%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP105^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:35 AM UTC 2026
% Result : Theorem 2.70s 1.59s
% Output : Refutation 2.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP105^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n005.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:25:59 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP105^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer ListSlice problem prob_61__5615344_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : ListSlice/prob_61__5615344_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 356 ( 115 unt; 52 typ; 0 def)
% 0.28/0.46 % Number of atoms : 866 ( 240 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 8 ( 2 avg)
% 0.28/0.46 % Number of connectives : 3266 ( 73 ~; 26 |; 47 &;2735 @)
% 0.28/0.46 % ( 0 <=>; 385 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 17 ( 6 avg)
% 0.28/0.46 % Number of types : 5 ( 4 usr)
% 0.28/0.46 % Number of type conns : 76 ( 76 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 51 ( 48 usr; 3 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 743 ( 25 ^; 647 !; 24 ?; 743 :)
% 0.28/0.46 % ( 47 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:15.739
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (7)
% 0.28/0.46 thf(ty_t_Option_Ooption,type,
% 0.28/0.46 option: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Real_Oreal,type,
% 0.28/0.46 real: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_List_Olist,type,
% 0.28/0.46 list: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Int_Oint,type,
% 0.28/0.46 int: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_a,type,
% 0.28/0.46 a: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (45)
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Nat_Osize,type,
% 0.28/0.46 size:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.28/0.46 zero:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.28/0.46 field:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.46 order:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring__1,type,
% 0.28/0.46 semiring_1:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.46 linorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.28/0.46 semiring_char_0:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.28/0.46 wellorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.28/0.46 division_ring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Parity_Osemiring__bits,type,
% 0.28/0.46 semiring_bits:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.28/0.46 semidom_divide:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.28/0.46 linordered_idom:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.28/0.46 linordered_field:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.28/0.46 linordered_semidom:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.28/0.46 linord1659791738miring:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 0.28/0.46 semiri134348788visors:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.28/0.46 archim1804426504_field:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.28/0.46 canoni770627133id_add:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.28/0.46 unique1598680935umeral:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
% 0.28/0.46 euclid1852923125th_nat:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.28/0.46 inverse_inverse:
% 0.28/0.46 !>[A: $tType] : ( A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/0.46 zero_zero:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_If,type,
% 0.28/0.46 if:
% 0.28/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice,type,
% 0.28/0.46 listSl1903280966_slice:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux,type,
% 0.28/0.46 listSl1123830174ce_aux:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > nat > nat > ( list @ ( list @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Obind,type,
% 0.28/0.46 bind:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Oconcat,type,
% 0.28/0.46 concat:
% 0.28/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Ocount__list,type,
% 0.28/0.46 count_list:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Ogen__length,type,
% 0.28/0.46 gen_length:
% 0.28/0.46 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Olist_ONil,type,
% 0.28/0.46 nil:
% 0.28/0.46 !>[A: $tType] : ( list @ A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Olist__ex1,type,
% 0.28/0.46 list_ex1:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Omap__filter,type,
% 0.28/0.46 map_filter:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Omaps,type,
% 0.28/0.46 maps:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Omember,type,
% 0.28/0.46 member:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Onull,type,
% 0.28/0.46 null:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.28/0.46 semiring_1_of_nat:
% 0.28/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.28/0.46 size_size:
% 0.28/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.46 ord_less:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.46 ord_less_eq:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Power_Opower__class_Opower,type,
% 0.28/0.46 power_power:
% 0.28/0.46 !>[A: $tType] : ( A > nat > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.28/0.46 divide_divide:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect,type,
% 0.28/0.46 collect:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member,type,
% 0.28/0.46 member2:
% 0.28/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_k,type,
% 0.28/0.46 k: nat ).
% 0.28/0.46
% 0.28/0.46 % Relevant facts (253)
% 0.28/0.46 thf(fact_0_list__slice__0,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A] :
% 0.28/0.46 ( ( listSl1903280966_slice @ A @ Xs @ ( zero_zero @ nat ) )
% 0.28/0.46 = ( nil @ ( list @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % list_slice_0
% 0.28/0.46 thf(fact_1_concat_Osimps_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 0.28/0.46 = ( nil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % concat.simps(1)
% 0.28/0.46 thf(fact_2_list__ex1__simps_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o] :
% 0.28/0.46 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % list_ex1_simps(1)
% 0.28/0.46 thf(fact_3_list__slice__less,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A,K: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K )
% 0.28/0.46 => ( ( listSl1903280966_slice @ A @ Xs @ K )
% 0.28/0.46 = ( nil @ ( list @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % list_slice_less
% 0.28/0.46 thf(fact_4_bind__simps_I1_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.28/0.46 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 0.28/0.46 = ( nil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % bind_simps(1)
% 0.28/0.46 thf(fact_5_member__rec_I2_J,axiom,
% 0.28/0.46 ! [A: $tType,Y: A] :
% 0.28/0.46 ~ ( member @ A @ ( nil @ A ) @ Y ) ).
% 0.28/0.46
% 0.28/0.46 % member_rec(2)
% 0.28/0.46 thf(fact_6_gen__length__code_I1_J,axiom,
% 0.28/0.46 ! [A: $tType,N: nat] :
% 0.28/0.46 ( ( gen_length @ A @ N @ ( nil @ A ) )
% 0.28/0.46 = N ) ).
% 0.28/0.46
% 0.28/0.46 % gen_length_code(1)
% 0.28/0.46 thf(fact_7_maps__simps_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.28/0.46 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 0.28/0.46 = ( nil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % maps_simps(2)
% 0.28/0.46 thf(fact_8_null__rec_I2_J,axiom,
% 0.28/0.46 ! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 % null_rec(2)
% 0.28/0.46 thf(fact_9_eq__Nil__null,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A] :
% 0.28/0.46 ( ( Xs
% 0.28/0.46 = ( nil @ A ) )
% 0.28/0.46 = ( null @ A @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_Nil_null
% 0.28/0.46 thf(fact_10_map__filter__simps_I2_J,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F: B > ( option @ A )] :
% 0.28/0.46 ( ( map_filter @ B @ A @ F @ ( nil @ B ) )
% 0.28/0.46 = ( nil @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % map_filter_simps(2)
% 0.28/0.46 thf(fact_11_length__0__conv,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A] :
% 0.28/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.28/0.46 = ( zero_zero @ nat ) )
% 0.28/0.46 = ( Xs
% 0.28/0.46 = ( nil @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % length_0_conv
% 0.28/0.46 thf(fact_12_length__greater__0__conv,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A] :
% 0.28/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.46 = ( Xs
% 0.28/0.46 != ( nil @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % length_greater_0_conv
% 0.28/0.46 thf(fact_13_list_Osize_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 0.28/0.46 = ( zero_zero @ nat ) ) ).
% 0.28/0.46
% 0.28/0.46 % list.size(3)
% 0.28/0.46 thf(fact_14_length__code,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( size_size @ ( list @ A ) )
% 0.28/0.46 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % length_code
% 0.28/0.46 thf(fact_15_length__induct,axiom,
% 0.28/0.46 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.28/0.46 ( ! [Xs2: list @ A] :
% 0.28/0.46 ( ! [Ys: list @ A] :
% 0.28/0.46 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.28/0.46 => ( P @ Ys ) )
% 0.28/0.46 => ( P @ Xs2 ) )
% 0.28/0.46 => ( P @ Xs ) ) ).
% 0.28/0.46
% 0.28/0.46 % length_induct
% 0.28/0.46 thf(fact_16_Ex__list__of__length,axiom,
% 0.28/0.46 ! [A: $tType,N: nat] :
% 0.28/0.46 ? [Xs2: list @ A] :
% 0.28/0.46 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.28/0.46 = N ) ).
% 0.28/0.46
% 0.28/0.46 % Ex_list_of_length
% 0.28/0.46 thf(fact_17_neq__if__length__neq,axiom,
% 0.28/0.46 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.28/0.46 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.28/0.46 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.28/0.46 => ( Xs != Ys2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % neq_if_length_neq
% 0.28/0.46 thf(fact_18_neq0__conv,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ( ( N
% 0.28/0.46 != ( zero_zero @ nat ) )
% 0.28/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/0.46
% 0.28/0.46 % neq0_conv
% 0.28/0.46 thf(fact_19_less__nat__zero__code,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_nat_zero_code
% 0.28/0.46 thf(fact_20_bot__nat__0_Onot__eq__extremum,axiom,
% 0.28/0.46 ! [A2: nat] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 != ( zero_zero @ nat ) )
% 0.28/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_nat_0.not_eq_extremum
% 0.28/0.46 thf(fact_21_not__gr__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N: A] :
% 0.28/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.28/0.46 = ( N
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_gr_zero
% 0.28/0.46 thf(fact_22_length__greater__imp__not__empty,axiom,
% 0.28/0.46 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.28/0.46 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.28/0.46 => ( Xs
% 0.28/0.46 != ( nil @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % length_greater_imp_not_empty
% 0.28/0.46 thf(fact_23_gr0I,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ( ( N
% 0.28/0.46 != ( zero_zero @ nat ) )
% 0.28/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/0.46
% 0.28/0.46 % gr0I
% 0.28/0.46 thf(fact_24_not__gr0,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.28/0.46 = ( N
% 0.28/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_gr0
% 0.28/0.46 thf(fact_25_not__less0,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_less0
% 0.28/0.46 thf(fact_26_less__zeroE,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_zeroE
% 0.28/0.46 thf(fact_27_gr__implies__not0,axiom,
% 0.28/0.46 ! [M: nat,N: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ M @ N )
% 0.28/0.46 => ( N
% 0.28/0.46 != ( zero_zero @ nat ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % gr_implies_not0
% 0.28/0.46 thf(fact_28_zero__reorient,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( zero @ A )
% 0.28/0.46 => ! [X: A] :
% 0.28/0.46 ( ( ( zero_zero @ A )
% 0.28/0.46 = X )
% 0.28/0.46 = ( X
% 0.28/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % zero_reorient
% 0.28/0.46 thf(fact_29_measure__induct__rule,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( wellorder @ B )
% 0.28/0.46 => ! [F: A > B,P: A > $o,A2: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ! [Y2: A] :
% 0.28/0.46 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.28/0.46 => ( P @ Y2 ) )
% 0.28/0.46 => ( P @ X2 ) )
% 0.28/0.46 => ( P @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % measure_induct_rule
% 0.28/0.46 thf(fact_30_measure__induct,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( wellorder @ B )
% 0.28/0.46 => ! [F: A > B,P: A > $o,A2: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ! [Y2: A] :
% 0.28/0.46 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.28/0.46 => ( P @ Y2 ) )
% 0.28/0.46 => ( P @ X2 ) )
% 0.28/0.46 => ( P @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % measure_induct
% 0.28/0.46 thf(fact_31_infinite__descent__measure,axiom,
% 0.28/0.46 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.28/0.46 ( ! [X2: A] :
% 0.28/0.46 ( ~ ( P @ X2 )
% 0.28/0.46 => ? [Y2: A] :
% 0.28/0.46 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X2 ) )
% 0.28/0.46 & ~ ( P @ Y2 ) ) )
% 0.28/0.46 => ( P @ X ) ) ).
% 0.28/0.46
% 0.28/0.46 % infinite_descent_measure
% 0.28/0.46 thf(fact_32_linorder__neqE__nat,axiom,
% 0.28/0.46 ! [X: nat,Y: nat] :
% 0.28/0.46 ( ( X != Y )
% 0.28/0.46 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.28/0.46 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % linorder_neqE_nat
% 0.28/0.46 thf(fact_33_infinite__descent,axiom,
% 0.28/0.46 ! [P: nat > $o,N: nat] :
% 0.28/0.46 ( ! [N2: nat] :
% 0.28/0.46 ( ~ ( P @ N2 )
% 0.28/0.46 => ? [M2: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.46 & ~ ( P @ M2 ) ) )
% 0.28/0.46 => ( P @ N ) ) ).
% 0.28/0.46
% 0.28/0.46 % infinite_descent
% 0.28/0.46 thf(fact_34_nat__less__induct,axiom,
% 0.28/0.46 ! [P: nat > $o,N: nat] :
% 0.28/0.46 ( ! [N2: nat] :
% 0.28/0.46 ( ! [M2: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.46 => ( P @ M2 ) )
% 0.28/0.46 => ( P @ N2 ) )
% 0.28/0.46 => ( P @ N ) ) ).
