46 thf(fact_207_zero__le__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_208_divide__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_209_divide__le__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_210_divide__neg__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_neg_neg
% 0.29/0.46 thf(fact_211_divide__neg__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_neg_pos
% 0.29/0.46 thf(fact_212_divide__pos__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_pos_neg
% 0.29/0.46 thf(fact_213_divide__pos__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_pos_pos
% 0.29/0.46 thf(fact_214_divide__less__0__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_215_divide__less__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_216_zero__less__divide__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_217_divide__strict__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_218_divide__strict__right__mono__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_219_frac__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [Y3: A,X: A,W: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y3 )
% 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 @ Y3 @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_le
% 0.29/0.46 thf(fact_220_frac__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: 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 @ Y3 )
% 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 @ Y3 @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_less
% 0.29/0.46 thf(fact_221_frac__less2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: 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 @ Y3 )
% 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 @ Y3 @ W ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % frac_less2
% 0.29/0.46 thf(fact_222_divide__le__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ 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_223_divide__nonneg__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_neg
% 0.29/0.46 thf(fact_224_divide__nonneg__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_pos
% 0.29/0.46 thf(fact_225_divide__nonpos__neg,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_neg
% 0.29/0.46 thf(fact_226_divide__nonpos__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.29/0.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_pos
% 0.29/0.46 thf(fact_227_div__neg__neg__trivial,axiom,
% 0.29/0.46 ! [K2: int,L: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
% 0.29/0.46 => ( ( ord_less @ int @ L @ K2 )
% 0.29/0.46 => ( ( divide_divide @ int @ K2 @ L )
% 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_228_div__pos__pos__trivial,axiom,
% 0.29/0.46 ! [K2: int,L: int] :
% 0.29/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
% 0.29/0.46 => ( ( ord_less @ int @ K2 @ L )
% 0.29/0.46 => ( ( divide_divide @ int @ K2 @ L )
% 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_229_x,axiom,
% 0.29/0.46 x = y ).
% 0.29/0.46
% 0.29/0.46 % x
% 0.29/0.46 thf(fact_230_div__by__Suc__0,axiom,
% 0.29/0.46 ! [M: nat] :
% 0.29/0.46 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 0.29/0.46 = M ) ).
% 0.29/0.46
% 0.29/0.46 % div_by_Suc_0
% 0.29/0.46 thf(fact_231_div__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 % div_less
% 0.29/0.46 thf(fact_232_div__le__mono,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_le_mono
% 0.29/0.46 thf(fact_233_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_234_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( divide_divide @ nat @ M @ N )
% 0.29/0.46 = ( zero_zero @ 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 % Euclidean_Division.div_eq_0_iff
% 0.29/0.46 thf(fact_235_Suc__div__le__mono,axiom,
% 0.29/0.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % Suc_div_le_mono
% 0.29/0.46 thf(fact_236_div__greater__zero__iff,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 0.29/0.46 = ( ( ord_less_eq @ nat @ N @ M )
% 0.29/0.46 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_greater_zero_iff
% 0.29/0.46 thf(fact_237_div__le__mono2,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K2: 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 @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % div_le_mono2
% 0.29/0.46 thf(fact_238_div__by__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ 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_239_div__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semidom_divide @ 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_240_linorder__neqE__linordered__idom,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( X != Y3 )
% 0.29/0.46 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.29/0.46 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_linordered_idom
% 0.29/0.46 thf(fact_241_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_242_zdiv__mono2,axiom,
% 0.29/0.46 ! [A2: int,B3: 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 ) @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ int @ B3 @ B )
% 0.29/0.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zdiv_mono2
% 0.29/0.46 thf(fact_243_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_244_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_245_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_246_div__positive,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( unique1598680935umeral @ 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 % div_positive
% 0.29/0.46 thf(fact_247_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( unique1598680935umeral @ 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 % unique_euclidean_semiring_numeral_class.div_less
% 0.29/0.46 thf(fact_248_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_249_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_250_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_251_pos__imp__zdiv__pos__iff,axiom,
% 0.29/0.46 ! [K2: int,I: int] :
% 0.29/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
% 0.29/0.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K2 ) )
% 0.29/0.46 = ( ord_less_eq @ int @ K2 @ I ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pos_imp_zdiv_pos_iff
% 0.29/0.46 thf(fact_252_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
% 0.29/0.46 % Type constructors (38)
% 0.29/0.46 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 0.29/0.46 unique1598680935umeral @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.29/0.46 semiri134348788visors @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.29/0.46 linordered_semidom @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.29/0.46 linordered_idom @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 0.29/0.46 semidom_divide @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
% 0.29/0.46 semiring_bits @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 0.29/0.46 monoid_mult @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.29/0.46 semiring_1 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Orderings_Oorder,axiom,
% 0.29/0.46 order @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 0.29/0.46 ring_char_0 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 0.29/0.46 ring_1 @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.29/0.46 zero @ int ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_1,axiom,
% 0.29/0.46 unique1598680935umeral @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.46 canoni770627133id_add @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_2,axiom,
% 0.29/0.46 semiri134348788visors @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_3,axiom,
% 0.29/0.46 linordered_semidom @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Osemidom__divide_4,axiom,
% 0.29/0.46 semidom_divide @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Parity_Osemiring__bits_5,axiom,
% 0.29/0.46 semiring_bits @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.29/0.46 wellorder @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Omonoid__mult_6,axiom,
% 0.29/0.46 monoid_mult @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_7,axiom,
% 0.29/0.46 semiring_1 @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 0.29/0.46 order @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ozero_9,axiom,
% 0.29/0.46 zero @ nat ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 0.29/0.46 order @ $o ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.29/0.46 archim1804426504_field @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_11,axiom,
% 0.29/0.46 semiri134348788visors @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_12,axiom,
% 0.29/0.46 linordered_semidom @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.29/0.46 linordered_field @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom_13,axiom,
% 0.29/0.46 linordered_idom @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Osemidom__divide_14,axiom,
% 0.29/0.46 semidom_divide @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.29/0.46 division_ring @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Omonoid__mult_15,axiom,
% 0.29/0.46 monoid_mult @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Osemiring__1_16,axiom,
% 0.29/0.46 semiring_1 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
% 0.29/0.46 order @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Nat_Oring__char__0_18,axiom,
% 0.29/0.46 ring_char_0 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Rings_Oring__1_19,axiom,
% 0.29/0.46 ring_1 @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.29/0.46 field @ real ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 0.29/0.46 zero @ real ).
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 ord_less @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ ( suc @ pm ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.48/1.43 ---- Embedded in THF ---
% 2.48/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.48/1.43 %%% Generated on Tue Feb 24 08:25:57 EST 2026
% 2.48/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.48/1.43 %%% Logic specification used:
% 2.48/1.43 %%% thf(spec, logic, $$dhol).
% 2.48/1.43
% 2.48/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2DTF15149.p
% 2.48/1.43 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.48/1.43 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.48/1.43 thf(ty_t_Int_Oint, type, int: $tType).
% 2.48/1.43 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Parity_Osemiring__bits, type, semiring_bits: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1598680935umeral: (!> [A:$tType]: ($o))).
% 2.48/1.43 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.48/1.43 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 2.48/1.43 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 2.48/1.43 thf(sy_c_NthRoot__Impl__Mirabelle__yrjxwmcnbt_Ofixed__root, type, nthRoo220209705d_root: (nat > (nat > $o))).
% 2.48/1.43 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.48/1.43 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.48/1.43 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.48/1.43 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.48/1.43 thf(sy_v_NY____, type, ny: real).
% 2.48/1.43 thf(sy_v_na____, type, na: int).
% 2.48/1.43 thf(sy_v_p, type, p: nat).
% 2.48/1.43 thf(sy_v_pm, type, pm: nat).
% 2.48/1.43 thf(sy_v_x, type, x: int).
% 2.48/1.43 thf(sy_v_y_H____, type, y: int).
% 2.48/1.43 thf(sy_v_ya____, type, ya: int).
% 2.48/1.43 thf(fact_0_n00, axiom, (na != (zero_zero @ int))).
% 2.48/1.43 thf(fact_1_of__int__less__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B:int,W:nat,X:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less @ int) @ (((power_power @ int) @ B) @ W)) @ X)))))))).
% 2.48/1.43 thf(fact_2_of__int__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B:int,W:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (((ord_less @ int) @ X) @ (((power_power @ int) @ B) @ W))))))))).
% 2.48/1.43 thf(fact_3_of__int__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z:int,N:nat]: ((((ring_1_of_int @ A) @ (((power_power @ int) @ Z) @ N)) = (((power_power @ A) @ ((ring_1_of_int @ A) @ Z)) @ N)))))))).
% 2.48/1.43 thf(fact_4_of__int__eq__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [B:int,W:nat,X:int]: ((((((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W) = ((ring_1_of_int @ A) @ X)) = ((((power_power @ int) @ B) @ W) = X)))))))).
% 2.48/1.43 thf(fact_5_of__int__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X:int,B:int,W:nat]: (((((ring_1_of_int @ A) @ X) = (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (X = (((power_power @ int) @ B) @ W))))))))).
% 2.48/1.43 thf(fact_6_of__int__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less @ int) @ W) @ Z)))))))).
% 2.48/1.43 thf(fact_7_n0, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ ((ring_1_of_int @ real) @ na))).
% 2.48/1.43 thf(fact_8_y0, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ ((ring_1_of_int @ real) @ ya))).
% 2.48/1.43 thf(fact_9_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W:int,Z:int]: (((((ring_1_of_int @ A) @ W) = ((ring_1_of_int @ A) @ Z)) = (W = Z)))))))).
% 2.48/1.43 thf(fact_10_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 2.48/1.43 thf(fact_11_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 2.48/1.43 thf(fact_12_y00, axiom, (ya != (zero_zero @ int))).
% 2.48/1.43 thf(fact_13__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062, axiom, ((((ord_less @ int) @ (zero_zero @ int)) @ ya) | (ya = (zero_zero @ int)))).
% 2.48/1.43 thf(fact_14_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_15_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 2.48/1.43 thf(fact_16_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 2.48/1.43 thf(fact_17_not_I1_J, axiom, (~ (((((ord_less @ int) @ ya) @ (zero_zero @ int)) | (((ord_less @ int) @ na) @ (zero_zero @ int)))))).
% 2.48/1.43 thf(fact_18_not_I2_J, axiom, (~ (((((ord_less @ int) @ ya) @ (zero_zero @ int)) | (((ord_less @ int) @ na) @ (zero_zero @ int)))))).
% 2.48/1.43 thf(fact_19__C1_Oprems_C_I2_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ ya)).
% 2.48/1.43 thf(fact_20__C1_Oprems_C_I3_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ na)).
% 2.48/1.43 thf(fact_21_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 2.48/1.43 thf(fact_22_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z)) = (Z = (zero_zero @ int))))))))).
% 2.48/1.43 thf(fact_23_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z:int]: (((((ring_1_of_int @ A) @ Z) = (zero_zero @ A)) = (Z = (zero_zero @ int))))))))).
% 2.48/1.43 thf(fact_24_of__int__less__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.48/1.43 thf(fact_25_of__int__0__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z)))))))).
% 2.48/1.43 thf(fact_26_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.48/1.43 thf(fact_27_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 2.48/1.43 thf(fact_28_strict__inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I) @ J) => ((! [I2:nat]: (((J = (suc @ I2)) => (P @ I2)))) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ J) => ((P @ (suc @ I2)) => (P @ I2))))) => (P @ I))))))).
% 2.48/1.43 thf(fact_29_less__Suc__induct, axiom, (! [I:nat,J:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I) @ J) => ((! [I2:nat]: (((P @ I2) @ (suc @ I2)))) => ((! [I2:nat,J2:nat,K:nat]: (((((ord_less @ nat) @ I2) @ J2) => ((((ord_less @ nat) @ J2) @ K) => (((P @ I2) @ J2) => (((P @ J2) @ K) => ((P @ I2) @ K))))))) => ((P @ I) @ J))))))).
% 2.48/1.43 thf(fact_30_less__trans__Suc, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ K2) => (((ord_less @ nat) @ (suc @ I)) @ K2)))))).
% 2.48/1.43 thf(fact_31_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_32_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 2.48/1.43 thf(fact_33_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M2:nat]: (((M = (suc @ M2)) & (((ord_less @ nat) @ N) @ M2)))))))).
% 2.48/1.43 thf(fact_34_All__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) => (P @ I3)))) = ((P @ N) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => (P @ I3))))))))).
% 2.48/1.43 thf(fact_35_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 2.48/1.43 thf(fact_36_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 2.48/1.43 thf(fact_37_Ex__less__Suc, axiom, (! [N:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) & (P @ I3)))) = ((P @ N) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N) & (P @ I3))))))))).
% 2.48/1.43 thf(fact_38_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 2.48/1.43 thf(fact_39_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 2.48/1.43 thf(fact_40_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 2.48/1.43 thf(fact_41_Suc__lessE, axiom, (! [I:nat,K2:nat]: (((((ord_less @ nat) @ (suc @ I)) @ K2) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K2 != (suc @ J2))))))))))).
% 2.48/1.43 thf(fact_42_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_43_Nat_OlessE, axiom, (! [I:nat,K2:nat]: (((((ord_less @ nat) @ I) @ K2) => ((K2 != (suc @ I)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K2 != (suc @ J2)))))))))))).
% 2.48/1.43 thf(fact_44_of__int__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z) => (((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.48/1.43 thf(fact_45_measure__induct__rule, axiom, (! [B2:$tType,A:$tType]: (((wellorder @ B2) => (! [F:(A > B2),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y:A]: (((((ord_less @ B2) @ (F @ Y)) @ (F @ X3)) => (P @ Y)))) => (P @ X3)))) => (P @ A2)))))))).
% 2.48/1.43 thf(fact_46_measure__induct, axiom, (! [B2:$tType,A:$tType]: (((wellorder @ B2) => (! [F:(A > B2),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y:A]: (((((ord_less @ B2) @ (F @ Y)) @ (F @ X3)) => (P @ Y)))) => (P @ X3)))) => (P @ A2)))))))).
% 2.48/1.43 thf(fact_47_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 2.48/1.43 thf(fact_48_Suc__inject, axiom, (! [X:nat,Y3:nat]: ((((suc @ X) = (suc @ Y3)) => (X = Y3))))).
% 2.48/1.43 thf(fact_49_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,M:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 2.48/1.43 thf(fact_50_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less @ nat) @ N) @ N3) => (((ord_less @ A) @ (F @ N)) @ (F @ N3)))))))))).
% 2.48/1.43 thf(fact_51_power__0__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ (zero_zero @ A)) @ (suc @ N)) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_52_NY0, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ ny)).
