%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT159^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n004.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 08:44:58 AM UTC 2026
% Result : Theorem 2.77s 1.55s
% Output : Refutation 2.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT159^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n004.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 06:09:56 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.43 ---- Original DTF file ---
% 0.16/0.43 thf(spec,logic,$$dhol).
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 % File : DAT159^1 : TPTP v9.2.1. Released v7.0.0.
% 0.16/0.43 % Domain : Data Structures
% 0.16/0.43 % Problem : Hamming stream 232
% 0.16/0.43 % Version : [Bla16] axioms : Especial.
% 0.16/0.43 % English :
% 0.16/0.43
% 0.16/0.43 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.16/0.43 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.16/0.43 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.16/0.43 % Source : [Bla16]
% 0.16/0.43 % Names : hamming_stream__232.p [Bla16]
% 0.16/0.43
% 0.16/0.43 % Status : Theorem
% 0.16/0.43 % Rating : 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% 0.16/0.43 % Syntax : Number of formulae : 366 ( 104 unt; 43 typ; 0 def)
% 0.16/0.43 % Number of atoms : 971 ( 212 equ; 0 cnn)
% 0.16/0.43 % Maximal formula atoms : 13 ( 3 avg)
% 0.16/0.43 % Number of connectives : 3600 ( 103 ~; 22 |; 46 &;2908 @)
% 0.16/0.43 % ( 0 <=>; 521 =>; 0 <=; 0 <~>)
% 0.16/0.43 % Maximal formula depth : 21 ( 8 avg)
% 0.16/0.43 % Number of types : 5 ( 4 usr)
% 0.16/0.43 % Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% 0.16/0.43 % Number of symbols : 42 ( 39 usr; 3 con; 0-4 aty)
% 0.16/0.43 % Number of variables : 888 ( 26 ^; 778 !; 49 ?; 888 :)
% 0.16/0.43 % ( 35 !>; 0 ?*; 0 @-; 0 @+)
% 0.16/0.43 % SPC : TH1_THM_EQU_NAR
% 0.16/0.43
% 0.16/0.43 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.16/0.43 % 2016-07-13 14:42:29.978
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 %----Could-be-implicit typings (5)
% 0.16/0.43 thf(ty_t_Code__Numeral_Onatural,type,
% 0.16/0.43 code_natural: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Real_Oreal,type,
% 0.16/0.43 real: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Nat_Onat,type,
% 0.16/0.43 nat: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_Int_Oint,type,
% 0.16/0.43 int: $tType ).
% 0.16/0.43
% 0.16/0.43 thf(ty_t_itself,type,
% 0.16/0.43 itself: $tType > $tType ).
% 0.16/0.43
% 0.16/0.43 %----Explicit typings (38)
% 0.16/0.43 thf(sy_cl_HOL_Otype,type,
% 0.16/0.43 type:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Groups_Ozero,type,
% 0.16/0.43 zero:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Fields_Ofield,type,
% 0.16/0.43 field:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Oord,type,
% 0.16/0.43 ord:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Oorder,type,
% 0.16/0.43 order:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Ono__bot,type,
% 0.16/0.43 no_bot:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Ono__top,type,
% 0.16/0.43 no_top:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Rings_Osemiring__1,type,
% 0.16/0.43 semiring_1:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Olinorder,type,
% 0.16/0.43 linorder:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Orderings_Opreorder,type,
% 0.16/0.43 preorder:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.16/0.43 zero_neq_one:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.43
% 0.16/0.43 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.16/0.43 semiring_char_0:
% 0.16/0.43 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Orderings_Owellorder,type,
% 0.16/0.44 wellorder:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Fields_Odivision__ring,type,
% 0.16/0.44 division_ring:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Orderings_Odense__order,type,
% 0.16/0.44 dense_order:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.16/0.44 linordered_idom:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Fields_Olinordered__field,type,
% 0.16/0.44 linordered_field:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.16/0.44 dense_linorder:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.16/0.44 linordered_semidom:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.16/0.44 semiri134348788visors:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.16/0.44 archim1804426504_field:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.16/0.44 canoni770627133id_add:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.16/0.44 condit1656338222tinuum:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.16/0.44 condit1037483654norder:
% 0.16/0.44 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 0.16/0.44 code_case_natural:
% 0.16/0.44 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 0.16/0.44 code_size_natural: code_natural > nat ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.16/0.44 inverse_inverse:
% 0.16/0.44 !>[A: $tType] : ( A > A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.16/0.44 zero_zero:
% 0.16/0.44 !>[A: $tType] : A ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth,type,
% 0.16/0.44 hammin1195055315smooth: nat > $o ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_If,type,
% 0.16/0.44 if:
% 0.16/0.44 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.16/0.44 semiring_1_of_nat:
% 0.16/0.44 !>[A: $tType] : ( nat > A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.16/0.44 size_size:
% 0.16/0.44 !>[A: $tType] : ( A > nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.16/0.44 ord_less:
% 0.16/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.16/0.44 ord_less_eq:
% 0.16/0.44 !>[A: $tType] : ( A > A > $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Power_Opower__class_Opower,type,
% 0.16/0.44 power_power:
% 0.16/0.44 !>[A: $tType] : ( A > nat > A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Pure_Otype,type,
% 0.16/0.44 type2:
% 0.16/0.44 !>[A: $tType] : ( itself @ A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 0.16/0.44 zero_neq_one_of_bool:
% 0.16/0.44 !>[A: $tType] : ( $o > A ) ).
% 0.16/0.44
% 0.16/0.44 thf(sy_v_n,type,
% 0.16/0.44 n: nat ).
% 0.16/0.44
% 0.16/0.44 %----Relevant facts (254)
% 0.16/0.44 thf(fact_0_smooth__0,axiom,
% 0.16/0.44 ~ ( hammin1195055315smooth @ ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % smooth_0
% 0.16/0.44 thf(fact_1_neq0__conv,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( N
% 0.16/0.44 != ( zero_zero @ nat ) )
% 0.16/0.44 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % neq0_conv
% 0.16/0.44 thf(fact_2_not__gr__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: A] :
% 0.16/0.44 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.16/0.44 = ( N
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_gr_zero
% 0.16/0.44 thf(fact_3_gr0I,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( N
% 0.16/0.44 != ( zero_zero @ nat ) )
% 0.16/0.44 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr0I
% 0.16/0.44 thf(fact_4_not__gr0,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.16/0.44 = ( N
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_gr0
% 0.16/0.44 thf(fact_5_not__less0,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_less0
% 0.16/0.44 thf(fact_6_less__zeroE,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_zeroE
% 0.16/0.44 thf(fact_7_gr__implies__not0,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M @ N )
% 0.16/0.44 => ( N
% 0.16/0.44 != ( zero_zero @ nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr_implies_not0
% 0.16/0.44 thf(fact_8_infinite__descent0,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ( P @ ( zero_zero @ nat ) )
% 0.16/0.44 => ( ! [N2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.16/0.44 => ( ~ ( P @ N2 )
% 0.16/0.44 => ? [M2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M2 @ N2 )
% 0.16/0.44 & ~ ( P @ M2 ) ) ) )
% 0.16/0.44 => ( P @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % infinite_descent0
% 0.16/0.44 thf(fact_9_less__nat__zero__code,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_nat_zero_code
% 0.16/0.44 thf(fact_10_infinite__descent0__measure,axiom,
% 0.16/0.44 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ( ( V @ X2 )
% 0.16/0.44 = ( zero_zero @ nat ) )
% 0.16/0.44 => ( P @ X2 ) )
% 0.16/0.44 => ( ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.16/0.44 => ( ~ ( P @ X2 )
% 0.16/0.44 => ? [Y: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.16/0.44 & ~ ( P @ Y ) ) ) )
% 0.16/0.44 => ( P @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % infinite_descent0_measure
% 0.16/0.44 thf(fact_11_less__numeral__extra_I3_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_numeral_extra(3)
% 0.16/0.44 thf(fact_12_zero__reorient,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( zero @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ( ( ( zero_zero @ A )
% 0.16/0.44 = X )
% 0.16/0.44 = ( X
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_reorient
% 0.16/0.44 thf(fact_13_infinite__descent__measure,axiom,
% 0.16/0.44 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ~ ( P @ X2 )
% 0.16/0.44 => ? [Y: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 0.16/0.44 & ~ ( P @ Y ) ) )
% 0.16/0.44 => ( P @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % infinite_descent_measure
% 0.16/0.44 thf(fact_14_measure__induct__rule,axiom,
% 0.16/0.44 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ! [Y: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
% 0.16/0.44 => ( P @ Y ) )
% 0.16/0.44 => ( P @ X2 ) )
% 0.16/0.44 => ( P @ A2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % measure_induct_rule
% 0.16/0.44 thf(fact_15_linorder__neqE__nat,axiom,
% 0.16/0.44 ! [X: nat,Y2: nat] :
% 0.16/0.44 ( ( X != Y2 )
% 0.16/0.44 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 0.16/0.44 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linorder_neqE_nat
% 0.16/0.44 thf(fact_16_infinite__descent,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ! [N2: nat] :
% 0.16/0.44 ( ~ ( P @ N2 )
% 0.16/0.44 => ? [M2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M2 @ N2 )
% 0.16/0.44 & ~ ( P @ M2 ) ) )
% 0.16/0.44 => ( P @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % infinite_descent
% 0.16/0.44 thf(fact_17_nat__less__induct,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ! [N2: nat] :
% 0.16/0.44 ( ! [M2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M2 @ N2 )
% 0.16/0.44 => ( P @ M2 ) )
% 0.16/0.44 => ( P @ N2 ) )
% 0.16/0.44 => ( P @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_less_induct
% 0.16/0.44 thf(fact_18_less__irrefl__nat,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.16/0.44
% 0.16/0.44 % less_irrefl_nat
% 0.16/0.44 thf(fact_19_measure__induct,axiom,
% 0.16/0.44 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ! [Y: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
% 0.16/0.44 => ( P @ Y ) )
% 0.16/0.44 => ( P @ X2 ) )
% 0.16/0.44 => ( P @ A2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % measure_induct
% 0.16/0.44 thf(fact_20_less__not__refl3,axiom,
% 0.16/0.44 ! [S: nat,T2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ S @ T2 )
% 0.16/0.44 => ( S != T2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_not_refl3
% 0.16/0.44 thf(fact_21_less__not__refl2,axiom,
% 0.16/0.44 ! [N: nat,M: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ N @ M )
% 0.16/0.44 => ( M != N ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_not_refl2
% 0.16/0.44 thf(fact_22_less__not__refl,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ~ ( ord_less @ nat @ N @ N ) ).
% 0.16/0.44
% 0.16/0.44 % less_not_refl
% 0.16/0.44 thf(fact_23_nat__neq__iff,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( M != N )
% 0.16/0.44 = ( ( ord_less @ nat @ M @ N )
% 0.16/0.44 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_neq_iff
% 0.16/0.44 thf(fact_24_zero__less__iff__neq__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.16/0.44 = ( N
% 0.16/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_iff_neq_zero
% 0.16/0.44 thf(fact_25_gr__implies__not__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: A,N: A] :
% 0.16/0.44 ( ( ord_less @ A @ M @ N )
% 0.16/0.44 => ( N
% 0.16/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr_implies_not_zero
% 0.16/0.44 thf(fact_26_not__less__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: A] :
% 0.16/0.44 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_less_zero
% 0.16/0.44 thf(fact_27_gr__zeroI,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: A] :
% 0.16/0.44 ( ( N
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % gr_zeroI
% 0.16/0.44 thf(fact_28_of__nat__0__less__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_0_less_iff
% 0.16/0.44 thf(fact_29_zero__natural_Orsp,axiom,
% 0.16/0.44 ( ( zero_zero @ nat )
% 0.16/0.44 = ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_natural.rsp
% 0.16/0.44 thf(fact_30_ex__gt__or__lt,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ? [B: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B )
% 0.16/0.44 | ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_gt_or_lt
% 0.16/0.44 thf(fact_31_linorder__neqE__linordered__idom,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( X != Y2 )
% 0.16/0.44 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linorder_neqE_linordered_idom
% 0.16/0.44 thf(fact_32_dual__order_Ostrict__implies__not__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ B2 @ A2 )
% 0.16/0.44 => ( A2 != B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_implies_not_eq
% 0.16/0.44 thf(fact_33_linordered__field__no__ub,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X3: A] :
% 0.16/0.44 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_field_no_ub
% 0.16/0.44 thf(fact_34_linordered__field__no__lb,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X3: A] :
% 0.16/0.44 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 0.16/0.44
% 0.16/0.44 % linordered_field_no_lb
% 0.16/0.44 thf(fact_35_order_Ostrict__implies__not__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( A2 != B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_implies_not_eq
% 0.16/0.44 thf(fact_36_dependent__wellorder__choice,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: ( A > B3 ) > A > B3 > $o] :
% 0.16/0.44 ( ! [R: B3,F2: A > B3,G: A > B3,X2: A] :
% 0.16/0.44 ( ! [Y: A] :
% 0.16/0.44 ( ( ord_less @ A @ Y @ X2 )
% 0.16/0.44 => ( ( F2 @ Y )
% 0.16/0.44 = ( G @ Y ) ) )
% 0.16/0.44 => ( ( P @ F2 @ X2 @ R )
% 0.16/0.44 = ( P @ G @ X2 @ R ) ) )
% 0.16/0.44 => ( ! [X2: A,F2: A > B3] :
% 0.16/0.44 ( ! [Y: A] :
% 0.16/0.44 ( ( ord_less @ A @ Y @ X2 )
% 0.16/0.44 => ( P @ F2 @ Y @ ( F2 @ Y ) ) )
% 0.16/0.44 => ? [X12: B3] : ( P @ F2 @ X2 @ X12 ) )
% 0.16/0.44 => ? [F2: A > B3] :
% 0.16/0.44 ! [X3: A] : ( P @ F2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dependent_wellorder_choice
% 0.16/0.44 thf(fact_37_not__less__iff__gr__or__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.44 = ( ( ord_less @ A @ Y2 @ X )
% 0.16/0.44 | ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_less_iff_gr_or_eq
% 0.16/0.44 thf(fact_38_dual__order_Ostrict__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A,C: A] :
% 0.16/0.44 ( ( ord_less @ A @ B2 @ A2 )
% 0.16/0.44 => ( ( ord_less @ A @ C @ B2 )
% 0.16/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_trans
% 0.16/0.44 thf(fact_39_of__nat__eq__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ( semiring_1_of_nat @ A @ M )
% 0.16/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 = ( M = N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_eq_iff
% 0.16/0.44 thf(fact_40_of__nat__0,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.16/0.44 = ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_0
% 0.16/0.44 thf(fact_41_of__nat__0__eq__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: nat] :
% 0.16/0.44 ( ( ( zero_zero @ A )
% 0.16/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 = ( ( zero_zero @ nat )
% 0.16/0.44 = N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_0_eq_iff
% 0.16/0.44 thf(fact_42_of__nat__eq__0__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat] :
% 0.16/0.44 ( ( ( semiring_1_of_nat @ A @ M )
% 0.16/0.44 = ( zero_zero @ A ) )
% 0.16/0.44 = ( M
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_eq_0_iff
% 0.16/0.44 thf(fact_43_of__nat__less__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_iff
% 0.16/0.44 thf(fact_44_of__nat__less__0__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat] :
% 0.16/0.44 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_0_iff
% 0.16/0.44 thf(fact_45_less__imp__of__nat__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M @ N )
% 0.16/0.44 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_of_nat_less
% 0.16/0.44 thf(fact_46_of__nat__less__imp__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_less_imp_less
% 0.16/0.44 thf(fact_47_ord__eq__less__subst,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( A2
% 0.16/0.44 = ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_eq_less_subst
% 0.16/0.44 thf(fact_48_ord__less__eq__subst,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > B3,C: B3] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ( F @ B2 )
% 0.16/0.44 = C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_less_eq_subst
% 0.16/0.44 thf(fact_49_order__less__subst1,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_less_subst1
% 0.16/0.44 thf(fact_50_order__less__subst2,axiom,
% 0.16/0.44 ! [A: $tType,C2: $tType] :
% 0.16/0.44 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_less_subst2
% 0.16/0.44 thf(fact_51_lt__ex,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % lt_ex
% 0.16/0.44 thf(fact_52_gt__ex,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( no_top @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.16/0.44
% 0.16/0.44 % gt_ex
% 0.16/0.44 thf(fact_53_neqE,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( X != Y2 )
% 0.16/0.44 => ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neqE
% 0.16/0.44 thf(fact_54_neq__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( X != Y2 )
% 0.16/0.44 = ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % neq_iff
% 0.16/0.44 thf(fact_55_order_Oasym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.asym
% 0.16/0.44 thf(fact_56_dense,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ? [Z: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Z )
% 0.16/0.44 & ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dense
% 0.16/0.44 thf(fact_57_less__imp__neq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( X != Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_neq
% 0.16/0.44 thf(fact_58_less__asym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_asym
% 0.16/0.44 thf(fact_59_less__asym_H,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_asym'
% 0.16/0.44 thf(fact_60_less__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.16/0.44 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_trans
% 0.16/0.44 thf(fact_61_less__linear,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 | ( X = Y2 )
% 0.16/0.44 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_linear
% 0.16/0.44 thf(fact_62_less__irrefl,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_irrefl
% 0.16/0.44 thf(fact_63_ord__eq__less__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( A2 = B2 )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_eq_less_trans
% 0.16/0.44 thf(fact_64_ord__less__eq__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( B2 = C )
% 0.16/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_less_eq_trans
% 0.16/0.44 thf(fact_65_dual__order_Oasym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ B2 @ A2 )
% 0.16/0.44 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.asym
% 0.16/0.44 thf(fact_66_less__imp__not__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( X != Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_not_eq
% 0.16/0.44 thf(fact_67_less__not__sym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_not_sym
% 0.16/0.44 thf(fact_68_less__induct,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > $o,A2: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ! [Y: A] :
% 0.16/0.44 ( ( ord_less @ A @ Y @ X2 )
% 0.16/0.44 => ( P @ Y ) )
% 0.16/0.44 => ( P @ X2 ) )
% 0.16/0.44 => ( P @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_induct
% 0.16/0.44 thf(fact_69_antisym__conv3,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Y2: A,X: A] :
% 0.16/0.44 ( ~ ( ord_less @ A @ Y2 @ X )
% 0.16/0.44 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.44 = ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % antisym_conv3
% 0.16/0.44 thf(fact_70_less__imp__not__eq2,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( Y2 != X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_not_eq2
% 0.16/0.44 thf(fact_71_less__imp__triv,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,P: $o] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less @ A @ Y2 @ X )
% 0.16/0.44 => P ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_triv
% 0.16/0.44 thf(fact_72_linorder__cases,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ( X != Y2 )
% 0.16/0.44 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linorder_cases
% 0.16/0.44 thf(fact_73_dual__order_Oirrefl,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.irrefl
% 0.16/0.44 thf(fact_74_order_Ostrict__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_trans
% 0.16/0.44 thf(fact_75_less__imp__not__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_not_less
% 0.16/0.44 thf(fact_76_zero__less__imp__eq__int,axiom,
% 0.16/0.44 ! [K: int] :
% 0.16/0.44 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.16/0.44 => ? [N2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.16/0.44 & ( K
% 0.16/0.44 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_imp_eq_int
% 0.16/0.44 thf(fact_77_reals__Archimedean2,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % reals_Archimedean2
% 0.16/0.44 thf(fact_78_natural_Osize_I1_J,axiom,
% 0.16/0.44 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 0.16/0.44 = ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % natural.size(1)
% 0.16/0.44 thf(fact_79_of__nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 0.16/0.44 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.16/0.44 | ( N
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_zero_less_power_iff
% 0.16/0.44 thf(fact_80_of__nat__le__0__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.16/0.44 = ( M
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_0_iff
% 0.16/0.44 thf(fact_81_natural_Osize_I3_J,axiom,
% 0.16/0.44 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 0.16/0.44 = ( zero_zero @ nat ) ) ).