% 0.28/0.46
% 0.28/0.46 % nat_less_induct
% 0.28/0.46 thf(fact_35_less__irrefl__nat,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.46
% 0.28/0.46 % less_irrefl_nat
% 0.28/0.46 thf(fact_36_less__not__refl3,axiom,
% 0.28/0.46 ! [S: nat,T: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ S @ T )
% 0.28/0.46 => ( S != T ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_not_refl3
% 0.28/0.46 thf(fact_37_less__not__refl2,axiom,
% 0.28/0.46 ! [N: nat,M: nat] :
% 0.28/0.46 ( ( ord_less @ nat @ N @ M )
% 0.28/0.46 => ( M != N ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_not_refl2
% 0.28/0.46 thf(fact_38_less__not__refl,axiom,
% 0.28/0.46 ! [N: nat] :
% 0.28/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.46
% 0.28/0.46 % less_not_refl
% 0.28/0.46 thf(fact_39_nat__neq__iff,axiom,
% 0.28/0.46 ! [M: nat,N: nat] :
% 0.28/0.46 ( ( M != N )
% 0.28/0.46 = ( ( ord_less @ nat @ M @ N )
% 0.28/0.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % nat_neq_iff
% 0.28/0.46 thf(fact_40_size__neq__size__imp__neq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( size @ A )
% 0.28/0.46 => ! [X: A,Y: A] :
% 0.28/0.46 ( ( ( size_size @ A @ X )
% 0.28/0.46 != ( size_size @ A @ Y ) )
% 0.28/0.46 => ( X != Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % size_neq_size_imp_neq
% 0.28/0.46 thf(fact_41_zero__less__iff__neq__zero,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( canoni770627133id_add @ A )
% 0.28/0.46 => ! [N: A] :
% 0.28/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.28/0.47 = ( N
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_iff_neq_zero
% 0.28/0.47 thf(fact_42_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_43_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X3: A] : ( member2 @ A @ X3 @ A3 ) )
% 0.28/0.47 = A3 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_44_Collect__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( P @ X2 )
% 0.28/0.47 = ( Q @ X2 ) )
% 0.28/0.47 => ( ( collect @ A @ P )
% 0.28/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_cong
% 0.28/0.47 thf(fact_45_ext,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( F @ X2 )
% 0.28/0.47 = ( G @ X2 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_46_gr__implies__not__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [M: A,N: A] :
% 0.28/0.47 ( ( ord_less @ A @ M @ N )
% 0.28/0.47 => ( N
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_implies_not_zero
% 0.28/0.47 thf(fact_47_not__less__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_less_zero
% 0.28/0.47 thf(fact_48_gr__zeroI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ( ( N
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_zeroI
% 0.28/0.47 thf(fact_49_infinite__descent0__measure,axiom,
% 0.28/0.47 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 0.28/0.47 ( ! [X2: A] :
% 0.28/0.47 ( ( ( V @ X2 )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( P @ X2 ) )
% 0.28/0.47 => ( ! [X2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 0.28/0.47 => ( ~ ( P @ X2 )
% 0.28/0.47 => ? [Y2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X2 ) )
% 0.28/0.47 & ~ ( P @ Y2 ) ) ) )
% 0.28/0.47 => ( P @ X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent0_measure
% 0.28/0.47 thf(fact_50_bot__nat__0_Oextremum__strict,axiom,
% 0.28/0.47 ! [A2: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum_strict
% 0.28/0.47 thf(fact_51_infinite__descent0,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.28/0.47 => ( ! [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 => ( ~ ( P @ N2 )
% 0.28/0.47 => ? [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 & ~ ( P @ M2 ) ) ) )
% 0.28/0.47 => ( P @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent0
% 0.28/0.47 thf(fact_52_gr__implies__gr0,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_implies_gr0
% 0.28/0.47 thf(fact_53_field__lbound__gt__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [D1: A,D2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 0.28/0.47 => ? [E: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.28/0.47 & ( ord_less @ A @ E @ D1 )
% 0.28/0.47 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % field_lbound_gt_zero
% 0.28/0.47 thf(fact_54_less__numeral__extra_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_numeral_extra(3)
% 0.28/0.47 thf(fact_55_list__slice__aux_Osimps_I1_J,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,K: nat] :
% 0.28/0.47 ( ( listSl1123830174ce_aux @ A @ Xs @ K @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( nil @ ( list @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_aux.simps(1)
% 0.28/0.47 thf(fact_56_of__nat__0__less__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0_less_iff
% 0.28/0.47 thf(fact_57_count__list_Osimps_I1_J,axiom,
% 0.28/0.47 ! [A: $tType,Y: A] :
% 0.28/0.47 ( ( count_list @ A @ ( nil @ A ) @ Y )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % count_list.simps(1)
% 0.28/0.47 thf(fact_58_interval__induct__rule,axiom,
% 0.28/0.47 ! [I: set @ nat,P: nat > $o,A2: nat] :
% 0.28/0.47 ( ! [X2: nat] :
% 0.28/0.47 ( ! [Y2: nat] :
% 0.28/0.47 ( ( ( member2 @ nat @ X2 @ I )
% 0.28/0.47 & ( member2 @ nat @ Y2 @ I )
% 0.28/0.47 & ( ord_less @ nat @ Y2 @ X2 ) )
% 0.28/0.47 => ( P @ Y2 ) )
% 0.28/0.47 => ( P @ X2 ) )
% 0.28/0.47 => ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % interval_induct_rule
% 0.28/0.47 thf(fact_59_interval__induct,axiom,
% 0.28/0.47 ! [I: set @ nat,P: nat > $o,A2: nat] :
% 0.28/0.47 ( ! [X2: nat] :
% 0.28/0.47 ( ! [Y2: nat] :
% 0.28/0.47 ( ( ( member2 @ nat @ X2 @ I )
% 0.28/0.47 & ( member2 @ nat @ Y2 @ I )
% 0.28/0.47 & ( ord_less @ nat @ Y2 @ X2 ) )
% 0.28/0.47 => ( P @ Y2 ) )
% 0.28/0.47 => ( P @ X2 ) )
% 0.28/0.47 => ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % interval_induct
% 0.28/0.47 thf(fact_60_zero__natural_Orsp,axiom,
% 0.28/0.47 ( ( zero_zero @ nat )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_natural.rsp
% 0.28/0.47 thf(fact_61_of__nat__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_iff
% 0.28/0.47 thf(fact_62_of__nat__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A )
% 0.28/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0
% 0.28/0.47 thf(fact_63_of__nat__0__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( ( zero_zero @ A )
% 0.28/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ( zero_zero @ nat )
% 0.28/0.47 = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0_eq_iff
% 0.28/0.47 thf(fact_64_of__nat__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A )
% 0.28/0.47 => ! [M: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( M
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_0_iff
% 0.28/0.47 thf(fact_65_of__nat__less__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_iff
% 0.28/0.47 thf(fact_66_list__slice__aux__length,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,K: nat,N: nat] :
% 0.28/0.47 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1123830174ce_aux @ A @ Xs @ K @ N ) )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_aux_length
% 0.28/0.47 thf(fact_67_of__nat__less__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat] :
% 0.28/0.47 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_0_iff
% 0.28/0.47 thf(fact_68_less__imp__of__nat__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_of_nat_less
% 0.28/0.47 thf(fact_69_of__nat__less__imp__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_imp_less
% 0.28/0.47 thf(fact_70_Collect__minI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( wellorder @ A )
% 0.28/0.47 => ! [K: A,I: set @ A,P: A > $o] :
% 0.28/0.47 ( ( member2 @ A @ K @ I )
% 0.28/0.47 => ( ( P @ K )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member2 @ A @ X2 @ I )
% 0.28/0.47 & ( P @ X2 )
% 0.28/0.47 & ! [Xa: A] :
% 0.28/0.47 ( ( member2 @ A @ Xa @ I )
% 0.28/0.47 => ( ( ord_less @ A @ Xa @ X2 )
% 0.28/0.47 => ~ ( P @ Xa ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_minI
% 0.28/0.47 thf(fact_71_Collect__minI__ex,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( wellorder @ A )
% 0.28/0.47 => ! [I: set @ A,P: A > $o] :
% 0.28/0.47 ( ? [X4: A] :
% 0.28/0.47 ( ( member2 @ A @ X4 @ I )
% 0.28/0.47 & ( P @ X4 ) )
% 0.28/0.47 => ? [X2: A] :
% 0.28/0.47 ( ( member2 @ A @ X2 @ I )
% 0.28/0.47 & ( P @ X2 )
% 0.28/0.47 & ! [Xa: A] :
% 0.28/0.47 ( ( member2 @ A @ Xa @ I )
% 0.28/0.47 => ( ( ord_less @ A @ Xa @ X2 )
% 0.28/0.47 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_minI_ex
% 0.28/0.47 thf(fact_72_zero__less__imp__eq__int,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 & ( K
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_imp_eq_int
% 0.28/0.47 thf(fact_73_pos__int__cases,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ~ ! [N2: nat] :
% 0.28/0.47 ( ( K
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N2 ) )
% 0.28/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pos_int_cases
% 0.28/0.47 thf(fact_74_reals__Archimedean2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A )
% 0.28/0.47 => ! [X: A] :
% 0.28/0.47 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % reals_Archimedean2
% 0.28/0.47 thf(fact_75_list__slice__def,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( listSl1903280966_slice @ A )
% 0.28/0.47 = ( ^ [Xs3: list @ A,K2: nat] : ( listSl1123830174ce_aux @ A @ Xs3 @ K2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ K2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_def
% 0.28/0.47 thf(fact_76_of__nat__zero__less__power__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [X: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 0.28/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.28/0.47 | ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_zero_less_power_iff
% 0.28/0.47 thf(fact_77_of__nat__le__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( M
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_le_0_iff
% 0.28/0.47 thf(fact_78_ex__inverse__of__nat__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A )
% 0.28/0.47 => ! [X: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_inverse_of_nat_less
% 0.28/0.47 thf(fact_79_bot__nat__0_Oextremum,axiom,
% 0.28/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum
% 0.28/0.47 thf(fact_80_le0,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/0.47
% 0.28/0.47 % le0
% 0.28/0.47 thf(fact_81_le__zero__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.28/0.47 = ( N
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_zero_eq
% 0.28/0.47 thf(fact_82_of__nat__le__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_le_iff
% 0.28/0.47 thf(fact_83_nat__zero__less__power__iff,axiom,
% 0.28/0.47 ! [X: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 0.28/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 0.28/0.47 | ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_zero_less_power_iff
% 0.28/0.47 thf(fact_84_of__nat__power,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 0.28/0.47 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_power
% 0.28/0.47 thf(fact_85_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A )
% 0.28/0.47 => ! [B2: nat,W: nat,X: nat] :
% 0.28/0.47 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
% 0.28/0.47 = ( semiring_1_of_nat @ A @ X ) )
% 0.28/0.47 = ( ( power_power @ nat @ B2 @ W )
% 0.28/0.47 = X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_of_nat_power_cancel_iff
% 0.28/0.47 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A )
% 0.28/0.47 => ! [X: nat,B2: nat,W: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ A @ X )
% 0.28/0.47 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 0.28/0.47 = ( X
% 0.28/0.47 = ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_power_eq_of_nat_cancel_iff
% 0.28/0.47 thf(fact_87_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [B2: nat,W: nat,X: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_le_of_nat_power_cancel_iff
% 0.28/0.47 thf(fact_88_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [X: nat,B2: nat,W: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_power_le_of_nat_cancel_iff
% 0.28/0.47 thf(fact_89_power__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri134348788visors @ A )
% 0.28/0.47 => ! [A2: A,N: nat] :
% 0.28/0.47 ( ( ( power_power @ A @ A2 @ N )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_eq_0_iff
% 0.28/0.47 thf(fact_90_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [X: nat,B2: nat,W: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 0.28/0.47 = ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_power_less_of_nat_cancel_iff
% 0.28/0.47 thf(fact_91_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [B2: nat,W: nat,X: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 0.28/0.47 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_of_nat_power_cancel_iff
% 0.28/0.47 thf(fact_92_power__mono__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,B2: A,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_mono_iff
% 0.28/0.47 thf(fact_93_nonneg__int__cases,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ~ ! [N2: nat] :
% 0.28/0.47 ( K
% 0.28/0.47 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nonneg_int_cases
% 0.28/0.47 thf(fact_94_zero__le__imp__eq__int,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( K
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le_imp_eq_int
% 0.28/0.47 thf(fact_95_zero__integer_Orsp,axiom,
% 0.28/0.47 ( ( zero_zero @ int )
% 0.28/0.47 = ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_integer.rsp
% 0.28/0.47 thf(fact_96_of__nat__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [I2: nat,J: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_mono
% 0.28/0.47 thf(fact_97_zle__int,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 0.28/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % zle_int
% 0.28/0.47 thf(fact_98_int__int__eq,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ int @ M )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N ) )
% 0.28/0.47 = ( M = N ) ) ).
% 0.28/0.47
% 0.28/0.47 % int_int_eq
% 0.28/0.47 thf(fact_99_real__arch__simple,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A )
% 0.28/0.47 => ! [X: A] :
% 0.28/0.47 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % real_arch_simple
% 0.28/0.47 thf(fact_100_power__less__imp__less__base,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,N: nat,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_less_imp_less_base
% 0.28/0.47 thf(fact_101_power__inverse,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A,N: nat] :
% 0.28/0.47 ( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
% 0.28/0.47 = ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_inverse
% 0.28/0.47 thf(fact_102_power__divide,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ! [A2: A,B2: A,N: nat] :
% 0.28/0.47 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
% 0.28/0.47 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_divide
% 0.28/0.47 thf(fact_103_power__not__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri134348788visors @ A )
% 0.28/0.47 => ! [A2: A,N: nat] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( power_power @ A @ A2 @ N )
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_not_zero
% 0.28/0.47 thf(fact_104_zero__le__power,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le_power
% 0.28/0.47 thf(fact_105_power__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,B2: A,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_mono
% 0.28/0.47 thf(fact_106_less__eq__int__code_I1_J,axiom,
% 0.28/0.47 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 0.28/0.47
% 0.28/0.47 % less_eq_int_code(1)
% 0.28/0.47 thf(fact_107_less__int__code_I1_J,axiom,
% 0.28/0.47 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_int_code(1)
% 0.28/0.47 thf(fact_108_power__eq__imp__eq__base,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,N: nat,B2: A] :
% 0.28/0.47 ( ( ( power_power @ A @ A2 @ N )
% 0.28/0.47 = ( power_power @ A @ B2 @ N ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_eq_imp_eq_base
% 0.28/0.47 thf(fact_109_power__eq__iff__eq__base,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [N: nat,A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ( power_power @ A @ A2 @ N )
% 0.28/0.47 = ( power_power @ A @ B2 @ N ) )
% 0.28/0.47 = ( A2 = B2 ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_eq_iff_eq_base
% 0.28/0.47 thf(fact_110_real__arch__inverse,axiom,
% 0.28/0.47 ! [E2: real] :
% 0.28/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 0.28/0.47 = ( ? [N3: nat] :
% 0.28/0.47 ( ( N3
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) )
% 0.28/0.47 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ E2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % real_arch_inverse
% 0.28/0.47 thf(fact_111_forall__pos__mono,axiom,
% 0.28/0.47 ! [P: real > $o,E2: real] :
% 0.28/0.47 ( ! [D: real,E: real] :
% 0.28/0.47 ( ( ord_less @ real @ D @ E )
% 0.28/0.47 => ( ( P @ D )
% 0.28/0.47 => ( P @ E ) ) )
% 0.28/0.47 => ( ! [N2: nat] :
% 0.28/0.47 ( ( N2
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
% 0.28/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 0.28/0.47 => ( P @ E2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % forall_pos_mono
% 0.28/0.47 thf(fact_112_zero__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A )
% 0.28/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le
% 0.28/0.47 thf(fact_113_le__numeral__extra_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_numeral_extra(3)
% 0.28/0.47 thf(fact_114_ge__less__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [A4: A,N3: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ X3 @ A4 )
% 0.28/0.47 => ( N3 != X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ge_less_neq_conv
% 0.28/0.47 thf(fact_115_less__ge__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [N3: A,A4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A4 @ X3 )
% 0.28/0.47 => ( N3 != X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_ge_neq_conv
% 0.28/0.47 thf(fact_116_greater__le__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less @ A )
% 0.28/0.47 = ( ^ [A4: A,N3: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X3 @ A4 )
% 0.28/0.47 => ( N3 != X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % greater_le_neq_conv
% 0.28/0.47 thf(fact_117_le__greater__neq__conv,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ( ( ord_less_eq @ A )
% 0.28/0.47 = ( ^ [N3: A,A4: A] :
% 0.28/0.47 ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ A @ A4 @ X3 )
% 0.28/0.47 => ( N3 != X3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_greater_neq_conv
% 0.28/0.47 thf(fact_118_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.28/0.47 ! [A2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/0.47 => ( A2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum_uniqueI
% 0.28/0.47 thf(fact_119_bot__nat__0_Oextremum__unique,axiom,
% 0.28/0.47 ! [A2: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bot_nat_0.extremum_unique
% 0.28/0.47 thf(fact_120_le__0__eq,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_0_eq
% 0.28/0.47 thf(fact_121_less__eq__nat_Osimps_I1_J,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_eq_nat.simps(1)
% 0.28/0.47 thf(fact_122_less__mono__imp__le__mono,axiom,
% 0.28/0.47 ! [F: nat > nat,I2: nat,J: nat] :
% 0.28/0.47 ( ! [I3: nat,J2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ J2 )
% 0.28/0.47 => ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_mono_imp_le_mono
% 0.28/0.47 thf(fact_123_le__neq__implies__less,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( M != N )
% 0.28/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_neq_implies_less
% 0.28/0.47 thf(fact_124_less__or__eq__imp__le,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 | ( M = N ) )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_or_eq_imp_le
% 0.28/0.47 thf(fact_125_le__eq__less__or__eq,axiom,
% 0.28/0.47 ( ( ord_less_eq @ nat )
% 0.28/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.28/0.47 | ( M3 = N3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_eq_less_or_eq
% 0.28/0.47 thf(fact_126_less__imp__le__nat,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_le_nat
% 0.28/0.47 thf(fact_127_nat__less__le,axiom,
% 0.28/0.47 ( ( ord_less @ nat )
% 0.28/0.47 = ( ^ [M3: nat,N3: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 0.28/0.47 & ( M3 != N3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_less_le
% 0.28/0.47 thf(fact_128_zero__less__power,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_power
% 0.28/0.47 thf(fact_129_nat__power__less__imp__less,axiom,
% 0.28/0.47 ! [I2: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N ) )
% 0.28/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_power_less_imp_less
% 0.28/0.47 thf(fact_130_power__strict__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A )
% 0.28/0.47 => ! [A2: A,B2: A,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % power_strict_mono
% 0.28/0.47 thf(fact_131_zero__power,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_power
% 0.28/0.47 thf(fact_132_of__nat__0__le__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord1659791738miring @ A )
% 0.28/0.47 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0_le_iff
% 0.28/0.47 thf(fact_133_ex__least__nat__le,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ( P @ N )
% 0.28/0.47 => ( ~ ( P @ ( zero_zero @ nat ) )
% 0.28/0.47 => ? [K3: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ K3 @ N )
% 0.28/0.47 & ! [I4: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I4 @ K3 )
% 0.28/0.47 => ~ ( P @ I4 ) )
% 0.28/0.47 & ( P @ K3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_least_nat_le
% 0.28/0.47 thf(fact_134_count__le__length,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.28/0.47
% 0.28/0.47 % count_le_length
% 0.28/0.47 thf(fact_135_list__slice__length,axiom,
% 0.28/0.47 ! [A: $tType,Xs: list @ A,K: nat] :
% 0.28/0.47 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1903280966_slice @ A @ Xs @ K ) )
% 0.28/0.47 = ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) ) ).