% 2.48/1.43 thf(fact_53_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: ((((((power_power @ A) @ A2) @ N) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 2.48/1.43 thf(fact_54_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_55_realpow__pos__nth2, axiom, (! [A2:real,N:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [R:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) & ((((power_power @ real) @ R) @ (suc @ N)) = A2)))))))).
% 2.48/1.43 thf(fact_56_zero__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.48/1.43 thf(fact_57_NY__def, axiom, (ny = (((divide_divide @ real) @ ((ring_1_of_int @ real) @ na)) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ ya)) @ pm)))).
% 2.48/1.43 thf(fact_58_ex__of__int__less, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [Z2:int]: ((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X))))))))).
% 2.48/1.43 thf(fact_59_ex__less__of__int, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [Z2:int]: ((((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ Z2)))))))))).
% 2.48/1.43 thf(fact_60_yyn, axiom, (((ord_less @ int) @ na) @ (((power_power @ int) @ ya) @ p))).
% 2.48/1.43 thf(fact_61__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom, (pm != (zero_zero @ nat))).
% 2.48/1.43 thf(fact_62_p0, axiom, (p != (zero_zero @ nat))).
% 2.48/1.43 thf(fact_63_pm0, axiom, (((ord_less @ nat) @ (zero_zero @ nat)) @ pm)).
% 2.48/1.43 thf(fact_64__C1_Oprems_C_I1_J, axiom, (((ord_less_eq @ int) @ na) @ (((power_power @ int) @ ya) @ p))).
% 2.48/1.43 thf(fact_65_p, axiom, (p = (suc @ pm))).
% 2.48/1.43 thf(fact_66_fixed__root__axioms, axiom, ((nthRoo220209705d_root @ p) @ pm)).
% 2.48/1.43 thf(fact_67_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_68_nat__power__eq__Suc__0__iff, axiom, (! [X:nat,M:nat]: ((((((power_power @ nat) @ X) @ M) = (suc @ (zero_zero @ nat))) = ((M = (zero_zero @ nat)) | (X = (suc @ (zero_zero @ nat)))))))).
% 2.48/1.43 thf(fact_69_power__Suc__0, axiom, (! [N:nat]: (((((power_power @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (suc @ (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_70_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.48/1.43 thf(fact_71_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_72_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.48/1.43 thf(fact_73_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_74_power__Suc0__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))).
% 2.48/1.43 thf(fact_75_zero__less__Suc, axiom, (! [N:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N))))).
% 2.48/1.43 thf(fact_76_less__Suc0, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (N = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_77_of__int__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ W) @ Z)))))))).
% 2.48/1.43 thf(fact_78_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)) = (((ord_less_eq @ A) @ A2) @ B))))))))))).
% 2.48/1.43 thf(fact_79_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z) @ (zero_zero @ int))))))))).
% 2.48/1.43 thf(fact_80_of__int__0__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z)))))))).
% 2.48/1.43 thf(fact_81_of__int__le__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B:int,W:nat,X:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) @ ((ring_1_of_int @ A) @ X)) = (((ord_less_eq @ int) @ (((power_power @ int) @ B) @ W)) @ X)))))))).
% 2.48/1.43 thf(fact_82_of__int__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:int,B:int,W:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B)) @ W)) = (((ord_less_eq @ int) @ X) @ (((power_power @ int) @ B) @ W))))))))).
% 2.48/1.43 thf(fact_83_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.48/1.43 thf(fact_84_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_85_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_86_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_87_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.48/1.43 thf(fact_88_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.48/1.43 thf(fact_89_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.48/1.43 thf(fact_90_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.48/1.43 thf(fact_91_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_92_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.48/1.43 thf(fact_93_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N))))).
% 2.48/1.43 thf(fact_94_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P @ M3))))))))) => (P @ N))))).
% 2.48/1.43 thf(fact_95_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N2) & (~ ((P @ M3)))))))))) => (P @ N)))))).
% 2.48/1.43 thf(fact_96_linorder__neqE__nat, axiom, (! [X:nat,Y3:nat]: (((X != Y3) => ((~ ((((ord_less @ nat) @ X) @ Y3))) => (((ord_less @ nat) @ Y3) @ X)))))).
% 2.48/1.43 thf(fact_97_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X3)) & (~ ((P @ Y))))))))) => (P @ X))))).
% 2.48/1.43 thf(fact_98_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_99_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X:A]: (((! [X3:A]: ((((V @ X3) = (zero_zero @ nat)) => (P @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X3)) => ((~ ((P @ X3))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X3)) & (~ ((P @ Y)))))))))) => (P @ X)))))).
% 2.48/1.43 thf(fact_100_nat__power__less__imp__less, axiom, (! [I:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.48/1.43 thf(fact_101_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N2))) @ (F @ N2)))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N3)) @ (F @ N)))))))))).
% 2.48/1.43 thf(fact_102_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F:(nat > A),N:nat,N3:nat]: (((! [N2:nat]: ((((ord_less_eq @ A) @ (F @ N2)) @ (F @ (suc @ N2))))) => ((((ord_less_eq @ nat) @ N) @ N3) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N3)))))))))).
% 2.48/1.43 thf(fact_103_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.48/1.43 thf(fact_104_power__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ A2) @ B)) @ N) = (((divide_divide @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N))))))))).
% 2.48/1.43 thf(fact_105_ex__le__of__int, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [Z2:int]: ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z2)))))))))).
% 2.48/1.43 thf(fact_106_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X_1:nat]: ((P @ X_1))) => (? [N2:nat]: (((~ ((P @ N2))) & (P @ (suc @ N2)))))))))).
% 2.48/1.43 thf(fact_107_power__gt__expt, axiom, (! [N:nat,K2:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N) => (((ord_less @ nat) @ K2) @ (((power_power @ nat) @ N) @ K2)))))).
% 2.48/1.43 thf(fact_108_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.48/1.43 thf(fact_109_nat_Odistinct_I1_J, axiom, (! [X2:nat]: (((zero_zero @ nat) != (suc @ X2))))).
% 2.48/1.43 thf(fact_110_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))).
% 2.48/1.43 thf(fact_111_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))).
% 2.48/1.43 thf(fact_112_nat_OdiscI, axiom, (! [Nat:nat,X2:nat]: (((Nat = (suc @ X2)) => (Nat != (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_113_nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((P @ N2) => (P @ (suc @ N2))))) => (P @ N)))))).
% 2.48/1.43 thf(fact_114_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N:nat]: (((! [X3:nat]: (((P @ X3) @ (zero_zero @ nat)))) => ((! [Y4:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y4)))) => ((! [X3:nat,Y4:nat]: ((((P @ X3) @ Y4) => ((P @ (suc @ X3)) @ (suc @ Y4))))) => ((P @ M) @ N))))))).
% 2.48/1.43 thf(fact_115_zero__induct, axiom, (! [P:(nat > $o),K2:nat]: (((P @ K2) => ((! [N2:nat]: (((P @ (suc @ N2)) => (P @ N2)))) => (P @ (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_116_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))).
% 2.48/1.43 thf(fact_117_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.48/1.43 thf(fact_118_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))).
% 2.48/1.43 thf(fact_119_old_Onat_Oexhaust, axiom, (! [Y3:nat]: (((Y3 != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y3 != (suc @ Nat4)))))))))).
% 2.48/1.43 thf(fact_120_old_Onat_Oinducts, axiom, (! [P:(nat > $o),Nat:nat]: (((P @ (zero_zero @ nat)) => ((! [Nat4:nat]: (((P @ Nat4) => (P @ (suc @ Nat4))))) => (P @ Nat)))))).
% 2.48/1.43 thf(fact_121_not0__implies__Suc, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (? [M4:nat]: ((N = (suc @ M4)))))))).
% 2.48/1.43 thf(fact_122_less__Suc__eq__0__disj, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((M = (zero_zero @ nat)) | (? [J3:nat]: (((M = (suc @ J3)) & (((ord_less @ nat) @ J3) @ N))))))))).
% 2.48/1.43 thf(fact_123_gr0__implies__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (? [M4:nat]: ((N = (suc @ M4)))))))).
% 2.48/1.43 thf(fact_124_All__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) => (P @ I3)))) = ((P @ (zero_zero @ nat)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => (P @ (suc @ I3)))))))))).
% 2.48/1.43 thf(fact_125_gr0__conv__Suc, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) = (? [M5:nat]: ((N = (suc @ M5)))))))).
% 2.48/1.43 thf(fact_126_Ex__less__Suc2, axiom, (! [N:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N)) & (P @ I3)))) = ((P @ (zero_zero @ nat)) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N) & (P @ (suc @ I3)))))))))).
% 2.48/1.43 thf(fact_127_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: ((((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (A2 = B))))))))))).
% 2.48/1.43 thf(fact_128_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,A2:A,B:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B) @ N)) = (A2 = B))))))))))).
% 2.48/1.43 thf(fact_129_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.48/1.43 thf(fact_130_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)))))))))).
% 2.48/1.43 thf(fact_131_of__int__nonneg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z))))))))).
% 2.48/1.43 thf(fact_132_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 2.48/1.43 thf(fact_133_power__strict__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B:A,N:nat]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B) @ N))))))))))).
% 2.48/1.43 thf(fact_134_power__inject__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: ((((((power_power @ A) @ A2) @ (suc @ N)) = (((power_power @ A) @ B) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (A2 = B)))))))))).
% 2.48/1.43 thf(fact_135_power__le__imp__le__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat,B:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ (suc @ N))) @ (((power_power @ A) @ B) @ (suc @ N))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B) => (((ord_less_eq @ A) @ A2) @ B))))))))).
% 2.48/1.43 thf(fact_136_realpow__pos__nth, axiom, (! [N:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [R:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) & ((((power_power @ real) @ R) @ N) = A2))))))))).
% 2.48/1.43 thf(fact_137_realpow__pos__nth__unique, axiom, (! [N:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) & (((((power_power @ real) @ X3) @ N) = A2) & (! [Y:real]: ((((((ord_less @ real) @ (zero_zero @ real)) @ Y) & ((((power_power @ real) @ Y) @ N) = A2)) => (Y = X3))))))))))))).
% 2.48/1.43 thf(fact_138_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_139_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.48/1.43 thf(fact_140_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_141_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_142_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_143_power__not__zero, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_144_zero__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A))))))))).
% 2.48/1.43 thf(fact_145_yyn_H, axiom, (((ord_less @ int) @ (((power_power @ int) @ y) @ p)) @ na)).
% 2.48/1.43 thf(fact_146_y_H0, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ y)).
% 2.48/1.43 thf(fact_147_compare__pow__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P2:nat,X:A,Y3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ P2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less @ A) @ (((power_power @ A) @ X) @ P2)) @ (((power_power @ A) @ Y3) @ P2)) = (((ord_less @ A) @ X) @ Y3))))))))))).
% 2.48/1.43 thf(fact_148_compare__pow__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P2:nat,X:A,Y3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ P2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ (((power_power @ A) @ X) @ P2)) @ (((power_power @ A) @ Y3) @ P2)) = (((ord_less_eq @ A) @ X) @ Y3))))))))))).
% 2.48/1.43 thf(fact_149_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_150_bits__div__by__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_151_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,C:A,B:A]: ((((((divide_divide @ A) @ A2) @ C) = (((divide_divide @ A) @ B) @ C)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 2.48/1.43 thf(fact_152_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B:A]: ((((((divide_divide @ A) @ A2) @ B) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B = (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_153_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C:A,A2:A,B:A]: ((((((divide_divide @ A) @ C) @ A2) = (((divide_divide @ A) @ C) @ B)) = ((C = (zero_zero @ A)) | (A2 = B))))))))).
% 2.48/1.43 thf(fact_154_bits__div__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_155_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.48/1.43 thf(fact_156_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.48/1.43 thf(fact_157_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 2.48/1.43 thf(fact_158_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.48/1.43 thf(fact_159_le__trans, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K2) => (((ord_less_eq @ nat) @ I) @ K2)))))).
% 2.48/1.43 thf(fact_160_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_161_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.48/1.43 thf(fact_162_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.48/1.43 thf(fact_163_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K2:nat,B:nat]: (((P @ K2) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B)))) => (? [X3:nat]: (((P @ X3) & (! [Y:nat]: (((P @ Y) => (((ord_less_eq @ nat) @ Y) @ X3)))))))))))).
% 2.48/1.43 thf(fact_164_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_165_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_166_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_167_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.48/1.43 thf(fact_168_transitive__stepwise__le, axiom, (! [M:nat,N:nat,R2:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M) @ N) => ((! [X3:nat]: (((R2 @ X3) @ X3))) => ((! [X3:nat,Y4:nat,Z2:nat]: ((((R2 @ X3) @ Y4) => (((R2 @ Y4) @ Z2) => ((R2 @ X3) @ Z2))))) => ((! [N2:nat]: (((R2 @ N2) @ (suc @ N2)))) => ((R2 @ M) @ N)))))))).
% 2.48/1.43 thf(fact_169_nat__induct__at__least, axiom, (! [M:nat,N:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M) @ N) => ((P @ M) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((P @ N2) => (P @ (suc @ N2)))))) => (P @ N))))))).
% 2.48/1.43 thf(fact_170_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M3:nat]: (((((ord_less_eq @ nat) @ (suc @ M3)) @ N2) => (P @ M3)))) => (P @ N2)))) => (P @ N))))).
% 2.48/1.43 thf(fact_171_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 2.48/1.43 thf(fact_172_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 2.48/1.43 thf(fact_173_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 2.48/1.43 thf(fact_174_Suc__le__D, axiom, (! [N:nat,M6:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M6) => (? [M4:nat]: ((M6 = (suc @ M4)))))))).
% 2.48/1.43 thf(fact_175_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 2.48/1.43 thf(fact_176_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 2.48/1.43 thf(fact_177_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_178_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))))))).
% 2.48/1.43 thf(fact_179_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.48/1.43 thf(fact_180_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_181_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M5:nat,N4:nat]: (((((ord_less @ nat) @ M5) @ N4) | (M5 = N4)))))).
% 2.48/1.43 thf(fact_182_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_183_nat__less__le, axiom, ((ord_less @ nat) = (^ [M5:nat,N4:nat]: (((((ord_less_eq @ nat) @ M5) @ N4) & (M5 != N4)))))).
% 2.48/1.43 thf(fact_184_fixed__root_Op0, axiom, (! [P2:nat,Pm:nat]: ((((nthRoo220209705d_root @ P2) @ Pm) => (P2 != (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_185_fixed__root_Ointro, axiom, (! [P2:nat,Pm:nat]: (((P2 = (suc @ Pm)) => ((nthRoo220209705d_root @ P2) @ Pm))))).
% 2.48/1.43 thf(fact_186_fixed__root__def, axiom, (nthRoo220209705d_root = (^ [P3:nat,Pm2:nat]: ((P3 = (suc @ Pm2)))))).