% 0.16/0.44
% 0.16/0.44 % natural.size(3)
% 0.16/0.44 thf(fact_82_of__bool__eq_I1_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( zero_neq_one_of_bool @ A @ $false )
% 0.16/0.44 = ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_bool_eq(1)
% 0.16/0.44 thf(fact_83_ex__inverse__of__nat__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.16/0.44 => ? [N2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.16/0.44 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_inverse_of_nat_less
% 0.16/0.44 thf(fact_84_order__refl,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_refl
% 0.16/0.44 thf(fact_85_le0,axiom,
% 0.16/0.44 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.16/0.44
% 0.16/0.44 % le0
% 0.16/0.44 thf(fact_86_inverse__inverse__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.16/0.44 = A2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_inverse_eq
% 0.16/0.44 thf(fact_87_inverse__eq__iff__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 = ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 = ( A2 = B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_eq_iff_eq
% 0.16/0.44 thf(fact_88_le__zero__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.16/0.44 = ( N
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_zero_eq
% 0.16/0.44 thf(fact_89_of__nat__le__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.16/0.44 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_le_iff
% 0.16/0.44 thf(fact_90_inverse__nonzero__iff__nonzero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 = ( zero_zero @ A ) )
% 0.16/0.44 = ( A2
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_nonzero_iff_nonzero
% 0.16/0.44 thf(fact_91_inverse__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.16/0.44 = ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_zero
% 0.16/0.44 thf(fact_92_nat__zero__less__power__iff,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.16/0.44 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.16/0.44 | ( N
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_zero_less_power_iff
% 0.16/0.44 thf(fact_93_of__nat__power,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [M: nat,N: nat] :
% 0.16/0.44 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 0.16/0.44 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_power
% 0.16/0.44 thf(fact_94_inverse__nonnegative__iff__nonnegative,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.16/0.44 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_nonnegative_iff_nonnegative
% 0.16/0.44 thf(fact_95_inverse__nonpositive__iff__nonpositive,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.16/0.44 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_nonpositive_iff_nonpositive
% 0.16/0.44 thf(fact_96_inverse__positive__iff__positive,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.16/0.44 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_positive_iff_positive
% 0.16/0.44 thf(fact_97_inverse__negative__iff__negative,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.16/0.44 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_negative_iff_negative
% 0.16/0.44 thf(fact_98_inverse__less__iff__less__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_less_iff_less_neg
% 0.16/0.44 thf(fact_99_inverse__less__iff__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_less_iff_less
% 0.16/0.44 thf(fact_100_inverse__le__iff__le__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_le_iff_le_neg
% 0.16/0.44 thf(fact_101_inverse__le__iff__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_le_iff_le
% 0.16/0.44 thf(fact_102_power__eq__0__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiri134348788visors @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat] :
% 0.16/0.44 ( ( ( power_power @ A @ A2 @ N )
% 0.16/0.44 = ( zero_zero @ A ) )
% 0.16/0.44 = ( ( A2
% 0.16/0.44 = ( zero_zero @ A ) )
% 0.16/0.44 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_eq_0_iff
% 0.16/0.44 thf(fact_103_nonneg__eq__int,axiom,
% 0.16/0.44 ! [Z2: int] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.16/0.44 => ~ ! [M3: nat] :
% 0.16/0.44 ( Z2
% 0.16/0.44 != ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nonneg_eq_int
% 0.16/0.44 thf(fact_104_nonneg__int__cases,axiom,
% 0.16/0.44 ! [K: int] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.16/0.44 => ~ ! [N2: nat] :
% 0.16/0.44 ( K
% 0.16/0.44 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nonneg_int_cases
% 0.16/0.44 thf(fact_105_zero__le__imp__eq__int,axiom,
% 0.16/0.44 ! [K: int] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.16/0.44 => ? [N2: nat] :
% 0.16/0.44 ( K
% 0.16/0.44 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_imp_eq_int
% 0.16/0.44 thf(fact_106_zero__integer_Orsp,axiom,
% 0.16/0.44 ( ( zero_zero @ int )
% 0.16/0.44 = ( zero_zero @ int ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_integer.rsp
% 0.16/0.44 thf(fact_107_inverse__eq__imp__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 = ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( A2 = B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_eq_imp_eq
% 0.16/0.44 thf(fact_108_dual__order_Oantisym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( A2 = B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.antisym
% 0.16/0.44 thf(fact_109_of__bool__eq__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P2: $o,Q: $o] :
% 0.16/0.44 ( ( ( zero_neq_one_of_bool @ A @ P2 )
% 0.16/0.44 = ( zero_neq_one_of_bool @ A @ Q ) )
% 0.16/0.44 = ( P2 = Q ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_bool_eq_iff
% 0.16/0.44 thf(fact_110_dual__order_Otrans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A,C: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.16/0.44 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.trans
% 0.16/0.44 thf(fact_111_linorder__wlog,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.16/0.44 ( ! [A3: A,B: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A3 @ B )
% 0.16/0.44 => ( P @ A3 @ B ) )
% 0.16/0.44 => ( ! [A3: A,B: A] :
% 0.16/0.44 ( ( P @ B @ A3 )
% 0.16/0.44 => ( P @ A3 @ B ) )
% 0.16/0.44 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linorder_wlog
% 0.16/0.44 thf(fact_112_dual__order_Orefl,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.refl
% 0.16/0.44 thf(fact_113_order__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.44 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_trans
% 0.16/0.44 thf(fact_114_order__class_Oorder_Oantisym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.16/0.44 => ( A2 = B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_class.order.antisym
% 0.16/0.44 thf(fact_115_ord__le__eq__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( B2 = C )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_le_eq_trans
% 0.16/0.44 thf(fact_116_ord__eq__le__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( ord @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( A2 = B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_eq_le_trans
% 0.16/0.44 thf(fact_117_antisym__conv,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Y2: A,X: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.44 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 = ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % antisym_conv
% 0.16/0.44 thf(fact_118_le__cases3,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.16/0.44 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_cases3
% 0.16/0.44 thf(fact_119_real__arch__simple,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] :
% 0.16/0.44 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % real_arch_simple
% 0.16/0.44 thf(fact_120_order_Otrans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.trans
% 0.16/0.44 thf(fact_121_le__cases,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_cases
% 0.16/0.44 thf(fact_122_eq__refl,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( X = Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % eq_refl
% 0.16/0.44 thf(fact_123_linear,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % linear
% 0.16/0.44 thf(fact_124_antisym,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.44 => ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % antisym
% 0.16/0.44 thf(fact_125_eq__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.16/0.44 = ( ^ [X4: A,Y5: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.16/0.44 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % eq_iff
% 0.16/0.44 thf(fact_126_ord__le__eq__subst,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > B3,C: B3] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ( F @ B2 )
% 0.16/0.44 = C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_le_eq_subst
% 0.16/0.44 thf(fact_127_ord__eq__le__subst,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( ord @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( A2
% 0.16/0.44 = ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ord_eq_le_subst
% 0.16/0.44 thf(fact_128_order__subst2,axiom,
% 0.16/0.44 ! [A: $tType,C2: $tType] :
% 0.16/0.44 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_subst2
% 0.16/0.44 thf(fact_129_order__subst1,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_subst1
% 0.16/0.44 thf(fact_130_le__fun__def,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 => ( ( ord_less_eq @ ( A > B3 ) )
% 0.16/0.44 = ( ^ [F3: A > B3,G2: A > B3] :
% 0.16/0.44 ! [X4: A] : ( ord_less_eq @ B3 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_fun_def
% 0.16/0.44 thf(fact_131_le__funI,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 => ! [F: A > B3,G3: A > B3] :
% 0.16/0.44 ( ! [X2: A] : ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G3 @ X2 ) )
% 0.16/0.44 => ( ord_less_eq @ ( A > B3 ) @ F @ G3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_funI
% 0.16/0.44 thf(fact_132_le__funE,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 => ! [F: A > B3,G3: A > B3,X: A] :
% 0.16/0.44 ( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
% 0.16/0.44 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_funE
% 0.16/0.44 thf(fact_133_le__funD,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 => ! [F: A > B3,G3: A > B3,X: A] :
% 0.16/0.44 ( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
% 0.16/0.44 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_funD
% 0.16/0.44 thf(fact_134_int__int__eq,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ( semiring_1_of_nat @ int @ M )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ N ) )
% 0.16/0.44 = ( M = N ) ) ).
% 0.16/0.44
% 0.16/0.44 % int_int_eq
% 0.16/0.44 thf(fact_135_zle__int,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.16/0.44 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % zle_int
% 0.16/0.44 thf(fact_136_power__not__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiri134348788visors @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat] :
% 0.16/0.44 ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ( power_power @ A @ A2 @ N )
% 0.16/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_not_zero
% 0.16/0.44 thf(fact_137_power__less__imp__less__base,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.16/0.44 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_less_imp_less_base
% 0.16/0.44 thf(fact_138_power__inverse,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat] :
% 0.16/0.44 ( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
% 0.16/0.44 = ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_inverse
% 0.16/0.44 thf(fact_139_zero__le__power,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le_power
% 0.16/0.44 thf(fact_140_power__mono,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_mono
% 0.16/0.44 thf(fact_141_less__eq__int__code_I1_J,axiom,
% 0.16/0.44 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.16/0.44
% 0.16/0.44 % less_eq_int_code(1)
% 0.16/0.44 thf(fact_142_less__int__code_I1_J,axiom,
% 0.16/0.44 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_int_code(1)
% 0.16/0.44 thf(fact_143_le__imp__inverse__le__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_imp_inverse_le_neg
% 0.16/0.44 thf(fact_144_inverse__le__imp__le__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_le_imp_le_neg
% 0.16/0.44 thf(fact_145_le__imp__inverse__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_imp_inverse_le
% 0.16/0.44 thf(fact_146_inverse__le__imp__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_le_imp_le
% 0.16/0.44 thf(fact_147_nonzero__imp__inverse__nonzero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nonzero_imp_inverse_nonzero
% 0.16/0.44 thf(fact_148_nonzero__inverse__inverse__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.16/0.44 = A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nonzero_inverse_inverse_eq
% 0.16/0.44 thf(fact_149_nonzero__inverse__eq__imp__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 = ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ( B2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( A2 = B2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nonzero_inverse_eq_imp_eq
% 0.16/0.44 thf(fact_150_inverse__zero__imp__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ( inverse_inverse @ A @ A2 )
% 0.16/0.44 = ( zero_zero @ A ) )
% 0.16/0.44 => ( A2
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_zero_imp_zero
% 0.16/0.44 thf(fact_151_field__class_Ofield__inverse__zero,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.16/0.44 = ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % field_class.field_inverse_zero
% 0.16/0.44 thf(fact_152_power__eq__imp__eq__base,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat,B2: A] :
% 0.16/0.44 ( ( ( power_power @ A @ A2 @ N )
% 0.16/0.44 = ( power_power @ A @ B2 @ N ) )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.16/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.16/0.44 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_eq_imp_eq_base
% 0.16/0.44 thf(fact_153_power__eq__iff__eq__base,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: nat,A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.16/0.44 => ( ( ( power_power @ A @ A2 @ N )
% 0.16/0.44 = ( power_power @ A @ B2 @ N ) )
% 0.16/0.44 = ( A2 = B2 ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_eq_iff_eq_base
% 0.16/0.44 thf(fact_154_zero__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_le
% 0.16/0.44 thf(fact_155_le__numeral__extra_I3_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_numeral_extra(3)
% 0.16/0.44 thf(fact_156_complete__interval,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,P: A > $o] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( P @ A2 )
% 0.16/0.44 => ( ~ ( P @ B2 )
% 0.16/0.44 => ? [C3: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.16/0.44 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.16/0.44 & ! [X3: A] :
% 0.16/0.44 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 0.16/0.44 & ( ord_less @ A @ X3 @ C3 ) )
% 0.16/0.44 => ( P @ X3 ) )
% 0.16/0.44 & ! [D: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.16/0.44 & ( ord_less @ A @ X2 @ D ) )
% 0.16/0.44 => ( P @ X2 ) )
% 0.16/0.44 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % complete_interval
% 0.16/0.44 thf(fact_157_order_Onot__eq__order__implies__strict,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( A2 != B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.not_eq_order_implies_strict
% 0.16/0.44 thf(fact_158_dual__order_Ostrict__implies__order,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ B2 @ A2 )
% 0.16/0.44 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_implies_order
% 0.16/0.44 thf(fact_159_dual__order_Ostrict__iff__order,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less @ A )
% 0.16/0.44 = ( ^ [B4: A,A4: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.16/0.44 & ( A4 != B4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_iff_order
% 0.16/0.44 thf(fact_160_dual__order_Oorder__iff__strict,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less_eq @ A )
% 0.16/0.44 = ( ^ [B4: A,A4: A] :
% 0.16/0.44 ( ( ord_less @ A @ B4 @ A4 )
% 0.16/0.44 | ( A4 = B4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.order_iff_strict
% 0.16/0.44 thf(fact_161_order_Ostrict__implies__order,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_implies_order
% 0.16/0.44 thf(fact_162_dense__le__bounded,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ! [W: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ W )
% 0.16/0.44 => ( ( ord_less @ A @ W @ Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dense_le_bounded
% 0.16/0.44 thf(fact_163_dense__ge__bounded,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Z2: A,X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z2 @ X )
% 0.16/0.44 => ( ! [W: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z2 @ W )
% 0.16/0.44 => ( ( ord_less @ A @ W @ X )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dense_ge_bounded
% 0.16/0.44 thf(fact_164_dual__order_Ostrict__trans2,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A,C: A] :
% 0.16/0.44 ( ( ord_less @ A @ B2 @ A2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.16/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_trans2
% 0.16/0.44 thf(fact_165_dual__order_Ostrict__trans1,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [B2: A,A2: A,C: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.16/0.44 => ( ( ord_less @ A @ C @ B2 )
% 0.16/0.44 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dual_order.strict_trans1
% 0.16/0.44 thf(fact_166_order_Ostrict__iff__order,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less @ A )
% 0.16/0.44 = ( ^ [A4: A,B4: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.16/0.44 & ( A4 != B4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_iff_order
% 0.16/0.44 thf(fact_167_order_Oorder__iff__strict,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less_eq @ A )
% 0.16/0.44 = ( ^ [A4: A,B4: A] :
% 0.16/0.44 ( ( ord_less @ A @ A4 @ B4 )
% 0.16/0.44 | ( A4 = B4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.order_iff_strict
% 0.16/0.44 thf(fact_168_order_Ostrict__trans2,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_trans2
% 0.16/0.44 thf(fact_169_order_Ostrict__trans1,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,C: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ C )
% 0.16/0.44 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order.strict_trans1
% 0.16/0.44 thf(fact_170_not__le__imp__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Y2: A,X: A] :
% 0.16/0.44 ( ~ ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.44 => ( ord_less @ A @ X @ Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_le_imp_less
% 0.16/0.44 thf(fact_171_less__le__not__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less @ A )
% 0.16/0.44 = ( ^ [X4: A,Y5: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.16/0.44 & ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_le_not_le
% 0.16/0.44 thf(fact_172_le__imp__less__or__eq,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 | ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_imp_less_or_eq
% 0.16/0.44 thf(fact_173_le__less__linear,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_less_linear
% 0.16/0.44 thf(fact_174_dense__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Y2: A,Z2: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dense_le
% 0.16/0.44 thf(fact_175_dense__ge,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Z2: A,Y2: A] :
% 0.16/0.44 ( ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z2 @ X2 )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % dense_ge
% 0.16/0.44 thf(fact_176_less__le__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 0.16/0.44 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_le_trans
% 0.16/0.44 thf(fact_177_le__less__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A,Z2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less @ A @ Y2 @ Z2 )
% 0.16/0.44 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_less_trans
% 0.16/0.44 thf(fact_178_antisym__conv2,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.44 = ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % antisym_conv2
% 0.16/0.44 thf(fact_179_antisym__conv1,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ X @ Y2 )
% 0.16/0.44 = ( X = Y2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % antisym_conv1
% 0.16/0.44 thf(fact_180_less__imp__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( preorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_le
% 0.16/0.44 thf(fact_181_le__neq__trans,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( A2 != B2 )
% 0.16/0.44 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_neq_trans
% 0.16/0.44 thf(fact_182_not__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 0.16/0.44 = ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_less
% 0.16/0.44 thf(fact_183_not__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
% 0.16/0.44 = ( ord_less @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % not_le
% 0.16/0.44 thf(fact_184_order__less__le__subst2,axiom,
% 0.16/0.44 ! [A: $tType,C2: $tType] :
% 0.16/0.44 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_less_le_subst2
% 0.16/0.44 thf(fact_185_order__less__le__subst1,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_less_le_subst1
% 0.16/0.44 thf(fact_186_order__le__less__subst2,axiom,
% 0.16/0.44 ! [A: $tType,C2: $tType] :
% 0.16/0.44 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.16/0.44 => ( ! [X2: A,Y3: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_le_less_subst2
% 0.16/0.44 thf(fact_187_order__le__less__subst1,axiom,
% 0.16/0.44 ! [A: $tType,B3: $tType] :
% 0.16/0.44 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 & ( order @ A @ ( type2 @ A ) ) )
% 0.16/0.44 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 0.16/0.44 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ B3 @ B2 @ C )
% 0.16/0.44 => ( ! [X2: B3,Y3: B3] :
% 0.16/0.44 ( ( ord_less @ B3 @ X2 @ Y3 )
% 0.16/0.44 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 0.16/0.44 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % order_le_less_subst1
% 0.16/0.44 thf(fact_188_less__le,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less @ A )
% 0.16/0.44 = ( ^ [X4: A,Y5: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 0.16/0.44 & ( X4 != Y5 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_le
% 0.16/0.44 thf(fact_189_le__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( order @ A @ ( type2 @ A ) )
% 0.16/0.44 => ( ( ord_less_eq @ A )
% 0.16/0.44 = ( ^ [X4: A,Y5: A] :