% 0.28/0.47
% 0.28/0.47 % list_slice_length
% 0.28/0.47 thf(fact_136_inverse__le__iff__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_le_iff_le
% 0.28/0.47 thf(fact_137_inverse__le__iff__le__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_le_iff_le_neg
% 0.28/0.47 thf(fact_138_div__less,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ( divide_divide @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_less
% 0.28/0.47 thf(fact_139_inverse__positive__iff__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_positive_iff_positive
% 0.28/0.47 thf(fact_140_inverse__negative__iff__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_negative_iff_negative
% 0.28/0.47 thf(fact_141_inverse__less__iff__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_iff_less_neg
% 0.28/0.47 thf(fact_142_inverse__eq__iff__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( A2 = B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_eq_iff_eq
% 0.28/0.47 thf(fact_143_inverse__inverse__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 = A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_inverse_eq
% 0.28/0.47 thf(fact_144_division__ring__divide__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % division_ring_divide_zero
% 0.28/0.47 thf(fact_145_divide__cancel__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ! [A2: A,C: A,B2: A] :
% 0.28/0.47 ( ( ( divide_divide @ A @ A2 @ C )
% 0.28/0.47 = ( divide_divide @ A @ B2 @ C ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_cancel_right
% 0.28/0.47 thf(fact_146_divide__cancel__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ! [C: A,A2: A,B2: A] :
% 0.28/0.47 ( ( ( divide_divide @ A @ C @ A2 )
% 0.28/0.47 = ( divide_divide @ A @ C @ B2 ) )
% 0.28/0.47 = ( ( C
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_cancel_left
% 0.28/0.47 thf(fact_147_divide__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( divide_divide @ A @ A2 @ B2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_eq_0_iff
% 0.28/0.47 thf(fact_148_inverse__nonzero__iff__nonzero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_nonzero_iff_nonzero
% 0.28/0.47 thf(fact_149_inverse__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_zero
% 0.28/0.47 thf(fact_150_inverse__divide,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_divide
% 0.28/0.47 thf(fact_151_inverse__nonnegative__iff__nonnegative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_nonnegative_iff_nonnegative
% 0.28/0.47 thf(fact_152_inverse__nonpositive__iff__nonpositive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_nonpositive_iff_nonpositive
% 0.28/0.47 thf(fact_153_inverse__less__iff__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_iff_less
% 0.28/0.47 thf(fact_154_div__neg__neg__trivial,axiom,
% 0.28/0.47 ! [K: int,L: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less @ int @ L @ K )
% 0.28/0.47 => ( ( divide_divide @ int @ K @ L )
% 0.28/0.47 = ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_neg_neg_trivial
% 0.28/0.47 thf(fact_155_div__pos__pos__trivial,axiom,
% 0.28/0.47 ! [K: int,L: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ( ( ord_less @ int @ K @ L )
% 0.28/0.47 => ( ( divide_divide @ int @ K @ L )
% 0.28/0.47 = ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_pos_pos_trivial
% 0.28/0.47 thf(fact_156_less__eq__real__def,axiom,
% 0.28/0.47 ( ( ord_less_eq @ real )
% 0.28/0.47 = ( ^ [X3: real,Y3: real] :
% 0.28/0.47 ( ( ord_less @ real @ X3 @ Y3 )
% 0.28/0.47 | ( X3 = Y3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_eq_real_def
% 0.28/0.47 thf(fact_157_le__refl,axiom,
% 0.28/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % le_refl
% 0.28/0.47 thf(fact_158_le__trans,axiom,
% 0.28/0.47 ! [I2: nat,J: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ I2 @ J )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.28/0.47 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_trans
% 0.28/0.47 thf(fact_159_eq__imp__le,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( M = N )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % eq_imp_le
% 0.28/0.47 thf(fact_160_le__antisym,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.28/0.47 => ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_antisym
% 0.28/0.47 thf(fact_161_nat__le__linear,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_le_linear
% 0.28/0.47 thf(fact_162_Nat_Oex__has__greatest__nat,axiom,
% 0.28/0.47 ! [P: nat > $o,K: nat,B2: nat] :
% 0.28/0.47 ( ( P @ K )
% 0.28/0.47 => ( ! [Y4: nat] :
% 0.28/0.47 ( ( P @ Y4 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 0.28/0.47 => ? [X2: nat] :
% 0.28/0.47 ( ( P @ X2 )
% 0.28/0.47 & ! [Y2: nat] :
% 0.28/0.47 ( ( P @ Y2 )
% 0.28/0.47 => ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat.ex_has_greatest_nat
% 0.28/0.47 thf(fact_163_real__of__nat__div4,axiom,
% 0.28/0.47 ! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % real_of_nat_div4
% 0.28/0.47 thf(fact_164_linordered__field__no__lb,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X4: A] :
% 0.28/0.47 ? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_no_lb
% 0.28/0.47 thf(fact_165_linordered__field__no__ub,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X4: A] :
% 0.28/0.47 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_no_ub
% 0.28/0.47 thf(fact_166_inverse__eq__imp__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( A2 = B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_eq_imp_eq
% 0.28/0.47 thf(fact_167_div__le__mono,axiom,
% 0.28/0.47 ! [M: nat,N: nat,K: nat] :
% 0.28/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_le_mono
% 0.28/0.47 thf(fact_168_div__le__dividend,axiom,
% 0.28/0.47 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 0.28/0.47
% 0.28/0.47 % div_le_dividend
% 0.28/0.47 thf(fact_169_nonzero__imp__inverse__nonzero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nonzero_imp_inverse_nonzero
% 0.28/0.47 thf(fact_170_nonzero__inverse__inverse__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 = A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nonzero_inverse_inverse_eq
% 0.28/0.47 thf(fact_171_nonzero__inverse__eq__imp__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( A2 = B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nonzero_inverse_eq_imp_eq
% 0.28/0.47 thf(fact_172_inverse__zero__imp__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 => ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_zero_imp_zero
% 0.28/0.47 thf(fact_173_field__class_Ofield__inverse__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A )
% 0.28/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % field_class.field_inverse_zero
% 0.28/0.47 thf(fact_174_divide__le__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.28/0.47 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_le_0_iff
% 0.28/0.47 thf(fact_175_divide__right__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_right_mono
% 0.28/0.47 thf(fact_176_zero__le__divide__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.28/0.47 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_le_divide_iff
% 0.28/0.47 thf(fact_177_divide__nonneg__nonneg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonneg_nonneg
% 0.28/0.47 thf(fact_178_divide__nonneg__nonpos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonneg_nonpos
% 0.28/0.47 thf(fact_179_divide__nonpos__nonneg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonpos_nonneg
% 0.28/0.47 thf(fact_180_divide__nonpos__nonpos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonpos_nonpos
% 0.28/0.47 thf(fact_181_divide__right__mono__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_right_mono_neg
% 0.28/0.47 thf(fact_182_divide__neg__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_neg_neg
% 0.28/0.47 thf(fact_183_divide__neg__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_neg_pos
% 0.28/0.47 thf(fact_184_divide__pos__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_pos_neg
% 0.28/0.47 thf(fact_185_divide__pos__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_pos_pos
% 0.28/0.47 thf(fact_186_divide__less__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_less_0_iff
% 0.28/0.47 thf(fact_187_divide__less__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,C: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ord_less @ A @ A2 @ B2 ) )
% 0.28/0.47 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ B2 @ A2 ) )
% 0.28/0.47 & ( C
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_less_cancel
% 0.28/0.47 thf(fact_188_zero__less__divide__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_divide_iff
% 0.28/0.47 thf(fact_189_divide__strict__right__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A,C: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_strict_right_mono
% 0.28/0.47 thf(fact_190_divide__strict__right__mono__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [B2: A,A2: A,C: A] :
% 0.28/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_strict_right_mono_neg
% 0.28/0.47 thf(fact_191_inverse__less__imp__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_imp_less
% 0.28/0.47 thf(fact_192_less__imp__inverse__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_inverse_less
% 0.28/0.47 thf(fact_193_inverse__less__imp__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_imp_less_neg
% 0.28/0.47 thf(fact_194_less__imp__inverse__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_inverse_less_neg
% 0.28/0.47 thf(fact_195_inverse__negative__imp__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_negative_imp_negative
% 0.28/0.47 thf(fact_196_inverse__positive__imp__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_positive_imp_positive
% 0.28/0.47 thf(fact_197_negative__imp__inverse__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % negative_imp_inverse_negative
% 0.28/0.47 thf(fact_198_positive__imp__inverse__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % positive_imp_inverse_positive
% 0.28/0.47 thf(fact_199_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( divide_divide @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 | ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Euclidean_Division.div_eq_0_iff
% 0.28/0.47 thf(fact_200_div__greater__zero__iff,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 0.28/0.47 = ( ( ord_less_eq @ nat @ N @ M )
% 0.28/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_greater_zero_iff
% 0.28/0.47 thf(fact_201_div__le__mono2,axiom,
% 0.28/0.47 ! [M: nat,N: nat,K: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.28/0.47 => ( ( ord_less_eq @ nat @ M @ N )
% 0.28/0.47 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_le_mono2
% 0.28/0.47 thf(fact_202_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( euclid1852923125th_nat @ A )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 0.28/0.47 = ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.28/0.47 thf(fact_203_divide__nonpos__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonpos_pos
% 0.28/0.47 thf(fact_204_divide__nonpos__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonpos_neg
% 0.28/0.47 thf(fact_205_divide__nonneg__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonneg_pos
% 0.28/0.47 thf(fact_206_divide__nonneg__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_nonneg_neg
% 0.28/0.47 thf(fact_207_divide__le__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,C: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 0.28/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) )
% 0.28/0.47 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divide_le_cancel
% 0.28/0.47 thf(fact_208_frac__less2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A,W: A,Z: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.28/0.47 => ( ( ord_less @ A @ W @ Z )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % frac_less2
% 0.28/0.47 thf(fact_209_frac__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [X: A,Y: A,W: A,Z: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.28/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.28/0.47 => ( ( ord_less_eq @ A @ W @ Z )
% 0.28/0.47 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % frac_less
% 0.28/0.47 thf(fact_210_frac__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [Y: A,X: A,W: A,Z: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.28/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 0.28/0.47 => ( ( ord_less_eq @ A @ W @ Z )
% 0.28/0.47 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % frac_le
% 0.28/0.47 thf(fact_211_le__imp__inverse__le__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_imp_inverse_le_neg
% 0.28/0.47 thf(fact_212_inverse__le__imp__le__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_le_imp_le_neg
% 0.28/0.47 thf(fact_213_le__imp__inverse__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % le_imp_inverse_le
% 0.28/0.47 thf(fact_214_inverse__le__imp__le,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_le_imp_le
% 0.28/0.47 thf(fact_215_bits__div__by__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_bits @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bits_div_by_0
% 0.28/0.47 thf(fact_216_bits__div__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_bits @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % bits_div_0
% 0.28/0.47 thf(fact_217_div__by__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semidom_divide @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_by_0
% 0.28/0.47 thf(fact_218_div__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semidom_divide @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_0
% 0.28/0.47 thf(fact_219_complete__real,axiom,
% 0.28/0.47 ! [S2: set @ real] :
% 0.28/0.47 ( ? [X4: real] : ( member2 @ real @ X4 @ S2 )
% 0.28/0.47 => ( ? [Z2: real] :
% 0.28/0.47 ! [X2: real] :
% 0.28/0.47 ( ( member2 @ real @ X2 @ S2 )
% 0.28/0.47 => ( ord_less_eq @ real @ X2 @ Z2 ) )
% 0.28/0.47 => ? [Y4: real] :
% 0.28/0.47 ( ! [X4: real] :
% 0.28/0.47 ( ( member2 @ real @ X4 @ S2 )
% 0.28/0.47 => ( ord_less_eq @ real @ X4 @ Y4 ) )
% 0.28/0.47 & ! [Z2: real] :
% 0.28/0.47 ( ! [X2: real] :
% 0.28/0.47 ( ( member2 @ real @ X2 @ S2 )
% 0.28/0.47 => ( ord_less_eq @ real @ X2 @ Z2 ) )
% 0.28/0.47 => ( ord_less_eq @ real @ Y4 @ Z2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % complete_real
% 0.28/0.47 thf(fact_220_linorder__neqE__linordered__idom,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_idom @ A )
% 0.28/0.47 => ! [X: A,Y: A] :
% 0.28/0.47 ( ( X != Y )
% 0.28/0.47 => ( ~ ( ord_less @ A @ X @ Y )
% 0.28/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_neqE_linordered_idom
% 0.28/0.47 thf(fact_221_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.28/0.47 ! [A2: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 0.28/0.47 = ( ( ord_less_eq @ int @ B2 @ A2 )
% 0.28/0.47 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nonneg1_imp_zdiv_pos_iff