% 2.48/1.43 thf(fact_187_fixed__root_Op, axiom, (! [P2:nat,Pm:nat]: ((((nthRoo220209705d_root @ P2) @ Pm) => (P2 = (suc @ Pm)))))).
% 2.48/1.43 thf(fact_188_nat__one__le__power, axiom, (! [I:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ I) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((power_power @ nat) @ I) @ N)))))).
% 2.48/1.43 thf(fact_189_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K:nat]: (((((ord_less_eq @ nat) @ K) @ N) & ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => (~ ((P @ I4)))))) & (P @ K)))))))))).
% 2.48/1.43 thf(fact_190_Suc__leI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (suc @ M)) @ N))))).
% 2.48/1.43 thf(fact_191_Suc__le__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) = (((ord_less @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_192_dec__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ I) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((ord_less @ nat) @ N2) @ J) => ((P @ N2) => (P @ (suc @ N2))))))) => (P @ J))))))).
% 2.48/1.43 thf(fact_193_inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ J) => ((! [N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((ord_less @ nat) @ N2) @ J) => ((P @ (suc @ N2)) => (P @ N2)))))) => (P @ I))))))).
% 2.48/1.43 thf(fact_194_Suc__le__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_195_le__less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 2.48/1.43 thf(fact_196_less__Suc__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.43 thf(fact_197_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N4:nat]: (((ord_less_eq @ nat) @ (suc @ N4)))))).
% 2.48/1.43 thf(fact_198_le__imp__less__Suc, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 2.48/1.43 thf(fact_199_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [X_12:A]: ((((ord_less @ A) @ X4) @ X_12))))))))).
% 2.48/1.43 thf(fact_200_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X4))))))))).
% 2.48/1.43 thf(fact_201_ex__least__nat__less, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K:nat]: (((((ord_less @ nat) @ K) @ N) & ((! [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ K) => (~ ((P @ I4)))))) & (P @ (suc @ K))))))))))).
% 2.48/1.43 thf(fact_202_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.43 thf(fact_203_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_204_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_205_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_206_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_207_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))))))))))).
% 2.48/1.43 thf(fact_208_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.43 thf(fact_209_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ B)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B)))))))))).
% 2.48/1.43 thf(fact_210_divide__neg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_211_divide__neg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_212_divide__pos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_213_divide__pos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_214_divide__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.43 thf(fact_215_divide__less__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ A2) @ B)) & (((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ B) @ A2)) & (C != (zero_zero @ A))))))))))).
% 2.48/1.43 thf(fact_216_zero__less__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ 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))))))))))).
% 2.48/1.43 thf(fact_217_divide__strict__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B) @ C)))))))))).
% 2.48/1.43 thf(fact_218_divide__strict__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.43 thf(fact_219_frac__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y3:A,X:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y3) => ((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y3) @ W)))))))))))).
% 2.48/1.43 thf(fact_220_frac__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y3) @ W)))))))))))).
% 2.48/1.43 thf(fact_221_frac__less2, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A,W:A,Z:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y3) @ W)))))))))))).
% 2.48/1.43 thf(fact_222_divide__le__cancel, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.43 thf(fact_223_divide__nonneg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_224_divide__nonneg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_225_divide__nonpos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y3)))))))))).
% 2.48/1.43 thf(fact_226_divide__nonpos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y3) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y3)) @ (zero_zero @ A)))))))))).
% 2.48/1.43 thf(fact_227_div__neg__neg__trivial, axiom, (! [K2:int,L:int]: (((((ord_less_eq @ int) @ K2) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K2) => ((((divide_divide @ int) @ K2) @ L) = (zero_zero @ int))))))).
% 2.48/1.43 thf(fact_228_div__pos__pos__trivial, axiom, (! [K2:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K2) => ((((ord_less @ int) @ K2) @ L) => ((((divide_divide @ int) @ K2) @ L) = (zero_zero @ int))))))).
% 2.48/1.43 thf(fact_229_x, axiom, (x = y)).
% 2.48/1.43 thf(fact_230_div__by__Suc__0, axiom, (! [M:nat]: (((((divide_divide @ nat) @ M) @ (suc @ (zero_zero @ nat))) = M)))).
% 2.48/1.43 thf(fact_231_div__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.48/1.43 thf(fact_232_div__le__mono, axiom, (! [M:nat,N:nat,K2:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K2)) @ (((divide_divide @ nat) @ N) @ K2)))))).
% 2.48/1.43 thf(fact_233_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 2.48/1.43 thf(fact_234_Euclidean__Division_Odiv__eq__0__iff, axiom, (! [M:nat,N:nat]: ((((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)) = ((((ord_less @ nat) @ M) @ N) | (N = (zero_zero @ nat))))))).
% 2.48/1.43 thf(fact_235_Suc__div__le__mono, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ (((divide_divide @ nat) @ (suc @ M)) @ N))))).
% 2.48/1.43 thf(fact_236_div__greater__zero__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ M) @ N)) = ((((ord_less_eq @ nat) @ N) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 2.48/1.43 thf(fact_237_div__le__mono2, axiom, (! [M:nat,N:nat,K2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ K2) @ N)) @ (((divide_divide @ nat) @ K2) @ M))))))).
% 2.48/1.43 thf(fact_238_div__by__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_239_div__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.48/1.43 thf(fact_240_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((X != Y3) => ((~ ((((ord_less @ A) @ X) @ Y3))) => (((ord_less @ A) @ Y3) @ X))))))))).
% 2.48/1.43 thf(fact_241_zdiv__mono1, axiom, (! [A2:int,A3:int,B:int]: (((((ord_less_eq @ int) @ A2) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (((divide_divide @ int) @ A3) @ B))))))).
% 2.48/1.43 thf(fact_242_zdiv__mono2, axiom, (! [A2:int,B3:int,B:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ B3) => ((((ord_less_eq @ int) @ B3) @ B) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (((divide_divide @ int) @ A2) @ B3)))))))).
% 2.48/1.43 thf(fact_243_pos__imp__zdiv__neg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int)) = (((ord_less @ int) @ A2) @ (zero_zero @ int))))))).
% 2.48/1.43 thf(fact_244_neg__imp__zdiv__neg__iff, axiom, (! [B:int,A2:int]: (((((ord_less @ int) @ B) @ (zero_zero @ int)) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int)) = (((ord_less @ int) @ (zero_zero @ int)) @ A2)))))).
% 2.48/1.43 thf(fact_245_div__neg__pos__less0, axiom, (! [A2:int,B:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B) => (((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B)) @ (zero_zero @ int))))))).
% 2.48/1.43 thf(fact_246_div__positive, axiom, (! [A:$tType]: (((unique1598680935umeral @ 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)))))))))).
% 2.48/1.43 thf(fact_247_unique__euclidean__semiring__numeral__class_Odiv__less, axiom, (! [A:$tType]: (((unique1598680935umeral @ 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)))))))))).
% 2.48/1.43 thf(fact_248_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))))))).
% 2.48/1.43 thf(fact_249_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)))))).
% 2.48/1.43 thf(fact_250_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))))))).
% 2.48/1.43 thf(fact_251_pos__imp__zdiv__pos__iff, axiom, (! [K2:int,I:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K2) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ I) @ K2)) = (((ord_less_eq @ int) @ K2) @ I)))))).
% 2.48/1.43 thf(fact_252_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))))))).
% 2.48/1.43 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1598680935umeral @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, (semidom_divide @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Parity_Osemiring__bits, axiom, (semiring_bits @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Orderings_Oorder, axiom, (order @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)).
% 2.48/1.43 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.48/1.43 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_1, axiom, (unique1598680935umeral @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_2, axiom, (semiri134348788visors @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_3, axiom, (linordered_semidom @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Rings_Osemidom__divide_4, axiom, (semidom_divide @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Parity_Osemiring__bits_5, axiom, (semiring_bits @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Groups_Omonoid__mult_6, axiom, (monoid_mult @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Rings_Osemiring__1_7, axiom, (semiring_1 @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Orderings_Oorder_8, axiom, (order @ nat)).
% 2.48/1.43 thf(tcon_Nat_Onat___Groups_Ozero_9, axiom, (zero @ nat)).
% 2.48/1.43 thf(tcon_HOL_Obool___Orderings_Oorder_10, axiom, (order @ $o)).
% 2.48/1.43 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, (archim1804426504_field @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_11, axiom, (semiri134348788visors @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_12, axiom, (linordered_semidom @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Olinordered__idom_13, axiom, (linordered_idom @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Osemidom__divide_14, axiom, (semidom_divide @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Groups_Omonoid__mult_15, axiom, (monoid_mult @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Osemiring__1_16, axiom, (semiring_1 @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Orderings_Oorder_17, axiom, (order @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Nat_Oring__char__0_18, axiom, (ring_char_0 @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Rings_Oring__1_19, axiom, (ring_1 @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 2.48/1.43 thf(tcon_Real_Oreal___Groups_Ozero_20, axiom, (zero @ real)).
% 2.48/1.43 thf(conj_0, conjecture, (((ord_less @ real) @ ((ring_1_of_int @ real) @ na)) @ (((power_power @ real) @ ((ring_1_of_int @ real) @ ya)) @ (suc @ pm)))).
% 2.48/1.43 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2DTF15149.p
% 2.48/1.43 ------------------------
% 2.48/1.46 ---- Cleaned THF ---
% 2.48/1.46 thf(ty_t_Real_Oreal,type,
% 2.48/1.46 real: $tType ).
% 2.48/1.46
% 2.48/1.46 thf(ty_t_Nat_Onat,type,
% 2.48/1.46 nat: $tType ).
% 2.48/1.46
% 2.48/1.46 thf(ty_t_Int_Oint,type,
% 2.48/1.46 int: $tType ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Groups_Ozero,type,
% 2.48/1.46 zero:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Fields_Ofield,type,
% 2.48/1.46 field:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Oring__1,type,
% 2.48/1.46 ring_1:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Nat_Oring__char__0,type,
% 2.48/1.46 ring_char_0:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Orderings_Oorder,type,
% 2.48/1.46 order:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Osemiring__1,type,
% 2.48/1.46 semiring_1:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Groups_Omonoid__mult,type,
% 2.48/1.46 monoid_mult:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Orderings_Owellorder,type,
% 2.48/1.46 wellorder:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Fields_Odivision__ring,type,
% 2.48/1.46 division_ring:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Parity_Osemiring__bits,type,
% 2.48/1.46 semiring_bits:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 2.48/1.46 semidom_divide:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.48/1.46 linordered_idom:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Fields_Olinordered__field,type,
% 2.48/1.46 linordered_field:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.48/1.46 linordered_semidom:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.48/1.46 semiri134348788visors:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.48/1.46 archim1804426504_field:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.48/1.46 canoni770627133id_add:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 2.48/1.46 unique1598680935umeral:
% 2.48/1.46 !>[A: $tType] : $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.48/1.46 zero_zero:
% 2.48/1.46 !>[A: $tType] : A ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 2.48/1.46 ring_1_of_int:
% 2.48/1.46 !>[A: $tType] : ( int > A ) ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Nat_OSuc,type,
% 2.48/1.46 suc: nat > nat ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_NthRoot__Impl__Mirabelle__yrjxwmcnbt_Ofixed__root,type,
% 2.48/1.46 nthRoo220209705d_root: nat > nat > $o ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.48/1.46 ord_less:
% 2.48/1.46 !>[A: $tType] : ( A > A > $o ) ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.48/1.46 ord_less_eq:
% 2.48/1.46 !>[A: $tType] : ( A > A > $o ) ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Power_Opower__class_Opower,type,
% 2.48/1.46 power_power:
% 2.48/1.46 !>[A: $tType] : ( A > nat > A ) ).
% 2.48/1.46
% 2.48/1.46 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.48/1.46 divide_divide:
% 2.48/1.46 !>[A: $tType] : ( A > A > A ) ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_NY____,type,
% 2.48/1.46 ny: real ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_na____,type,
% 2.48/1.46 na: int ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_p,type,
% 2.48/1.46 p: nat ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_pm,type,
% 2.48/1.46 pm: nat ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_x,type,
% 2.48/1.46 x: int ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_y_H____,type,
% 2.48/1.46 y: int ).
% 2.48/1.46
% 2.48/1.46 thf(sy_v_ya____,type,
% 2.48/1.46 ya: int ).
% 2.48/1.46
% 2.48/1.46 thf(fact_0_n00,axiom,
% 2.48/1.46 ( na
% 2.48/1.46 != ( zero_zero @ int ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_1_of__int__less__of__int__power__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [B: int,W: nat,X: int] :
% 2.48/1.46 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 2.48/1.46 = ( ord_less @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_2_of__int__power__less__of__int__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [X: int,B: int,W: nat] :
% 2.48/1.46 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.48/1.46 = ( ord_less @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_3_of__int__power,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_1 @ A )
% 2.48/1.46 => ! [Z: int,N: nat] :
% 2.48/1.46 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
% 2.48/1.46 = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_4_of__int__eq__of__int__power__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_char_0 @ A )
% 2.48/1.46 => ! [B: int,W: nat,X: int] :
% 2.48/1.46 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W )
% 2.48/1.46 = ( ring_1_of_int @ A @ X ) )
% 2.48/1.46 = ( ( power_power @ int @ B @ W )
% 2.48/1.46 = X ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_5_of__int__power__eq__of__int__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_char_0 @ A )
% 2.48/1.46 => ! [X: int,B: int,W: nat] :
% 2.48/1.46 ( ( ( ring_1_of_int @ A @ X )
% 2.48/1.46 = ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.48/1.46 = ( X
% 2.48/1.46 = ( power_power @ int @ B @ W ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_6_of__int__less__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [W: int,Z: int] :
% 2.48/1.46 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( ord_less @ int @ W @ Z ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_7_n0,axiom,
% 2.48/1.46 ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ na ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_8_y0,axiom,
% 2.48/1.46 ord_less @ real @ ( zero_zero @ real ) @ ( ring_1_of_int @ real @ ya ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_9_of__int__eq__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_char_0 @ A )
% 2.48/1.46 => ! [W: int,Z: int] :
% 2.48/1.46 ( ( ( ring_1_of_int @ A @ W )
% 2.48/1.46 = ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( W = Z ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_10_nat_Oinject,axiom,
% 2.48/1.46 ! [X2: nat,Y2: nat] :
% 2.48/1.46 ( ( ( suc @ X2 )
% 2.48/1.46 = ( suc @ Y2 ) )
% 2.48/1.46 = ( X2 = Y2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_11_old_Onat_Oinject,axiom,
% 2.48/1.46 ! [Nat: nat,Nat2: nat] :
% 2.48/1.46 ( ( ( suc @ Nat )
% 2.48/1.46 = ( suc @ Nat2 ) )
% 2.48/1.46 = ( Nat = Nat2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_12_y00,axiom,
% 2.48/1.46 ( ya
% 2.48/1.46 != ( zero_zero @ int ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_13__092_060open_0620_A_060_Ay_A_092_060or_062_Ay_A_061_A0_092_060close_062,axiom,
% 2.48/1.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ ya )
% 2.48/1.46 | ( ya
% 2.48/1.46 = ( zero_zero @ int ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_14_Suc__less__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.48/1.46 = ( ord_less @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_15_Suc__mono,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_16_lessI,axiom,
% 2.48/1.46 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_17_not_I1_J,axiom,
% 2.48/1.46 ~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
% 2.48/1.46 | ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_18_not_I2_J,axiom,
% 2.48/1.46 ~ ( ( ord_less @ int @ ya @ ( zero_zero @ int ) )
% 2.48/1.46 | ( ord_less @ int @ na @ ( zero_zero @ int ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_19__C1_Oprems_C_I2_J,axiom,
% 2.48/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ya ).