% 0.16/0.44 ( ( ord_less @ A @ X4 @ Y5 )
% 0.16/0.44 | ( X4 = Y5 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_less
% 0.16/0.44 thf(fact_190_leI,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [X: A,Y2: A] :
% 0.16/0.44 ( ~ ( ord_less @ A @ X @ Y2 )
% 0.16/0.44 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % leI
% 0.16/0.44 thf(fact_191_leD,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [Y2: A,X: A] :
% 0.16/0.44 ( ( ord_less_eq @ A @ Y2 @ X )
% 0.16/0.44 => ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % leD
% 0.16/0.44 thf(fact_192_le__0__eq,axiom,
% 0.16/0.44 ! [N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.16/0.44 = ( N
% 0.16/0.44 = ( zero_zero @ nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_0_eq
% 0.16/0.44 thf(fact_193_less__eq__nat_Osimps_I1_J,axiom,
% 0.16/0.44 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.16/0.44
% 0.16/0.44 % less_eq_nat.simps(1)
% 0.16/0.44 thf(fact_194_less__mono__imp__le__mono,axiom,
% 0.16/0.44 ! [F: nat > nat,I: nat,J: nat] :
% 0.16/0.44 ( ! [I2: nat,J2: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ I2 @ J2 )
% 0.16/0.44 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.16/0.44 => ( ( ord_less_eq @ nat @ I @ J )
% 0.16/0.44 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_mono_imp_le_mono
% 0.16/0.44 thf(fact_195_le__neq__implies__less,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.44 => ( ( M != N )
% 0.16/0.44 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_neq_implies_less
% 0.16/0.44 thf(fact_196_less__or__eq__imp__le,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ( ord_less @ nat @ M @ N )
% 0.16/0.44 | ( M = N ) )
% 0.16/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_or_eq_imp_le
% 0.16/0.44 thf(fact_197_le__eq__less__or__eq,axiom,
% 0.16/0.44 ( ( ord_less_eq @ nat )
% 0.16/0.44 = ( ^ [M4: nat,N3: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M4 @ N3 )
% 0.16/0.44 | ( M4 = N3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_eq_less_or_eq
% 0.16/0.44 thf(fact_198_less__imp__le__nat,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ M @ N )
% 0.16/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_le_nat
% 0.16/0.44 thf(fact_199_nat__less__le,axiom,
% 0.16/0.44 ( ( ord_less @ nat )
% 0.16/0.44 = ( ^ [M4: nat,N3: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 0.16/0.44 & ( M4 != N3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_less_le
% 0.16/0.44 thf(fact_200_zero__less__power,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_less_power
% 0.16/0.44 thf(fact_201_nat__power__less__imp__less,axiom,
% 0.16/0.44 ! [I: nat,M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 0.16/0.44 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 0.16/0.44 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_power_less_imp_less
% 0.16/0.44 thf(fact_202_power__strict__mono,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A,N: nat] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.16/0.44 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % power_strict_mono
% 0.16/0.44 thf(fact_203_positive__imp__inverse__positive,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % positive_imp_inverse_positive
% 0.16/0.44 thf(fact_204_negative__imp__inverse__negative,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % negative_imp_inverse_negative
% 0.16/0.44 thf(fact_205_inverse__positive__imp__positive,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.16/0.44 => ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_positive_imp_positive
% 0.16/0.44 thf(fact_206_inverse__negative__imp__negative,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ( A2
% 0.16/0.44 != ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_negative_imp_negative
% 0.16/0.44 thf(fact_207_less__imp__inverse__less__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_inverse_less_neg
% 0.16/0.44 thf(fact_208_inverse__less__imp__less__neg,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.16/0.44 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_less_imp_less_neg
% 0.16/0.44 thf(fact_209_less__imp__inverse__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ A2 @ B2 )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_imp_inverse_less
% 0.16/0.44 thf(fact_210_inverse__less__imp__less,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [A2: A,B2: A] :
% 0.16/0.44 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.16/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.16/0.44 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % inverse_less_imp_less
% 0.16/0.44 thf(fact_211_zero__power,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.16/0.44 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.16/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % zero_power
% 0.16/0.44 thf(fact_212_of__nat__0__le__iff,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % of_nat_0_le_iff
% 0.16/0.44 thf(fact_213_ex__least__nat__le,axiom,
% 0.16/0.44 ! [P: nat > $o,N: nat] :
% 0.16/0.44 ( ~ ( P @ ( zero_zero @ nat ) )
% 0.16/0.44 => ( ( P @ N )
% 0.16/0.44 => ? [K2: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.16/0.44 & ! [I3: nat] :
% 0.16/0.44 ( ( ord_less @ nat @ I3 @ K2 )
% 0.16/0.44 => ~ ( P @ I3 ) )
% 0.16/0.44 & ( P @ K2 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_least_nat_le
% 0.16/0.44 thf(fact_214_transfer__int__nat__relations_I2_J,axiom,
% 0.16/0.44 ! [X: nat,Y2: nat] :
% 0.16/0.44 ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 0.16/0.44 = ( ord_less @ nat @ X @ Y2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_relations(2)
% 0.16/0.44 thf(fact_215_transfer__int__nat__numerals_I1_J,axiom,
% 0.16/0.44 ( ( zero_zero @ int )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_numerals(1)
% 0.16/0.44 thf(fact_216_natural_Osimps_I4_J,axiom,
% 0.16/0.44 ! [T: $tType,F1: T,F22: code_natural > T] :
% 0.16/0.44 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.16/0.44 = F1 ) ).
% 0.16/0.44
% 0.16/0.44 % natural.simps(4)
% 0.16/0.44 thf(fact_217_transfer__int__nat__quantifiers_I2_J,axiom,
% 0.16/0.44 ! [P: int > $o] :
% 0.16/0.44 ( ( ? [X4: int] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.16/0.44 & ( P @ X4 ) ) )
% 0.16/0.44 = ( ? [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_quantifiers(2)
% 0.16/0.44 thf(fact_218_transfer__int__nat__quantifiers_I1_J,axiom,
% 0.16/0.44 ! [P: int > $o] :
% 0.16/0.44 ( ( ! [X4: int] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 0.16/0.44 => ( P @ X4 ) ) )
% 0.16/0.44 = ( ! [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_quantifiers(1)
% 0.16/0.44 thf(fact_219_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 0.16/0.44 ! [Z2: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % Nat_Transfer.transfer_nat_int_function_closures(9)
% 0.16/0.44 thf(fact_220_less__fun__def,axiom,
% 0.16/0.44 ! [B3: $tType,A: $tType] :
% 0.16/0.44 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.16/0.44 => ( ( ord_less @ ( A > B3 ) )
% 0.16/0.44 = ( ^ [F3: A > B3,G2: A > B3] :
% 0.16/0.44 ( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
% 0.16/0.44 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % less_fun_def
% 0.16/0.44 thf(fact_221_nat__le__linear,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.44 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.16/0.44
% 0.16/0.44 % nat_le_linear
% 0.16/0.44 thf(fact_222_le__antisym,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ M @ N )
% 0.16/0.44 => ( ( ord_less_eq @ nat @ N @ M )
% 0.16/0.44 => ( M = N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_antisym
% 0.16/0.44 thf(fact_223_eq__imp__le,axiom,
% 0.16/0.44 ! [M: nat,N: nat] :
% 0.16/0.44 ( ( M = N )
% 0.16/0.44 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.16/0.44
% 0.16/0.44 % eq_imp_le
% 0.16/0.44 thf(fact_224_le__trans,axiom,
% 0.16/0.44 ! [I: nat,J: nat,K: nat] :
% 0.16/0.44 ( ( ord_less_eq @ nat @ I @ J )
% 0.16/0.44 => ( ( ord_less_eq @ nat @ J @ K )
% 0.16/0.44 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % le_trans
% 0.16/0.44 thf(fact_225_le__refl,axiom,
% 0.16/0.44 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.16/0.44
% 0.16/0.44 % le_refl
% 0.16/0.44 thf(fact_226_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.16/0.44 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.16/0.44
% 0.16/0.44 % Nat_Transfer.transfer_nat_int_function_closures(5)
% 0.16/0.44 thf(fact_227_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
% 0.16/0.44 ! [X: int,N: nat] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.16/0.44 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( power_power @ int @ X @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Nat_Transfer.transfer_nat_int_function_closures(4)
% 0.16/0.44 thf(fact_228_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
% 0.16/0.44 ! [X: nat,N: nat] :
% 0.16/0.44 ( ( power_power @ int @ ( semiring_1_of_nat @ int @ X ) @ N )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ ( power_power @ nat @ X @ N ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % Nat_Transfer.transfer_int_nat_functions(4)
% 0.16/0.44 thf(fact_229_transfer__int__nat__relations_I3_J,axiom,
% 0.16/0.44 ! [X: nat,Y2: nat] :
% 0.16/0.44 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 0.16/0.44 = ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_relations(3)
% 0.16/0.44 thf(fact_230_transfer__int__nat__relations_I1_J,axiom,
% 0.16/0.44 ! [X: nat,Y2: nat] :
% 0.16/0.44 ( ( ( semiring_1_of_nat @ int @ X )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ Y2 ) )
% 0.16/0.44 = ( X = Y2 ) ) ).
% 0.16/0.44
% 0.16/0.44 % transfer_int_nat_relations(1)
% 0.16/0.44 thf(fact_231_int__if__cong,axiom,
% 0.16/0.44 ! [P: $o,X: nat,Y2: nat] :
% 0.16/0.44 ( ( P
% 0.16/0.44 => ( ( semiring_1_of_nat @ int @ X )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
% 0.16/0.44 & ( ~ P
% 0.16/0.44 => ( ( semiring_1_of_nat @ int @ Y2 )
% 0.16/0.44 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % int_if_cong
% 0.16/0.44 thf(fact_232_imp__le__cong,axiom,
% 0.16/0.44 ! [X: int,X5: int,P: $o,P3: $o] :
% 0.16/0.44 ( ( X = X5 )
% 0.16/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.16/0.44 => ( P = P3 ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.16/0.44 => P )
% 0.16/0.44 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.16/0.44 => P3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % imp_le_cong
% 0.16/0.44 thf(fact_233_conj__le__cong,axiom,
% 0.16/0.44 ! [X: int,X5: int,P: $o,P3: $o] :
% 0.16/0.44 ( ( X = X5 )
% 0.16/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.16/0.44 => ( P = P3 ) )
% 0.16/0.44 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 0.16/0.44 & P )
% 0.16/0.44 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 0.16/0.44 & P3 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % conj_le_cong
% 0.16/0.44 thf(fact_234_pinf_I1_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.16/0.44 ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z4 @ X2 )
% 0.16/0.44 => ( ( P @ X2 )
% 0.16/0.44 = ( P3 @ X2 ) ) )
% 0.16/0.44 => ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z4 @ X2 )
% 0.16/0.44 => ( ( Q2 @ X2 )
% 0.16/0.44 = ( Q3 @ X2 ) ) )
% 0.16/0.44 => ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( ( ( P @ X3 )
% 0.16/0.44 & ( Q2 @ X3 ) )
% 0.16/0.44 = ( ( P3 @ X3 )
% 0.16/0.44 & ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(1)
% 0.16/0.44 thf(fact_235_pinf_I2_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.16/0.44 ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z4 @ X2 )
% 0.16/0.44 => ( ( P @ X2 )
% 0.16/0.44 = ( P3 @ X2 ) ) )
% 0.16/0.44 => ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z4 @ X2 )
% 0.16/0.44 => ( ( Q2 @ X2 )
% 0.16/0.44 = ( Q3 @ X2 ) ) )
% 0.16/0.44 => ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( ( ( P @ X3 )
% 0.16/0.44 | ( Q2 @ X3 ) )
% 0.16/0.44 = ( ( P3 @ X3 )
% 0.16/0.44 | ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(2)
% 0.16/0.44 thf(fact_236_pinf_I3_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( X3 != T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(3)
% 0.16/0.44 thf(fact_237_pinf_I4_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( X3 != T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(4)
% 0.16/0.44 thf(fact_238_pinf_I5_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ~ ( ord_less @ A @ X3 @ T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(5)
% 0.16/0.44 thf(fact_239_pinf_I7_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( ord_less @ A @ T2 @ X3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(7)
% 0.16/0.44 thf(fact_240_pinf_I11_J,axiom,
% 0.16/0.44 ! [C2: $tType,D2: $tType] :
% 0.16/0.44 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 => ! [F4: D2] :
% 0.16/0.44 ? [Z: C2] :
% 0.16/0.44 ! [X3: C2] :
% 0.16/0.44 ( ( ord_less @ C2 @ Z @ X3 )
% 0.16/0.44 => ( F4 = F4 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(11)
% 0.16/0.44 thf(fact_241_minf_I1_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.16/0.44 ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Z4 )
% 0.16/0.44 => ( ( P @ X2 )
% 0.16/0.44 = ( P3 @ X2 ) ) )
% 0.16/0.44 => ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Z4 )
% 0.16/0.44 => ( ( Q2 @ X2 )
% 0.16/0.44 = ( Q3 @ X2 ) ) )
% 0.16/0.44 => ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( ( ( P @ X3 )
% 0.16/0.44 & ( Q2 @ X3 ) )
% 0.16/0.44 = ( ( P3 @ X3 )
% 0.16/0.44 & ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(1)
% 0.16/0.44 thf(fact_242_minf_I2_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 0.16/0.44 ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Z4 )
% 0.16/0.44 => ( ( P @ X2 )
% 0.16/0.44 = ( P3 @ X2 ) ) )
% 0.16/0.44 => ( ? [Z4: A] :
% 0.16/0.44 ! [X2: A] :
% 0.16/0.44 ( ( ord_less @ A @ X2 @ Z4 )
% 0.16/0.44 => ( ( Q2 @ X2 )
% 0.16/0.44 = ( Q3 @ X2 ) ) )
% 0.16/0.44 => ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( ( ( P @ X3 )
% 0.16/0.44 | ( Q2 @ X3 ) )
% 0.16/0.44 = ( ( P3 @ X3 )
% 0.16/0.44 | ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(2)
% 0.16/0.44 thf(fact_243_minf_I3_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( X3 != T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(3)
% 0.16/0.44 thf(fact_244_minf_I4_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( X3 != T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(4)
% 0.16/0.44 thf(fact_245_minf_I5_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( ord_less @ A @ X3 @ T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(5)
% 0.16/0.44 thf(fact_246_minf_I7_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ~ ( ord_less @ A @ T2 @ X3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(7)
% 0.16/0.44 thf(fact_247_minf_I11_J,axiom,
% 0.16/0.44 ! [C2: $tType,D2: $tType] :
% 0.16/0.44 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.16/0.44 => ! [F4: D2] :
% 0.16/0.44 ? [Z: C2] :
% 0.16/0.44 ! [X3: C2] :
% 0.16/0.44 ( ( ord_less @ C2 @ X3 @ Z )
% 0.16/0.44 => ( F4 = F4 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(11)
% 0.16/0.44 thf(fact_248_minf_I8_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(8)
% 0.16/0.44 thf(fact_249_minf_I6_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ X3 @ Z )
% 0.16/0.44 => ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % minf(6)
% 0.16/0.44 thf(fact_250_pinf_I8_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(8)
% 0.16/0.44 thf(fact_251_pinf_I6_J,axiom,
% 0.16/0.44 ! [A: $tType] :
% 0.16/0.44 ( ( linorder @ A @ ( type2 @ A ) )
% 0.16/0.44 => ! [T2: A] :
% 0.16/0.44 ? [Z: A] :
% 0.16/0.44 ! [X3: A] :
% 0.16/0.44 ( ( ord_less @ A @ Z @ X3 )
% 0.16/0.44 => ~ ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % pinf(6)
% 0.16/0.44 thf(fact_252_ex__has__greatest__nat,axiom,
% 0.16/0.44 ! [A: $tType,P: A > $o,K: A,M: A > nat,B2: nat] :
% 0.16/0.44 ( ( P @ K )
% 0.16/0.44 => ( ! [Y3: A] :
% 0.16/0.44 ( ( P @ Y3 )
% 0.16/0.44 => ( ord_less @ nat @ ( M @ Y3 ) @ B2 ) )
% 0.16/0.44 => ? [X2: A] :
% 0.16/0.44 ( ( P @ X2 )
% 0.16/0.44 & ! [Y: A] :
% 0.16/0.44 ( ( P @ Y )
% 0.16/0.44 => ( ord_less_eq @ nat @ ( M @ Y ) @ ( M @ X2 ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % ex_has_greatest_nat
% 0.16/0.44 thf(fact_253_realpow__pos__nth__unique,axiom,
% 0.16/0.44 ! [N: nat,A2: real] :
% 0.16/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.16/0.44 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.16/0.44 => ? [X2: real] :
% 0.16/0.44 ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
% 0.16/0.44 & ( ( power_power @ real @ X2 @ N )
% 0.16/0.44 = A2 )
% 0.16/0.44 & ! [Y: real] :
% 0.16/0.44 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
% 0.16/0.44 & ( ( power_power @ real @ Y @ N )
% 0.16/0.44 = A2 ) )
% 0.16/0.44 => ( Y = X2 ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 % realpow_pos_nth_unique
% 0.16/0.44
% 0.16/0.44 %----Type constructors (64)
% 0.16/0.44 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.16/0.44 ! [A5: $tType,A6: $tType] :
% 0.16/0.44 ( ( preorder @ A6 @ ( type2 @ A6 ) )
% 0.16/0.44 => ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.16/0.44 ! [A5: $tType,A6: $tType] :
% 0.16/0.44 ( ( order @ A6 @ ( type2 @ A6 ) )
% 0.16/0.44 => ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_fun___Orderings_Oord,axiom,
% 0.16/0.44 ! [A5: $tType,A6: $tType] :
% 0.16/0.44 ( ( ord @ A6 @ ( type2 @ A6 ) )
% 0.16/0.44 => ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.16/0.44 condit1037483654norder @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.16/0.44 semiri134348788visors @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.16/0.44 linordered_semidom @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.16/0.44 linordered_idom @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.16/0.44 semiring_char_0 @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 0.16/0.44 zero_neq_one @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 0.16/0.44 preorder @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.16/0.44 linorder @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.16/0.44 semiring_1 @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 0.16/0.44 no_top @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 0.16/0.44 no_bot @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 0.16/0.44 order @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 0.16/0.44 ord @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.16/0.44 zero @ int @ ( type2 @ int ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,axiom,
% 0.16/0.44 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.16/0.44 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 0.16/0.44 semiri134348788visors @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 0.16/0.44 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.16/0.44 wellorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
% 0.16/0.44 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_8,axiom,
% 0.16/0.44 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 0.16/0.44 preorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 0.16/0.44 linorder @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 0.16/0.44 semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Ono__top_12,axiom,
% 0.16/0.44 no_top @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Oorder_13,axiom,
% 0.16/0.44 order @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Orderings_Oord_14,axiom,
% 0.16/0.44 ord @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Nat_Onat___Groups_Ozero_15,axiom,
% 0.16/0.44 zero @ nat @ ( type2 @ nat ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
% 0.16/0.44 preorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_HOL_Obool___Orderings_Olinorder_17,axiom,
% 0.16/0.44 linorder @ $o @ ( type2 @ $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 0.16/0.44 order @ $o @ ( type2 @ $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 0.16/0.44 ord @ $o @ ( type2 @ $o ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_20,axiom,
% 0.16/0.44 condit1037483654norder @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 0.16/0.44 condit1656338222tinuum @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.16/0.44 archim1804426504_field @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_21,axiom,
% 0.16/0.44 semiri134348788visors @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_22,axiom,
% 0.16/0.44 linordered_semidom @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 0.16/0.44 dense_linorder @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.16/0.44 linordered_field @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Rings_Olinordered__idom_23,axiom,
% 0.16/0.44 linordered_idom @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 0.16/0.44 dense_order @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.16/0.44 division_ring @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_24,axiom,
% 0.16/0.44 semiring_char_0 @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_25,axiom,
% 0.16/0.44 zero_neq_one @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Opreorder_26,axiom,
% 0.16/0.44 preorder @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Olinorder_27,axiom,
% 0.16/0.44 linorder @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Rings_Osemiring__1_28,axiom,
% 0.16/0.44 semiring_1 @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Ono__top_29,axiom,
% 0.16/0.44 no_top @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Ono__bot_30,axiom,
% 0.16/0.44 no_bot @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Oorder_31,axiom,
% 0.16/0.44 order @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Orderings_Oord_32,axiom,
% 0.16/0.44 ord @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.16/0.44 field @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Real_Oreal___Groups_Ozero_33,axiom,
% 0.16/0.44 zero @ real @ ( type2 @ real ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_34,axiom,
% 0.16/0.44 semiri134348788visors @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_35,axiom,
% 0.16/0.44 zero_neq_one @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_36,axiom,
% 0.16/0.44 preorder @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_37,axiom,
% 0.16/0.44 linorder @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_38,axiom,
% 0.16/0.44 semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_39,axiom,
% 0.16/0.44 order @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_40,axiom,
% 0.16/0.44 ord @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_41,axiom,
% 0.16/0.44 zero @ code_natural @ ( type2 @ code_natural ) ).