% 0.28/0.47 thf(fact_222_pos__imp__zdiv__nonneg__iff,axiom,
% 0.28/0.47 ! [B2: int,A2: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 0.28/0.47 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pos_imp_zdiv_nonneg_iff
% 0.28/0.47 thf(fact_223_div__neg__pos__less0,axiom,
% 0.28/0.47 ! [A2: int,B2: int] :
% 0.28/0.47 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.28/0.47 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_neg_pos_less0
% 0.28/0.47 thf(fact_224_neg__imp__zdiv__neg__iff,axiom,
% 0.28/0.47 ! [B2: int,A2: int] :
% 0.28/0.47 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
% 0.28/0.47 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % neg_imp_zdiv_neg_iff
% 0.28/0.47 thf(fact_225_pos__imp__zdiv__neg__iff,axiom,
% 0.28/0.47 ! [B2: int,A2: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.28/0.47 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
% 0.28/0.47 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pos_imp_zdiv_neg_iff
% 0.28/0.47 thf(fact_226_zdiv__int,axiom,
% 0.28/0.47 ! [A2: nat,B2: nat] :
% 0.28/0.47 ( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
% 0.28/0.47 = ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_int
% 0.28/0.47 thf(fact_227_div__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( unique1598680935umeral @ A )
% 0.28/0.47 => ! [B2: A,A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_positive
% 0.28/0.47 thf(fact_228_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( unique1598680935umeral @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( divide_divide @ A @ A2 @ B2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % unique_euclidean_semiring_numeral_class.div_less
% 0.28/0.47 thf(fact_229_zdiv__mono1,axiom,
% 0.28/0.47 ! [A2: int,A5: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ A2 @ A5 )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A5 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_mono1
% 0.28/0.47 thf(fact_230_zdiv__mono2,axiom,
% 0.28/0.47 ! [A2: int,B3: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ int @ B3 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_mono2
% 0.28/0.47 thf(fact_231_zdiv__eq__0__iff,axiom,
% 0.28/0.47 ! [I2: int,K: int] :
% 0.28/0.47 ( ( ( divide_divide @ int @ I2 @ K )
% 0.28/0.47 = ( zero_zero @ int ) )
% 0.28/0.47 = ( ( K
% 0.28/0.47 = ( zero_zero @ int ) )
% 0.28/0.47 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
% 0.28/0.47 & ( ord_less @ int @ I2 @ K ) )
% 0.28/0.47 | ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
% 0.28/0.47 & ( ord_less @ int @ K @ I2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_eq_0_iff
% 0.28/0.47 thf(fact_232_zdiv__mono1__neg,axiom,
% 0.28/0.47 ! [A2: int,A5: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ A2 @ A5 )
% 0.28/0.47 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A5 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_mono1_neg
% 0.28/0.47 thf(fact_233_zdiv__mono2__neg,axiom,
% 0.28/0.47 ! [A2: int,B3: int,B2: int] :
% 0.28/0.47 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
% 0.28/0.47 => ( ( ord_less_eq @ int @ B3 @ B2 )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B3 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zdiv_mono2_neg
% 0.28/0.47 thf(fact_234_div__int__pos__iff,axiom,
% 0.28/0.47 ! [K: int,L: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
% 0.28/0.47 = ( ( K
% 0.28/0.47 = ( zero_zero @ int ) )
% 0.28/0.47 | ( L
% 0.28/0.47 = ( zero_zero @ int ) )
% 0.28/0.47 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
% 0.28/0.47 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.28/0.47 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_int_pos_iff
% 0.28/0.47 thf(fact_235_div__positive__int,axiom,
% 0.28/0.47 ! [L: int,K: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ L @ K )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.28/0.47 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_positive_int
% 0.28/0.47 thf(fact_236_div__nonneg__neg__le0,axiom,
% 0.28/0.47 ! [A2: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_nonneg_neg_le0
% 0.28/0.47 thf(fact_237_div__nonpos__pos__le0,axiom,
% 0.28/0.47 ! [A2: int,B2: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 0.28/0.47 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_nonpos_pos_le0
% 0.28/0.47 thf(fact_238_pos__imp__zdiv__pos__iff,axiom,
% 0.28/0.47 ! [K: int,I2: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K ) )
% 0.28/0.47 = ( ord_less_eq @ int @ K @ I2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % pos_imp_zdiv_pos_iff
% 0.28/0.47 thf(fact_239_neg__imp__zdiv__nonneg__iff,axiom,
% 0.28/0.47 ! [B2: int,A2: int] :
% 0.28/0.47 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 0.28/0.47 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 0.28/0.47 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % neg_imp_zdiv_nonneg_iff
% 0.28/0.47 thf(fact_240_realpow__pos__nth__unique,axiom,
% 0.28/0.47 ! [N: nat,A2: real] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.28/0.47 => ? [X2: real] :
% 0.28/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
% 0.28/0.47 & ( ( power_power @ real @ X2 @ N )
% 0.28/0.47 = A2 )
% 0.28/0.47 & ! [Y2: real] :
% 0.28/0.47 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
% 0.28/0.47 & ( ( power_power @ real @ Y2 @ N )
% 0.28/0.47 = A2 ) )
% 0.28/0.47 => ( Y2 = X2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % realpow_pos_nth_unique
% 0.28/0.47 thf(fact_241_realpow__pos__nth,axiom,
% 0.28/0.47 ! [N: nat,A2: real] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 0.28/0.47 => ? [R: real] :
% 0.28/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 0.28/0.47 & ( ( power_power @ real @ R @ N )
% 0.28/0.47 = A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % realpow_pos_nth
% 0.28/0.47 thf(fact_242_nat__int__comparison_I3_J,axiom,
% 0.28/0.47 ( ( ord_less_eq @ nat )
% 0.28/0.47 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_int_comparison(3)
% 0.28/0.47 thf(fact_243_nat__int__comparison_I2_J,axiom,
% 0.28/0.47 ( ( ord_less @ nat )
% 0.28/0.47 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_int_comparison(2)
% 0.28/0.47 thf(fact_244_verit__la__disequality,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linorder @ A )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( A2 = B2 )
% 0.28/0.47 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.47 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % verit_la_disequality
% 0.28/0.47 thf(fact_245_verit__comp__simplify1_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( order @ A )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % verit_comp_simplify1(1)
% 0.28/0.47 thf(fact_246_verit__la__generic,axiom,
% 0.28/0.47 ! [A2: int,X: int] :
% 0.28/0.47 ( ( ord_less_eq @ int @ A2 @ X )
% 0.28/0.47 | ( A2 = X )
% 0.28/0.47 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % verit_la_generic
% 0.28/0.47 thf(fact_247_int__if,axiom,
% 0.28/0.47 ! [P: $o,A2: nat,B2: nat] :
% 0.28/0.47 ( ( P
% 0.28/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 0.28/0.47 & ( ~ P
% 0.28/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % int_if
% 0.28/0.47 thf(fact_248_nat__int__comparison_I1_J,axiom,
% 0.28/0.47 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 0.28/0.47 = ( ^ [A4: nat,B4: nat] :
% 0.28/0.47 ( ( semiring_1_of_nat @ int @ A4 )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_int_comparison(1)
% 0.28/0.47 thf(fact_249_verit__comp__simplify1_I3_J,axiom,
% 0.28/0.47 ! [B: $tType] :
% 0.28/0.47 ( ( linorder @ B )
% 0.28/0.47 => ! [B3: B,A5: B] :
% 0.28/0.47 ( ( ~ ( ord_less_eq @ B @ B3 @ A5 ) )
% 0.28/0.47 = ( ord_less @ B @ A5 @ B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % verit_comp_simplify1(3)
% 0.28/0.47 thf(fact_250_int__ops_I1_J,axiom,
% 0.28/0.47 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % int_ops(1)
% 0.28/0.47 thf(fact_251_div__gr__imp__gr__divisor,axiom,
% 0.28/0.47 ! [X: nat,N: nat,M: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ X @ ( divide_divide @ nat @ N @ M ) )
% 0.28/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_gr_imp_gr_divisor
% 0.28/0.47 thf(fact_252_div__eq__0__conv,axiom,
% 0.28/0.47 ! [N: nat,M: nat] :
% 0.28/0.47 ( ( ( divide_divide @ nat @ N @ M )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 = ( ( M
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % div_eq_0_conv
% 0.28/0.47
% 0.28/0.47 % Type constructors (47)
% 0.28/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.47 ! [A6: $tType,A7: $tType] :
% 0.28/0.47 ( ( order @ A7 )
% 0.28/0.47 => ( order @ ( A6 > A7 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 0.28/0.47 euclid1852923125th_nat @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 0.28/0.47 unique1598680935umeral @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 0.28/0.47 semiri134348788visors @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.28/0.47 linord1659791738miring @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.28/0.47 linordered_semidom @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.28/0.47 linordered_idom @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 0.28/0.47 semidom_divide @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
% 0.28/0.47 semiring_bits @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.28/0.47 semiring_char_0 @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.28/0.47 linorder @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.28/0.47 semiring_1 @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Orderings_Oorder_1,axiom,
% 0.28/0.47 order @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.28/0.47 zero @ int ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_2,axiom,
% 0.28/0.47 euclid1852923125th_nat @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_3,axiom,
% 0.28/0.47 unique1598680935umeral @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.28/0.47 canoni770627133id_add @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4,axiom,
% 0.28/0.47 semiri134348788visors @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5,axiom,
% 0.28/0.47 linord1659791738miring @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 0.28/0.47 linordered_semidom @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemidom__divide_7,axiom,
% 0.28/0.47 semidom_divide @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Parity_Osemiring__bits_8,axiom,
% 0.28/0.47 semiring_bits @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.28/0.47 wellorder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9,axiom,
% 0.28/0.47 semiring_char_0 @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 0.28/0.47 linorder @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 0.28/0.47 semiring_1 @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
% 0.28/0.47 order @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ozero_13,axiom,
% 0.28/0.47 zero @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.28/0.47 size @ nat ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 0.28/0.47 ! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 0.28/0.47 linorder @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 0.28/0.47 order @ $o ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_List_Olist___Nat_Osize_17,axiom,
% 0.28/0.47 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.28/0.47 archim1804426504_field @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_18,axiom,
% 0.28/0.47 semiri134348788visors @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19,axiom,
% 0.28/0.47 linord1659791738miring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_20,axiom,
% 0.28/0.47 linordered_semidom @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.28/0.47 linordered_field @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__idom_21,axiom,
% 0.28/0.47 linordered_idom @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemidom__divide_22,axiom,
% 0.28/0.47 semidom_divide @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.28/0.47 division_ring @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_23,axiom,
% 0.28/0.47 semiring_char_0 @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_24,axiom,
% 0.28/0.47 linorder @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__1_25,axiom,
% 0.28/0.47 semiring_1 @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_26,axiom,
% 0.28/0.47 order @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.28/0.47 field @ real ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 0.28/0.47 zero @ real ).
% 0.28/0.47
% 0.28/0.47 % Helper facts (3)
% 0.28/0.47 thf(help_If_3_1_T,axiom,
% 0.28/0.47 ! [P: $o] :
% 0.28/0.47 ( ( P = $true )
% 0.28/0.47 | ( P = $false ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_2_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X: A,Y: A] :
% 0.28/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.28/0.47 = Y ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_1_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X: A,Y: A] :
% 0.28/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.28/0.47 = X ) ).
% 0.28/0.47
% 0.28/0.47 % Conjectures (1)
% 0.28/0.47 thf(conj_0,conjecture,
% 0.28/0.47 ( ( listSl1903280966_slice @ a @ ( nil @ a ) @ k )
% 0.28/0.47 = ( nil @ ( list @ a ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.48/1.48 ---- Embedded in THF ---
% 2.48/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.48/1.48 %%% Generated on Tue Feb 24 08:26:00 EST 2026
% 2.48/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.48/1.48 %%% Logic specification used:
% 2.48/1.48 %%% thf(spec, logic, $$dhol).
% 2.48/1.48
% 2.48/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2DTF7234.p
% 2.48/1.48 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 2.48/1.48 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.48/1.48 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.48/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.48/1.48 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.48/1.48 thf(ty_t_Int_Oint, type, int: $tType).
% 2.48/1.48 thf(ty_tf_a, type, a: $tType).
% 2.48/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Parity_Osemiring__bits, type, semiring_bits: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri134348788visors: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1598680935umeral: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat, type, euclid1852923125th_nat: (!> [A:$tType]: ($o))).
% 2.48/1.48 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 2.48/1.48 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.48/1.48 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.48/1.48 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice, type, listSl1903280966_slice: (!> [A:$tType]: (((list @ A) > (nat > (list @ (list @ A))))))).
% 2.48/1.48 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux, type, listSl1123830174ce_aux: (!> [A:$tType]: (((list @ A) > (nat > (nat > (list @ (list @ A)))))))).