% 2.48/1.46
% 2.48/1.46 thf(fact_20__C1_Oprems_C_I3_J,axiom,
% 2.48/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ na ).
% 2.48/1.46
% 2.48/1.46 thf(fact_21_of__int__0,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_1 @ A )
% 2.48/1.46 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_22_of__int__0__eq__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_char_0 @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ( zero_zero @ A )
% 2.48/1.46 = ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( Z
% 2.48/1.46 = ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_23_of__int__eq__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( ring_char_0 @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ( ring_1_of_int @ A @ Z )
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 = ( Z
% 2.48/1.46 = ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_24_of__int__less__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.48/1.46 = ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_25_of__int__0__less__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_26_less__int__code_I1_J,axiom,
% 2.48/1.46 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_27_not__less__less__Suc__eq,axiom,
% 2.48/1.46 ! [N: nat,M: nat] :
% 2.48/1.46 ( ~ ( ord_less @ nat @ N @ M )
% 2.48/1.46 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.48/1.46 = ( N = M ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_28_strict__inc__induct,axiom,
% 2.48/1.46 ! [I: nat,J: nat,P: nat > $o] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ J )
% 2.48/1.46 => ( ! [I2: nat] :
% 2.48/1.46 ( ( J
% 2.48/1.46 = ( suc @ I2 ) )
% 2.48/1.46 => ( P @ I2 ) )
% 2.48/1.46 => ( ! [I2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I2 @ J )
% 2.48/1.46 => ( ( P @ ( suc @ I2 ) )
% 2.48/1.46 => ( P @ I2 ) ) )
% 2.48/1.46 => ( P @ I ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_29_less__Suc__induct,axiom,
% 2.48/1.46 ! [I: nat,J: nat,P: nat > nat > $o] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ J )
% 2.48/1.46 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 2.48/1.46 => ( ! [I2: nat,J2: nat,K: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I2 @ J2 )
% 2.48/1.46 => ( ( ord_less @ nat @ J2 @ K )
% 2.48/1.46 => ( ( P @ I2 @ J2 )
% 2.48/1.46 => ( ( P @ J2 @ K )
% 2.48/1.46 => ( P @ I2 @ K ) ) ) ) )
% 2.48/1.46 => ( P @ I @ J ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_30_less__trans__Suc,axiom,
% 2.48/1.46 ! [I: nat,J: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ J )
% 2.48/1.46 => ( ( ord_less @ nat @ J @ K2 )
% 2.48/1.46 => ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_31_Suc__less__SucD,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 2.48/1.46 => ( ord_less @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_32_less__antisym,axiom,
% 2.48/1.46 ! [N: nat,M: nat] :
% 2.48/1.46 ( ~ ( ord_less @ nat @ N @ M )
% 2.48/1.46 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.48/1.46 => ( M = N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_33_Suc__less__eq2,axiom,
% 2.48/1.46 ! [N: nat,M: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 2.48/1.46 = ( ? [M2: nat] :
% 2.48/1.46 ( ( M
% 2.48/1.46 = ( suc @ M2 ) )
% 2.48/1.46 & ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_34_All__less__Suc,axiom,
% 2.48/1.46 ! [N: nat,P: nat > $o] :
% 2.48/1.46 ( ( ! [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 2.48/1.46 => ( P @ I3 ) ) )
% 2.48/1.46 = ( ( P @ N )
% 2.48/1.46 & ! [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ N )
% 2.48/1.46 => ( P @ I3 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_35_not__less__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 2.48/1.46 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_36_less__Suc__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 = ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 | ( M = N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_37_Ex__less__Suc,axiom,
% 2.48/1.46 ! [N: nat,P: nat > $o] :
% 2.48/1.46 ( ( ? [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 2.48/1.46 & ( P @ I3 ) ) )
% 2.48/1.46 = ( ( P @ N )
% 2.48/1.46 | ? [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ N )
% 2.48/1.46 & ( P @ I3 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_38_less__SucI,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_39_less__SucE,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 => ( ~ ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( M = N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_40_Suc__lessI,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ( ( suc @ M )
% 2.48/1.46 != N )
% 2.48/1.46 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_41_Suc__lessE,axiom,
% 2.48/1.46 ! [I: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ I ) @ K2 )
% 2.48/1.46 => ~ ! [J2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ J2 )
% 2.48/1.46 => ( K2
% 2.48/1.46 != ( suc @ J2 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_42_Suc__lessD,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 2.48/1.46 => ( ord_less @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_43_Nat_OlessE,axiom,
% 2.48/1.46 ! [I: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ K2 )
% 2.48/1.46 => ( ( K2
% 2.48/1.46 != ( suc @ I ) )
% 2.48/1.46 => ~ ! [J2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I @ J2 )
% 2.48/1.46 => ( K2
% 2.48/1.46 != ( suc @ J2 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_44_of__int__pos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_45_measure__induct__rule,axiom,
% 2.48/1.46 ! [B2: $tType,A: $tType] :
% 2.48/1.46 ( ( wellorder @ B2 )
% 2.48/1.46 => ! [F: A > B2,P: A > $o,A2: A] :
% 2.48/1.46 ( ! [X3: A] :
% 2.48/1.46 ( ! [Y: A] :
% 2.48/1.46 ( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
% 2.48/1.46 => ( P @ Y ) )
% 2.48/1.46 => ( P @ X3 ) )
% 2.48/1.46 => ( P @ A2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_46_measure__induct,axiom,
% 2.48/1.46 ! [B2: $tType,A: $tType] :
% 2.48/1.46 ( ( wellorder @ B2 )
% 2.48/1.46 => ! [F: A > B2,P: A > $o,A2: A] :
% 2.48/1.46 ( ! [X3: A] :
% 2.48/1.46 ( ! [Y: A] :
% 2.48/1.46 ( ( ord_less @ B2 @ ( F @ Y ) @ ( F @ X3 ) )
% 2.48/1.46 => ( P @ Y ) )
% 2.48/1.46 => ( P @ X3 ) )
% 2.48/1.46 => ( P @ A2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_47_n__not__Suc__n,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( N
% 2.48/1.46 != ( suc @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_48_Suc__inject,axiom,
% 2.48/1.46 ! [X: nat,Y3: nat] :
% 2.48/1.46 ( ( ( suc @ X )
% 2.48/1.46 = ( suc @ Y3 ) )
% 2.48/1.46 => ( X = Y3 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_49_lift__Suc__mono__less__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( order @ A )
% 2.48/1.46 => ! [F: nat > A,N: nat,M: nat] :
% 2.48/1.46 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.48/1.46 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 2.48/1.46 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_50_lift__Suc__mono__less,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( order @ A )
% 2.48/1.46 => ! [F: nat > A,N: nat,N3: nat] :
% 2.48/1.46 ( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.48/1.46 => ( ( ord_less @ nat @ N @ N3 )
% 2.48/1.46 => ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_51_power__0__Suc,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiring_1 @ A )
% 2.48/1.46 => ! [N: nat] :
% 2.48/1.46 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_52_NY0,axiom,
% 2.48/1.46 ord_less @ real @ ( zero_zero @ real ) @ ny ).
% 2.48/1.46
% 2.48/1.46 thf(fact_53_power__eq__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiri134348788visors @ A )
% 2.48/1.46 => ! [A2: A,N: nat] :
% 2.48/1.46 ( ( ( power_power @ A @ A2 @ N )
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 = ( ( A2
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_54_not__gr__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [N: A] :
% 2.48/1.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.48/1.46 = ( N
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_55_realpow__pos__nth2,axiom,
% 2.48/1.46 ! [A2: real,N: nat] :
% 2.48/1.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.48/1.46 => ? [R: real] :
% 2.48/1.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 2.48/1.46 & ( ( power_power @ real @ R @ ( suc @ N ) )
% 2.48/1.46 = A2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_56_zero__less__power,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_57_NY__def,axiom,
% 2.48/1.46 ( ny
% 2.48/1.46 = ( divide_divide @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ pm ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_58_ex__of__int__less,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( archim1804426504_field @ A )
% 2.48/1.46 => ! [X: A] :
% 2.48/1.46 ? [Z2: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_59_ex__less__of__int,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( archim1804426504_field @ A )
% 2.48/1.46 => ! [X: A] :
% 2.48/1.46 ? [Z2: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_60_yyn,axiom,
% 2.48/1.46 ord_less @ int @ na @ ( power_power @ int @ ya @ p ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_61__092_060open_062pm_A_061_A0_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
% 2.48/1.46 ( pm
% 2.48/1.46 != ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_62_p0,axiom,
% 2.48/1.46 ( p
% 2.48/1.46 != ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_63_pm0,axiom,
% 2.48/1.46 ord_less @ nat @ ( zero_zero @ nat ) @ pm ).
% 2.48/1.46
% 2.48/1.46 thf(fact_64__C1_Oprems_C_I1_J,axiom,
% 2.48/1.46 ord_less_eq @ int @ na @ ( power_power @ int @ ya @ p ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_65_p,axiom,
% 2.48/1.46 ( p
% 2.48/1.46 = ( suc @ pm ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_66_fixed__root__axioms,axiom,
% 2.48/1.46 nthRoo220209705d_root @ p @ pm ).
% 2.48/1.46
% 2.48/1.46 thf(fact_67_le__zero__eq,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [N: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.48/1.46 = ( N
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_68_nat__power__eq__Suc__0__iff,axiom,
% 2.48/1.46 ! [X: nat,M: nat] :
% 2.48/1.46 ( ( ( power_power @ nat @ X @ M )
% 2.48/1.46 = ( suc @ ( zero_zero @ nat ) ) )
% 2.48/1.46 = ( ( M
% 2.48/1.46 = ( zero_zero @ nat ) )
% 2.48/1.46 | ( X
% 2.48/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_69_power__Suc__0,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 2.48/1.46 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_70_neq0__conv,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( N
% 2.48/1.46 != ( zero_zero @ nat ) )
% 2.48/1.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_71_less__nat__zero__code,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_72_bot__nat__0_Onot__eq__extremum,axiom,
% 2.48/1.46 ! [A2: nat] :
% 2.48/1.46 ( ( A2
% 2.48/1.46 != ( zero_zero @ nat ) )
% 2.48/1.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_73_nat__zero__less__power__iff,axiom,
% 2.48/1.46 ! [X: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 2.48/1.46 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.48/1.46 | ( N
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_74_power__Suc0__right,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( monoid_mult @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
% 2.48/1.46 = A2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_75_zero__less__Suc,axiom,
% 2.48/1.46 ! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_76_less__Suc0,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 2.48/1.46 = ( N
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_77_of__int__le__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [W: int,Z: int] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( ord_less_eq @ int @ W @ Z ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_78_power__mono__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,B: A,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 2.48/1.46 = ( ord_less_eq @ A @ A2 @ B ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_79_of__int__le__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
% 2.48/1.46 = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_80_of__int__0__le__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
% 2.48/1.46 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_81_of__int__le__of__int__power__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [B: int,W: nat,X: int] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) @ ( ring_1_of_int @ A @ X ) )
% 2.48/1.46 = ( ord_less_eq @ int @ ( power_power @ int @ B @ W ) @ X ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_82_of__int__power__le__of__int__cancel__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [X: int,B: int,W: nat] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B ) @ W ) )
% 2.48/1.46 = ( ord_less_eq @ int @ X @ ( power_power @ int @ B @ W ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_83_gr0I,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( N
% 2.48/1.46 != ( zero_zero @ nat ) )
% 2.48/1.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_84_not__gr0,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.48/1.46 = ( N
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_85_not__less0,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_86_less__zeroE,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_87_nat__neq__iff,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( M != N )
% 2.48/1.46 = ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_88_less__not__refl,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ N @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_89_less__not__refl2,axiom,
% 2.48/1.46 ! [N: nat,M: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ N @ M )
% 2.48/1.46 => ( M != N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_90_less__not__refl3,axiom,
% 2.48/1.46 ! [S: nat,T: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ S @ T )
% 2.48/1.46 => ( S != T ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_91_gr__implies__not0,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( N
% 2.48/1.46 != ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_92_less__irrefl__nat,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ N @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_93_nat__less__induct,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ! [N2: nat] :
% 2.48/1.46 ( ! [M3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M3 @ N2 )
% 2.48/1.46 => ( P @ M3 ) )
% 2.48/1.46 => ( P @ N2 ) )
% 2.48/1.46 => ( P @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_94_infinite__descent,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ! [N2: nat] :
% 2.48/1.46 ( ~ ( P @ N2 )
% 2.48/1.46 => ? [M3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M3 @ N2 )
% 2.48/1.46 & ~ ( P @ M3 ) ) )
% 2.48/1.46 => ( P @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_95_infinite__descent0,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.48/1.46 => ( ~ ( P @ N2 )
% 2.48/1.46 => ? [M3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M3 @ N2 )
% 2.48/1.46 & ~ ( P @ M3 ) ) ) )
% 2.48/1.46 => ( P @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_96_linorder__neqE__nat,axiom,
% 2.48/1.46 ! [X: nat,Y3: nat] :
% 2.48/1.46 ( ( X != Y3 )
% 2.48/1.46 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 2.48/1.46 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_97_infinite__descent__measure,axiom,
% 2.48/1.46 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 2.48/1.46 ( ! [X3: A] :
% 2.48/1.46 ( ~ ( P @ X3 )
% 2.48/1.46 => ? [Y: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
% 2.48/1.46 & ~ ( P @ Y ) ) )
% 2.48/1.46 => ( P @ X ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_98_bot__nat__0_Oextremum__strict,axiom,
% 2.48/1.46 ! [A2: nat] :
% 2.48/1.46 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_99_infinite__descent0__measure,axiom,
% 2.48/1.46 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 2.48/1.46 ( ! [X3: A] :
% 2.48/1.46 ( ( ( V @ X3 )
% 2.48/1.46 = ( zero_zero @ nat ) )
% 2.48/1.46 => ( P @ X3 ) )
% 2.48/1.46 => ( ! [X3: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 2.48/1.46 => ( ~ ( P @ X3 )
% 2.48/1.46 => ? [Y: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
% 2.48/1.46 & ~ ( P @ Y ) ) ) )
% 2.48/1.46 => ( P @ X ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_100_nat__power__less__imp__less,axiom,
% 2.48/1.46 ! [I: nat,M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 2.48/1.46 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 2.48/1.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_101_lift__Suc__antimono__le,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( order @ A )
% 2.48/1.46 => ! [F: nat > A,N: nat,N3: nat] :
% 2.48/1.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_102_lift__Suc__mono__le,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( order @ A )
% 2.48/1.46 => ! [F: nat > A,N: nat,N3: nat] :
% 2.48/1.46 ( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ N @ N3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_103_zero__le,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_104_power__divide,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( field @ A )
% 2.48/1.46 => ! [A2: A,B: A,N: nat] :
% 2.48/1.46 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B ) @ N )