% 0.16/0.44
% 0.16/0.44 %----Helper facts (3)
% 0.16/0.44 thf(help_If_3_1_T,axiom,
% 0.16/0.44 ! [P: $o] :
% 0.16/0.44 ( ( P = $true )
% 0.16/0.44 | ( P = $false ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(help_If_2_1_T,axiom,
% 0.16/0.44 ! [A: $tType,X: A,Y2: A] :
% 0.16/0.44 ( ( if @ A @ $false @ X @ Y2 )
% 0.16/0.44 = Y2 ) ).
% 0.16/0.44
% 0.16/0.44 thf(help_If_1_1_T,axiom,
% 0.16/0.44 ! [A: $tType,X: A,Y2: A] :
% 0.16/0.44 ( ( if @ A @ $true @ X @ Y2 )
% 0.16/0.44 = X ) ).
% 0.16/0.44
% 0.16/0.44 %----Conjectures (2)
% 0.16/0.44 thf(conj_0,hypothesis,
% 0.16/0.44 hammin1195055315smooth @ n ).
% 0.16/0.44
% 0.16/0.44 thf(conj_1,conjecture,
% 0.16/0.44 ord_less @ nat @ ( zero_zero @ nat ) @ n ).
% 0.16/0.44
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 ------------------------
% 2.63/1.44 ---- Embedded in THF ---
% 2.63/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.63/1.44 %%% Generated on Tue Feb 24 06:09:57 EST 2026
% 2.63/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 2.63/1.44 %%% Logic specification used:
% 2.63/1.44 %%% thf(spec, logic, $$dhol).
% 2.63/1.44
% 2.63/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2DTF32658.p
% 2.63/1.44 thf(ty_t_Code__Numeral_Onatural, type, code_natural: $tType).
% 2.63/1.44 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.63/1.44 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.63/1.44 thf(ty_t_Int_Oint, type, int: $tType).
% 2.63/1.44 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.63/1.44 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.63/1.44 thf(sy_c_Code__Numeral_Onatural_Ocase__natural, type, code_case_natural: (!> [T:$tType]: ((T > ((code_natural > T) > (code_natural > T)))))).
% 2.63/1.44 thf(sy_c_Code__Numeral_Onatural_Osize__natural, type, code_size_natural: (code_natural > nat)).
% 2.63/1.44 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 2.63/1.44 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.63/1.44 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth, type, hammin1195055315smooth: (nat > $o)).
% 2.63/1.44 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.63/1.44 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.63/1.44 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.63/1.44 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.63/1.44 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.63/1.44 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.63/1.44 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.63/1.44 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))).
% 2.63/1.44 thf(sy_v_n, type, n: nat).
% 2.63/1.44 thf(fact_0_smooth__0, axiom, (~ ((hammin1195055315smooth @ (zero_zero @ nat))))).
% 2.63/1.44 thf(fact_1_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.63/1.44 thf(fact_2_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_3_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.63/1.44 thf(fact_4_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.63/1.44 thf(fact_5_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.63/1.44 thf(fact_6_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.63/1.44 thf(fact_7_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.63/1.44 thf(fact_8_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 2.63/1.44 thf(fact_9_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.63/1.44 thf(fact_10_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X:A]: (((! [X2:A]: ((((V @ X2) = (zero_zero @ nat)) => (P @ X2)))) => ((! [X2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X2)) => ((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y)))))))))) => (P @ X)))))).
% 2.63/1.44 thf(fact_11_less__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.63/1.44 thf(fact_12_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type2 @ A)) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_13_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y:A]: (((((ord_less @ nat) @ (V @ Y)) @ (V @ X2)) & (~ ((P @ Y))))))))) => (P @ X))))).
% 2.63/1.44 thf(fact_14_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ nat) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2))))).
% 2.63/1.44 thf(fact_15_linorder__neqE__nat, axiom, (! [X:nat,Y2:nat]: (((X != Y2) => ((~ ((((ord_less @ nat) @ X) @ Y2))) => (((ord_less @ nat) @ Y2) @ X)))))).
% 2.63/1.44 thf(fact_16_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.63/1.44 thf(fact_17_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.63/1.44 thf(fact_18_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.63/1.44 thf(fact_19_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ nat) @ (F @ Y)) @ (F @ X2)) => (P @ Y)))) => (P @ X2)))) => (P @ A2))))).
% 2.63/1.44 thf(fact_20_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))).
% 2.63/1.44 thf(fact_21_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.63/1.44 thf(fact_22_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.63/1.44 thf(fact_23_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.63/1.44 thf(fact_24_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_25_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_26_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.63/1.44 thf(fact_27_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.63/1.44 thf(fact_28_of__nat__0__less__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 2.63/1.44 thf(fact_29_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.63/1.44 thf(fact_30_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B:A]: (((((ord_less @ A) @ A2) @ B) | (((ord_less @ A) @ B) @ A2)))))))))).
% 2.63/1.44 thf(fact_31_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.63/1.44 thf(fact_32_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.63/1.44 thf(fact_33_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X3:A]: ((? [X1:A]: ((((ord_less @ A) @ X3) @ X1))))))))).
% 2.63/1.44 thf(fact_34_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X3:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X3))))))))).
% 2.63/1.44 thf(fact_35_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.63/1.44 thf(fact_36_dependent__wellorder__choice, axiom, (! [B3:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:((A > B3) > (A > (B3 > $o)))]: (((! [R:B3,F2:(A > B3),G:(A > B3),X2:A]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X2) => ((F2 @ Y) = (G @ Y))))) => ((((P @ F2) @ X2) @ R) = (((P @ G) @ X2) @ R))))) => ((! [X2:A,F2:(A > B3)]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X2) => (((P @ F2) @ Y) @ (F2 @ Y))))) => (? [X12:B3]: ((((P @ F2) @ X2) @ X12)))))) => (? [F2:(A > B3)]: ((! [X3:A]: ((((P @ F2) @ X3) @ (F2 @ X3)))))))))))))).
% 2.63/1.44 thf(fact_37_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) = ((((ord_less @ A) @ Y2) @ X) | (X = Y2))))))))).
% 2.63/1.44 thf(fact_38_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.63/1.44 thf(fact_39_of__nat__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.63/1.44 thf(fact_40_of__nat__0, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.63/1.44 thf(fact_41_of__nat__0__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.63/1.44 thf(fact_42_of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.63/1.44 thf(fact_43_of__nat__less__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 2.63/1.44 thf(fact_44_of__nat__less__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 2.63/1.44 thf(fact_45_less__imp__of__nat__less, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.63/1.44 thf(fact_46_of__nat__less__imp__less, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) => (((ord_less @ nat) @ M) @ N)))))))).
% 2.63/1.44 thf(fact_47_ord__eq__less__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((A2 = (F @ B2)) => ((((ord_less @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less @ B3) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_48_ord__less__eq__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B3),C:B3]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ B3) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ B3) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_49_order__less__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less @ B3) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_50_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_51_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X))))))))).
% 2.63/1.44 thf(fact_52_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.63/1.44 thf(fact_53_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.63/1.44 thf(fact_54_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X != Y2) = ((((ord_less @ A) @ X) @ Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.63/1.44 thf(fact_55_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_56_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (? [Z:A]: (((((ord_less @ A) @ X) @ Z) & (((ord_less @ A) @ Z) @ Y2))))))))))).
% 2.63/1.44 thf(fact_57_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.63/1.44 thf(fact_58_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.63/1.44 thf(fact_59_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_60_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.63/1.44 thf(fact_61_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) | ((X = Y2) | (((ord_less @ A) @ Y2) @ X))))))))).
% 2.63/1.44 thf(fact_62_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.63/1.44 thf(fact_63_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_64_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_65_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.63/1.44 thf(fact_66_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (X != Y2)))))))).
% 2.63/1.44 thf(fact_67_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.63/1.44 thf(fact_68_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y:A]: (((((ord_less @ A) @ Y) @ X2) => (P @ Y)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.63/1.44 thf(fact_69_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((~ ((((ord_less @ A) @ Y2) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y2))) = (X = Y2))))))))).
% 2.63/1.44 thf(fact_70_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (Y2 != X)))))))).
% 2.63/1.44 thf(fact_71_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,P:$o]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ X) => P)))))))).
% 2.63/1.44 thf(fact_72_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => ((X != Y2) => (((ord_less @ A) @ Y2) @ X))))))))).
% 2.63/1.44 thf(fact_73_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.63/1.44 thf(fact_74_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_75_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (~ ((((ord_less @ A) @ Y2) @ X)))))))))).
% 2.63/1.44 thf(fact_76_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (K = ((semiring_1_of_nat @ int) @ N2))))))))).
% 2.63/1.44 thf(fact_77_reals__Archimedean2, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [N2:nat]: ((((ord_less @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.63/1.44 thf(fact_78_natural_Osize_I1_J, axiom, ((code_size_natural @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 2.63/1.44 thf(fact_79_of__nat__zero__less__power__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [X:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ X)) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat)))))))))).
% 2.63/1.44 thf(fact_80_of__nat__le__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.63/1.44 thf(fact_81_natural_Osize_I3_J, axiom, (((size_size @ code_natural) @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 2.63/1.44 thf(fact_82_of__bool__eq_I1_J, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (((zero_neq_one_of_bool @ A) @ $false) = (zero_zero @ A)))))).
% 2.63/1.44 thf(fact_83_ex__inverse__of__nat__less, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ X))))))))))).
% 2.63/1.44 thf(fact_84_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.63/1.44 thf(fact_85_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.63/1.44 thf(fact_86_inverse__inverse__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: ((((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2))))))).
% 2.63/1.44 thf(fact_87_inverse__eq__iff__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) = (A2 = B2)))))))).
% 2.63/1.44 thf(fact_88_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_89_of__nat__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less_eq @ nat) @ M) @ N)))))))).
% 2.63/1.44 thf(fact_90_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_91_inverse__zero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.63/1.44 thf(fact_92_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 2.63/1.44 thf(fact_93_of__nat__power, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((power_power @ nat) @ M) @ N)) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N)))))))).
% 2.63/1.44 thf(fact_94_inverse__nonnegative__iff__nonnegative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.63/1.44 thf(fact_95_inverse__nonpositive__iff__nonpositive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_96_inverse__positive__iff__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.63/1.44 thf(fact_97_inverse__negative__iff__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_98_inverse__less__iff__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_99_inverse__less__iff__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_100_inverse__le__iff__le__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_101_inverse__le__iff__le, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))))).
% 2.63/1.44 thf(fact_102_power__eq__0__iff, axiom, (! [A:$tType]: ((((semiri134348788visors @ A) @ (type2 @ A)) => (! [A2:A,N:nat]: ((((((power_power @ A) @ A2) @ N) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 2.63/1.44 thf(fact_103_nonneg__eq__int, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (~ ((! [M3:nat]: ((Z2 != ((semiring_1_of_nat @ int) @ M3)))))))))).
% 2.63/1.44 thf(fact_104_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: ((K != ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.63/1.44 thf(fact_105_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: ((K = ((semiring_1_of_nat @ int) @ N2)))))))).
% 2.63/1.44 thf(fact_106_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 2.63/1.44 thf(fact_107_inverse__eq__imp__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => (A2 = B2)))))))).
% 2.63/1.44 thf(fact_108_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.63/1.44 thf(fact_109_of__bool__eq__iff, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (! [P2:$o,Q:$o]: (((((zero_neq_one_of_bool @ A) @ P2) = ((zero_neq_one_of_bool @ A) @ Q)) = (P2 = Q)))))))).
% 2.63/1.44 thf(fact_110_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.63/1.44 thf(fact_111_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B:A]: (((((ord_less_eq @ A) @ A3) @ B) => ((P @ A3) @ B)))) => ((! [A3:A,B:A]: ((((P @ B) @ A3) => ((P @ A3) @ B)))) => ((P @ A2) @ B2))))))))).
% 2.63/1.44 thf(fact_112_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.63/1.44 thf(fact_113_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.63/1.44 thf(fact_114_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.63/1.44 thf(fact_115_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_116_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_117_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.63/1.44 thf(fact_118_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z2)))) => (((((ord_less_eq @ A) @ Y2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y2)))) => (((((ord_less_eq @ A) @ Z2) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X)))) => (((((ord_less_eq @ A) @ Y2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y2))))))))))))))))).
% 2.63/1.44 thf(fact_119_real__arch__simple, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type2 @ A)) => (! [X:A]: ((? [N2:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.63/1.44 thf(fact_120_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_121_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_122_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((X = Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.63/1.44 thf(fact_123_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_124_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X) => (X = Y2))))))))).
% 2.63/1.44 thf(fact_125_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X4))))))))).
% 2.63/1.44 thf(fact_126_ord__le__eq__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B3),C:B3]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ B3) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ B3) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_127_ord__eq__le__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less_eq @ B3) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_128_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_129_order__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less_eq @ B3) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_130_le__fun__def, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => ((ord_less_eq @ (A > B3)) = (^ [F3:(A > B3),G2:(A > B3)]: ((! [X4:A]: ((((ord_less_eq @ B3) @ (F3 @ X4)) @ (G2 @ X4))))))))))).
% 2.63/1.44 thf(fact_131_le__funI, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G3:(A > B3)]: (((! [X2:A]: ((((ord_less_eq @ B3) @ (F @ X2)) @ (G3 @ X2)))) => (((ord_less_eq @ (A > B3)) @ F) @ G3)))))))).
% 2.63/1.44 thf(fact_132_le__funE, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G3:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G3) => (((ord_less_eq @ B3) @ (F @ X)) @ (G3 @ X))))))))).
% 2.63/1.44 thf(fact_133_le__funD, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G3:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G3) => (((ord_less_eq @ B3) @ (F @ X)) @ (G3 @ X))))))))).
% 2.63/1.44 thf(fact_134_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.63/1.44 thf(fact_135_zle__int, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.63/1.44 thf(fact_136_power__not__zero, axiom, (! [A:$tType]: ((((semiri134348788visors @ A) @ (type2 @ A)) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_137_power__less__imp__less__base, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,N:nat,B2:A]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.63/1.44 thf(fact_138_power__inverse, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A,N:nat]: (((((power_power @ A) @ ((inverse_inverse @ A) @ A2)) @ N) = ((inverse_inverse @ A) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.63/1.44 thf(fact_139_zero__le__power, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.63/1.44 thf(fact_140_power__mono, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N)))))))))).
% 2.63/1.44 thf(fact_141_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.63/1.44 thf(fact_142_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.63/1.44 thf(fact_143_le__imp__inverse__le__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.63/1.44 thf(fact_144_inverse__le__imp__le__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.63/1.44 thf(fact_145_le__imp__inverse__le, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.63/1.44 thf(fact_146_inverse__le__imp__le, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.63/1.44 thf(fact_147_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A2) != (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_148_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2)))))))).
% 2.63/1.44 thf(fact_149_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => ((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (A2 = B2)))))))))).
% 2.63/1.44 thf(fact_150_inverse__zero__imp__zero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_151_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.63/1.44 thf(fact_152_power__eq__imp__eq__base, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,N:nat,B2:A]: ((((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B2) @ N)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (A2 = B2))))))))))).
% 2.63/1.44 thf(fact_153_power__eq__iff__eq__base, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:nat,A2:A,B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((((power_power @ A) @ A2) @ N) = (((power_power @ A) @ B2) @ N)) = (A2 = B2))))))))))).