% 2.48/1.48 thf(sy_c_List_Obind, type, bind: (!> [A:$tType,B:$tType]: (((list @ A) > ((A > (list @ B)) > (list @ B)))))).
% 2.48/1.48 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))).
% 2.48/1.48 thf(sy_c_List_Ocount__list, type, count_list: (!> [A:$tType]: (((list @ A) > (A > nat))))).
% 2.48/1.48 thf(sy_c_List_Ogen__length, type, gen_length: (!> [A:$tType]: ((nat > ((list @ A) > nat))))).
% 2.48/1.48 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.48/1.48 thf(sy_c_List_Olist__ex1, type, list_ex1: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 2.48/1.48 thf(sy_c_List_Omap__filter, type, map_filter: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > (list @ B)))))).
% 2.48/1.48 thf(sy_c_List_Omaps, type, maps: (!> [A:$tType,B:$tType]: (((A > (list @ B)) > ((list @ A) > (list @ B)))))).
% 2.48/1.48 thf(sy_c_List_Omember, type, member: (!> [A:$tType]: (((list @ A) > (A > $o))))).
% 2.48/1.48 thf(sy_c_List_Onull, type, null: (!> [A:$tType]: (((list @ A) > $o)))).
% 2.48/1.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.48/1.48 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.48/1.48 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.48/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.48/1.48 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 2.48/1.48 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.48/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.48/1.48 thf(sy_c_member, type, member2: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.48/1.48 thf(sy_v_k, type, k: nat).
% 2.48/1.48 thf(fact_0_list__slice__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((listSl1903280966_slice @ A) @ Xs) @ (zero_zero @ nat)) = (nil @ (list @ A)))))).
% 2.48/1.48 thf(fact_1_concat_Osimps_I1_J, axiom, (! [A:$tType]: ((((concat @ A) @ (nil @ (list @ A))) = (nil @ A))))).
% 2.48/1.48 thf(fact_2_list__ex1__simps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((~ ((((list_ex1 @ A) @ P) @ (nil @ A))))))).
% 2.48/1.48 thf(fact_3_list__slice__less, axiom, (! [A:$tType,Xs:(list @ A),K:nat]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K) => ((((listSl1903280966_slice @ A) @ Xs) @ K) = (nil @ (list @ A))))))).
% 2.48/1.48 thf(fact_4_bind__simps_I1_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((bind @ B) @ A) @ (nil @ B)) @ F) = (nil @ A))))).
% 2.48/1.48 thf(fact_5_member__rec_I2_J, axiom, (! [A:$tType,Y:A]: ((~ ((((member @ A) @ (nil @ A)) @ Y)))))).
% 2.48/1.48 thf(fact_6_gen__length__code_I1_J, axiom, (! [A:$tType,N:nat]: (((((gen_length @ A) @ N) @ (nil @ A)) = N)))).
% 2.48/1.48 thf(fact_7_maps__simps_I2_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((maps @ B) @ A) @ F) @ (nil @ B)) = (nil @ A))))).
% 2.48/1.48 thf(fact_8_null__rec_I2_J, axiom, (! [B:$tType]: (((null @ B) @ (nil @ B))))).
% 2.48/1.48 thf(fact_9_eq__Nil__null, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs = (nil @ A)) = ((null @ A) @ Xs))))).
% 2.48/1.48 thf(fact_10_map__filter__simps_I2_J, axiom, (! [B:$tType,A:$tType,F:(B > (option @ A))]: ((((((map_filter @ B) @ A) @ F) @ (nil @ B)) = (nil @ A))))).
% 2.48/1.48 thf(fact_11_length__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = (zero_zero @ nat)) = (Xs = (nil @ A)))))).
% 2.48/1.48 thf(fact_12_length__greater__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)) = (Xs != (nil @ A)))))).
% 2.48/1.48 thf(fact_13_list_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (list @ A)) @ (nil @ A)) = (zero_zero @ nat))))).
% 2.48/1.48 thf(fact_14_length__code, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = ((gen_length @ A) @ (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_15_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys)) @ ((size_size @ (list @ A)) @ Xs2)) => (P @ Ys)))) => (P @ Xs2)))) => (P @ Xs))))).
% 2.48/1.48 thf(fact_16_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 2.48/1.48 thf(fact_17_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys2)) => (Xs != Ys2))))).
% 2.48/1.48 thf(fact_18_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.48/1.48 thf(fact_19_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_20_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.48/1.48 thf(fact_21_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_22_length__greater__imp__not__empty, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (Xs != (nil @ A)))))).
% 2.48/1.48 thf(fact_23_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.48/1.48 thf(fact_24_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_25_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_26_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_27_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_28_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_29_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X2)) => (P @ Y2)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.48/1.48 thf(fact_30_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X2)) => (P @ Y2)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.48/1.48 thf(fact_31_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X2)) & (~ ((P @ Y2))))))))) => (P @ X))))).
% 2.48/1.48 thf(fact_32_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 2.48/1.48 thf(fact_33_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.48/1.48 thf(fact_34_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.48/1.48 thf(fact_35_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.48/1.48 thf(fact_36_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.48/1.48 thf(fact_37_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.48/1.48 thf(fact_38_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.48/1.48 thf(fact_39_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.48/1.48 thf(fact_40_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X:A,Y:A]: (((((size_size @ A) @ X) != ((size_size @ A) @ Y)) => (X != Y)))))))).
% 2.48/1.48 thf(fact_41_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_42_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member2 @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.48/1.48 thf(fact_43_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member2 @ A) @ X3) @ A3)))) = A3)))).
% 2.48/1.48 thf(fact_44_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.48/1.48 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.48/1.48 thf(fact_46_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_47_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_48_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.48/1.48 thf(fact_49_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))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X2)) & (~ ((P @ Y2)))))))))) => (P @ X)))))).
% 2.48/1.48 thf(fact_50_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_51_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 2.48/1.48 thf(fact_52_gr__implies__gr0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.48/1.48 thf(fact_53_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D2))))))))))))).
% 2.48/1.48 thf(fact_54_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_55_list__slice__aux_Osimps_I1_J, axiom, (! [A:$tType,Xs:(list @ A),K:nat]: ((((((listSl1123830174ce_aux @ A) @ Xs) @ K) @ (zero_zero @ nat)) = (nil @ (list @ A)))))).
% 2.48/1.48 thf(fact_56_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 2.48/1.48 thf(fact_57_count__list_Osimps_I1_J, axiom, (! [A:$tType,Y:A]: (((((count_list @ A) @ (nil @ A)) @ Y) = (zero_zero @ nat))))).
% 2.48/1.48 thf(fact_58_interval__induct__rule, axiom, (! [I:(set @ nat),P:(nat > $o),A2:nat]: (((! [X2:nat]: (((! [Y2:nat]: ((((((member2 @ nat) @ X2) @ I) & ((((member2 @ nat) @ Y2) @ I) & (((ord_less @ nat) @ Y2) @ X2))) => (P @ Y2)))) => (P @ X2)))) => (P @ A2))))).
% 2.48/1.48 thf(fact_59_interval__induct, axiom, (! [I:(set @ nat),P:(nat > $o),A2:nat]: (((! [X2:nat]: (((! [Y2:nat]: ((((((member2 @ nat) @ X2) @ I) & ((((member2 @ nat) @ Y2) @ I) & (((ord_less @ nat) @ Y2) @ X2))) => (P @ Y2)))) => (P @ X2)))) => (P @ A2))))).
% 2.48/1.48 thf(fact_60_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 2.48/1.48 thf(fact_61_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.48/1.48 thf(fact_62_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.48/1.48 thf(fact_63_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.48/1.48 thf(fact_64_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.48/1.48 thf(fact_65_of__nat__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ 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)))))))).
% 2.48/1.48 thf(fact_66_list__slice__aux__length, axiom, (! [A:$tType,Xs:(list @ A),K:nat,N:nat]: ((((size_size @ (list @ (list @ A))) @ ((((listSl1123830174ce_aux @ A) @ Xs) @ K) @ N)) = N)))).
% 2.48/1.48 thf(fact_67_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_68_less__imp__of__nat__less, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.48/1.48 thf(fact_69_of__nat__less__imp__less, axiom, (! [A:$tType]: (((linord1659791738miring @ 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)))))))).
% 2.48/1.48 thf(fact_70_Collect__minI, axiom, (! [A:$tType]: (((wellorder @ A) => (! [K:A,I:(set @ A),P:(A > $o)]: (((((member2 @ A) @ K) @ I) => ((P @ K) => (? [X2:A]: (((((member2 @ A) @ X2) @ I) & ((P @ X2) & (! [Xa:A]: (((((member2 @ A) @ Xa) @ I) => ((((ord_less @ A) @ Xa) @ X2) => (~ ((P @ Xa))))))))))))))))))).
% 2.48/1.48 thf(fact_71_Collect__minI__ex, axiom, (! [A:$tType]: (((wellorder @ A) => (! [I:(set @ A),P:(A > $o)]: (((? [X4:A]: (((((member2 @ A) @ X4) @ I) & (P @ X4)))) => (? [X2:A]: (((((member2 @ A) @ X2) @ I) & ((P @ X2) & (! [Xa:A]: (((((member2 @ A) @ Xa) @ I) => ((((ord_less @ A) @ Xa) @ X2) => (~ ((P @ Xa)))))))))))))))))).
% 2.48/1.48 thf(fact_72_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (K = ((semiring_1_of_nat @ int) @ N2))))))))).
% 2.48/1.48 thf(fact_73_pos__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: (((K = ((semiring_1_of_nat @ int) @ N2)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))))))))).
% 2.48/1.48 thf(fact_74_reals__Archimedean2, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [N2:nat]: ((((ord_less @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.48/1.48 thf(fact_75_list__slice__def, axiom, (! [A:$tType]: (((listSl1903280966_slice @ A) = (^ [Xs3:(list @ A),K2:nat]: (((((listSl1123830174ce_aux @ A) @ Xs3) @ K2) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ K2)))))))).
% 2.48/1.48 thf(fact_76_of__nat__zero__less__power__iff, axiom, (! [A:$tType]: (((linordered_semidom @ 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)))))))))).
% 2.48/1.48 thf(fact_77_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.48/1.48 thf(fact_78_ex__inverse__of__nat__less, axiom, (! [A:$tType]: (((archim1804426504_field @ 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))))))))))).
% 2.48/1.48 thf(fact_79_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.48/1.48 thf(fact_80_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.48/1.48 thf(fact_81_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_82_of__nat__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ 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)))))))).
% 2.48/1.48 thf(fact_83_nat__zero__less__power__iff, axiom, (! [X:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X) | (N = (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_84_of__nat__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((power_power @ nat) @ M) @ N)) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ M)) @ N)))))))).
% 2.48/1.48 thf(fact_85_of__nat__eq__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [B2:nat,W:nat,X:nat]: ((((((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W) = ((semiring_1_of_nat @ A) @ X)) = ((((power_power @ nat) @ B2) @ W) = X)))))))).
% 2.48/1.48 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [X:nat,B2:nat,W:nat]: (((((semiring_1_of_nat @ A) @ X) = (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W)) = (X = (((power_power @ nat) @ B2) @ W))))))))).
% 2.48/1.48 thf(fact_87_of__nat__le__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W:nat,X:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ W)) @ X)))))))).
% 2.48/1.48 thf(fact_88_of__nat__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B2:nat,W:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W)) = (((ord_less_eq @ nat) @ X) @ (((power_power @ nat) @ B2) @ W))))))))).
% 2.48/1.48 thf(fact_89_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: ((((((power_power @ A) @ A2) @ N) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 2.48/1.48 thf(fact_90_of__nat__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X:nat,B2:nat,W:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W)) = (((ord_less @ nat) @ X) @ (((power_power @ nat) @ B2) @ W))))))))).
% 2.48/1.48 thf(fact_91_of__nat__less__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W:nat,X:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W)) @ ((semiring_1_of_nat @ A) @ X)) = (((ord_less @ nat) @ (((power_power @ nat) @ B2) @ W)) @ X)))))))).
% 2.48/1.48 thf(fact_92_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less_eq @ A) @ A2) @ B2))))))))))).
% 2.48/1.48 thf(fact_93_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: ((K != ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.48/1.48 thf(fact_94_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: ((K = ((semiring_1_of_nat @ int) @ N2)))))))).
% 2.48/1.48 thf(fact_95_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 2.48/1.48 thf(fact_96_of__nat__mono, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [I2:nat,J:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I2)) @ ((semiring_1_of_nat @ A) @ J))))))))).
% 2.48/1.48 thf(fact_97_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))))).
% 2.48/1.48 thf(fact_98_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.48/1.48 thf(fact_99_real__arch__simple, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X:A]: ((? [N2:nat]: ((((ord_less_eq @ A) @ X) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.48/1.48 thf(fact_100_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ 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))))))))).
% 2.48/1.48 thf(fact_101_power__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,N:nat]: (((((power_power @ A) @ ((inverse_inverse @ A) @ A2)) @ N) = ((inverse_inverse @ A) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.48/1.48 thf(fact_102_power__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ N) = (((divide_divide @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N))))))))).
% 2.48/1.48 thf(fact_103_power__not__zero, axiom, (! [A:$tType]: (((semiri134348788visors @ A) => (! [A2:A,N:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N) != (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_104_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.48/1.48 thf(fact_105_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N:nat]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N)))))))))).
% 2.48/1.48 thf(fact_106_less__eq__int__code_I1_J, axiom, (((ord_less_eq @ int) @ (zero_zero @ int)) @ (zero_zero @ int))).
% 2.48/1.48 thf(fact_107_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.48/1.48 thf(fact_108_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ 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))))))))))).
% 2.48/1.48 thf(fact_109_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ 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))))))))))).
% 2.48/1.48 thf(fact_110_real__arch__inverse, axiom, (! [E2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E2) = (? [N3:nat]: (((N3 != (zero_zero @ nat)) & ((((ord_less @ real) @ (zero_zero @ real)) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N3))) & (((ord_less @ real) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N3))) @ E2))))))))).
% 2.48/1.48 thf(fact_111_forall__pos__mono, axiom, (! [P:(real > $o),E2:real]: (((! [D:real,E:real]: (((((ord_less @ real) @ D) @ E) => ((P @ D) => (P @ E))))) => ((! [N2:nat]: (((N2 != (zero_zero @ nat)) => (P @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N2)))))) => ((((ord_less @ real) @ (zero_zero @ real)) @ E2) => (P @ E2))))))).