% 2.48/1.46 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_105_ex__le__of__int,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( archim1804426504_field @ A )
% 2.48/1.46 => ! [X: A] :
% 2.48/1.46 ? [Z2: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_106_exists__least__lemma,axiom,
% 2.48/1.46 ! [P: nat > $o] :
% 2.48/1.46 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( ? [X_1: nat] : ( P @ X_1 )
% 2.48/1.46 => ? [N2: nat] :
% 2.48/1.46 ( ~ ( P @ N2 )
% 2.48/1.46 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_107_power__gt__expt,axiom,
% 2.48/1.46 ! [N: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 2.48/1.46 => ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_108_less__eq__int__code_I1_J,axiom,
% 2.48/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_109_nat_Odistinct_I1_J,axiom,
% 2.48/1.46 ! [X2: nat] :
% 2.48/1.46 ( ( zero_zero @ nat )
% 2.48/1.46 != ( suc @ X2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
% 2.48/1.46 ! [Nat3: nat] :
% 2.48/1.46 ( ( suc @ Nat3 )
% 2.48/1.46 != ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_111_old_Onat_Odistinct_I1_J,axiom,
% 2.48/1.46 ! [Nat2: nat] :
% 2.48/1.46 ( ( zero_zero @ nat )
% 2.48/1.46 != ( suc @ Nat2 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_112_nat_OdiscI,axiom,
% 2.48/1.46 ! [Nat: nat,X2: nat] :
% 2.48/1.46 ( ( Nat
% 2.48/1.46 = ( suc @ X2 ) )
% 2.48/1.46 => ( Nat
% 2.48/1.46 != ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_113_nat__induct,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( P @ N2 )
% 2.48/1.46 => ( P @ ( suc @ N2 ) ) )
% 2.48/1.46 => ( P @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_114_diff__induct,axiom,
% 2.48/1.46 ! [P: nat > nat > $o,M: nat,N: nat] :
% 2.48/1.46 ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
% 2.48/1.46 => ( ! [X3: nat,Y4: nat] :
% 2.48/1.46 ( ( P @ X3 @ Y4 )
% 2.48/1.46 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 2.48/1.46 => ( P @ M @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_115_zero__induct,axiom,
% 2.48/1.46 ! [P: nat > $o,K2: nat] :
% 2.48/1.46 ( ( P @ K2 )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( P @ ( suc @ N2 ) )
% 2.48/1.46 => ( P @ N2 ) )
% 2.48/1.46 => ( P @ ( zero_zero @ nat ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_116_Suc__neq__Zero,axiom,
% 2.48/1.46 ! [M: nat] :
% 2.48/1.46 ( ( suc @ M )
% 2.48/1.46 != ( zero_zero @ nat ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_117_Zero__neq__Suc,axiom,
% 2.48/1.46 ! [M: nat] :
% 2.48/1.46 ( ( zero_zero @ nat )
% 2.48/1.46 != ( suc @ M ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_118_Zero__not__Suc,axiom,
% 2.48/1.46 ! [M: nat] :
% 2.48/1.46 ( ( zero_zero @ nat )
% 2.48/1.46 != ( suc @ M ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_119_old_Onat_Oexhaust,axiom,
% 2.48/1.46 ! [Y3: nat] :
% 2.48/1.46 ( ( Y3
% 2.48/1.46 != ( zero_zero @ nat ) )
% 2.48/1.46 => ~ ! [Nat4: nat] :
% 2.48/1.46 ( Y3
% 2.48/1.46 != ( suc @ Nat4 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_120_old_Onat_Oinducts,axiom,
% 2.48/1.46 ! [P: nat > $o,Nat: nat] :
% 2.48/1.46 ( ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( ! [Nat4: nat] :
% 2.48/1.46 ( ( P @ Nat4 )
% 2.48/1.46 => ( P @ ( suc @ Nat4 ) ) )
% 2.48/1.46 => ( P @ Nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_121_not0__implies__Suc,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( N
% 2.48/1.46 != ( zero_zero @ nat ) )
% 2.48/1.46 => ? [M4: nat] :
% 2.48/1.46 ( N
% 2.48/1.46 = ( suc @ M4 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_122_less__Suc__eq__0__disj,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 = ( ( M
% 2.48/1.46 = ( zero_zero @ nat ) )
% 2.48/1.46 | ? [J3: nat] :
% 2.48/1.46 ( ( M
% 2.48/1.46 = ( suc @ J3 ) )
% 2.48/1.46 & ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_123_gr0__implies__Suc,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ? [M4: nat] :
% 2.48/1.46 ( N
% 2.48/1.46 = ( suc @ M4 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_124_All__less__Suc2,axiom,
% 2.48/1.46 ! [N: nat,P: nat > $o] :
% 2.48/1.46 ( ( ! [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 2.48/1.46 => ( P @ I3 ) ) )
% 2.48/1.46 = ( ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 & ! [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ N )
% 2.48/1.46 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_125_gr0__conv__Suc,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 = ( ? [M5: nat] :
% 2.48/1.46 ( N
% 2.48/1.46 = ( suc @ M5 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_126_Ex__less__Suc2,axiom,
% 2.48/1.46 ! [N: nat,P: nat > $o] :
% 2.48/1.46 ( ( ? [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ ( suc @ N ) )
% 2.48/1.46 & ( P @ I3 ) ) )
% 2.48/1.46 = ( ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 | ? [I3: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I3 @ N )
% 2.48/1.46 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_127_power__eq__imp__eq__base,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat,B: A] :
% 2.48/1.46 ( ( ( power_power @ A @ A2 @ N )
% 2.48/1.46 = ( power_power @ A @ B @ N ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( A2 = B ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_128_power__eq__iff__eq__base,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [N: nat,A2: A,B: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ( ( power_power @ A @ A2 @ N )
% 2.48/1.46 = ( power_power @ A @ B @ N ) )
% 2.48/1.46 = ( A2 = B ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_129_zero__le__power,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_130_power__mono,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,B: A,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ A @ A2 @ B )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_131_of__int__nonneg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [Z: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_132_power__less__imp__less__base,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat,B: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_133_power__strict__mono,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,B: A,N: nat] :
% 2.48/1.46 ( ( ord_less @ A @ A2 @ B )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B @ N ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_134_power__inject__base,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat,B: A] :
% 2.48/1.46 ( ( ( power_power @ A @ A2 @ ( suc @ N ) )
% 2.48/1.46 = ( power_power @ A @ B @ ( suc @ N ) ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( A2 = B ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_135_power__le__imp__le__base,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_semidom @ A )
% 2.48/1.46 => ! [A2: A,N: nat,B: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B @ ( suc @ N ) ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ord_less_eq @ A @ A2 @ B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_136_realpow__pos__nth,axiom,
% 2.48/1.46 ! [N: nat,A2: real] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.48/1.46 => ? [R: real] :
% 2.48/1.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 2.48/1.46 & ( ( power_power @ real @ R @ N )
% 2.48/1.46 = A2 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_137_realpow__pos__nth__unique,axiom,
% 2.48/1.46 ! [N: nat,A2: real] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.48/1.46 => ? [X3: real] :
% 2.48/1.46 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
% 2.48/1.46 & ( ( power_power @ real @ X3 @ N )
% 2.48/1.46 = A2 )
% 2.48/1.46 & ! [Y: real] :
% 2.48/1.46 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
% 2.48/1.46 & ( ( power_power @ real @ Y @ N )
% 2.48/1.46 = A2 ) )
% 2.48/1.46 => ( Y = X3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_138_zero__reorient,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( zero @ A )
% 2.48/1.46 => ! [X: A] :
% 2.48/1.46 ( ( ( zero_zero @ A )
% 2.48/1.46 = X )
% 2.48/1.46 = ( X
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_139_gr__zeroI,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [N: A] :
% 2.48/1.46 ( ( N
% 2.48/1.46 != ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_140_not__less__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [N: A] :
% 2.48/1.46 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_141_gr__implies__not__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [M: A,N: A] :
% 2.48/1.46 ( ( ord_less @ A @ M @ N )
% 2.48/1.46 => ( N
% 2.48/1.46 != ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_142_zero__less__iff__neq__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( canoni770627133id_add @ A )
% 2.48/1.46 => ! [N: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.48/1.46 = ( N
% 2.48/1.46 != ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_143_power__not__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiri134348788visors @ A )
% 2.48/1.46 => ! [A2: A,N: nat] :
% 2.48/1.46 ( ( A2
% 2.48/1.46 != ( zero_zero @ A ) )
% 2.48/1.46 => ( ( power_power @ A @ A2 @ N )
% 2.48/1.46 != ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_144_zero__power,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiring_1 @ A )
% 2.48/1.46 => ! [N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.48/1.46 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_145_yyn_H,axiom,
% 2.48/1.46 ord_less @ int @ ( power_power @ int @ y @ p ) @ na ).
% 2.48/1.46
% 2.48/1.46 thf(fact_146_y_H0,axiom,
% 2.48/1.46 ord_less_eq @ int @ ( zero_zero @ int ) @ y ).
% 2.48/1.46
% 2.48/1.46 thf(fact_147_compare__pow__less__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [P2: nat,X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ( ord_less @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
% 2.48/1.46 = ( ord_less @ A @ X @ Y3 ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_148_compare__pow__le__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [P2: nat,X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ P2 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( power_power @ A @ X @ P2 ) @ ( power_power @ A @ Y3 @ P2 ) )
% 2.48/1.46 = ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_149_division__ring__divide__zero,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( division_ring @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_150_bits__div__by__0,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiring_bits @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_151_divide__cancel__right,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( field @ A )
% 2.48/1.46 => ! [A2: A,C: A,B: A] :
% 2.48/1.46 ( ( ( divide_divide @ A @ A2 @ C )
% 2.48/1.46 = ( divide_divide @ A @ B @ C ) )
% 2.48/1.46 = ( ( C
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 | ( A2 = B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_152_divide__eq__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( field @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ( divide_divide @ A @ A2 @ B )
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 = ( ( A2
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 | ( B
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_153_divide__cancel__left,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( field @ A )
% 2.48/1.46 => ! [C: A,A2: A,B: A] :
% 2.48/1.46 ( ( ( divide_divide @ A @ C @ A2 )
% 2.48/1.46 = ( divide_divide @ A @ C @ B ) )
% 2.48/1.46 = ( ( C
% 2.48/1.46 = ( zero_zero @ A ) )
% 2.48/1.46 | ( A2 = B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_154_bits__div__0,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semiring_bits @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_155_bot__nat__0_Oextremum,axiom,
% 2.48/1.46 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_156_le0,axiom,
% 2.48/1.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_157_Suc__le__mono,axiom,
% 2.48/1.46 ! [N: nat,M: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 2.48/1.46 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_158_le__refl,axiom,
% 2.48/1.46 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_159_le__trans,axiom,
% 2.48/1.46 ! [I: nat,J: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I @ J )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ J @ K2 )
% 2.48/1.46 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_160_eq__imp__le,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( M = N )
% 2.48/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_161_le__antisym,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ N @ M )
% 2.48/1.46 => ( M = N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_162_nat__le__linear,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_163_Nat_Oex__has__greatest__nat,axiom,
% 2.48/1.46 ! [P: nat > $o,K2: nat,B: nat] :
% 2.48/1.46 ( ( P @ K2 )
% 2.48/1.46 => ( ! [Y4: nat] :
% 2.48/1.46 ( ( P @ Y4 )
% 2.48/1.46 => ( ord_less_eq @ nat @ Y4 @ B ) )
% 2.48/1.46 => ? [X3: nat] :
% 2.48/1.46 ( ( P @ X3 )
% 2.48/1.46 & ! [Y: nat] :
% 2.48/1.46 ( ( P @ Y )
% 2.48/1.46 => ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_164_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.48/1.46 ! [A2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.48/1.46 => ( A2
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_165_bot__nat__0_Oextremum__unique,axiom,
% 2.48/1.46 ! [A2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.48/1.46 = ( A2
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_166_le__0__eq,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.48/1.46 = ( N
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_167_less__eq__nat_Osimps_I1_J,axiom,
% 2.48/1.46 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_168_transitive__stepwise__le,axiom,
% 2.48/1.46 ! [M: nat,N: nat,R2: nat > nat > $o] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
% 2.48/1.46 => ( ! [X3: nat,Y4: nat,Z2: nat] :
% 2.48/1.46 ( ( R2 @ X3 @ Y4 )
% 2.48/1.46 => ( ( R2 @ Y4 @ Z2 )
% 2.48/1.46 => ( R2 @ X3 @ Z2 ) ) )
% 2.48/1.46 => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
% 2.48/1.46 => ( R2 @ M @ N ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_169_nat__induct__at__least,axiom,
% 2.48/1.46 ! [M: nat,N: nat,P: nat > $o] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ( P @ M )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N2 )
% 2.48/1.46 => ( ( P @ N2 )
% 2.48/1.46 => ( P @ ( suc @ N2 ) ) ) )
% 2.48/1.46 => ( P @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_170_full__nat__induct,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ! [N2: nat] :
% 2.48/1.46 ( ! [M3: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
% 2.48/1.46 => ( P @ M3 ) )
% 2.48/1.46 => ( P @ N2 ) )
% 2.48/1.46 => ( P @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_171_not__less__eq__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 2.48/1.46 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_172_Suc__n__not__le__n,axiom,
% 2.48/1.46 ! [N: nat] :
% 2.48/1.46 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_173_le__Suc__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 = ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 | ( M
% 2.48/1.46 = ( suc @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_174_Suc__le__D,axiom,
% 2.48/1.46 ! [N: nat,M6: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
% 2.48/1.46 => ? [M4: nat] :
% 2.48/1.46 ( M6
% 2.48/1.46 = ( suc @ M4 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_175_le__SucI,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_176_le__SucE,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( M
% 2.48/1.46 = ( suc @ N ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_177_Suc__leD,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_178_less__mono__imp__le__mono,axiom,
% 2.48/1.46 ! [F: nat > nat,I: nat,J: nat] :
% 2.48/1.46 ( ! [I2: nat,J2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I2 @ J2 )
% 2.48/1.46 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ I @ J )
% 2.48/1.46 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_179_le__neq__implies__less,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ( M != N )
% 2.48/1.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_180_less__or__eq__imp__le,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 | ( M = N ) )
% 2.48/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_181_le__eq__less__or__eq,axiom,
% 2.48/1.46 ( ( ord_less_eq @ nat )
% 2.48/1.46 = ( ^ [M5: nat,N4: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M5 @ N4 )
% 2.48/1.46 | ( M5 = N4 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_182_less__imp__le__nat,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_183_nat__less__le,axiom,
% 2.48/1.46 ( ( ord_less @ nat )
% 2.48/1.46 = ( ^ [M5: nat,N4: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M5 @ N4 )
% 2.48/1.46 & ( M5 != N4 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_184_fixed__root_Op0,axiom,
% 2.48/1.46 ! [P2: nat,Pm: nat] :
% 2.48/1.46 ( ( nthRoo220209705d_root @ P2 @ Pm )
% 2.48/1.46 => ( P2
% 2.48/1.46 != ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_185_fixed__root_Ointro,axiom,
% 2.48/1.46 ! [P2: nat,Pm: nat] :
% 2.48/1.46 ( ( P2
% 2.48/1.46 = ( suc @ Pm ) )
% 2.48/1.46 => ( nthRoo220209705d_root @ P2 @ Pm ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_186_fixed__root__def,axiom,
% 2.48/1.46 ( nthRoo220209705d_root
% 2.48/1.46 = ( ^ [P3: nat,Pm2: nat] :