% 2.63/1.44 thf(fact_154_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.63/1.44 thf(fact_155_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.63/1.44 thf(fact_156_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X3:A]: ((((((ord_less_eq @ A) @ A2) @ X3) & (((ord_less @ A) @ X3) @ C3)) => (P @ X3)))) & (! [D:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D)) => (P @ X2)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 2.63/1.44 thf(fact_157_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.63/1.44 thf(fact_158_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.63/1.44 thf(fact_159_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.63/1.44 thf(fact_160_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.63/1.44 thf(fact_161_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.63/1.44 thf(fact_162_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y2) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.63/1.44 thf(fact_163_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X:A,Y2:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y2) @ W))))) => (((ord_less_eq @ A) @ Y2) @ Z2))))))))).
% 2.63/1.44 thf(fact_164_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.63/1.44 thf(fact_165_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.63/1.44 thf(fact_166_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))).
% 2.63/1.44 thf(fact_167_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))).
% 2.63/1.44 thf(fact_168_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_169_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.63/1.44 thf(fact_170_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y2) @ X))) => (((ord_less @ A) @ X) @ Y2)))))))).
% 2.63/1.44 thf(fact_171_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (~ ((((ord_less_eq @ A) @ Y5) @ X4))))))))))).
% 2.63/1.44 thf(fact_172_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ X) @ Y2) | (X = Y2))))))))).
% 2.63/1.44 thf(fact_173_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) | (((ord_less @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_174_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y2:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.63/1.44 thf(fact_175_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y2:A]: (((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ Y2) @ X2)))) => (((ord_less_eq @ A) @ Y2) @ Z2)))))))).
% 2.63/1.44 thf(fact_176_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less @ A) @ X) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.63/1.44 thf(fact_177_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((((ord_less @ A) @ Y2) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.63/1.44 thf(fact_178_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => ((~ ((((ord_less @ A) @ X) @ Y2))) = (X = Y2))))))))).
% 2.63/1.44 thf(fact_179_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => ((((ord_less_eq @ A) @ X) @ Y2) = (X = Y2))))))))).
% 2.63/1.44 thf(fact_180_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (((ord_less_eq @ A) @ X) @ Y2)))))))).
% 2.63/1.44 thf(fact_181_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.63/1.44 thf(fact_182_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) = (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_183_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y2))) = (((ord_less @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_184_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less @ A) @ X2) @ Y3) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_185_order__less__le__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less_eq @ B3) @ X2) @ Y3) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_186_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.63/1.44 thf(fact_187_order__le__less__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B2:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B3) @ B2) @ C) => ((! [X2:B3,Y3:B3]: (((((ord_less @ B3) @ X2) @ Y3) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.63/1.44 thf(fact_188_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y5:A]: (((((ord_less_eq @ A) @ X4) @ Y5) & (X4 != Y5))))))))).
% 2.63/1.44 thf(fact_189_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X4:A,Y5:A]: (((((ord_less @ A) @ X4) @ Y5) | (X4 = Y5))))))))).
% 2.63/1.44 thf(fact_190_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y2:A]: (((~ ((((ord_less @ A) @ X) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X)))))))).
% 2.63/1.44 thf(fact_191_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y2:A,X:A]: (((((ord_less_eq @ A) @ Y2) @ X) => (~ ((((ord_less @ A) @ X) @ Y2)))))))))).
% 2.63/1.44 thf(fact_192_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.63/1.44 thf(fact_193_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.63/1.44 thf(fact_194_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))))))).
% 2.63/1.44 thf(fact_195_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.63/1.44 thf(fact_196_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.63/1.44 thf(fact_197_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M4:nat,N3:nat]: (((((ord_less @ nat) @ M4) @ N3) | (M4 = N3)))))).
% 2.63/1.44 thf(fact_198_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.63/1.44 thf(fact_199_nat__less__le, axiom, ((ord_less @ nat) = (^ [M4:nat,N3:nat]: (((((ord_less_eq @ nat) @ M4) @ N3) & (M4 != N3)))))).
% 2.63/1.44 thf(fact_200_zero__less__power, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.63/1.44 thf(fact_201_nat__power__less__imp__less, axiom, (! [I:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.63/1.44 thf(fact_202_power__strict__mono, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A,N:nat]: (((((ord_less @ A) @ A2) @ B2) => ((((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) @ B2) @ N))))))))))).
% 2.63/1.44 thf(fact_203_positive__imp__inverse__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2))))))))).
% 2.63/1.44 thf(fact_204_negative__imp__inverse__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_205_inverse__positive__imp__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))).
% 2.63/1.44 thf(fact_206_inverse__negative__imp__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))).
% 2.63/1.44 thf(fact_207_less__imp__inverse__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.63/1.44 thf(fact_208_inverse__less__imp__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2))))))))).
% 2.63/1.44 thf(fact_209_less__imp__inverse__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.63/1.44 thf(fact_210_inverse__less__imp__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ B2) @ A2))))))))).
% 2.63/1.44 thf(fact_211_zero__power, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type2 @ A)) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A))))))))).
% 2.63/1.44 thf(fact_212_of__nat__0__le__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.63/1.44 thf(fact_213_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((~ ((P @ (zero_zero @ nat)))) => ((P @ N) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P @ I3)))))) & (P @ K2)))))))))).
% 2.63/1.44 thf(fact_214_transfer__int__nat__relations_I2_J, axiom, (! [X:nat,Y2:nat]: (((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y2)) = (((ord_less @ nat) @ X) @ Y2))))).
% 2.63/1.44 thf(fact_215_transfer__int__nat__numerals_I1_J, axiom, ((zero_zero @ int) = ((semiring_1_of_nat @ int) @ (zero_zero @ nat)))).
% 2.63/1.44 thf(fact_216_natural_Osimps_I4_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T)]: ((((((code_case_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 2.63/1.44 thf(fact_217_transfer__int__nat__quantifiers_I2_J, axiom, (! [P:(int > $o)]: (((? [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) & (P @ X4)))) = (? [X4:nat]: ((P @ ((semiring_1_of_nat @ int) @ X4)))))))).
% 2.63/1.44 thf(fact_218_transfer__int__nat__quantifiers_I1_J, axiom, (! [P:(int > $o)]: (((! [X4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X4) => (P @ X4)))) = (! [X4:nat]: ((P @ ((semiring_1_of_nat @ int) @ X4)))))))).
% 2.63/1.44 thf(fact_219_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, axiom, (! [Z2:nat]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((semiring_1_of_nat @ int) @ Z2))))).
% 2.63/1.44 thf(fact_220_less__fun__def, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => ((ord_less @ (A > B3)) = (^ [F3:(A > B3),G2:(A > B3)]: (((((ord_less_eq @ (A > B3)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B3)) @ G2) @ F3))))))))))).
% 2.63/1.44 thf(fact_221_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.63/1.44 thf(fact_222_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.63/1.44 thf(fact_223_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.63/1.44 thf(fact_224_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.63/1.44 thf(fact_225_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.63/1.44 thf(fact_226_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.63/1.44 thf(fact_227_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J, axiom, (! [X:int,N:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((power_power @ int) @ X) @ N)))))).
% 2.63/1.44 thf(fact_228_Nat__Transfer_Otransfer__int__nat__functions_I4_J, axiom, (! [X:nat,N:nat]: (((((power_power @ int) @ ((semiring_1_of_nat @ int) @ X)) @ N) = ((semiring_1_of_nat @ int) @ (((power_power @ nat) @ X) @ N)))))).
% 2.63/1.44 thf(fact_229_transfer__int__nat__relations_I3_J, axiom, (! [X:nat,Y2:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ X)) @ ((semiring_1_of_nat @ int) @ Y2)) = (((ord_less_eq @ nat) @ X) @ Y2))))).
% 2.63/1.44 thf(fact_230_transfer__int__nat__relations_I1_J, axiom, (! [X:nat,Y2:nat]: (((((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ Y2)) = (X = Y2))))).
% 2.63/1.44 thf(fact_231_int__if__cong, axiom, (! [P:$o,X:nat,Y2:nat]: (((P => (((semiring_1_of_nat @ int) @ X) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y2)))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ Y2) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X) @ Y2)))))))).
% 2.63/1.44 thf(fact_232_imp__le__cong, axiom, (! [X:int,X5:int,P:$o,P3:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P = P3)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => P3))))))).
% 2.63/1.44 thf(fact_233_conj__le__cong, axiom, (! [X:int,X5:int,P:$o,P3:$o]: (((X = X5) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) => (P = P3)) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & P) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ X5) & P3))))))).
% 2.63/1.44 thf(fact_234_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P3 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((P @ X3) & (Q2 @ X3)) = ((P3 @ X3) & (Q3 @ X3)))))))))))))))).
% 2.63/1.44 thf(fact_235_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P3 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((P @ X3) | (Q2 @ X3)) = ((P3 @ X3) | (Q3 @ X3)))))))))))))))).
% 2.63/1.44 thf(fact_236_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (X3 != T2)))))))))))).
% 2.63/1.44 thf(fact_237_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (X3 != T2)))))))))))).
% 2.63/1.44 thf(fact_238_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (~ ((((ord_less @ A) @ X3) @ T2)))))))))))))).
% 2.63/1.44 thf(fact_239_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((ord_less @ A) @ T2) @ X3)))))))))))).
% 2.63/1.44 thf(fact_240_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D2]: ((? [Z:C2]: ((! [X3:C2]: (((((ord_less @ C2) @ Z) @ X3) => (F4 = F4)))))))))))).
% 2.63/1.44 thf(fact_241_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P3 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (((P @ X3) & (Q2 @ X3)) = ((P3 @ X3) & (Q3 @ X3)))))))))))))))).
% 2.63/1.44 thf(fact_242_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:(A > $o),Q2:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P3 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q2 @ X2) = (Q3 @ X2))))))) => (? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (((P @ X3) | (Q2 @ X3)) = ((P3 @ X3) | (Q3 @ X3)))))))))))))))).
% 2.63/1.44 thf(fact_243_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (X3 != T2)))))))))))).
% 2.63/1.44 thf(fact_244_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (X3 != T2)))))))))))).
% 2.63/1.44 thf(fact_245_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (((ord_less @ A) @ X3) @ T2)))))))))))).
% 2.63/1.44 thf(fact_246_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (~ ((((ord_less @ A) @ T2) @ X3)))))))))))))).
% 2.63/1.44 thf(fact_247_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D2]: ((? [Z:C2]: ((! [X3:C2]: (((((ord_less @ C2) @ X3) @ Z) => (F4 = F4)))))))))))).
% 2.63/1.44 thf(fact_248_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (~ ((((ord_less_eq @ A) @ T2) @ X3)))))))))))))).
% 2.63/1.44 thf(fact_249_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z) => (((ord_less_eq @ A) @ X3) @ T2)))))))))))).
% 2.63/1.44 thf(fact_250_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (((ord_less_eq @ A) @ T2) @ X3)))))))))))).
% 2.63/1.44 thf(fact_251_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z:A]: ((! [X3:A]: (((((ord_less @ A) @ Z) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ T2)))))))))))))).
% 2.63/1.44 thf(fact_252_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat),B2:nat]: (((P @ K) => ((! [Y3:A]: (((P @ Y3) => (((ord_less @ nat) @ (M @ Y3)) @ B2)))) => (? [X2:A]: (((P @ X2) & (! [Y:A]: (((P @ Y) => (((ord_less_eq @ nat) @ (M @ Y)) @ (M @ X2))))))))))))).
% 2.63/1.44 thf(fact_253_realpow__pos__nth__unique, axiom, (! [N:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [X2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X2) & (((((power_power @ real) @ X2) @ N) = A2) & (! [Y:real]: ((((((ord_less @ real) @ (zero_zero @ real)) @ Y) & ((((power_power @ real) @ Y) @ N) = A2)) => (Y = X2))))))))))))).
% 2.63/1.44 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A5:$tType,A6:$tType]: ((((preorder @ A6) @ (type2 @ A6)) => ((preorder @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.63/1.44 thf(tcon_fun___Orderings_Oorder, axiom, (! [A5:$tType,A6:$tType]: ((((order @ A6) @ (type2 @ A6)) => ((order @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.63/1.44 thf(tcon_fun___Orderings_Oord, axiom, (! [A5:$tType,A6:$tType]: ((((ord @ A6) @ (type2 @ A6)) => ((ord @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.63/1.44 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, ((semiri134348788visors @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, ((linordered_idom @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, ((semiring_char_0 @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Rings_Ozero__neq__one, axiom, ((zero_neq_one @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Opreorder_1, axiom, ((preorder @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, ((linorder @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, ((semiring_1 @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, ((no_top @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, ((no_bot @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Oorder_2, axiom, ((order @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Orderings_Oord_3, axiom, ((ord @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Int_Oint___Groups_Ozero, axiom, ((zero @ int) @ (type2 @ int))).
% 2.63/1.44 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5, axiom, ((semiri134348788visors @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7, axiom, ((semiring_char_0 @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_8, axiom, ((zero_neq_one @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Opreorder_9, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Olinorder_10, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Rings_Osemiring__1_11, axiom, ((semiring_1 @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Ono__top_12, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Oorder_13, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Orderings_Oord_14, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_Nat_Onat___Groups_Ozero_15, axiom, ((zero @ nat) @ (type2 @ nat))).
% 2.63/1.44 thf(tcon_HOL_Obool___Orderings_Opreorder_16, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.63/1.44 thf(tcon_HOL_Obool___Orderings_Olinorder_17, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.63/1.44 thf(tcon_HOL_Obool___Orderings_Oorder_18, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.63/1.44 thf(tcon_HOL_Obool___Orderings_Oord_19, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.63/1.44 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_20, axiom, ((condit1037483654norder @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum, axiom, ((condit1656338222tinuum @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, ((archim1804426504_field @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_21, axiom, ((semiri134348788visors @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_22, axiom, ((linordered_semidom @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Odense__linorder, axiom, ((dense_linorder @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, ((linordered_field @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Rings_Olinordered__idom_23, axiom, ((linordered_idom @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Odense__order, axiom, ((dense_order @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, ((division_ring @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_24, axiom, ((semiring_char_0 @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_25, axiom, ((zero_neq_one @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Opreorder_26, axiom, ((preorder @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Olinorder_27, axiom, ((linorder @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Rings_Osemiring__1_28, axiom, ((semiring_1 @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Ono__top_29, axiom, ((no_top @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Ono__bot_30, axiom, ((no_bot @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Oorder_31, axiom, ((order @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Orderings_Oord_32, axiom, ((ord @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Fields_Ofield, axiom, ((field @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Real_Oreal___Groups_Ozero_33, axiom, ((zero @ real) @ (type2 @ real))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_34, axiom, ((semiri134348788visors @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_35, axiom, ((zero_neq_one @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_36, axiom, ((preorder @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_37, axiom, ((linorder @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_38, axiom, ((semiring_1 @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_39, axiom, ((order @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_40, axiom, ((ord @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_41, axiom, ((zero @ code_natural) @ (type2 @ code_natural))).
% 2.63/1.44 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.63/1.44 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y2:A]: ((((((if @ A) @ $false) @ X) @ Y2) = Y2)))).
% 2.63/1.44 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y2:A]: ((((((if @ A) @ $true) @ X) @ Y2) = X)))).
% 2.63/1.44 thf(conj_0, hypothesis, (hammin1195055315smooth @ n)).
% 2.63/1.44 thf(conj_1, conjecture, (((ord_less @ nat) @ (zero_zero @ nat)) @ n)).
% 2.63/1.44 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2DTF32658.p
% 2.63/1.44 ------------------------
% 2.63/1.48 ---- Cleaned THF ---
% 2.63/1.49 thf(ty_t_Code__Numeral_Onatural,type,
% 2.63/1.49 code_natural: $tType ).
% 2.63/1.49
% 2.63/1.49 thf(ty_t_Real_Oreal,type,
% 2.63/1.49 real: $tType ).
% 2.63/1.49
% 2.63/1.49 thf(ty_t_Nat_Onat,type,
% 2.63/1.49 nat: $tType ).
% 2.63/1.49
% 2.63/1.49 thf(ty_t_Int_Oint,type,
% 2.63/1.49 int: $tType ).
% 2.63/1.49
% 2.63/1.49 thf(ty_t_itself,type,
% 2.63/1.49 itself: $tType > $tType ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_HOL_Otype,type,
% 2.63/1.49 type:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Groups_Ozero,type,
% 2.63/1.49 zero:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Fields_Ofield,type,
% 2.63/1.49 field:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Oord,type,
% 2.63/1.49 ord:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Oorder,type,
% 2.63/1.49 order:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Ono__bot,type,
% 2.63/1.49 no_bot:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Ono__top,type,
% 2.63/1.49 no_top:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Rings_Osemiring__1,type,
% 2.63/1.49 semiring_1:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Olinorder,type,
% 2.63/1.49 linorder:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Opreorder,type,
% 2.63/1.49 preorder:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.63/1.49 zero_neq_one:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.63/1.49 semiring_char_0:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Owellorder,type,
% 2.63/1.49 wellorder:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Fields_Odivision__ring,type,
% 2.63/1.49 division_ring:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Odense__order,type,
% 2.63/1.49 dense_order:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.63/1.49 linordered_idom:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Fields_Olinordered__field,type,
% 2.63/1.49 linordered_field:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.63/1.49 dense_linorder:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.63/1.49 linordered_semidom:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.63/1.49 semiri134348788visors:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.63/1.49 archim1804426504_field:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.63/1.49 canoni770627133id_add:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.63/1.49 condit1656338222tinuum:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.63/1.49 condit1037483654norder:
% 2.63/1.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 2.63/1.49 code_case_natural:
% 2.63/1.49 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
% 2.63/1.49 code_size_natural: code_natural > nat ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 2.63/1.49 inverse_inverse:
% 2.63/1.49 !>[A: $tType] : ( A > A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.63/1.49 zero_zero:
% 2.63/1.49 !>[A: $tType] : A ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth,type,
% 2.63/1.49 hammin1195055315smooth: nat > $o ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_If,type,
% 2.63/1.49 if:
% 2.63/1.49 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.63/1.49 semiring_1_of_nat:
% 2.63/1.49 !>[A: $tType] : ( nat > A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.63/1.49 size_size:
% 2.63/1.49 !>[A: $tType] : ( A > nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.63/1.49 ord_less:
% 2.63/1.49 !>[A: $tType] : ( A > A > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.63/1.49 ord_less_eq:
% 2.63/1.49 !>[A: $tType] : ( A > A > $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Power_Opower__class_Opower,type,
% 2.63/1.49 power_power:
% 2.63/1.49 !>[A: $tType] : ( A > nat > A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Pure_Otype,type,
% 2.63/1.49 type2:
% 2.63/1.49 !>[A: $tType] : ( itself @ A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 2.63/1.49 zero_neq_one_of_bool:
% 2.63/1.49 !>[A: $tType] : ( $o > A ) ).