% 2.48/1.48 thf(fact_112_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.48/1.48 thf(fact_113_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.48/1.48 thf(fact_114_ge__less__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A4:A,N3:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ A4) => (N3 != X3))))))))))).
% 2.48/1.48 thf(fact_115_less__ge__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [N3:A,A4:A]: ((! [X3:A]: (((((ord_less_eq @ A) @ A4) @ X3) => (N3 != X3))))))))))).
% 2.48/1.48 thf(fact_116_greater__le__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A4:A,N3:A]: ((! [X3:A]: (((((ord_less_eq @ A) @ X3) @ A4) => (N3 != X3))))))))))).
% 2.48/1.48 thf(fact_117_le__greater__neq__conv, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [N3:A,A4:A]: ((! [X3:A]: (((((ord_less @ A) @ A4) @ X3) => (N3 != X3))))))))))).
% 2.48/1.48 thf(fact_118_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_119_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_120_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_121_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.48/1.48 thf(fact_122_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I2:nat,J:nat]: (((! [I3:nat,J2:nat]: (((((ord_less @ nat) @ I3) @ J2) => (((ord_less @ nat) @ (F @ I3)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I2) @ J) => (((ord_less_eq @ nat) @ (F @ I2)) @ (F @ J))))))).
% 2.48/1.48 thf(fact_123_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.48/1.48 thf(fact_124_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.48 thf(fact_125_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less @ nat) @ M3) @ N3) | (M3 = N3)))))).
% 2.48/1.48 thf(fact_126_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.48 thf(fact_127_nat__less__le, axiom, ((ord_less @ nat) = (^ [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) & (M3 != N3)))))).
% 2.48/1.48 thf(fact_128_zero__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N))))))))).
% 2.48/1.48 thf(fact_129_nat__power__less__imp__less, axiom, (! [I2:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I2) => ((((ord_less @ nat) @ (((power_power @ nat) @ I2) @ M)) @ (((power_power @ nat) @ I2) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.48/1.48 thf(fact_130_power__strict__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N:nat]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N)) @ (((power_power @ A) @ B2) @ N))))))))))).
% 2.48/1.48 thf(fact_131_zero__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_132_of__nat__0__le__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)))))))).
% 2.48/1.48 thf(fact_133_ex__least__nat__le, axiom, (! [P:(nat > $o),N:nat]: (((P @ N) => ((~ ((P @ (zero_zero @ nat)))) => (? [K3:nat]: (((((ord_less_eq @ nat) @ K3) @ N) & ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K3) => (~ ((P @ I4)))))) & (P @ K3)))))))))).
% 2.48/1.48 thf(fact_134_count__le__length, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((ord_less_eq @ nat) @ (((count_list @ A) @ Xs) @ X)) @ ((size_size @ (list @ A)) @ Xs))))).
% 2.48/1.48 thf(fact_135_list__slice__length, axiom, (! [A:$tType,Xs:(list @ A),K:nat]: ((((size_size @ (list @ (list @ A))) @ (((listSl1903280966_slice @ A) @ Xs) @ K)) = (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K))))).
% 2.48/1.48 thf(fact_136_inverse__le__iff__le, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_137_inverse__le__iff__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))))).
% 2.48/1.48 thf(fact_138_div__less, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => ((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)))))).
% 2.48/1.48 thf(fact_139_inverse__positive__iff__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.48/1.48 thf(fact_140_inverse__negative__iff__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_141_inverse__less__iff__less__neg, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_142_inverse__eq__iff__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) = (A2 = B2)))))))).
% 2.48/1.48 thf(fact_143_inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2))))))).
% 2.48/1.48 thf(fact_144_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_145_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,C:A,B2:A]: ((((((divide_divide @ A) @ A2) @ C) = (((divide_divide @ A) @ B2) @ C)) = ((C = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.48/1.48 thf(fact_146_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C:A,A2:A,B2:A]: ((((((divide_divide @ A) @ C) @ A2) = (((divide_divide @ A) @ C) @ B2)) = ((C = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.48/1.48 thf(fact_147_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_148_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_149_inverse__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.48/1.48 thf(fact_150_inverse__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ B2) @ A2)))))))).
% 2.48/1.48 thf(fact_151_inverse__nonnegative__iff__nonnegative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.48/1.48 thf(fact_152_inverse__nonpositive__iff__nonpositive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_153_inverse__less__iff__less, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_154_div__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_155_div__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_156_less__eq__real__def, axiom, ((ord_less_eq @ real) = (^ [X3:real,Y3:real]: (((((ord_less @ real) @ X3) @ Y3) | (X3 = Y3)))))).
% 2.48/1.48 thf(fact_157_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.48/1.48 thf(fact_158_le__trans, axiom, (! [I2:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I2) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I2) @ K)))))).
% 2.48/1.48 thf(fact_159_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.48 thf(fact_160_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.48/1.48 thf(fact_161_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.48/1.48 thf(fact_162_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (? [X2:nat]: (((P @ X2) & (! [Y2:nat]: (((P @ Y2) => (((ord_less_eq @ nat) @ Y2) @ X2)))))))))))).
% 2.48/1.48 thf(fact_163_real__of__nat__div4, axiom, (! [N:nat,X:nat]: ((((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ (((divide_divide @ nat) @ N) @ X))) @ (((divide_divide @ real) @ ((semiring_1_of_nat @ real) @ N)) @ ((semiring_1_of_nat @ real) @ X)))))).
% 2.48/1.48 thf(fact_164_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X4))))))))).
% 2.48/1.48 thf(fact_165_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X4:A]: ((? [X_1:A]: ((((ord_less @ A) @ X4) @ X_1))))))))).
% 2.48/1.48 thf(fact_166_inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => (A2 = B2)))))))).
% 2.48/1.48 thf(fact_167_div__le__mono, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N) @ K)))))).
% 2.48/1.48 thf(fact_168_div__le__dividend, axiom, (! [M:nat,N:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N)) @ M)))).
% 2.48/1.48 thf(fact_169_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A2) != (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_170_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2)))))))).
% 2.48/1.48 thf(fact_171_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => ((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (A2 = B2)))))))))).
% 2.48/1.48 thf(fact_172_inverse__zero__imp__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_173_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: (((field @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.48/1.48 thf(fact_174_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 2.48/1.48 thf(fact_175_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B2) @ C)))))))))).
% 2.48/1.48 thf(fact_176_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))))))))))).
% 2.48/1.48 thf(fact_177_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_178_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_179_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_180_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_181_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C)) @ (((divide_divide @ A) @ A2) @ C)))))))))).
% 2.48/1.48 thf(fact_182_divide__neg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_183_divide__neg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_184_divide__pos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_185_divide__pos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_186_divide__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 2.48/1.48 thf(fact_187_divide__less__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B2) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ A2) @ B2)) & (((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2)) & (C != (zero_zero @ A))))))))))).
% 2.48/1.48 thf(fact_188_zero__less__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))).
% 2.48/1.48 thf(fact_189_divide__strict__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B2) @ C)))))))))).
% 2.48/1.48 thf(fact_190_divide__strict__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B2) @ C)))))))))).
% 2.48/1.48 thf(fact_191_inverse__less__imp__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ B2) @ A2))))))))).
% 2.48/1.48 thf(fact_192_less__imp__inverse__less, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_193_inverse__less__imp__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2))))))))).
% 2.48/1.48 thf(fact_194_less__imp__inverse__less__neg, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_195_inverse__negative__imp__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_196_inverse__positive__imp__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))).
% 2.48/1.48 thf(fact_197_negative__imp__inverse__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A))))))))).
% 2.48/1.48 thf(fact_198_positive__imp__inverse__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2))))))))).
% 2.48/1.48 thf(fact_199_Euclidean__Division_Odiv__eq__0__iff, axiom, (! [M:nat,N:nat]: ((((((divide_divide @ nat) @ M) @ N) = (zero_zero @ nat)) = ((((ord_less @ nat) @ M) @ N) | (N = (zero_zero @ nat))))))).
% 2.48/1.48 thf(fact_200_div__greater__zero__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ M) @ N)) = ((((ord_less_eq @ nat) @ N) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 2.48/1.48 thf(fact_201_div__le__mono2, axiom, (! [M:nat,N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ K) @ N)) @ (((divide_divide @ nat) @ K) @ M))))))).
% 2.48/1.48 thf(fact_202_unique__euclidean__semiring__with__nat__class_Oof__nat__div, axiom, (! [A:$tType]: (((euclid1852923125th_nat @ A) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((divide_divide @ nat) @ M) @ N)) = (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.48/1.48 thf(fact_203_divide__nonpos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_204_divide__nonpos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_205_divide__nonneg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X) @ Y)))))))))).
% 2.48/1.48 thf(fact_206_divide__nonneg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_207_divide__le__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C)) @ (((divide_divide @ A) @ B2) @ C)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C) => (((ord_less_eq @ A) @ A2) @ B2)) & ((((ord_less @ A) @ C) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2)))))))))).
% 2.48/1.48 thf(fact_208_frac__less2, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A,W:A,Z:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 2.48/1.48 thf(fact_209_frac__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 2.48/1.48 thf(fact_210_frac__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,X:A,W:A,Z:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W) => ((((ord_less_eq @ A) @ W) @ Z) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X) @ Z)) @ (((divide_divide @ A) @ Y) @ W)))))))))))).
% 2.48/1.48 thf(fact_211_le__imp__inverse__le__neg, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_212_inverse__le__imp__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.48/1.48 thf(fact_213_le__imp__inverse__le, axiom, (! [A:$tType]: (((linordered_field @ 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)))))))))).
% 2.48/1.48 thf(fact_214_inverse__le__imp__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.48/1.48 thf(fact_215_bits__div__by__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_216_bits__div__0, axiom, (! [A:$tType]: (((semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_217_div__by__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_218_div__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.48/1.48 thf(fact_219_complete__real, axiom, (! [S2:(set @ real)]: (((? [X4:real]: ((((member2 @ real) @ X4) @ S2))) => ((? [Z2:real]: ((! [X2:real]: (((((member2 @ real) @ X2) @ S2) => (((ord_less_eq @ real) @ X2) @ Z2)))))) => (? [Y4:real]: (((! [X4:real]: (((((member2 @ real) @ X4) @ S2) => (((ord_less_eq @ real) @ X4) @ Y4)))) & (! [Z2:real]: (((! [X2:real]: (((((member2 @ real) @ X2) @ S2) => (((ord_less_eq @ real) @ X2) @ Z2)))) => (((ord_less_eq @ real) @ Y4) @ Z2)))))))))))).
% 2.48/1.48 thf(fact_220_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.48/1.48 thf(fact_221_nonneg1__imp__zdiv__pos__iff, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = ((((ord_less_eq @ int) @ B2) @ A2) & (((ord_less @ int) @ (zero_zero @ int)) @ B2))))))).
% 2.48/1.48 thf(fact_222_pos__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A2)))))).
% 2.48/1.48 thf(fact_223_div__neg__pos__less0, axiom, (! [A2:int,B2:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_224_neg__imp__zdiv__neg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ (zero_zero @ int)) @ A2)))))).
% 2.48/1.48 thf(fact_225_pos__imp__zdiv__neg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ A2) @ (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_226_zdiv__int, axiom, (! [A2:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ A2) @ B2)) = (((divide_divide @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 2.48/1.48 thf(fact_227_div__positive, axiom, (! [A:$tType]: (((unique1598680935umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)))))))))).
% 2.48/1.48 thf(fact_228_unique__euclidean__semiring__numeral__class_Odiv__less, axiom, (! [A:$tType]: (((unique1598680935umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ B2) => ((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)))))))))).
% 2.48/1.48 thf(fact_229_zdiv__mono1, axiom, (! [A2:int,A5:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ A5) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (((divide_divide @ int) @ A5) @ B2))))))).
% 2.48/1.48 thf(fact_230_zdiv__mono2, axiom, (! [A2:int,B3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ B3) => ((((ord_less_eq @ int) @ B3) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (((divide_divide @ int) @ A2) @ B3)))))))).
% 2.48/1.48 thf(fact_231_zdiv__eq__0__iff, axiom, (! [I2:int,K:int]: ((((((divide_divide @ int) @ I2) @ K) = (zero_zero @ int)) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I2) & (((ord_less @ int) @ I2) @ K)) | ((((ord_less_eq @ int) @ I2) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I2)))))))).
% 2.48/1.48 thf(fact_232_zdiv__mono1__neg, axiom, (! [A2:int,A5:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ A5) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A5) @ B2)) @ (((divide_divide @ int) @ A2) @ B2))))))).
% 2.48/1.48 thf(fact_233_zdiv__mono2__neg, axiom, (! [A2:int,B3:int,B2:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B3) => ((((ord_less_eq @ int) @ B3) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B3)) @ (((divide_divide @ int) @ A2) @ B2)))))))).
% 2.48/1.48 thf(fact_234_div__int__pos__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L)) = ((K = (zero_zero @ int)) | ((L = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)) | ((((ord_less @ int) @ K) @ (zero_zero @ int)) & (((ord_less @ int) @ L) @ (zero_zero @ int)))))))))).
% 2.48/1.48 thf(fact_235_div__positive__int, axiom, (! [L:int,K:int]: (((((ord_less_eq @ int) @ L) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L))))))).
% 2.48/1.48 thf(fact_236_div__nonneg__neg__le0, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_237_div__nonpos__pos__le0, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_238_pos__imp__zdiv__pos__iff, axiom, (! [K:int,I2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ I2) @ K)) = (((ord_less_eq @ int) @ K) @ I2)))))).
% 2.48/1.48 thf(fact_239_neg__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = (((ord_less_eq @ int) @ A2) @ (zero_zero @ int))))))).
% 2.48/1.48 thf(fact_240_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) & (! [Y2:real]: ((((((ord_less @ real) @ (zero_zero @ real)) @ Y2) & ((((power_power @ real) @ Y2) @ N) = A2)) => (Y2 = X2))))))))))))).
% 2.48/1.48 thf(fact_241_realpow__pos__nth, axiom, (! [N:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [R:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) & ((((power_power @ real) @ R) @ N) = A2))))))))).
% 2.48/1.48 thf(fact_242_nat__int__comparison_I3_J, axiom, ((ord_less_eq @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))).
% 2.48/1.48 thf(fact_243_nat__int__comparison_I2_J, axiom, ((ord_less @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))).
% 2.48/1.48 thf(fact_244_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))).