% 2.48/1.46 ( P3
% 2.48/1.46 = ( suc @ Pm2 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_187_fixed__root_Op,axiom,
% 2.48/1.46 ! [P2: nat,Pm: nat] :
% 2.48/1.46 ( ( nthRoo220209705d_root @ P2 @ Pm )
% 2.48/1.46 => ( P2
% 2.48/1.46 = ( suc @ Pm ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_188_nat__one__le__power,axiom,
% 2.48/1.46 ! [I: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
% 2.48/1.46 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_189_ex__least__nat__le,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ( P @ N )
% 2.48/1.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ? [K: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ K @ N )
% 2.48/1.46 & ! [I4: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ I4 @ K )
% 2.48/1.46 => ~ ( P @ I4 ) )
% 2.48/1.46 & ( P @ K ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_190_Suc__leI,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_191_Suc__le__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.48/1.46 = ( ord_less @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_192_dec__induct,axiom,
% 2.48/1.46 ! [I: nat,J: nat,P: nat > $o] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I @ J )
% 2.48/1.46 => ( ( P @ I )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 2.48/1.46 => ( ( ord_less @ nat @ N2 @ J )
% 2.48/1.46 => ( ( P @ N2 )
% 2.48/1.46 => ( P @ ( suc @ N2 ) ) ) ) )
% 2.48/1.46 => ( P @ J ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_193_inc__induct,axiom,
% 2.48/1.46 ! [I: nat,J: nat,P: nat > $o] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I @ J )
% 2.48/1.46 => ( ( P @ J )
% 2.48/1.46 => ( ! [N2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I @ N2 )
% 2.48/1.46 => ( ( ord_less @ nat @ N2 @ J )
% 2.48/1.46 => ( ( P @ ( suc @ N2 ) )
% 2.48/1.46 => ( P @ N2 ) ) ) )
% 2.48/1.46 => ( P @ I ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_194_Suc__le__lessD,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 2.48/1.46 => ( ord_less @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_195_le__less__Suc__eq,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 2.48/1.46 = ( N = M ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_196_less__Suc__eq__le,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 2.48/1.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_197_less__eq__Suc__le,axiom,
% 2.48/1.46 ( ( ord_less @ nat )
% 2.48/1.46 = ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_198_le__imp__less__Suc,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_199_linordered__field__no__ub,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X4: A] :
% 2.48/1.46 ? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_200_linordered__field__no__lb,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X4: A] :
% 2.48/1.46 ? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_201_ex__least__nat__less,axiom,
% 2.48/1.46 ! [P: nat > $o,N: nat] :
% 2.48/1.46 ( ( P @ N )
% 2.48/1.46 => ( ~ ( P @ ( zero_zero @ nat ) )
% 2.48/1.46 => ? [K: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ K @ N )
% 2.48/1.46 & ! [I4: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ I4 @ K )
% 2.48/1.46 => ~ ( P @ I4 ) )
% 2.48/1.46 & ( P @ ( suc @ K ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_202_divide__right__mono__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A,C: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ A2 @ B )
% 2.48/1.46 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ B @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_203_divide__nonpos__nonpos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_204_divide__nonpos__nonneg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_205_divide__nonneg__nonpos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_206_divide__nonneg__nonneg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_207_zero__le__divide__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 2.48/1.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
% 2.48/1.46 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_208_divide__right__mono,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A,C: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ A2 @ B )
% 2.48/1.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_209_divide__le__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 2.48/1.46 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 & ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
% 2.48/1.46 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_210_divide__neg__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_211_divide__neg__pos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_212_divide__pos__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_213_divide__pos__pos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_214_divide__less__0__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B ) @ ( zero_zero @ A ) )
% 2.48/1.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
% 2.48/1.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_215_divide__less__cancel,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,C: A,B: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 2.48/1.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.48/1.46 => ( ord_less @ A @ A2 @ B ) )
% 2.48/1.46 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less @ A @ B @ A2 ) )
% 2.48/1.46 & ( C
% 2.48/1.46 != ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_216_zero__less__divide__iff,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) )
% 2.48/1.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 & ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
% 2.48/1.46 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 & ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_217_divide__strict__right__mono,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,B: A,C: A] :
% 2.48/1.46 ( ( ord_less @ A @ A2 @ B )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_218_divide__strict__right__mono__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [B: A,A2: A,C: A] :
% 2.48/1.46 ( ( ord_less @ A @ B @ A2 )
% 2.48/1.46 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_219_frac__le,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [Y3: A,X: A,W: A,Z: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ( ord_less_eq @ A @ X @ Y3 )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.48/1.46 => ( ( ord_less_eq @ A @ W @ Z )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_220_frac__less,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A,W: A,Z: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less @ A @ X @ Y3 )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.48/1.46 => ( ( ord_less_eq @ A @ W @ Z )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_221_frac__less2,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A,W: A,Z: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less_eq @ A @ X @ Y3 )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.48/1.46 => ( ( ord_less @ A @ W @ Z )
% 2.48/1.46 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_222_divide__le__cancel,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [A2: A,C: A,B: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B @ C ) )
% 2.48/1.46 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.48/1.46 => ( ord_less_eq @ A @ A2 @ B ) )
% 2.48/1.46 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ B @ A2 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_223_divide__nonneg__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_224_divide__nonneg__pos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_225_divide__nonpos__neg,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_226_divide__nonpos__pos,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_field @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.48/1.46 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
% 2.48/1.46 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_227_div__neg__neg__trivial,axiom,
% 2.48/1.46 ! [K2: int,L: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
% 2.48/1.46 => ( ( ord_less @ int @ L @ K2 )
% 2.48/1.46 => ( ( divide_divide @ int @ K2 @ L )
% 2.48/1.46 = ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_228_div__pos__pos__trivial,axiom,
% 2.48/1.46 ! [K2: int,L: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
% 2.48/1.46 => ( ( ord_less @ int @ K2 @ L )
% 2.48/1.46 => ( ( divide_divide @ int @ K2 @ L )
% 2.48/1.46 = ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_229_x,axiom,
% 2.48/1.46 x = y ).
% 2.48/1.46
% 2.48/1.46 thf(fact_230_div__by__Suc__0,axiom,
% 2.48/1.46 ! [M: nat] :
% 2.48/1.46 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
% 2.48/1.46 = M ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_231_div__less,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 => ( ( divide_divide @ nat @ M @ N )
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_232_div__le__mono,axiom,
% 2.48/1.46 ! [M: nat,N: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_233_div__le__dividend,axiom,
% 2.48/1.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_234_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ( divide_divide @ nat @ M @ N )
% 2.48/1.46 = ( zero_zero @ nat ) )
% 2.48/1.46 = ( ( ord_less @ nat @ M @ N )
% 2.48/1.46 | ( N
% 2.48/1.46 = ( zero_zero @ nat ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_235_Suc__div__le__mono,axiom,
% 2.48/1.46 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_236_div__greater__zero__iff,axiom,
% 2.48/1.46 ! [M: nat,N: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 2.48/1.46 = ( ( ord_less_eq @ nat @ N @ M )
% 2.48/1.46 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_237_div__le__mono2,axiom,
% 2.48/1.46 ! [M: nat,N: nat,K2: nat] :
% 2.48/1.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.48/1.46 => ( ( ord_less_eq @ nat @ M @ N )
% 2.48/1.46 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_238_div__by__0,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semidom_divide @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_239_div__0,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( semidom_divide @ A )
% 2.48/1.46 => ! [A2: A] :
% 2.48/1.46 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_240_linorder__neqE__linordered__idom,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( linordered_idom @ A )
% 2.48/1.46 => ! [X: A,Y3: A] :
% 2.48/1.46 ( ( X != Y3 )
% 2.48/1.46 => ( ~ ( ord_less @ A @ X @ Y3 )
% 2.48/1.46 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_241_zdiv__mono1,axiom,
% 2.48/1.46 ! [A2: int,A3: int,B: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ A2 @ A3 )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.48/1.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A3 @ B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_242_zdiv__mono2,axiom,
% 2.48/1.46 ! [A2: int,B3: int,B: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
% 2.48/1.46 => ( ( ord_less_eq @ int @ B3 @ B )
% 2.48/1.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_243_pos__imp__zdiv__neg__iff,axiom,
% 2.48/1.46 ! [B: int,A2: int] :
% 2.48/1.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.48/1.46 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 2.48/1.46 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_244_neg__imp__zdiv__neg__iff,axiom,
% 2.48/1.46 ! [B: int,A2: int] :
% 2.48/1.46 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 2.48/1.46 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) )
% 2.48/1.46 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_245_div__neg__pos__less0,axiom,
% 2.48/1.46 ! [A2: int,B: int] :
% 2.48/1.46 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.48/1.46 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_246_div__positive,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( unique1598680935umeral @ A )
% 2.48/1.46 => ! [B: A,A2: A] :
% 2.48/1.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
% 2.48/1.46 => ( ( ord_less_eq @ A @ B @ A2 )
% 2.48/1.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_247_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 2.48/1.46 ! [A: $tType] :
% 2.48/1.46 ( ( unique1598680935umeral @ A )
% 2.48/1.46 => ! [A2: A,B: A] :
% 2.48/1.46 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.48/1.46 => ( ( ord_less @ A @ A2 @ B )
% 2.48/1.46 => ( ( divide_divide @ A @ A2 @ B )
% 2.48/1.46 = ( zero_zero @ A ) ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_248_nonneg1__imp__zdiv__pos__iff,axiom,
% 2.48/1.46 ! [A2: int,B: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 2.48/1.46 = ( ( ord_less_eq @ int @ B @ A2 )
% 2.48/1.46 & ( ord_less @ int @ ( zero_zero @ int ) @ B ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_249_pos__imp__zdiv__nonneg__iff,axiom,
% 2.48/1.46 ! [B: int,A2: int] :
% 2.48/1.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.48/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 2.48/1.46 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_250_neg__imp__zdiv__nonneg__iff,axiom,
% 2.48/1.46 ! [B: int,A2: int] :
% 2.48/1.46 ( ( ord_less @ int @ B @ ( zero_zero @ int ) )
% 2.48/1.46 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B ) )
% 2.48/1.46 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_251_pos__imp__zdiv__pos__iff,axiom,
% 2.48/1.46 ! [K2: int,I: int] :
% 2.48/1.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K2 ) )
% 2.48/1.46 = ( ord_less_eq @ int @ K2 @ I ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(fact_252_div__nonpos__pos__le0,axiom,
% 2.48/1.46 ! [A2: int,B: int] :
% 2.48/1.46 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 2.48/1.46 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B )
% 2.48/1.46 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B ) @ ( zero_zero @ int ) ) ) ) ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 2.48/1.46 unique1598680935umeral @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.48/1.46 semiri134348788visors @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.48/1.46 linordered_semidom @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.48/1.46 linordered_idom @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 2.48/1.46 semidom_divide @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
% 2.48/1.46 semiring_bits @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 2.48/1.46 monoid_mult @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.48/1.46 semiring_1 @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Orderings_Oorder,axiom,
% 2.48/1.46 order @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 2.48/1.46 ring_char_0 @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 2.48/1.46 ring_1 @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.48/1.46 zero @ int ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_1,axiom,
% 2.48/1.46 unique1598680935umeral @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.48/1.46 canoni770627133id_add @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_2,axiom,
% 2.48/1.46 semiri134348788visors @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_3,axiom,
% 2.48/1.46 linordered_semidom @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Rings_Osemidom__divide_4,axiom,
% 2.48/1.46 semidom_divide @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Parity_Osemiring__bits_5,axiom,
% 2.48/1.46 semiring_bits @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.48/1.46 wellorder @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Groups_Omonoid__mult_6,axiom,
% 2.48/1.46 monoid_mult @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_7,axiom,
% 2.48/1.46 semiring_1 @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 2.48/1.46 order @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Nat_Onat___Groups_Ozero_9,axiom,
% 2.48/1.46 zero @ nat ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
% 2.48/1.46 order @ $o ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.48/1.46 archim1804426504_field @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_11,axiom,
% 2.48/1.46 semiri134348788visors @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_12,axiom,
% 2.48/1.46 linordered_semidom @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.48/1.46 linordered_field @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Olinordered__idom_13,axiom,
% 2.48/1.46 linordered_idom @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Osemidom__divide_14,axiom,
% 2.48/1.46 semidom_divide @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.48/1.46 division_ring @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Groups_Omonoid__mult_15,axiom,
% 2.48/1.46 monoid_mult @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Osemiring__1_16,axiom,
% 2.48/1.46 semiring_1 @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
% 2.48/1.46 order @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Nat_Oring__char__0_18,axiom,
% 2.48/1.46 ring_char_0 @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Rings_Oring__1_19,axiom,
% 2.48/1.46 ring_1 @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.48/1.46 field @ real ).
% 2.48/1.46
% 2.48/1.46 thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
% 2.48/1.46 zero @ real ).
% 2.48/1.46
% 2.48/1.46 thf(conj_0,conjecture,
% 2.48/1.46 ord_less @ real @ ( ring_1_of_int @ real @ na ) @ ( power_power @ real @ ( ring_1_of_int @ real @ ya ) @ ( suc @ pm ) ) ).