% 2.63/1.49
% 2.63/1.49 thf(sy_v_n,type,
% 2.63/1.49 n: nat ).
% 2.63/1.49
% 2.63/1.49 thf(fact_0_smooth__0,axiom,
% 2.63/1.49 ~ ( hammin1195055315smooth @ ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_1_neq0__conv,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ( ( N
% 2.63/1.49 != ( zero_zero @ nat ) )
% 2.63/1.49 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_2_not__gr__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: A] :
% 2.63/1.49 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.63/1.49 = ( N
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_3_gr0I,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ( ( N
% 2.63/1.49 != ( zero_zero @ nat ) )
% 2.63/1.49 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_4_not__gr0,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.63/1.49 = ( N
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_5_not__less0,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_6_less__zeroE,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_7_gr__implies__not0,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M @ N )
% 2.63/1.49 => ( N
% 2.63/1.49 != ( zero_zero @ nat ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_8_infinite__descent0,axiom,
% 2.63/1.49 ! [P: nat > $o,N: nat] :
% 2.63/1.49 ( ( P @ ( zero_zero @ nat ) )
% 2.63/1.49 => ( ! [N2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.63/1.49 => ( ~ ( P @ N2 )
% 2.63/1.49 => ? [M2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M2 @ N2 )
% 2.63/1.49 & ~ ( P @ M2 ) ) ) )
% 2.63/1.49 => ( P @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_9_less__nat__zero__code,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_10_infinite__descent0__measure,axiom,
% 2.63/1.49 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ( ( V @ X2 )
% 2.63/1.49 = ( zero_zero @ nat ) )
% 2.63/1.49 => ( P @ X2 ) )
% 2.63/1.49 => ( ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 2.63/1.49 => ( ~ ( P @ X2 )
% 2.63/1.49 => ? [Y: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 2.63/1.49 & ~ ( P @ Y ) ) ) )
% 2.63/1.49 => ( P @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_11_less__numeral__extra_I3_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_12_zero__reorient,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( zero @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ( ( ( zero_zero @ A )
% 2.63/1.49 = X )
% 2.63/1.49 = ( X
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_13_infinite__descent__measure,axiom,
% 2.63/1.49 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ~ ( P @ X2 )
% 2.63/1.49 => ? [Y: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
% 2.63/1.49 & ~ ( P @ Y ) ) )
% 2.63/1.49 => ( P @ X ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_14_measure__induct__rule,axiom,
% 2.63/1.49 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ! [Y: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
% 2.63/1.49 => ( P @ Y ) )
% 2.63/1.49 => ( P @ X2 ) )
% 2.63/1.49 => ( P @ A2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_15_linorder__neqE__nat,axiom,
% 2.63/1.49 ! [X: nat,Y2: nat] :
% 2.63/1.49 ( ( X != Y2 )
% 2.63/1.49 => ( ~ ( ord_less @ nat @ X @ Y2 )
% 2.63/1.49 => ( ord_less @ nat @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_16_infinite__descent,axiom,
% 2.63/1.49 ! [P: nat > $o,N: nat] :
% 2.63/1.49 ( ! [N2: nat] :
% 2.63/1.49 ( ~ ( P @ N2 )
% 2.63/1.49 => ? [M2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M2 @ N2 )
% 2.63/1.49 & ~ ( P @ M2 ) ) )
% 2.63/1.49 => ( P @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_17_nat__less__induct,axiom,
% 2.63/1.49 ! [P: nat > $o,N: nat] :
% 2.63/1.49 ( ! [N2: nat] :
% 2.63/1.49 ( ! [M2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M2 @ N2 )
% 2.63/1.49 => ( P @ M2 ) )
% 2.63/1.49 => ( P @ N2 ) )
% 2.63/1.49 => ( P @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_18_less__irrefl__nat,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ~ ( ord_less @ nat @ N @ N ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_19_measure__induct,axiom,
% 2.63/1.49 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ! [Y: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
% 2.63/1.49 => ( P @ Y ) )
% 2.63/1.49 => ( P @ X2 ) )
% 2.63/1.49 => ( P @ A2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_20_less__not__refl3,axiom,
% 2.63/1.49 ! [S: nat,T2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ S @ T2 )
% 2.63/1.49 => ( S != T2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_21_less__not__refl2,axiom,
% 2.63/1.49 ! [N: nat,M: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ N @ M )
% 2.63/1.49 => ( M != N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_22_less__not__refl,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ~ ( ord_less @ nat @ N @ N ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_23_nat__neq__iff,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( M != N )
% 2.63/1.49 = ( ( ord_less @ nat @ M @ N )
% 2.63/1.49 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_24_zero__less__iff__neq__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.63/1.49 = ( N
% 2.63/1.49 != ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_25_gr__implies__not__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: A,N: A] :
% 2.63/1.49 ( ( ord_less @ A @ M @ N )
% 2.63/1.49 => ( N
% 2.63/1.49 != ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_26_not__less__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: A] :
% 2.63/1.49 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_27_gr__zeroI,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: A] :
% 2.63/1.49 ( ( N
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_28_of__nat__0__less__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_29_zero__natural_Orsp,axiom,
% 2.63/1.49 ( ( zero_zero @ nat )
% 2.63/1.49 = ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_30_ex__gt__or__lt,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ? [B: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B )
% 2.63/1.49 | ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_31_linorder__neqE__linordered__idom,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( X != Y2 )
% 2.63/1.49 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_32_dual__order_Ostrict__implies__not__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ B2 @ A2 )
% 2.63/1.49 => ( A2 != B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_33_linordered__field__no__ub,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X3: A] :
% 2.63/1.49 ? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_34_linordered__field__no__lb,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X3: A] :
% 2.63/1.49 ? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_35_order_Ostrict__implies__not__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( A2 != B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_36_dependent__wellorder__choice,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: ( A > B3 ) > A > B3 > $o] :
% 2.63/1.49 ( ! [R: B3,F2: A > B3,G: A > B3,X2: A] :
% 2.63/1.49 ( ! [Y: A] :
% 2.63/1.49 ( ( ord_less @ A @ Y @ X2 )
% 2.63/1.49 => ( ( F2 @ Y )
% 2.63/1.49 = ( G @ Y ) ) )
% 2.63/1.49 => ( ( P @ F2 @ X2 @ R )
% 2.63/1.49 = ( P @ G @ X2 @ R ) ) )
% 2.63/1.49 => ( ! [X2: A,F2: A > B3] :
% 2.63/1.49 ( ! [Y: A] :
% 2.63/1.49 ( ( ord_less @ A @ Y @ X2 )
% 2.63/1.49 => ( P @ F2 @ Y @ ( F2 @ Y ) ) )
% 2.63/1.49 => ? [X12: B3] : ( P @ F2 @ X2 @ X12 ) )
% 2.63/1.49 => ? [F2: A > B3] :
% 2.63/1.49 ! [X3: A] : ( P @ F2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_37_not__less__iff__gr__or__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.63/1.49 = ( ( ord_less @ A @ Y2 @ X )
% 2.63/1.49 | ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_38_dual__order_Ostrict__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A,C: A] :
% 2.63/1.49 ( ( ord_less @ A @ B2 @ A2 )
% 2.63/1.49 => ( ( ord_less @ A @ C @ B2 )
% 2.63/1.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_39_of__nat__eq__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ( semiring_1_of_nat @ A @ M )
% 2.63/1.49 = ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 = ( M = N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_40_of__nat__0,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.63/1.49 = ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_41_of__nat__0__eq__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: nat] :
% 2.63/1.49 ( ( ( zero_zero @ A )
% 2.63/1.49 = ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 = ( ( zero_zero @ nat )
% 2.63/1.49 = N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_42_of__nat__eq__0__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat] :
% 2.63/1.49 ( ( ( semiring_1_of_nat @ A @ M )
% 2.63/1.49 = ( zero_zero @ A ) )
% 2.63/1.49 = ( M
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_43_of__nat__less__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 = ( ord_less @ nat @ M @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_44_of__nat__less__0__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat] :
% 2.63/1.49 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_45_less__imp__of__nat__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M @ N )
% 2.63/1.49 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_46_of__nat__less__imp__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_47_ord__eq__less__subst,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( ord @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( A2
% 2.63/1.49 = ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_48_ord__less__eq__subst,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( ord @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > B3,C: B3] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ( F @ B2 )
% 2.63/1.49 = C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_49_order__less__subst1,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_50_order__less__subst2,axiom,
% 2.63/1.49 ! [A: $tType,C2: $tType] :
% 2.63/1.49 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_51_lt__ex,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_52_gt__ex,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( no_top @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_53_neqE,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( X != Y2 )
% 2.63/1.49 => ( ~ ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_54_neq__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( X != Y2 )
% 2.63/1.49 = ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 | ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_55_order_Oasym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_56_dense,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ? [Z: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Z )
% 2.63/1.49 & ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_57_less__imp__neq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( X != Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_58_less__asym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_59_less__asym_H,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_60_less__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.63/1.49 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_61_less__linear,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 | ( X = Y2 )
% 2.63/1.49 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_62_less__irrefl,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_63_ord__eq__less__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( ord @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( A2 = B2 )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_64_ord__less__eq__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( ord @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( B2 = C )
% 2.63/1.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_65_dual__order_Oasym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ B2 @ A2 )
% 2.63/1.49 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_66_less__imp__not__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( X != Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_67_less__not__sym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_68_less__induct,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > $o,A2: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ! [Y: A] :
% 2.63/1.49 ( ( ord_less @ A @ Y @ X2 )
% 2.63/1.49 => ( P @ Y ) )
% 2.63/1.49 => ( P @ X2 ) )
% 2.63/1.49 => ( P @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_69_antisym__conv3,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Y2: A,X: A] :
% 2.63/1.49 ( ~ ( ord_less @ A @ Y2 @ X )
% 2.63/1.49 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.63/1.49 = ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_70_less__imp__not__eq2,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( Y2 != X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_71_less__imp__triv,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,P: $o] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less @ A @ Y2 @ X )
% 2.63/1.49 => P ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_72_linorder__cases,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ( X != Y2 )
% 2.63/1.49 => ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_73_dual__order_Oirrefl,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_74_order_Ostrict__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_75_less__imp__not__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_76_zero__less__imp__eq__int,axiom,
% 2.63/1.49 ! [K: int] :
% 2.63/1.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.63/1.49 => ? [N2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.63/1.49 & ( K
% 2.63/1.49 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_77_reals__Archimedean2,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_78_natural_Osize_I1_J,axiom,
% 2.63/1.49 ( ( code_size_natural @ ( zero_zero @ code_natural ) )
% 2.63/1.49 = ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_79_of__nat__zero__less__power__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 2.63/1.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.63/1.49 | ( N
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_80_of__nat__le__0__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.63/1.49 = ( M
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_81_natural_Osize_I3_J,axiom,
% 2.63/1.49 ( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
% 2.63/1.49 = ( zero_zero @ nat ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_82_of__bool__eq_I1_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( zero_neq_one_of_bool @ A @ $false )
% 2.63/1.49 = ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_83_ex__inverse__of__nat__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.63/1.49 => ? [N2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.63/1.49 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_84_order__refl,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_85_le0,axiom,
% 2.63/1.49 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_86_inverse__inverse__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 2.63/1.49 = A2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_87_inverse__eq__iff__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 = ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 = ( A2 = B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_88_le__zero__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.63/1.49 = ( N
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_89_of__nat__le__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.63/1.49 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_90_inverse__nonzero__iff__nonzero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 = ( zero_zero @ A ) )
% 2.63/1.49 = ( A2
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_91_inverse__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 2.63/1.49 = ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_92_nat__zero__less__power__iff,axiom,
% 2.63/1.49 ! [X: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 2.63/1.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.63/1.49 | ( N
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_93_of__nat__power,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [M: nat,N: nat] :
% 2.63/1.49 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 2.63/1.49 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_94_inverse__nonnegative__iff__nonnegative,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.63/1.49 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_95_inverse__nonpositive__iff__nonpositive,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.63/1.49 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_96_inverse__positive__iff__positive,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.63/1.49 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_97_inverse__negative__iff__negative,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.63/1.49 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_98_inverse__less__iff__less__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_99_inverse__less__iff__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_100_inverse__le__iff__le__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_101_inverse__le__iff__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_102_power__eq__0__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiri134348788visors @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat] :
% 2.63/1.49 ( ( ( power_power @ A @ A2 @ N )
% 2.63/1.49 = ( zero_zero @ A ) )
% 2.63/1.49 = ( ( A2
% 2.63/1.49 = ( zero_zero @ A ) )
% 2.63/1.49 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_103_nonneg__eq__int,axiom,
% 2.63/1.49 ! [Z2: int] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 2.63/1.49 => ~ ! [M3: nat] :
% 2.63/1.49 ( Z2
% 2.63/1.49 != ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_104_nonneg__int__cases,axiom,
% 2.63/1.49 ! [K: int] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.63/1.49 => ~ ! [N2: nat] :
% 2.63/1.49 ( K
% 2.63/1.49 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_105_zero__le__imp__eq__int,axiom,
% 2.63/1.49 ! [K: int] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.63/1.49 => ? [N2: nat] :
% 2.63/1.49 ( K
% 2.63/1.49 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_106_zero__integer_Orsp,axiom,
% 2.63/1.49 ( ( zero_zero @ int )
% 2.63/1.49 = ( zero_zero @ int ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_107_inverse__eq__imp__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 = ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( A2 = B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_108_dual__order_Oantisym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( A2 = B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_109_of__bool__eq__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P2: $o,Q: $o] :
% 2.63/1.49 ( ( ( zero_neq_one_of_bool @ A @ P2 )
% 2.63/1.49 = ( zero_neq_one_of_bool @ A @ Q ) )
% 2.63/1.49 = ( P2 = Q ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_110_dual__order_Otrans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A,C: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.63/1.49 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_111_linorder__wlog,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.63/1.49 ( ! [A3: A,B: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A3 @ B )
% 2.63/1.49 => ( P @ A3 @ B ) )
% 2.63/1.49 => ( ! [A3: A,B: A] :
% 2.63/1.49 ( ( P @ B @ A3 )
% 2.63/1.49 => ( P @ A3 @ B ) )
% 2.63/1.49 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_112_dual__order_Orefl,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_113_order__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.63/1.49 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_114_order__class_Oorder_Oantisym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.63/1.49 => ( A2 = B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_115_ord__le__eq__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( ord @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( B2 = C )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_116_ord__eq__le__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( ord @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( A2 = B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_117_antisym__conv,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Y2: A,X: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.63/1.49 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 = ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_118_le__cases3,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ A @ Y2 @ X )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ Y2 @ X ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.63/1.49 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_119_real__arch__simple,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( archim1804426504_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] :
% 2.63/1.49 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_120_order_Otrans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_121_le__cases,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ~ ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_122_eq__refl,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( X = Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_123_linear,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 | ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_124_antisym,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ Y2 @ X )
% 2.63/1.49 => ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_125_eq__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.63/1.49 = ( ^ [X4: A,Y5: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.63/1.49 & ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_126_ord__le__eq__subst,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( ord @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > B3,C: B3] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ( F @ B2 )
% 2.63/1.49 = C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_127_ord__eq__le__subst,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( ord @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( A2
% 2.63/1.49 = ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_128_order__subst2,axiom,
% 2.63/1.49 ! [A: $tType,C2: $tType] :
% 2.63/1.49 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_129_order__subst1,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_130_le__fun__def,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 => ( ( ord_less_eq @ ( A > B3 ) )
% 2.63/1.49 = ( ^ [F3: A > B3,G2: A > B3] :
% 2.63/1.49 ! [X4: A] : ( ord_less_eq @ B3 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_131_le__funI,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 => ! [F: A > B3,G3: A > B3] :
% 2.63/1.49 ( ! [X2: A] : ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G3 @ X2 ) )
% 2.63/1.49 => ( ord_less_eq @ ( A > B3 ) @ F @ G3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_132_le__funE,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 => ! [F: A > B3,G3: A > B3,X: A] :
% 2.63/1.49 ( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
% 2.63/1.49 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_133_le__funD,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 => ! [F: A > B3,G3: A > B3,X: A] :
% 2.63/1.49 ( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
% 2.63/1.49 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_134_int__int__eq,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ( semiring_1_of_nat @ int @ M )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ N ) )
% 2.63/1.49 = ( M = N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_135_zle__int,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.63/1.49 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_136_power__not__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiri134348788visors @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat] :
% 2.63/1.49 ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ( power_power @ A @ A2 @ N )
% 2.63/1.49 != ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_137_power__less__imp__less__base,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.63/1.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_138_power__inverse,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat] :
% 2.63/1.49 ( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
% 2.63/1.49 = ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_139_zero__le__power,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_140_power__mono,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_141_less__eq__int__code_I1_J,axiom,
% 2.63/1.49 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_142_less__int__code_I1_J,axiom,
% 2.63/1.49 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_143_le__imp__inverse__le__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_144_inverse__le__imp__le__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_145_le__imp__inverse__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_146_inverse__le__imp__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_147_nonzero__imp__inverse__nonzero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 != ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_148_nonzero__inverse__inverse__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 2.63/1.49 = A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_149_nonzero__inverse__eq__imp__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 = ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ( B2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( A2 = B2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_150_inverse__zero__imp__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ( inverse_inverse @ A @ A2 )
% 2.63/1.49 = ( zero_zero @ A ) )
% 2.63/1.49 => ( A2
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_151_field__class_Ofield__inverse__zero,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 2.63/1.49 = ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_152_power__eq__imp__eq__base,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat,B2: A] :
% 2.63/1.49 ( ( ( power_power @ A @ A2 @ N )
% 2.63/1.49 = ( power_power @ A @ B2 @ N ) )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.63/1.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.63/1.49 => ( A2 = B2 ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_153_power__eq__iff__eq__base,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: nat,A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.63/1.49 => ( ( ( power_power @ A @ A2 @ N )
% 2.63/1.49 = ( power_power @ A @ B2 @ N ) )
% 2.63/1.49 = ( A2 = B2 ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_154_zero__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_155_le__numeral__extra_I3_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_156_complete__interval,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,P: A > $o] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( P @ A2 )
% 2.63/1.49 => ( ~ ( P @ B2 )
% 2.63/1.49 => ? [C3: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.63/1.49 & ( ord_less_eq @ A @ C3 @ B2 )