% 2.48/1.48 thf(fact_245_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.48/1.48 thf(fact_246_verit__la__generic, axiom, (! [A2:int,X:int]: (((((ord_less_eq @ int) @ A2) @ X) | ((A2 = X) | (((ord_less_eq @ int) @ X) @ A2)))))).
% 2.48/1.48 thf(fact_247_int__if, axiom, (! [P:$o,A2:nat,B2:nat]: (((P => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B2)) = ((semiring_1_of_nat @ int) @ A2))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B2)) = ((semiring_1_of_nat @ int) @ B2))))))).
% 2.48/1.48 thf(fact_248_nat__int__comparison_I1_J, axiom, ((^ [Y5:nat,Z3:nat]: ((Y5 = Z3))) = (^ [A4:nat,B4:nat]: ((((semiring_1_of_nat @ int) @ A4) = ((semiring_1_of_nat @ int) @ B4)))))).
% 2.48/1.48 thf(fact_249_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B3:B,A5:B]: (((~ ((((ord_less_eq @ B) @ B3) @ A5))) = (((ord_less @ B) @ A5) @ B3)))))))).
% 2.48/1.48 thf(fact_250_int__ops_I1_J, axiom, (((semiring_1_of_nat @ int) @ (zero_zero @ nat)) = (zero_zero @ int))).
% 2.48/1.48 thf(fact_251_div__gr__imp__gr__divisor, axiom, (! [X:nat,N:nat,M:nat]: (((((ord_less @ nat) @ X) @ (((divide_divide @ nat) @ N) @ M)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.48/1.48 thf(fact_252_div__eq__0__conv, axiom, (! [N:nat,M:nat]: ((((((divide_divide @ nat) @ N) @ M) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (((ord_less @ nat) @ N) @ M)))))).
% 2.48/1.48 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: (((order @ A7) => (order @ (A6 > A7)))))).
% 2.48/1.48 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat, axiom, (euclid1852923125th_nat @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1598680935umeral @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri134348788visors @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, (semidom_divide @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Parity_Osemiring__bits, axiom, (semiring_bits @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Orderings_Oorder_1, axiom, (order @ int)).
% 2.48/1.48 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.48/1.48 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_2, axiom, (euclid1852923125th_nat @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_3, axiom, (unique1598680935umeral @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4, axiom, (semiri134348788visors @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5, axiom, (linord1659791738miring @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6, axiom, (linordered_semidom @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Rings_Osemidom__divide_7, axiom, (semidom_divide @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Parity_Osemiring__bits_8, axiom, (semiring_bits @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9, axiom, (semiring_char_0 @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Orderings_Olinorder_10, axiom, (linorder @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Rings_Osemiring__1_11, axiom, (semiring_1 @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Orderings_Oorder_12, axiom, (order @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Groups_Ozero_13, axiom, (zero @ nat)).
% 2.48/1.48 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.48/1.48 thf(tcon_Set_Oset___Orderings_Oorder_14, axiom, (! [A6:$tType]: ((order @ (set @ A6))))).
% 2.48/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder_15, axiom, (linorder @ $o)).
% 2.48/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_16, axiom, (order @ $o)).
% 2.48/1.48 thf(tcon_List_Olist___Nat_Osize_17, axiom, (! [A6:$tType]: ((size @ (list @ A6))))).
% 2.48/1.48 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, (archim1804426504_field @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_18, axiom, (semiri134348788visors @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19, axiom, (linord1659791738miring @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_20, axiom, (linordered_semidom @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Olinordered__idom_21, axiom, (linordered_idom @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Osemidom__divide_22, axiom, (semidom_divide @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_23, axiom, (semiring_char_0 @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Orderings_Olinorder_24, axiom, (linorder @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Rings_Osemiring__1_25, axiom, (semiring_1 @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Orderings_Oorder_26, axiom, (order @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 2.48/1.48 thf(tcon_Real_Oreal___Groups_Ozero_27, axiom, (zero @ real)).
% 2.48/1.48 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.48/1.48 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.48/1.48 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.48/1.48 thf(conj_0, conjecture, ((((listSl1903280966_slice @ a) @ (nil @ a)) @ k) = (nil @ (list @ a)))).
% 2.48/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2DTF7234.p
% 2.48/1.48 ------------------------
% 2.70/1.53 ---- Cleaned THF ---
% 2.70/1.53 thf(ty_t_Option_Ooption,type,
% 2.70/1.53 option: $tType > $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_t_Real_Oreal,type,
% 2.70/1.53 real: $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_t_List_Olist,type,
% 2.70/1.53 list: $tType > $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_t_Set_Oset,type,
% 2.70/1.53 set: $tType > $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_t_Nat_Onat,type,
% 2.70/1.53 nat: $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_t_Int_Oint,type,
% 2.70/1.53 int: $tType ).
% 2.70/1.53
% 2.70/1.53 thf(ty_tf_a,type,
% 2.70/1.53 a: $tType ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_HOL_Otype,type,
% 2.70/1.53 type:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Nat_Osize,type,
% 2.70/1.53 size:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Groups_Ozero,type,
% 2.70/1.53 zero:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Fields_Ofield,type,
% 2.70/1.53 field:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Orderings_Oorder,type,
% 2.70/1.53 order:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Osemiring__1,type,
% 2.70/1.53 semiring_1:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Orderings_Olinorder,type,
% 2.70/1.53 linorder:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.70/1.53 semiring_char_0:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Orderings_Owellorder,type,
% 2.70/1.53 wellorder:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Fields_Odivision__ring,type,
% 2.70/1.53 division_ring:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Parity_Osemiring__bits,type,
% 2.70/1.53 semiring_bits:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Osemidom__divide,type,
% 2.70/1.53 semidom_divide:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.70/1.53 linordered_idom:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Fields_Olinordered__field,type,
% 2.70/1.53 linordered_field:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.70/1.53 linordered_semidom:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.70/1.53 linord1659791738miring:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
% 2.70/1.53 semiri134348788visors:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.70/1.53 archim1804426504_field:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.70/1.53 canoni770627133id_add:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 2.70/1.53 unique1598680935umeral:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
% 2.70/1.53 euclid1852923125th_nat:
% 2.70/1.53 !>[A: $tType] : $o ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 2.70/1.53 inverse_inverse:
% 2.70/1.53 !>[A: $tType] : ( A > A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.70/1.53 zero_zero:
% 2.70/1.53 !>[A: $tType] : A ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_If,type,
% 2.70/1.53 if:
% 2.70/1.53 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice,type,
% 2.70/1.53 listSl1903280966_slice:
% 2.70/1.53 !>[A: $tType] : ( ( list @ A ) > nat > ( list @ ( list @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_ListSlice__Mirabelle__zurbwwxtke_Olist__slice__aux,type,
% 2.70/1.53 listSl1123830174ce_aux:
% 2.70/1.53 !>[A: $tType] : ( ( list @ A ) > nat > nat > ( list @ ( list @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Obind,type,
% 2.70/1.53 bind:
% 2.70/1.53 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Oconcat,type,
% 2.70/1.53 concat:
% 2.70/1.53 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Ocount__list,type,
% 2.70/1.53 count_list:
% 2.70/1.53 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Ogen__length,type,
% 2.70/1.53 gen_length:
% 2.70/1.53 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Olist_ONil,type,
% 2.70/1.53 nil:
% 2.70/1.53 !>[A: $tType] : ( list @ A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Olist__ex1,type,
% 2.70/1.53 list_ex1:
% 2.70/1.53 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Omap__filter,type,
% 2.70/1.53 map_filter:
% 2.70/1.53 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Omaps,type,
% 2.70/1.53 maps:
% 2.70/1.53 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Omember,type,
% 2.70/1.53 member:
% 2.70/1.53 !>[A: $tType] : ( ( list @ A ) > A > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_List_Onull,type,
% 2.70/1.53 null:
% 2.70/1.53 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.70/1.53 semiring_1_of_nat:
% 2.70/1.53 !>[A: $tType] : ( nat > A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.70/1.53 size_size:
% 2.70/1.53 !>[A: $tType] : ( A > nat ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.70/1.53 ord_less:
% 2.70/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.70/1.53 ord_less_eq:
% 2.70/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Power_Opower__class_Opower,type,
% 2.70/1.53 power_power:
% 2.70/1.53 !>[A: $tType] : ( A > nat > A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.70/1.53 divide_divide:
% 2.70/1.53 !>[A: $tType] : ( A > A > A ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_Set_OCollect,type,
% 2.70/1.53 collect:
% 2.70/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_c_member,type,
% 2.70/1.53 member2:
% 2.70/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.70/1.53
% 2.70/1.53 thf(sy_v_k,type,
% 2.70/1.53 k: nat ).
% 2.70/1.53
% 2.70/1.53 thf(fact_0_list__slice__0,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A] :
% 2.70/1.53 ( ( listSl1903280966_slice @ A @ Xs @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( nil @ ( list @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_1_concat_Osimps_I1_J,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 2.70/1.53 = ( nil @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_2_list__ex1__simps_I1_J,axiom,
% 2.70/1.53 ! [A: $tType,P: A > $o] :
% 2.70/1.53 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_3_list__slice__less,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,K: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K )
% 2.70/1.53 => ( ( listSl1903280966_slice @ A @ Xs @ K )
% 2.70/1.53 = ( nil @ ( list @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_4_bind__simps_I1_J,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 2.70/1.53 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 2.70/1.53 = ( nil @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_5_member__rec_I2_J,axiom,
% 2.70/1.53 ! [A: $tType,Y: A] :
% 2.70/1.53 ~ ( member @ A @ ( nil @ A ) @ Y ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_6_gen__length__code_I1_J,axiom,
% 2.70/1.53 ! [A: $tType,N: nat] :
% 2.70/1.53 ( ( gen_length @ A @ N @ ( nil @ A ) )
% 2.70/1.53 = N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_7_maps__simps_I2_J,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 2.70/1.53 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 2.70/1.53 = ( nil @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_8_null__rec_I2_J,axiom,
% 2.70/1.53 ! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_9_eq__Nil__null,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A] :
% 2.70/1.53 ( ( Xs
% 2.70/1.53 = ( nil @ A ) )
% 2.70/1.53 = ( null @ A @ Xs ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_10_map__filter__simps_I2_J,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType,F: B > ( option @ A )] :
% 2.70/1.53 ( ( map_filter @ B @ A @ F @ ( nil @ B ) )
% 2.70/1.53 = ( nil @ A ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_11_length__0__conv,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A] :
% 2.70/1.53 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.70/1.53 = ( zero_zero @ nat ) )
% 2.70/1.53 = ( Xs
% 2.70/1.53 = ( nil @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_12_length__greater__0__conv,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.70/1.53 = ( Xs
% 2.70/1.53 != ( nil @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_13_list_Osize_I3_J,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
% 2.70/1.53 = ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_14_length__code,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( size_size @ ( list @ A ) )
% 2.70/1.53 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_15_length__induct,axiom,
% 2.70/1.53 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.70/1.53 ( ! [Xs2: list @ A] :
% 2.70/1.53 ( ! [Ys: list @ A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.70/1.53 => ( P @ Ys ) )
% 2.70/1.53 => ( P @ Xs2 ) )
% 2.70/1.53 => ( P @ Xs ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_16_Ex__list__of__length,axiom,
% 2.70/1.53 ! [A: $tType,N: nat] :
% 2.70/1.53 ? [Xs2: list @ A] :
% 2.70/1.53 ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.70/1.53 = N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_17_neq__if__length__neq,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.70/1.53 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.70/1.53 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.70/1.53 => ( Xs != Ys2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_18_neq0__conv,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ( ( N
% 2.70/1.53 != ( zero_zero @ nat ) )
% 2.70/1.53 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_19_less__nat__zero__code,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_20_bot__nat__0_Onot__eq__extremum,axiom,
% 2.70/1.53 ! [A2: nat] :
% 2.70/1.53 ( ( A2
% 2.70/1.53 != ( zero_zero @ nat ) )
% 2.70/1.53 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_21_not__gr__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [N: A] :
% 2.70/1.53 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.70/1.53 = ( N
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_22_length__greater__imp__not__empty,axiom,
% 2.70/1.53 ! [A: $tType,N: nat,Xs: list @ A] :
% 2.70/1.53 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 2.70/1.53 => ( Xs
% 2.70/1.53 != ( nil @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_23_gr0I,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ( ( N
% 2.70/1.53 != ( zero_zero @ nat ) )
% 2.70/1.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_24_not__gr0,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.70/1.53 = ( N
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_25_not__less0,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_26_less__zeroE,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_27_gr__implies__not0,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 => ( N
% 2.70/1.53 != ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_28_zero__reorient,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( zero @ A )
% 2.70/1.53 => ! [X: A] :
% 2.70/1.53 ( ( ( zero_zero @ A )
% 2.70/1.53 = X )
% 2.70/1.53 = ( X
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_29_measure__induct__rule,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType] :
% 2.70/1.53 ( ( wellorder @ B )
% 2.70/1.53 => ! [F: A > B,P: A > $o,A2: A] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ! [Y2: A] :
% 2.70/1.53 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.70/1.53 => ( P @ Y2 ) )
% 2.70/1.53 => ( P @ X2 ) )
% 2.70/1.53 => ( P @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_30_measure__induct,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType] :
% 2.70/1.53 ( ( wellorder @ B )
% 2.70/1.53 => ! [F: A > B,P: A > $o,A2: A] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ! [Y2: A] :
% 2.70/1.53 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.70/1.53 => ( P @ Y2 ) )
% 2.70/1.53 => ( P @ X2 ) )
% 2.70/1.53 => ( P @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_31_infinite__descent__measure,axiom,
% 2.70/1.53 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ~ ( P @ X2 )
% 2.70/1.53 => ? [Y2: A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X2 ) )
% 2.70/1.53 & ~ ( P @ Y2 ) ) )
% 2.70/1.53 => ( P @ X ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_32_linorder__neqE__nat,axiom,
% 2.70/1.53 ! [X: nat,Y: nat] :
% 2.70/1.53 ( ( X != Y )
% 2.70/1.53 => ( ~ ( ord_less @ nat @ X @ Y )
% 2.70/1.53 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_33_infinite__descent,axiom,
% 2.70/1.53 ! [P: nat > $o,N: nat] :
% 2.70/1.53 ( ! [N2: nat] :
% 2.70/1.53 ( ~ ( P @ N2 )