% 2.48/1.46 ------------------------
% 2.48/1.48 % (15251)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/2Mi)
% 2.48/1.48 % (15248)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/183Mi)
% 2.48/1.48 % (15251)Instruction limit reached!
% 2.48/1.48 % (15251)------------------------------
% 2.48/1.48 % (15251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.48 % (15251)Termination reason: Unknown
% 2.48/1.48 % (15251)Termination phase: shuffling
% 2.48/1.48
% 2.48/1.48 % (15251)Memory used [KB]: 1279
% 2.48/1.48 % (15251)Time elapsed: 0.002 s
% 2.48/1.48 % (15251)Instructions burned: 2 (million)
% 2.48/1.48 % (15251)------------------------------
% 2.48/1.48 % (15251)------------------------------
% 2.48/1.48 % (15254)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/18Mi)
% 2.48/1.48 % (15252)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_15149 for (2999ds/2Mi)
% 2.48/1.49 % (15249)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_15149 for (2999ds/4Mi)
% 2.48/1.49 % (15252)Instruction limit reached!
% 2.48/1.49 % (15252)------------------------------
% 2.48/1.49 % (15252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.49 % (15252)Termination reason: Unknown
% 2.48/1.49 % (15252)Termination phase: shuffling
% 2.48/1.49
% 2.48/1.49 % (15252)Memory used [KB]: 1279
% 2.48/1.49 % (15252)Time elapsed: 0.003 s
% 2.48/1.49 % (15252)Instructions burned: 3 (million)
% 2.48/1.49 % (15252)------------------------------
% 2.48/1.49 % (15252)------------------------------
% 2.48/1.49 % (15250)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_15149 for (2999ds/27Mi)
% 2.48/1.49 % (15254)Instruction limit reached!
% 2.48/1.49 % (15254)------------------------------
% 2.48/1.49 % (15254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.49 % (15254)Termination reason: Unknown
% 2.48/1.49 % (15254)Termination phase: Property scanning
% 2.48/1.49
% 2.48/1.49 % (15254)Memory used [KB]: 1535
% 2.48/1.49 % (15254)Time elapsed: 0.009 s
% 2.48/1.49 % (15254)Instructions burned: 21 (million)
% 2.48/1.49 % (15254)------------------------------
% 2.48/1.49 % (15254)------------------------------
% 2.48/1.49 % (15253)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_15149 for (2999ds/275Mi)
% 2.48/1.49 % (15255)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/3Mi)
% 2.48/1.49 % (15249)Instruction limit reached!
% 2.48/1.49 % (15249)------------------------------
% 2.48/1.49 % (15249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.49 % (15249)Termination reason: Unknown
% 2.48/1.49 % (15249)Termination phase: shuffling
% 2.48/1.49
% 2.48/1.49 % (15249)Memory used [KB]: 1407
% 2.48/1.49 % (15249)Time elapsed: 0.005 s
% 2.48/1.49 % (15249)Instructions burned: 5 (million)
% 2.48/1.49 % (15249)------------------------------
% 2.48/1.49 % (15249)------------------------------
% 2.48/1.49 % (15255)Instruction limit reached!
% 2.48/1.49 % (15255)------------------------------
% 2.48/1.49 % (15255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.48/1.49 % (15255)Termination reason: Unknown
% 2.48/1.49 % (15255)Termination phase: shuffling
% 2.48/1.49
% 2.48/1.49 % (15255)Memory used [KB]: 1279
% 2.48/1.49 % (15255)Time elapsed: 0.002 s
% 2.48/1.49 % (15255)Instructions burned: 3 (million)
% 2.48/1.49 % (15255)------------------------------
% 2.48/1.49 % (15255)------------------------------
% 2.55/1.50 % (15257)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_15149 for (2999ds/15Mi)
% 2.55/1.50 % (15256)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_15149 for (2999ds/37Mi)
% 2.55/1.50 % (15250)Instruction limit reached!
% 2.55/1.50 % (15250)------------------------------
% 2.55/1.50 % (15250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (15250)Termination reason: Unknown
% 2.55/1.50 % (15250)Termination phase: Preprocessing 1
% 2.55/1.50
% 2.55/1.50 % (15250)Memory used [KB]: 1663
% 2.55/1.50 % (15250)Time elapsed: 0.018 s
% 2.55/1.50 % (15250)Instructions burned: 28 (million)
% 2.55/1.50 % (15250)------------------------------
% 2.55/1.50 % (15250)------------------------------
% 2.55/1.50 % (15257)Instruction limit reached!
% 2.55/1.50 % (15257)------------------------------
% 2.55/1.50 % (15257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (15257)Termination reason: Unknown
% 2.55/1.50 % (15257)Termination phase: Property scanning
% 2.55/1.50
% 2.55/1.50 % (15257)Memory used [KB]: 1535
% 2.55/1.50 % (15257)Time elapsed: 0.008 s
% 2.55/1.50 % (15257)Instructions burned: 17 (million)
% 2.55/1.50 % (15257)------------------------------
% 2.55/1.50 % (15257)------------------------------
% 2.55/1.51 % (15259)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_15149 for (2999ds/1041Mi)
% 2.55/1.51 % (15258)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_15149 for (2999ds/3Mi)
% 2.55/1.51 % (15258)Instruction limit reached!
% 2.55/1.51 % (15258)------------------------------
% 2.55/1.51 % (15258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (15258)Termination reason: Unknown
% 2.55/1.51 % (15258)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (15258)Memory used [KB]: 1279
% 2.55/1.51 % (15258)Time elapsed: 0.004 s
% 2.55/1.51 % (15258)Instructions burned: 3 (million)
% 2.55/1.51 % (15258)------------------------------
% 2.55/1.51 % (15258)------------------------------
% 2.55/1.51 % (15256)Instruction limit reached!
% 2.55/1.51 % (15256)------------------------------
% 2.55/1.51 % (15256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (15256)Termination reason: Unknown
% 2.55/1.51 % (15256)Termination phase: Property scanning
% 2.55/1.51
% 2.55/1.51 % (15256)Memory used [KB]: 1791
% 2.55/1.51 % (15256)Time elapsed: 0.013 s
% 2.55/1.51 % (15256)Instructions burned: 37 (million)
% 2.55/1.51 % (15256)------------------------------
% 2.55/1.51 % (15256)------------------------------
% 2.55/1.51 % (15261)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/16Mi)
% 2.55/1.52 % (15261)Instruction limit reached!
% 2.55/1.52 % (15261)------------------------------
% 2.55/1.52 % (15261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (15261)Termination reason: Unknown
% 2.55/1.52 % (15261)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (15261)Memory used [KB]: 1663
% 2.55/1.52 % (15261)Time elapsed: 0.007 s
% 2.55/1.52 % (15261)Instructions burned: 17 (million)
% 2.55/1.52 % (15261)------------------------------
% 2.55/1.52 % (15261)------------------------------
% 2.55/1.52 % (15260)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/7Mi)
% 2.55/1.52 % (15263)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_15149 for (2999ds/3Mi)
% 2.55/1.52 % (15263)Instruction limit reached!
% 2.55/1.52 % (15263)------------------------------
% 2.55/1.52 % (15263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (15263)Termination reason: Unknown
% 2.55/1.52 % (15263)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (15263)Memory used [KB]: 1407
% 2.55/1.52 % (15263)Time elapsed: 0.003 s
% 2.55/1.52 % (15263)Instructions burned: 5 (million)
% 2.55/1.52 % (15263)------------------------------
% 2.55/1.52 % (15263)------------------------------
% 2.55/1.52 % (15260)Instruction limit reached!
% 2.55/1.52 % (15260)------------------------------
% 2.55/1.52 % (15260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (15260)Termination reason: Unknown
% 2.55/1.52 % (15260)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (15260)Memory used [KB]: 1407
% 2.55/1.52 % (15260)Time elapsed: 0.007 s
% 2.55/1.52 % (15260)Instructions burned: 8 (million)
% 2.55/1.52 % (15260)------------------------------
% 2.55/1.52 % (15260)------------------------------
% 2.55/1.53 % (15262)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_15149 for (2999ds/3Mi)
% 2.55/1.53 % (15262)Instruction limit reached!
% 2.55/1.53 % (15262)------------------------------
% 2.55/1.53 % (15262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (15262)Termination reason: Unknown
% 2.55/1.53 % (15262)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (15262)Memory used [KB]: 1279
% 2.55/1.53 % (15262)Time elapsed: 0.004 s
% 2.55/1.53 % (15262)Instructions burned: 3 (million)
% 2.55/1.53 % (15262)------------------------------
% 2.55/1.53 % (15262)------------------------------
% 2.55/1.53 % (15264)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/7Mi)
% 2.55/1.53 % (15264)Instruction limit reached!
% 2.55/1.53 % (15264)------------------------------
% 2.55/1.53 % (15264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (15264)Termination reason: Unknown
% 2.55/1.53 % (15264)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (15264)Memory used [KB]: 1407
% 2.55/1.53 % (15264)Time elapsed: 0.005 s
% 2.55/1.53 % (15264)Instructions burned: 9 (million)
% 2.55/1.53 % (15264)------------------------------
% 2.55/1.53 % (15264)------------------------------
% 2.55/1.53 % (15253)Refutation not found, incomplete strategy
% 2.55/1.53 % (15253)------------------------------
% 2.55/1.53 % (15253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (15253)Termination reason: Refutation not found, incomplete strategy
% 2.55/1.53
% 2.55/1.53
% 2.55/1.53 % (15253)Memory used [KB]: 6652
% 2.55/1.53 % (15253)Time elapsed: 0.048 s
% 2.55/1.53 % (15253)Instructions burned: 80 (million)
% 2.55/1.53 % (15253)------------------------------
% 2.55/1.53 % (15253)------------------------------
% 2.55/1.53 % (15265)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/3Mi)
% 2.55/1.53 % (15265)Instruction limit reached!
% 2.55/1.53 % (15265)------------------------------
% 2.55/1.53 % (15265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (15265)Termination reason: Unknown
% 2.55/1.53 % (15265)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (15265)Memory used [KB]: 1279
% 2.55/1.53 % (15265)Time elapsed: 0.002 s
% 2.55/1.53 % (15265)Instructions burned: 3 (million)
% 2.55/1.53 % (15265)------------------------------
% 2.55/1.53 % (15265)------------------------------
% 2.55/1.54 % (15248)Instruction limit reached!
% 2.55/1.54 % (15248)------------------------------
% 2.55/1.54 % (15248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (15248)Termination reason: Unknown
% 2.55/1.54 % (15248)Termination phase: Saturation
% 2.55/1.54
% 2.55/1.54 % (15248)Memory used [KB]: 6780
% 2.55/1.54 % (15248)Time elapsed: 0.062 s
% 2.55/1.54 % (15248)Instructions burned: 184 (million)
% 2.55/1.54 % (15248)------------------------------
% 2.55/1.54 % (15248)------------------------------
% 2.55/1.54 % (15266)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/4Mi)
% 2.55/1.54 % (15268)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/710Mi)
% 2.55/1.54 % (15270)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/902Mi)
% 2.55/1.55 % (15266)Instruction limit reached!
% 2.55/1.55 % (15266)------------------------------
% 2.55/1.55 % (15266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (15266)Termination reason: Unknown
% 2.55/1.55 % (15266)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (15266)Memory used [KB]: 1407
% 2.55/1.55 % (15266)Time elapsed: 0.005 s
% 2.55/1.55 % (15266)Instructions burned: 5 (million)
% 2.55/1.55 % (15266)------------------------------
% 2.55/1.55 % (15266)------------------------------
% 2.55/1.55 % (15267)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/18Mi)
% 2.55/1.55 % (15271)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/21Mi)
% 2.55/1.55 % (15269)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/6Mi)
% 2.55/1.55 % (15269)Instruction limit reached!
% 2.55/1.55 % (15269)------------------------------
% 2.55/1.55 % (15269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.55 % (15269)Termination reason: Unknown
% 2.55/1.55 % (15269)Termination phase: shuffling
% 2.55/1.55
% 2.55/1.55 % (15269)Memory used [KB]: 1407
% 2.55/1.55 % (15269)Time elapsed: 0.004 s
% 2.55/1.55 % (15269)Instructions burned: 6 (million)
% 2.55/1.55 % (15269)------------------------------
% 2.55/1.55 % (15269)------------------------------
% 2.55/1.56 % (15267)Instruction limit reached!
% 2.55/1.56 % (15267)------------------------------
% 2.55/1.56 % (15267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.56 % (15267)Termination reason: Unknown
% 2.55/1.56 % (15267)Termination phase: Property scanning
% 2.55/1.56
% 2.55/1.56 % (15267)Memory used [KB]: 1663
% 2.55/1.56 % (15267)Time elapsed: 0.012 s
% 2.55/1.56 % (15267)Instructions burned: 19 (million)
% 2.55/1.56 % (15267)------------------------------
% 2.55/1.56 % (15267)------------------------------
% 2.55/1.56 % (15271)Instruction limit reached!
% 2.55/1.56 % (15271)------------------------------
% 2.55/1.56 % (15271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.56 % (15271)Termination reason: Unknown
% 2.55/1.56 % (15271)Termination phase: Property scanning
% 2.55/1.56
% 2.55/1.56 % (15271)Memory used [KB]: 1663
% 2.55/1.56 % (15271)Time elapsed: 0.008 s
% 2.55/1.56 % (15271)Instructions burned: 21 (million)
% 2.55/1.56 % (15271)------------------------------
% 2.55/1.56 % (15271)------------------------------
% 2.55/1.56 % (15272)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/5Mi)
% 2.55/1.57 % (15273)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/6Mi)
% 2.55/1.57 % (15272)Instruction limit reached!
% 2.55/1.57 % (15272)------------------------------
% 2.55/1.57 % (15272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.57 % (15272)Termination reason: Unknown
% 2.55/1.57 % (15272)Termination phase: shuffling
% 2.55/1.57
% 2.55/1.57 % (15272)Memory used [KB]: 1407
% 2.55/1.57 % (15272)Time elapsed: 0.005 s
% 2.55/1.57 % (15272)Instructions burned: 6 (million)
% 2.55/1.57 % (15272)------------------------------
% 2.55/1.57 % (15272)------------------------------
% 2.55/1.57 % (15275)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/779Mi)
% 2.55/1.57 % (15273)Instruction limit reached!
% 2.55/1.57 % (15273)------------------------------
% 2.55/1.57 % (15273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.57 % (15273)Termination reason: Unknown
% 2.55/1.57 % (15273)Termination phase: shuffling
% 2.55/1.57
% 2.55/1.57 % (15273)Memory used [KB]: 1407
% 2.55/1.57 % (15273)Time elapsed: 0.004 s
% 2.55/1.57 % (15273)Instructions burned: 8 (million)
% 2.55/1.57 % (15273)------------------------------
% 2.55/1.57 % (15273)------------------------------
% 2.55/1.57 % (15274)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_15149 for (2999ds/377Mi)
% 2.55/1.58 % (15270)First to succeed.