% 2.63/1.49 & ! [X3: A] :
% 2.63/1.49 ( ( ( ord_less_eq @ A @ A2 @ X3 )
% 2.63/1.49 & ( ord_less @ A @ X3 @ C3 ) )
% 2.63/1.49 => ( P @ X3 ) )
% 2.63/1.49 & ! [D: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.63/1.49 & ( ord_less @ A @ X2 @ D ) )
% 2.63/1.49 => ( P @ X2 ) )
% 2.63/1.49 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_157_order_Onot__eq__order__implies__strict,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( A2 != B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_158_dual__order_Ostrict__implies__order,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ B2 @ A2 )
% 2.63/1.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_159_dual__order_Ostrict__iff__order,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less @ A )
% 2.63/1.49 = ( ^ [B4: A,A4: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.63/1.49 & ( A4 != B4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_160_dual__order_Oorder__iff__strict,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less_eq @ A )
% 2.63/1.49 = ( ^ [B4: A,A4: A] :
% 2.63/1.49 ( ( ord_less @ A @ B4 @ A4 )
% 2.63/1.49 | ( A4 = B4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_161_order_Ostrict__implies__order,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_162_dense__le__bounded,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ! [W: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ W )
% 2.63/1.49 => ( ( ord_less @ A @ W @ Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_163_dense__ge__bounded,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Z2: A,X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z2 @ X )
% 2.63/1.49 => ( ! [W: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z2 @ W )
% 2.63/1.49 => ( ( ord_less @ A @ W @ X )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ W ) ) )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_164_dual__order_Ostrict__trans2,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A,C: A] :
% 2.63/1.49 ( ( ord_less @ A @ B2 @ A2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.63/1.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_165_dual__order_Ostrict__trans1,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [B2: A,A2: A,C: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.63/1.49 => ( ( ord_less @ A @ C @ B2 )
% 2.63/1.49 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_166_order_Ostrict__iff__order,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less @ A )
% 2.63/1.49 = ( ^ [A4: A,B4: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.63/1.49 & ( A4 != B4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_167_order_Oorder__iff__strict,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less_eq @ A )
% 2.63/1.49 = ( ^ [A4: A,B4: A] :
% 2.63/1.49 ( ( ord_less @ A @ A4 @ B4 )
% 2.63/1.49 | ( A4 = B4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_168_order_Ostrict__trans2,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_169_order_Ostrict__trans1,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,C: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ C )
% 2.63/1.49 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_170_not__le__imp__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Y2: A,X: A] :
% 2.63/1.49 ( ~ ( ord_less_eq @ A @ Y2 @ X )
% 2.63/1.49 => ( ord_less @ A @ X @ Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_171_less__le__not__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less @ A )
% 2.63/1.49 = ( ^ [X4: A,Y5: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.63/1.49 & ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_172_le__imp__less__or__eq,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 | ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_173_le__less__linear,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 | ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_174_dense__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Y2: A,Z2: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_175_dense__ge,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Z2: A,Y2: A] :
% 2.63/1.49 ( ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z2 @ X2 )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ X2 ) )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_176_less__le__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ Y2 @ Z2 )
% 2.63/1.49 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_177_le__less__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A,Z2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less @ A @ Y2 @ Z2 )
% 2.63/1.49 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_178_antisym__conv2,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.63/1.49 = ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_179_antisym__conv1,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ X @ Y2 )
% 2.63/1.49 = ( X = Y2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_180_less__imp__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( preorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_181_le__neq__trans,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( A2 != B2 )
% 2.63/1.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_182_not__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ~ ( ord_less @ A @ X @ Y2 ) )
% 2.63/1.49 = ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_183_not__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
% 2.63/1.49 = ( ord_less @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_184_order__less__le__subst2,axiom,
% 2.63/1.49 ! [A: $tType,C2: $tType] :
% 2.63/1.49 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_185_order__less__le__subst1,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less_eq @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less_eq @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_186_order__le__less__subst2,axiom,
% 2.63/1.49 ! [A: $tType,C2: $tType] :
% 2.63/1.49 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.63/1.49 => ( ! [X2: A,Y3: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_187_order__le__less__subst1,axiom,
% 2.63/1.49 ! [A: $tType,B3: $tType] :
% 2.63/1.49 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 & ( order @ A @ ( type2 @ A ) ) )
% 2.63/1.49 => ! [A2: A,F: B3 > A,B2: B3,C: B3] :
% 2.63/1.49 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ B3 @ B2 @ C )
% 2.63/1.49 => ( ! [X2: B3,Y3: B3] :
% 2.63/1.49 ( ( ord_less @ B3 @ X2 @ Y3 )
% 2.63/1.49 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
% 2.63/1.49 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_188_less__le,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less @ A )
% 2.63/1.49 = ( ^ [X4: A,Y5: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ X4 @ Y5 )
% 2.63/1.49 & ( X4 != Y5 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_189_le__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( order @ A @ ( type2 @ A ) )
% 2.63/1.49 => ( ( ord_less_eq @ A )
% 2.63/1.49 = ( ^ [X4: A,Y5: A] :
% 2.63/1.49 ( ( ord_less @ A @ X4 @ Y5 )
% 2.63/1.49 | ( X4 = Y5 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_190_leI,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [X: A,Y2: A] :
% 2.63/1.49 ( ~ ( ord_less @ A @ X @ Y2 )
% 2.63/1.49 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_191_leD,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [Y2: A,X: A] :
% 2.63/1.49 ( ( ord_less_eq @ A @ Y2 @ X )
% 2.63/1.49 => ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_192_le__0__eq,axiom,
% 2.63/1.49 ! [N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.63/1.49 = ( N
% 2.63/1.49 = ( zero_zero @ nat ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_193_less__eq__nat_Osimps_I1_J,axiom,
% 2.63/1.49 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_194_less__mono__imp__le__mono,axiom,
% 2.63/1.49 ! [F: nat > nat,I: nat,J: nat] :
% 2.63/1.49 ( ! [I2: nat,J2: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ I2 @ J2 )
% 2.63/1.49 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.63/1.49 => ( ( ord_less_eq @ nat @ I @ J )
% 2.63/1.49 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_195_le__neq__implies__less,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ M @ N )
% 2.63/1.49 => ( ( M != N )
% 2.63/1.49 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_196_less__or__eq__imp__le,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ( ord_less @ nat @ M @ N )
% 2.63/1.49 | ( M = N ) )
% 2.63/1.49 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_197_le__eq__less__or__eq,axiom,
% 2.63/1.49 ( ( ord_less_eq @ nat )
% 2.63/1.49 = ( ^ [M4: nat,N3: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M4 @ N3 )
% 2.63/1.49 | ( M4 = N3 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_198_less__imp__le__nat,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ M @ N )
% 2.63/1.49 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_199_nat__less__le,axiom,
% 2.63/1.49 ( ( ord_less @ nat )
% 2.63/1.49 = ( ^ [M4: nat,N3: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 2.63/1.49 & ( M4 != N3 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_200_zero__less__power,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_201_nat__power__less__imp__less,axiom,
% 2.63/1.49 ! [I: nat,M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
% 2.63/1.49 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
% 2.63/1.49 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_202_power__strict__mono,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A,N: nat] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.63/1.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_203_positive__imp__inverse__positive,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_204_negative__imp__inverse__negative,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_205_inverse__positive__imp__positive,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.63/1.49 => ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_206_inverse__negative__imp__negative,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ( A2
% 2.63/1.49 != ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_207_less__imp__inverse__less__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_208_inverse__less__imp__less__neg,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.63/1.49 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_209_less__imp__inverse__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ A2 @ B2 )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_210_inverse__less__imp__less,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [A2: A,B2: A] :
% 2.63/1.49 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.63/1.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.63/1.49 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_211_zero__power,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( semiring_1 @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.63/1.49 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.63/1.49 = ( zero_zero @ A ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_212_of__nat__0__le__iff,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_213_ex__least__nat__le,axiom,
% 2.63/1.49 ! [P: nat > $o,N: nat] :
% 2.63/1.49 ( ~ ( P @ ( zero_zero @ nat ) )
% 2.63/1.49 => ( ( P @ N )
% 2.63/1.49 => ? [K2: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.63/1.49 & ! [I3: nat] :
% 2.63/1.49 ( ( ord_less @ nat @ I3 @ K2 )
% 2.63/1.49 => ~ ( P @ I3 ) )
% 2.63/1.49 & ( P @ K2 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_214_transfer__int__nat__relations_I2_J,axiom,
% 2.63/1.49 ! [X: nat,Y2: nat] :
% 2.63/1.49 ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 2.63/1.49 = ( ord_less @ nat @ X @ Y2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_215_transfer__int__nat__numerals_I1_J,axiom,
% 2.63/1.49 ( ( zero_zero @ int )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_216_natural_Osimps_I4_J,axiom,
% 2.63/1.49 ! [T: $tType,F1: T,F22: code_natural > T] :
% 2.63/1.49 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 2.63/1.49 = F1 ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_217_transfer__int__nat__quantifiers_I2_J,axiom,
% 2.63/1.49 ! [P: int > $o] :
% 2.63/1.49 ( ( ? [X4: int] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.63/1.49 & ( P @ X4 ) ) )
% 2.63/1.49 = ( ? [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_218_transfer__int__nat__quantifiers_I1_J,axiom,
% 2.63/1.49 ! [P: int > $o] :
% 2.63/1.49 ( ( ! [X4: int] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
% 2.63/1.49 => ( P @ X4 ) ) )
% 2.63/1.49 = ( ! [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_219_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 2.63/1.49 ! [Z2: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_220_less__fun__def,axiom,
% 2.63/1.49 ! [B3: $tType,A: $tType] :
% 2.63/1.49 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 2.63/1.49 => ( ( ord_less @ ( A > B3 ) )
% 2.63/1.49 = ( ^ [F3: A > B3,G2: A > B3] :
% 2.63/1.49 ( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
% 2.63/1.49 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_221_nat__le__linear,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ M @ N )
% 2.63/1.49 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_222_le__antisym,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ M @ N )
% 2.63/1.49 => ( ( ord_less_eq @ nat @ N @ M )
% 2.63/1.49 => ( M = N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_223_eq__imp__le,axiom,
% 2.63/1.49 ! [M: nat,N: nat] :
% 2.63/1.49 ( ( M = N )
% 2.63/1.49 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_224_le__trans,axiom,
% 2.63/1.49 ! [I: nat,J: nat,K: nat] :
% 2.63/1.49 ( ( ord_less_eq @ nat @ I @ J )
% 2.63/1.49 => ( ( ord_less_eq @ nat @ J @ K )
% 2.63/1.49 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_225_le__refl,axiom,
% 2.63/1.49 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_226_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 2.63/1.49 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_227_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
% 2.63/1.49 ! [X: int,N: nat] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.63/1.49 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( power_power @ int @ X @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_228_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
% 2.63/1.49 ! [X: nat,N: nat] :
% 2.63/1.49 ( ( power_power @ int @ ( semiring_1_of_nat @ int @ X ) @ N )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ ( power_power @ nat @ X @ N ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_229_transfer__int__nat__relations_I3_J,axiom,
% 2.63/1.49 ! [X: nat,Y2: nat] :
% 2.63/1.49 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
% 2.63/1.49 = ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_230_transfer__int__nat__relations_I1_J,axiom,
% 2.63/1.49 ! [X: nat,Y2: nat] :
% 2.63/1.49 ( ( ( semiring_1_of_nat @ int @ X )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ Y2 ) )
% 2.63/1.49 = ( X = Y2 ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_231_int__if__cong,axiom,
% 2.63/1.49 ! [P: $o,X: nat,Y2: nat] :
% 2.63/1.49 ( ( P
% 2.63/1.49 => ( ( semiring_1_of_nat @ int @ X )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
% 2.63/1.49 & ( ~ P
% 2.63/1.49 => ( ( semiring_1_of_nat @ int @ Y2 )
% 2.63/1.49 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_232_imp__le__cong,axiom,
% 2.63/1.49 ! [X: int,X5: int,P: $o,P3: $o] :
% 2.63/1.49 ( ( X = X5 )
% 2.63/1.49 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.63/1.49 => ( P = P3 ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.63/1.49 => P )
% 2.63/1.49 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.63/1.49 => P3 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_233_conj__le__cong,axiom,
% 2.63/1.49 ! [X: int,X5: int,P: $o,P3: $o] :
% 2.63/1.49 ( ( X = X5 )
% 2.63/1.49 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.63/1.49 => ( P = P3 ) )
% 2.63/1.49 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
% 2.63/1.49 & P )
% 2.63/1.49 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
% 2.63/1.49 & P3 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_234_pinf_I1_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.63/1.49 ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z4 @ X2 )
% 2.63/1.49 => ( ( P @ X2 )
% 2.63/1.49 = ( P3 @ X2 ) ) )
% 2.63/1.49 => ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z4 @ X2 )
% 2.63/1.49 => ( ( Q2 @ X2 )
% 2.63/1.49 = ( Q3 @ X2 ) ) )
% 2.63/1.49 => ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( ( ( P @ X3 )
% 2.63/1.49 & ( Q2 @ X3 ) )
% 2.63/1.49 = ( ( P3 @ X3 )
% 2.63/1.49 & ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_235_pinf_I2_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.63/1.49 ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z4 @ X2 )
% 2.63/1.49 => ( ( P @ X2 )
% 2.63/1.49 = ( P3 @ X2 ) ) )
% 2.63/1.49 => ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z4 @ X2 )
% 2.63/1.49 => ( ( Q2 @ X2 )
% 2.63/1.49 = ( Q3 @ X2 ) ) )
% 2.63/1.49 => ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( ( ( P @ X3 )
% 2.63/1.49 | ( Q2 @ X3 ) )
% 2.63/1.49 = ( ( P3 @ X3 )
% 2.63/1.49 | ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_236_pinf_I3_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( X3 != T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_237_pinf_I4_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( X3 != T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_238_pinf_I5_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ~ ( ord_less @ A @ X3 @ T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_239_pinf_I7_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( ord_less @ A @ T2 @ X3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_240_pinf_I11_J,axiom,
% 2.63/1.49 ! [C2: $tType,D2: $tType] :
% 2.63/1.49 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 => ! [F4: D2] :
% 2.63/1.49 ? [Z: C2] :
% 2.63/1.49 ! [X3: C2] :
% 2.63/1.49 ( ( ord_less @ C2 @ Z @ X3 )
% 2.63/1.49 => ( F4 = F4 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_241_minf_I1_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.63/1.49 ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Z4 )
% 2.63/1.49 => ( ( P @ X2 )
% 2.63/1.49 = ( P3 @ X2 ) ) )
% 2.63/1.49 => ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Z4 )
% 2.63/1.49 => ( ( Q2 @ X2 )
% 2.63/1.49 = ( Q3 @ X2 ) ) )
% 2.63/1.49 => ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( ( ( P @ X3 )
% 2.63/1.49 & ( Q2 @ X3 ) )
% 2.63/1.49 = ( ( P3 @ X3 )
% 2.63/1.49 & ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_242_minf_I2_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
% 2.63/1.49 ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Z4 )
% 2.63/1.49 => ( ( P @ X2 )
% 2.63/1.49 = ( P3 @ X2 ) ) )
% 2.63/1.49 => ( ? [Z4: A] :
% 2.63/1.49 ! [X2: A] :
% 2.63/1.49 ( ( ord_less @ A @ X2 @ Z4 )
% 2.63/1.49 => ( ( Q2 @ X2 )
% 2.63/1.49 = ( Q3 @ X2 ) ) )
% 2.63/1.49 => ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( ( ( P @ X3 )
% 2.63/1.49 | ( Q2 @ X3 ) )
% 2.63/1.49 = ( ( P3 @ X3 )
% 2.63/1.49 | ( Q3 @ X3 ) ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_243_minf_I3_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( X3 != T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_244_minf_I4_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( X3 != T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_245_minf_I5_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( ord_less @ A @ X3 @ T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_246_minf_I7_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ~ ( ord_less @ A @ T2 @ X3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_247_minf_I11_J,axiom,
% 2.63/1.49 ! [C2: $tType,D2: $tType] :
% 2.63/1.49 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.63/1.49 => ! [F4: D2] :
% 2.63/1.49 ? [Z: C2] :
% 2.63/1.49 ! [X3: C2] :
% 2.63/1.49 ( ( ord_less @ C2 @ X3 @ Z )
% 2.63/1.49 => ( F4 = F4 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_248_minf_I8_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_249_minf_I6_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ X3 @ Z )
% 2.63/1.49 => ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_250_pinf_I8_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_251_pinf_I6_J,axiom,
% 2.63/1.49 ! [A: $tType] :
% 2.63/1.49 ( ( linorder @ A @ ( type2 @ A ) )
% 2.63/1.49 => ! [T2: A] :
% 2.63/1.49 ? [Z: A] :
% 2.63/1.49 ! [X3: A] :
% 2.63/1.49 ( ( ord_less @ A @ Z @ X3 )
% 2.63/1.49 => ~ ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_252_ex__has__greatest__nat,axiom,
% 2.63/1.49 ! [A: $tType,P: A > $o,K: A,M: A > nat,B2: nat] :
% 2.63/1.49 ( ( P @ K )
% 2.63/1.49 => ( ! [Y3: A] :
% 2.63/1.49 ( ( P @ Y3 )
% 2.63/1.49 => ( ord_less @ nat @ ( M @ Y3 ) @ B2 ) )
% 2.63/1.49 => ? [X2: A] :
% 2.63/1.49 ( ( P @ X2 )
% 2.63/1.49 & ! [Y: A] :
% 2.63/1.49 ( ( P @ Y )
% 2.63/1.49 => ( ord_less_eq @ nat @ ( M @ Y ) @ ( M @ X2 ) ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(fact_253_realpow__pos__nth__unique,axiom,
% 2.63/1.49 ! [N: nat,A2: real] :
% 2.63/1.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.63/1.49 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.63/1.49 => ? [X2: real] :
% 2.63/1.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
% 2.63/1.49 & ( ( power_power @ real @ X2 @ N )
% 2.63/1.49 = A2 )
% 2.63/1.49 & ! [Y: real] :
% 2.63/1.49 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
% 2.63/1.49 & ( ( power_power @ real @ Y @ N )
% 2.63/1.49 = A2 ) )
% 2.63/1.49 => ( Y = X2 ) ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.63/1.49 ! [A5: $tType,A6: $tType] :
% 2.63/1.49 ( ( preorder @ A6 @ ( type2 @ A6 ) )
% 2.63/1.49 => ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.63/1.49 ! [A5: $tType,A6: $tType] :
% 2.63/1.49 ( ( order @ A6 @ ( type2 @ A6 ) )
% 2.63/1.49 => ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_fun___Orderings_Oord,axiom,
% 2.63/1.49 ! [A5: $tType,A6: $tType] :
% 2.63/1.49 ( ( ord @ A6 @ ( type2 @ A6 ) )
% 2.63/1.49 => ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.63/1.49 condit1037483654norder @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.63/1.49 semiri134348788visors @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.63/1.49 linordered_semidom @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.63/1.49 linordered_idom @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.63/1.49 semiring_char_0 @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 2.63/1.49 zero_neq_one @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
% 2.63/1.49 preorder @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.63/1.49 linorder @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.63/1.49 semiring_1 @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 2.63/1.49 no_top @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 2.63/1.49 no_bot @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
% 2.63/1.49 order @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
% 2.63/1.49 ord @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.63/1.49 zero @ int @ ( type2 @ int ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,axiom,
% 2.63/1.49 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.63/1.49 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
% 2.63/1.49 semiri134348788visors @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 2.63/1.49 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.63/1.49 wellorder @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
% 2.63/1.49 semiring_char_0 @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_8,axiom,
% 2.63/1.49 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
% 2.63/1.49 preorder @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 2.63/1.49 linorder @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 2.63/1.49 semiring_1 @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Ono__top_12,axiom,
% 2.63/1.49 no_top @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Oorder_13,axiom,
% 2.63/1.49 order @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Orderings_Oord_14,axiom,
% 2.63/1.49 ord @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Nat_Onat___Groups_Ozero_15,axiom,
% 2.63/1.49 zero @ nat @ ( type2 @ nat ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
% 2.63/1.49 preorder @ $o @ ( type2 @ $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder_17,axiom,
% 2.63/1.49 linorder @ $o @ ( type2 @ $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
% 2.63/1.49 order @ $o @ ( type2 @ $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 2.63/1.49 ord @ $o @ ( type2 @ $o ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_20,axiom,
% 2.63/1.49 condit1037483654norder @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
% 2.63/1.49 condit1656338222tinuum @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.63/1.49 archim1804426504_field @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_21,axiom,
% 2.63/1.49 semiri134348788visors @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_22,axiom,
% 2.63/1.49 linordered_semidom @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
% 2.63/1.49 dense_linorder @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.63/1.49 linordered_field @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Rings_Olinordered__idom_23,axiom,
% 2.63/1.49 linordered_idom @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
% 2.63/1.49 dense_order @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.63/1.49 division_ring @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_24,axiom,
% 2.63/1.49 semiring_char_0 @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_25,axiom,
% 2.63/1.49 zero_neq_one @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Opreorder_26,axiom,
% 2.63/1.49 preorder @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Olinorder_27,axiom,
% 2.63/1.49 linorder @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Rings_Osemiring__1_28,axiom,
% 2.63/1.49 semiring_1 @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Ono__top_29,axiom,
% 2.63/1.49 no_top @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Ono__bot_30,axiom,
% 2.63/1.49 no_bot @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Oorder_31,axiom,
% 2.63/1.49 order @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Orderings_Oord_32,axiom,
% 2.63/1.49 ord @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.63/1.49 field @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Real_Oreal___Groups_Ozero_33,axiom,
% 2.63/1.49 zero @ real @ ( type2 @ real ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_34,axiom,
% 2.63/1.49 semiri134348788visors @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_35,axiom,
% 2.63/1.49 zero_neq_one @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_36,axiom,
% 2.63/1.49 preorder @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_37,axiom,
% 2.63/1.49 linorder @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_38,axiom,
% 2.63/1.49 semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_39,axiom,
% 2.63/1.49 order @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_40,axiom,
% 2.63/1.49 ord @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_41,axiom,
% 2.63/1.49 zero @ code_natural @ ( type2 @ code_natural ) ).
% 2.63/1.49
% 2.63/1.49 thf(help_If_3_1_T,axiom,
% 2.63/1.49 ! [P: $o] :
% 2.63/1.49 ( ( P = $true )
% 2.63/1.49 | ( P = $false ) ) ).
% 2.63/1.49
% 2.63/1.49 thf(help_If_2_1_T,axiom,
% 2.63/1.49 ! [A: $tType,X: A,Y2: A] :
% 2.63/1.49 ( ( if @ A @ $false @ X @ Y2 )
% 2.63/1.49 = Y2 ) ).
% 2.63/1.49
% 2.63/1.49 thf(help_If_1_1_T,axiom,
% 2.63/1.49 ! [A: $tType,X: A,Y2: A] :
% 2.63/1.49 ( ( if @ A @ $true @ X @ Y2 )
% 2.63/1.49 = X ) ).
% 2.63/1.49
% 2.63/1.49 thf(conj_0,hypothesis,
% 2.63/1.49 hammin1195055315smooth @ n ).
% 2.63/1.49
% 2.63/1.49 thf(conj_1,conjecture,
% 2.63/1.49 ord_less @ nat @ ( zero_zero @ nat ) @ n ).
% 2.63/1.49 ------------------------
% 2.77/1.52 % (32760)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32658 for (2999ds/183Mi)
% 2.77/1.52 % (32762)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_32658 for (2999ds/27Mi)
% 2.77/1.52 % (32765)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_32658 for (2999ds/275Mi)
% 2.77/1.52 % (32764)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_32658 for (2999ds/2Mi)
% 2.77/1.52 % (32763)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32658 for (2999ds/2Mi)
% 2.77/1.52 % (32766)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32658 for (2999ds/18Mi)
% 2.77/1.53 % (32763)Instruction limit reached!
% 2.77/1.53 % (32763)------------------------------
% 2.77/1.53 % (32763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (32763)Termination reason: Unknown
% 2.77/1.53 % (32763)Termination phase: shuffling
% 2.77/1.53 % (32764)Instruction limit reached!
% 2.77/1.53 % (32764)------------------------------
% 2.77/1.53 % (32764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (32764)Termination reason: Unknown
% 2.77/1.53 % (32764)Termination phase: shuffling
% 2.77/1.53
% 2.77/1.53 % (32764)Memory used [KB]: 1407
% 2.77/1.53 % (32764)Time elapsed: 0.003 s
% 2.77/1.53 % (32764)Instructions burned: 3 (million)
% 2.77/1.53 % (32764)------------------------------
% 2.77/1.53 % (32764)------------------------------
% 2.77/1.53
% 2.77/1.53 % (32763)Memory used [KB]: 1279
% 2.77/1.53 % (32763)Time elapsed: 0.003 s
% 2.77/1.53 % (32763)Instructions burned: 3 (million)
% 2.77/1.53 % (32763)------------------------------
% 2.77/1.53 % (32763)------------------------------
% 2.77/1.53 % (32761)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_32658 for (2999ds/4Mi)
% 2.77/1.53 % (32761)Instruction limit reached!
% 2.77/1.53 % (32761)------------------------------
% 2.77/1.53 % (32761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (32761)Termination reason: Unknown
% 2.77/1.53 % (32761)Termination phase: shuffling
% 2.77/1.53
% 2.77/1.53 % (32761)Memory used [KB]: 1407
% 2.77/1.53 % (32761)Time elapsed: 0.003 s
% 2.77/1.53 % (32761)Instructions burned: 4 (million)
% 2.77/1.53 % (32761)------------------------------
% 2.77/1.53 % (32761)------------------------------
% 2.77/1.53 % (32766)Instruction limit reached!
% 2.77/1.53 % (32766)------------------------------
% 2.77/1.53 % (32766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.53 % (32766)Termination reason: Unknown
% 2.77/1.53 % (32766)Termination phase: shuffling
% 2.77/1.53
% 2.77/1.53 % (32766)Memory used [KB]: 1791
% 2.77/1.53 % (32766)Time elapsed: 0.011 s
% 2.77/1.53 % (32766)Instructions burned: 18 (million)
% 2.77/1.53 % (32766)------------------------------
% 2.77/1.53 % (32766)------------------------------
% 2.77/1.54 % (32762)Instruction limit reached!
% 2.77/1.54 % (32762)------------------------------
% 2.77/1.54 % (32762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.54 % (32762)Termination reason: Unknown
% 2.77/1.54 % (32762)Termination phase: Property scanning
% 2.77/1.54
% 2.77/1.54 % (32762)Memory used [KB]: 1791
% 2.77/1.54 % (32762)Time elapsed: 0.015 s
% 2.77/1.54 % (32762)Instructions burned: 27 (million)
% 2.77/1.54 % (32762)------------------------------
% 2.77/1.54 % (32762)------------------------------
% 2.77/1.54 % (32767)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32658 for (2999ds/3Mi)
% 2.77/1.54 % (300)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_32658 for (2999ds/37Mi)
% 2.77/1.54 % (301)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_32658 for (2999ds/15Mi)
% 2.77/1.55 % (32767)Instruction limit reached!
% 2.77/1.55 % (32767)------------------------------
% 2.77/1.55 % (32767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (32767)Termination reason: Unknown
% 2.77/1.55 % (32767)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (32767)Memory used [KB]: 1279
% 2.77/1.55 % (32767)Time elapsed: 0.003 s
% 2.77/1.55 % (32767)Instructions burned: 3 (million)
% 2.77/1.55 % (32767)------------------------------
% 2.77/1.55 % (32767)------------------------------
% 2.77/1.55 % (32765)First to succeed.
% 2.77/1.55 % (32765)Refutation found. Thanks to Tanya!
% 2.77/1.55 % SZS status Theorem for DTF2THF_32658
% 2.77/1.55 % SZS output start Proof for DTF2THF_32658
% 2.77/1.55 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.77/1.55 thf(type_def_7, type, code_natural: $tType).
% 2.77/1.55 thf(type_def_8, type, nat: $tType).
% 2.77/1.55 thf(type_def_9, type, int: $tType).
% 2.77/1.55 thf(type_def_10, type, real: $tType).
% 2.77/1.55 thf(func_def_0, type, code_natural: $tType).
% 2.77/1.55 thf(func_def_1, type, real: $tType).
% 2.77/1.55 thf(func_def_2, type, nat: $tType).
% 2.77/1.55 thf(func_def_3, type, int: $tType).
% 2.77/1.55 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.77/1.55 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_6, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_7, type, field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_8, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_9, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_10, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_11, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_12, type, semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_13, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_14, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_15, type, zero_neq_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_16, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_17, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_18, type, division_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_19, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_20, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_21, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_22, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_23, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_24, type, semiri134348788visors: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_25, type, archim1804426504_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_27, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_28, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.77/1.55 thf(func_def_29, type, code_case_natural: !>[X0: $tType]:(X0 > (code_natural > X0) > code_natural > X0)).
% 2.77/1.55 thf(func_def_30, type, code_size_natural: code_natural > nat).
% 2.77/1.55 thf(func_def_31, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 2.77/1.55 thf(func_def_32, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.77/1.55 thf(func_def_33, type, hammin1195055315smooth: nat > $o).
% 2.77/1.55 thf(func_def_34, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.77/1.55 thf(func_def_35, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.77/1.55 thf(func_def_36, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.77/1.55 thf(func_def_37, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.55 thf(func_def_38, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.77/1.55 thf(func_def_39, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 2.77/1.55 thf(func_def_40, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.77/1.55 thf(func_def_41, type, zero_neq_one_of_bool: !>[X0: $tType]:($o > X0)).
% 2.77/1.55 thf(func_def_42, type, n: nat).
% 2.77/1.55 thf(func_def_55, type, sK0: nat > (nat > $o) > nat).
% 2.77/1.55 thf(func_def_56, type, sK1: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.77/1.55 thf(func_def_57, type, sK2: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.77/1.55 thf(func_def_58, type, sK3: (nat > $o) > nat).
% 2.77/1.55 thf(func_def_59, type, sK4: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.77/1.55 thf(func_def_60, type, sK5: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.77/1.55 thf(func_def_61, type, sK6: (nat > nat) > nat).
% 2.77/1.55 thf(func_def_62, type, sK7: (nat > nat) > nat).
% 2.77/1.55 thf(func_def_63, type, sK8: (nat > $o) > nat).
% 2.77/1.55 thf(func_def_64, type, sK9: (nat > $o) > nat).
% 2.77/1.55 thf(func_def_65, type, sK10: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.77/1.55 thf(func_def_66, type, sK11: !>[X0: $tType]:(nat > (X0 > nat) > (X0 > $o) > X0)).
% 2.77/1.55 thf(func_def_67, type, sK12: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.77/1.55 thf(func_def_69, type, ph14: !>[X0: $tType]:(X0)).
% 2.77/1.55 thf(f1131,plain,(
% 2.77/1.55 $false),
% 2.77/1.55 inference(subsumption_resolution,[status(thm)],[f1113,f1065])).
% 2.77/1.55 thf(f1065,plain,(
% 2.77/1.55 ((hammin1195055315smooth @ n) = $true)),
% 2.77/1.55 inference(cnf_transformation,[status(thm)],[f383])).
% 2.77/1.55 thf(f383,plain,(
% 2.77/1.55 ((hammin1195055315smooth @ n) = $true)),
% 2.77/1.55 inference(fool_elimination,[status(thm)],[f382])).
% 2.77/1.55 thf(f382,plain,(
% 2.77/1.55 (hammin1195055315smooth @ n)),
% 2.77/1.55 inference(rectify,[status(thm)],[f322])).
% 2.77/1.55 thf(f322,axiom,(
% 2.77/1.55 (hammin1195055315smooth @ n)),
% 2.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2THF_32658.p',conj_0)).
% 2.77/1.55 thf(f1113,plain,(
% 2.77/1.55 ((hammin1195055315smooth @ n) != $true)),
% 2.77/1.55 inference(superposition,[status(thm)],[f1030,f1112])).
% 2.77/1.55 thf(f1112,plain,(
% 2.77/1.55 (zero_zero @ nat = n)),
% 2.77/1.55 inference(trivial_inequality_removal,[status(thm)],[f1102])).
% 2.77/1.55 thf(f1102,plain,(
% 2.77/1.55 (zero_zero @ nat = n) | ($true != $true)),
% 2.77/1.55 inference(superposition,[status(thm)],[f1039,f1040])).
% 2.77/1.55 thf(f1040,plain,(
% 2.77/1.55 ( ! [X0 : nat] : (((ord_less @ nat @ zero_zero @ nat @ X0) = $true) | (zero_zero @ nat = X0)) )),
% 2.77/1.55 inference(cnf_transformation,[status(thm)],[f995])).
% 2.77/1.55 thf(f995,plain,(
% 2.77/1.55 ! [X0 : nat] : ((zero_zero @ nat = X0) | ((ord_less @ nat @ zero_zero @ nat @ X0) = $true))),
% 2.77/1.55 inference(ennf_transformation,[status(thm)],[f877])).
% 2.77/1.55 thf(f877,plain,(
% 2.77/1.55 ! [X0 : nat] : ((zero_zero @ nat != X0) => ((ord_less @ nat @ zero_zero @ nat @ X0) = $true))),
% 2.77/1.55 inference(fool_elimination,[status(thm)],[f876])).
% 2.77/1.55 thf(f876,plain,(
% 2.77/1.55 ! [X0 : nat] : ((zero_zero @ nat != X0) => (ord_less @ nat @ zero_zero @ nat @ X0))),
% 2.77/1.55 inference(rectify,[status(thm)],[f4])).
% 2.77/1.55 thf(f4,axiom,(
% 2.77/1.55 ! [X2 : nat] : ((zero_zero @ nat != X2) => (ord_less @ nat @ zero_zero @ nat @ X2))),
% 2.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2THF_32658.p',fact_3_gr0I)).
% 2.77/1.55 thf(f1039,plain,(
% 2.77/1.55 ((ord_less @ nat @ zero_zero @ nat @ n) != $true)),
% 2.77/1.55 inference(cnf_transformation,[status(thm)],[f952])).
% 2.77/1.55 thf(f952,plain,(
% 2.77/1.55 ((ord_less @ nat @ zero_zero @ nat @ n) != $true)),
% 2.77/1.55 inference(flattening,[status(thm)],[f405])).
% 2.77/1.55 thf(f405,plain,(
% 2.77/1.55 ~((ord_less @ nat @ zero_zero @ nat @ n) = $true)),
% 2.77/1.55 inference(fool_elimination,[status(thm)],[f404])).
% 2.77/1.55 thf(f404,plain,(
% 2.77/1.55 ~(ord_less @ nat @ zero_zero @ nat @ n)),
% 2.77/1.55 inference(rectify,[status(thm)],[f324])).
% 2.77/1.55 thf(f324,negated_conjecture,(
% 2.77/1.55 ~(ord_less @ nat @ zero_zero @ nat @ n)),
% 2.77/1.55 inference(negated_conjecture,[status(cth)],[f323])).
% 2.77/1.55 thf(f323,conjecture,(
% 2.77/1.55 (ord_less @ nat @ zero_zero @ nat @ n)),
% 2.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2THF_32658.p',conj_1)).
% 2.77/1.55 thf(f1030,plain,(
% 2.77/1.55 ((hammin1195055315smooth @ zero_zero @ nat) != $true)),
% 2.77/1.55 inference(cnf_transformation,[status(thm)],[f958])).
% 2.77/1.55 thf(f958,plain,(
% 2.77/1.55 ((hammin1195055315smooth @ zero_zero @ nat) != $true)),
% 2.77/1.55 inference(flattening,[status(thm)],[f663])).
% 2.77/1.55 thf(f663,plain,(
% 2.77/1.55 ~((hammin1195055315smooth @ zero_zero @ nat) = $true)),
% 2.77/1.55 inference(fool_elimination,[status(thm)],[f662])).
% 2.77/1.55 thf(f662,plain,(
% 2.77/1.55 ~(hammin1195055315smooth @ zero_zero @ nat)),
% 2.77/1.55 inference(rectify,[status(thm)],[f1])).
% 2.77/1.55 thf(f1,axiom,(
% 2.77/1.55 ~(hammin1195055315smooth @ zero_zero @ nat)),
% 2.77/1.55 file('/export/starexec/sandbox2/tmp/tmp.pMGclSoXjV/DTF2THF_32658.p',fact_0_smooth__0)).
% 2.77/1.55 % SZS output end Proof for DTF2THF_32658
% 2.77/1.55 % (32765)------------------------------
% 2.77/1.55 % (32765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (32765)Termination reason: Refutation
% 2.77/1.55
% 2.77/1.55 % (32765)Memory used [KB]: 6396
% 2.77/1.55 % (32765)Time elapsed: 0.025 s
% 2.77/1.55 % (32765)Instructions burned: 46 (million)
% 2.77/1.55 % (32765)------------------------------
% 2.77/1.55 % (32765)------------------------------
% 2.77/1.55 % (32759)Success in time 0.041 s
%------------------------------------------------------------------------------