% 2.55/1.58 % (15277)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_15149 for (2998ds/879Mi)
% 2.55/1.58 % (15270)Refutation found. Thanks to Tanya!
% 2.55/1.58 % SZS status Theorem for DTF2THF_15149
% 2.55/1.58 % SZS output start Proof for DTF2THF_15149
% 2.55/1.58 thf(type_def_5, type, int: $tType).
% 2.55/1.58 thf(type_def_7, type, nat: $tType).
% 2.55/1.58 thf(type_def_8, type, real: $tType).
% 2.55/1.58 thf(func_def_0, type, real: $tType).
% 2.55/1.58 thf(func_def_1, type, nat: $tType).
% 2.55/1.58 thf(func_def_2, type, int: $tType).
% 2.55/1.58 thf(func_def_3, type, zero: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_4, type, field: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_5, type, ring_1: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_6, type, ring_char_0: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_8, type, semiring_1: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_9, type, monoid_mult: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_10, type, wellorder: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_11, type, division_ring: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_12, type, semiring_bits: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_13, type, semidom_divide: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_14, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_15, type, linordered_field: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_16, type, linordered_semidom: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_17, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_18, type, archim1804426504_field: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_19, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_20, type, unique1598680935umeral: !>[X0: $tType]:($o)).
% 2.55/1.58 thf(func_def_21, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.55/1.58 thf(func_def_22, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 2.55/1.58 thf(func_def_23, type, suc: nat > nat).
% 2.55/1.58 thf(func_def_24, type, nthRoo220209705d_root: nat > nat > $o).
% 2.55/1.58 thf(func_def_25, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.55/1.58 thf(func_def_26, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.55/1.58 thf(func_def_27, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 2.55/1.58 thf(func_def_28, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.55/1.58 thf(func_def_29, type, ny: real).
% 2.55/1.58 thf(func_def_30, type, na: int).
% 2.55/1.58 thf(func_def_31, type, p: nat).
% 2.55/1.58 thf(func_def_32, type, pm: nat).
% 2.55/1.58 thf(func_def_33, type, x: int).
% 2.55/1.58 thf(func_def_34, type, y: int).
% 2.55/1.58 thf(func_def_35, type, ya: int).
% 2.55/1.58 thf(func_def_46, type, sK0: nat > nat > nat).
% 2.55/1.58 thf(func_def_47, type, sK1: !>[X0: $tType]:(X0 > int)).
% 2.55/1.58 thf(func_def_48, type, sK2: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_49, type, sK3: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_50, type, sK4: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_51, type, sK5: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_52, type, sK6: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_53, type, sK7: nat > nat > nat).
% 2.55/1.58 thf(func_def_54, type, sK8: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_55, type, sK9: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_56, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 2.55/1.58 thf(func_def_57, type, sK11: real > nat > real).
% 2.55/1.58 thf(func_def_58, type, sK12: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_59, type, sK13: nat > nat > nat).
% 2.55/1.58 thf(func_def_60, type, sK14: (nat > nat) > nat).
% 2.55/1.58 thf(func_def_61, type, sK15: (nat > nat) > nat).
% 2.55/1.58 thf(func_def_62, type, sK16: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.55/1.58 thf(func_def_63, type, sK17: nat > nat).
% 2.55/1.58 thf(func_def_64, type, sK18: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_65, type, sK19: nat > (nat > $o) > nat).
% 2.55/1.58 thf(func_def_66, type, sK20: !>[X0: $tType]:(X0 > int)).
% 2.55/1.58 thf(func_def_67, type, sK21: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_68, type, sK22: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_69, type, sK23: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_70, type, sK24: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_71, type, sK25: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_72, type, sK26: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_73, type, sK27: nat > nat).
% 2.55/1.58 thf(func_def_74, type, sK28: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_75, type, sK29: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_76, type, sK30: nat > nat).
% 2.55/1.58 thf(func_def_77, type, sK31: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_78, type, sK32: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_79, type, sK33: nat > nat).
% 2.55/1.58 thf(func_def_80, type, sK34: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.55/1.58 thf(func_def_81, type, sK35: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.55/1.58 thf(func_def_82, type, sK36: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_83, type, sK37: nat > real > real).
% 2.55/1.58 thf(func_def_84, type, sK38: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_85, type, sK39: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_86, type, sK40: nat > (nat > $o) > nat).
% 2.55/1.58 thf(func_def_87, type, sK41: !>[X0: $tType]:(X0 > X0)).
% 2.55/1.58 thf(func_def_88, type, sK42: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_89, type, sK43: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_90, type, sK44: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_91, type, sK45: (nat > nat > $o) > nat).
% 2.55/1.58 thf(func_def_92, type, sK46: !>[X0: $tType]:(X0 > X0)).
% 2.55/1.58 thf(func_def_93, type, sK47: (nat > $o) > nat > nat > nat).
% 2.55/1.58 thf(func_def_94, type, sK48: (nat > $o) > nat > nat > nat).
% 2.55/1.58 thf(func_def_95, type, sK49: !>[X0: $tType]:((nat > X0) > nat)).
% 2.55/1.58 thf(func_def_96, type, sK50: !>[X0: $tType, X1: $tType]:((X1 > $o) > (X1 > X0) > X1)).
% 2.55/1.58 thf(func_def_97, type, sK51: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_98, type, sK52: nat > (nat > $o) > nat).
% 2.55/1.58 thf(func_def_99, type, sK53: nat > nat > nat).
% 2.55/1.58 thf(func_def_100, type, sK54: nat > nat).
% 2.55/1.58 thf(func_def_101, type, sK55: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_102, type, sK56: (nat > $o) > nat > nat).
% 2.55/1.58 thf(func_def_103, type, sK57: real > nat > real).
% 2.55/1.58 thf(func_def_104, type, sK58: !>[X0: $tType]:((nat > X0) > nat)).
% 2.55/1.58 thf(func_def_105, type, sK59: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_106, type, sK60: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_107, type, sK61: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_108, type, sK62: (nat > $o) > nat).
% 2.55/1.58 thf(func_def_109, type, sK63: !>[X0: $tType]:((nat > X0) > nat)).
% 2.55/1.58 thf(func_def_110, type, sK64: !>[X0: $tType]:(X0 > int)).
% 2.55/1.58 thf(func_def_111, type, sK65: !>[X0: $tType]:((nat > X0) > nat)).
% 2.55/1.58 thf(func_def_113, type, ph67: !>[X0: $tType]:(X0)).
% 2.55/1.58 thf(f2093,plain,(
% 2.55/1.58 $false),
% 2.55/1.58 inference(trivial_inequality_removal,[status(thm)],[f2091])).
% 2.55/1.58 thf(f2091,plain,(
% 2.55/1.58 ($true != $true)),
% 2.55/1.58 inference(superposition,[status(thm)],[f1828,f1619])).
% 2.55/1.58 thf(f1619,plain,(
% 2.55/1.58 ((ord_less @ int @ na @ (power_power @ int @ ya @ p)) = $true)),
% 2.55/1.58 inference(cnf_transformation,[status(thm)],[f332])).
% 2.55/1.58 thf(f332,plain,(
% 2.55/1.58 ((ord_less @ int @ na @ (power_power @ int @ ya @ p)) = $true)),
% 2.55/1.58 inference(fool_elimination,[status(thm)],[f331])).
% 2.55/1.58 thf(f331,plain,(
% 2.55/1.58 (ord_less @ int @ na @ (power_power @ int @ ya @ p))),
% 2.55/1.58 inference(rectify,[status(thm)],[f61])).
% 2.55/1.58 thf(f61,axiom,(
% 2.55/1.58 (ord_less @ int @ na @ (power_power @ int @ ya @ p))),
% 2.55/1.58 file('/export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2THF_15149.p',fact_60_yyn)).
% 2.55/1.58 thf(f1828,plain,(
% 2.55/1.58 ((ord_less @ int @ na @ (power_power @ int @ ya @ p)) != $true)),
% 2.55/1.58 inference(trivial_inequality_removal,[status(thm)],[f1827])).
% 2.55/1.58 thf(f1827,plain,(
% 2.55/1.58 ($true != $true) | ((ord_less @ int @ na @ (power_power @ int @ ya @ p)) != $true)),
% 2.55/1.58 inference(forward_demodulation,[status(thm)],[f1820,f1393])).
% 2.55/1.58 thf(f1393,plain,(
% 2.55/1.58 (linordered_idom @ real = $true)),
% 2.55/1.58 inference(cnf_transformation,[status(thm)],[f644])).
% 2.55/1.58 thf(f644,plain,(
% 2.55/1.58 (linordered_idom @ real = $true)),
% 2.55/1.58 inference(fool_elimination,[status(thm)],[f643])).
% 2.55/1.58 thf(f643,plain,(
% 2.55/1.58 linordered_idom @ real),
% 2.55/1.58 inference(rectify,[status(thm)],[f282])).
% 2.55/1.58 thf(f282,axiom,(
% 2.55/1.58 linordered_idom @ real),
% 2.55/1.58 file('/export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2THF_15149.p',tcon_Real_Oreal___Rings_Olinordered__idom_13)).
% 2.55/1.58 thf(f1820,plain,(
% 2.55/1.58 (linordered_idom @ real != $true) | ((ord_less @ int @ na @ (power_power @ int @ ya @ p)) != $true)),
% 2.55/1.58 inference(superposition,[status(thm)],[f1816,f1613])).
% 2.55/1.58 thf(f1613,plain,(
% 2.55/1.58 ( ! [X0 : $tType,X2 : int,X3 : nat,X1 : int] : (((ord_less @ int @ X1 @ (power_power @ int @ X2 @ X3)) = (ord_less @ X0 @ (ring_1_of_int @ X0 @ X1) @ (power_power @ X0 @ (ring_1_of_int @ X0 @ X2) @ X3))) | (linordered_idom @ X0 != $true)) )),
% 2.55/1.58 inference(cnf_transformation,[status(thm)],[f1034])).
% 2.55/1.58 thf(f1034,plain,(
% 2.55/1.58 ! [X0 : $tType] : ((linordered_idom @ X0 != $true) | ! [X1 : int,X2 : int,X3 : nat] : ((ord_less @ int @ X1 @ (power_power @ int @ X2 @ X3)) = (ord_less @ X0 @ (ring_1_of_int @ X0 @ X1) @ (power_power @ X0 @ (ring_1_of_int @ X0 @ X2) @ X3))))),
% 2.55/1.58 inference(ennf_transformation,[status(thm)],[f788])).
% 2.55/1.58 thf(f788,plain,(
% 2.55/1.58 ! [X0 : $tType] : ((linordered_idom @ X0 = $true) => ! [X1 : int,X2 : int,X3 : nat] : ((ord_less @ int @ X1 @ (power_power @ int @ X2 @ X3)) = (ord_less @ X0 @ (ring_1_of_int @ X0 @ X1) @ (power_power @ X0 @ (ring_1_of_int @ X0 @ X2) @ X3))))),
% 2.55/1.58 inference(fool_elimination,[status(thm)],[f787])).
% 2.55/1.58 thf(f787,plain,(
% 2.55/1.58 ! [X0 : $tType] : (linordered_idom @ X0 => ! [X1 : int,X2 : int,X3 : nat] : ((ord_less @ int @ X1 @ (power_power @ int @ X2 @ X3)) = (ord_less @ X0 @ (ring_1_of_int @ X0 @ X1) @ (power_power @ X0 @ (ring_1_of_int @ X0 @ X2) @ X3))))),
% 2.55/1.58 inference(rectify,[status(thm)],[f3])).
% 2.55/1.58 thf(f3,axiom,(
% 2.55/1.58 ! [X0 : $tType] : (linordered_idom @ X0 => ! [X3 : int,X1 : int,X2 : nat] : ((ord_less @ X0 @ (ring_1_of_int @ X0 @ X3) @ (power_power @ X0 @ (ring_1_of_int @ X0 @ X1) @ X2)) = (ord_less @ int @ X3 @ (power_power @ int @ X1 @ X2))))),
% 2.55/1.58 file('/export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2THF_15149.p',fact_2_of__int__power__less__of__int__cancel__iff)).
% 2.55/1.58 thf(f1816,plain,(
% 2.55/1.58 ((ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ p)) != $true)),
% 2.55/1.58 inference(forward_demodulation,[status(thm)],[f1431,f1485])).
% 2.55/1.58 thf(f1485,plain,(
% 2.55/1.58 (p = (suc @ pm))),
% 2.55/1.58 inference(cnf_transformation,[status(thm)],[f66])).
% 2.55/1.58 thf(f66,axiom,(
% 2.55/1.58 (p = (suc @ pm))),
% 2.55/1.58 file('/export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2THF_15149.p',fact_65_p)).
% 2.55/1.58 thf(f1431,plain,(
% 2.55/1.58 ((ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm))) != $true)),
% 2.55/1.58 inference(cnf_transformation,[status(thm)],[f850])).
% 2.55/1.58 thf(f850,plain,(
% 2.55/1.58 ((ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm))) != $true)),
% 2.55/1.58 inference(flattening,[status(thm)],[f558])).
% 2.55/1.58 thf(f558,plain,(
% 2.55/1.58 ~((ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm))) = $true)),
% 2.55/1.58 inference(fool_elimination,[status(thm)],[f557])).
% 2.55/1.58 thf(f557,plain,(
% 2.55/1.58 ~(ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm)))),
% 2.55/1.58 inference(rectify,[status(thm)],[f293])).
% 2.55/1.58 thf(f293,negated_conjecture,(
% 2.55/1.58 ~(ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm)))),
% 2.55/1.58 inference(negated_conjecture,[status(cth)],[f292])).
% 2.55/1.58 thf(f292,conjecture,(
% 2.55/1.58 (ord_less @ real @ (ring_1_of_int @ real @ na) @ (power_power @ real @ (ring_1_of_int @ real @ ya) @ (suc @ pm)))),
% 2.55/1.58 file('/export/starexec/sandbox/tmp/tmp.PxC9NsNnOd/DTF2THF_15149.p',conj_0)).
% 2.55/1.58 % SZS output end Proof for DTF2THF_15149
% 2.55/1.58 % (15270)------------------------------
% 2.55/1.58 % (15270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.58 % (15270)Termination reason: Refutation
% 2.55/1.58
% 2.55/1.58 % (15270)Memory used [KB]: 6908
% 2.55/1.58 % (15270)Time elapsed: 0.035 s
% 2.55/1.58 % (15270)Instructions burned: 113 (million)
% 2.55/1.58 % (15270)------------------------------
% 2.55/1.58 % (15270)------------------------------
% 2.55/1.58 % (15247)Success in time 0.12 s
%------------------------------------------------------------------------------