% 2.70/1.53 => ? [M2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M2 @ N2 )
% 2.70/1.53 & ~ ( P @ M2 ) ) )
% 2.70/1.53 => ( P @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_34_nat__less__induct,axiom,
% 2.70/1.53 ! [P: nat > $o,N: nat] :
% 2.70/1.53 ( ! [N2: nat] :
% 2.70/1.53 ( ! [M2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M2 @ N2 )
% 2.70/1.53 => ( P @ M2 ) )
% 2.70/1.53 => ( P @ N2 ) )
% 2.70/1.53 => ( P @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_35_less__irrefl__nat,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ N @ N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_36_less__not__refl3,axiom,
% 2.70/1.53 ! [S: nat,T: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ S @ T )
% 2.70/1.53 => ( S != T ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_37_less__not__refl2,axiom,
% 2.70/1.53 ! [N: nat,M: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ N @ M )
% 2.70/1.53 => ( M != N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_38_less__not__refl,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ N @ N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_39_nat__neq__iff,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( M != N )
% 2.70/1.53 = ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_40_size__neq__size__imp__neq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( size @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ( size_size @ A @ X )
% 2.70/1.53 != ( size_size @ A @ Y ) )
% 2.70/1.53 => ( X != Y ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_41_zero__less__iff__neq__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [N: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.70/1.53 = ( N
% 2.70/1.53 != ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_42_mem__Collect__eq,axiom,
% 2.70/1.53 ! [A: $tType,A2: A,P: A > $o] :
% 2.70/1.53 ( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
% 2.70/1.53 = ( P @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_43_Collect__mem__eq,axiom,
% 2.70/1.53 ! [A: $tType,A3: set @ A] :
% 2.70/1.53 ( ( collect @ A
% 2.70/1.53 @ ^ [X3: A] : ( member2 @ A @ X3 @ A3 ) )
% 2.70/1.53 = A3 ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_44_Collect__cong,axiom,
% 2.70/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ( P @ X2 )
% 2.70/1.53 = ( Q @ X2 ) )
% 2.70/1.53 => ( ( collect @ A @ P )
% 2.70/1.53 = ( collect @ A @ Q ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_45_ext,axiom,
% 2.70/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ( F @ X2 )
% 2.70/1.53 = ( G @ X2 ) )
% 2.70/1.53 => ( F = G ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_46_gr__implies__not__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [M: A,N: A] :
% 2.70/1.53 ( ( ord_less @ A @ M @ N )
% 2.70/1.53 => ( N
% 2.70/1.53 != ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_47_not__less__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [N: A] :
% 2.70/1.53 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_48_gr__zeroI,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [N: A] :
% 2.70/1.53 ( ( N
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_49_infinite__descent0__measure,axiom,
% 2.70/1.53 ! [A: $tType,V: A > nat,P: A > $o,X: A] :
% 2.70/1.53 ( ! [X2: A] :
% 2.70/1.53 ( ( ( V @ X2 )
% 2.70/1.53 = ( zero_zero @ nat ) )
% 2.70/1.53 => ( P @ X2 ) )
% 2.70/1.53 => ( ! [X2: A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
% 2.70/1.53 => ( ~ ( P @ X2 )
% 2.70/1.53 => ? [Y2: A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X2 ) )
% 2.70/1.53 & ~ ( P @ Y2 ) ) ) )
% 2.70/1.53 => ( P @ X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_50_bot__nat__0_Oextremum__strict,axiom,
% 2.70/1.53 ! [A2: nat] :
% 2.70/1.53 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_51_infinite__descent0,axiom,
% 2.70/1.53 ! [P: nat > $o,N: nat] :
% 2.70/1.53 ( ( P @ ( zero_zero @ nat ) )
% 2.70/1.53 => ( ! [N2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.70/1.53 => ( ~ ( P @ N2 )
% 2.70/1.53 => ? [M2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M2 @ N2 )
% 2.70/1.53 & ~ ( P @ M2 ) ) ) )
% 2.70/1.53 => ( P @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_52_gr__implies__gr0,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_53_field__lbound__gt__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [D1: A,D2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
% 2.70/1.53 => ? [E: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 2.70/1.53 & ( ord_less @ A @ E @ D1 )
% 2.70/1.53 & ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_54_less__numeral__extra_I3_J,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_55_list__slice__aux_Osimps_I1_J,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,K: nat] :
% 2.70/1.53 ( ( listSl1123830174ce_aux @ A @ Xs @ K @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( nil @ ( list @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_56_of__nat__0__less__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_57_count__list_Osimps_I1_J,axiom,
% 2.70/1.53 ! [A: $tType,Y: A] :
% 2.70/1.53 ( ( count_list @ A @ ( nil @ A ) @ Y )
% 2.70/1.53 = ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_58_interval__induct__rule,axiom,
% 2.70/1.53 ! [I: set @ nat,P: nat > $o,A2: nat] :
% 2.70/1.53 ( ! [X2: nat] :
% 2.70/1.53 ( ! [Y2: nat] :
% 2.70/1.53 ( ( ( member2 @ nat @ X2 @ I )
% 2.70/1.53 & ( member2 @ nat @ Y2 @ I )
% 2.70/1.53 & ( ord_less @ nat @ Y2 @ X2 ) )
% 2.70/1.53 => ( P @ Y2 ) )
% 2.70/1.53 => ( P @ X2 ) )
% 2.70/1.53 => ( P @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_59_interval__induct,axiom,
% 2.70/1.53 ! [I: set @ nat,P: nat > $o,A2: nat] :
% 2.70/1.53 ( ! [X2: nat] :
% 2.70/1.53 ( ! [Y2: nat] :
% 2.70/1.53 ( ( ( member2 @ nat @ X2 @ I )
% 2.70/1.53 & ( member2 @ nat @ Y2 @ I )
% 2.70/1.53 & ( ord_less @ nat @ Y2 @ X2 ) )
% 2.70/1.53 => ( P @ Y2 ) )
% 2.70/1.53 => ( P @ X2 ) )
% 2.70/1.53 => ( P @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_60_zero__natural_Orsp,axiom,
% 2.70/1.53 ( ( zero_zero @ nat )
% 2.70/1.53 = ( zero_zero @ nat ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_61_of__nat__eq__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_char_0 @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ( semiring_1_of_nat @ A @ M )
% 2.70/1.53 = ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 = ( M = N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_62_of__nat__0,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_1 @ A )
% 2.70/1.53 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_63_of__nat__0__eq__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_char_0 @ A )
% 2.70/1.53 => ! [N: nat] :
% 2.70/1.53 ( ( ( zero_zero @ A )
% 2.70/1.53 = ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 = ( ( zero_zero @ nat )
% 2.70/1.53 = N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_64_of__nat__eq__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_char_0 @ A )
% 2.70/1.53 => ! [M: nat] :
% 2.70/1.53 ( ( ( semiring_1_of_nat @ A @ M )
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 = ( M
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_65_of__nat__less__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 = ( ord_less @ nat @ M @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_66_list__slice__aux__length,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,K: nat,N: nat] :
% 2.70/1.53 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1123830174ce_aux @ A @ Xs @ K @ N ) )
% 2.70/1.53 = N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_67_of__nat__less__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat] :
% 2.70/1.53 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_68_less__imp__of__nat__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_69_of__nat__less__imp__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_70_Collect__minI,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( wellorder @ A )
% 2.70/1.53 => ! [K: A,I: set @ A,P: A > $o] :
% 2.70/1.53 ( ( member2 @ A @ K @ I )
% 2.70/1.53 => ( ( P @ K )
% 2.70/1.53 => ? [X2: A] :
% 2.70/1.53 ( ( member2 @ A @ X2 @ I )
% 2.70/1.53 & ( P @ X2 )
% 2.70/1.53 & ! [Xa: A] :
% 2.70/1.53 ( ( member2 @ A @ Xa @ I )
% 2.70/1.53 => ( ( ord_less @ A @ Xa @ X2 )
% 2.70/1.53 => ~ ( P @ Xa ) ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_71_Collect__minI__ex,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( wellorder @ A )
% 2.70/1.53 => ! [I: set @ A,P: A > $o] :
% 2.70/1.53 ( ? [X4: A] :
% 2.70/1.53 ( ( member2 @ A @ X4 @ I )
% 2.70/1.53 & ( P @ X4 ) )
% 2.70/1.53 => ? [X2: A] :
% 2.70/1.53 ( ( member2 @ A @ X2 @ I )
% 2.70/1.53 & ( P @ X2 )
% 2.70/1.53 & ! [Xa: A] :
% 2.70/1.53 ( ( member2 @ A @ Xa @ I )
% 2.70/1.53 => ( ( ord_less @ A @ Xa @ X2 )
% 2.70/1.53 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_72_zero__less__imp__eq__int,axiom,
% 2.70/1.53 ! [K: int] :
% 2.70/1.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ? [N2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.70/1.53 & ( K
% 2.70/1.53 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_73_pos__int__cases,axiom,
% 2.70/1.53 ! [K: int] :
% 2.70/1.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ~ ! [N2: nat] :
% 2.70/1.53 ( ( K
% 2.70/1.53 = ( semiring_1_of_nat @ int @ N2 ) )
% 2.70/1.53 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_74_reals__Archimedean2,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( archim1804426504_field @ A )
% 2.70/1.53 => ! [X: A] :
% 2.70/1.53 ? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_75_list__slice__def,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( listSl1903280966_slice @ A )
% 2.70/1.53 = ( ^ [Xs3: list @ A,K2: nat] : ( listSl1123830174ce_aux @ A @ Xs3 @ K2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ K2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_76_of__nat__zero__less__power__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [X: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
% 2.70/1.53 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.70/1.53 | ( N
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_77_of__nat__le__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 2.70/1.53 = ( M
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_78_ex__inverse__of__nat__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( archim1804426504_field @ A )
% 2.70/1.53 => ! [X: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ? [N2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.70/1.53 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_79_bot__nat__0_Oextremum,axiom,
% 2.70/1.53 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_80_le0,axiom,
% 2.70/1.53 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_81_le__zero__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [N: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.70/1.53 = ( N
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_82_of__nat__le__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.70/1.53 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_83_nat__zero__less__power__iff,axiom,
% 2.70/1.53 ! [X: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
% 2.70/1.53 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
% 2.70/1.53 | ( N
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_84_of__nat__power,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_1 @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
% 2.70/1.53 = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_85_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_char_0 @ A )
% 2.70/1.53 => ! [B2: nat,W: nat,X: nat] :
% 2.70/1.53 ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
% 2.70/1.53 = ( semiring_1_of_nat @ A @ X ) )
% 2.70/1.53 = ( ( power_power @ nat @ B2 @ W )
% 2.70/1.53 = X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_86_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_char_0 @ A )
% 2.70/1.53 => ! [X: nat,B2: nat,W: nat] :
% 2.70/1.53 ( ( ( semiring_1_of_nat @ A @ X )
% 2.70/1.53 = ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 2.70/1.53 = ( X
% 2.70/1.53 = ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_87_of__nat__le__of__nat__power__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [B2: nat,W: nat,X: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.70/1.53 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_88_of__nat__power__le__of__nat__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [X: nat,B2: nat,W: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 2.70/1.53 = ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_89_power__eq__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiri134348788visors @ A )
% 2.70/1.53 => ! [A2: A,N: nat] :
% 2.70/1.53 ( ( ( power_power @ A @ A2 @ N )
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 = ( ( A2
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_90_of__nat__power__less__of__nat__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [X: nat,B2: nat,W: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
% 2.70/1.53 = ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_91_of__nat__less__of__nat__power__cancel__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [B2: nat,W: nat,X: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
% 2.70/1.53 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_92_power__mono__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,B2: A,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 2.70/1.53 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_93_nonneg__int__cases,axiom,
% 2.70/1.53 ! [K: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ~ ! [N2: nat] :
% 2.70/1.53 ( K
% 2.70/1.53 != ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_94_zero__le__imp__eq__int,axiom,
% 2.70/1.53 ! [K: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ? [N2: nat] :
% 2.70/1.53 ( K
% 2.70/1.53 = ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_95_zero__integer_Orsp,axiom,
% 2.70/1.53 ( ( zero_zero @ int )
% 2.70/1.53 = ( zero_zero @ int ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_96_of__nat__mono,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [I2: nat,J: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.70/1.53 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_97_zle__int,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
% 2.70/1.53 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_98_int__int__eq,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ( semiring_1_of_nat @ int @ M )
% 2.70/1.53 = ( semiring_1_of_nat @ int @ N ) )
% 2.70/1.53 = ( M = N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_99_real__arch__simple,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( archim1804426504_field @ A )
% 2.70/1.53 => ! [X: A] :
% 2.70/1.53 ? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_100_power__less__imp__less__base,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,N: nat,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_101_power__inverse,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A,N: nat] :
% 2.70/1.53 ( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
% 2.70/1.53 = ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_102_power__divide,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ! [A2: A,B2: A,N: nat] :
% 2.70/1.53 ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
% 2.70/1.53 = ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_103_power__not__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiri134348788visors @ A )
% 2.70/1.53 => ! [A2: A,N: nat] :
% 2.70/1.53 ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ( power_power @ A @ A2 @ N )
% 2.70/1.53 != ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_104_zero__le__power,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_105_power__mono,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,B2: A,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_106_less__eq__int__code_I1_J,axiom,
% 2.70/1.53 ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_107_less__int__code_I1_J,axiom,
% 2.70/1.53 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_108_power__eq__imp__eq__base,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,N: nat,B2: A] :
% 2.70/1.53 ( ( ( power_power @ A @ A2 @ N )
% 2.70/1.53 = ( power_power @ A @ B2 @ N ) )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( A2 = B2 ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_109_power__eq__iff__eq__base,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [N: nat,A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ( power_power @ A @ A2 @ N )
% 2.70/1.53 = ( power_power @ A @ B2 @ N ) )
% 2.70/1.53 = ( A2 = B2 ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_110_real__arch__inverse,axiom,
% 2.70/1.53 ! [E2: real] :
% 2.70/1.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 2.70/1.53 = ( ? [N3: nat] :
% 2.70/1.53 ( ( N3
% 2.70/1.53 != ( zero_zero @ nat ) )
% 2.70/1.53 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) )
% 2.70/1.53 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ E2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_111_forall__pos__mono,axiom,
% 2.70/1.53 ! [P: real > $o,E2: real] :
% 2.70/1.53 ( ! [D: real,E: real] :
% 2.70/1.53 ( ( ord_less @ real @ D @ E )
% 2.70/1.53 => ( ( P @ D )
% 2.70/1.53 => ( P @ E ) ) )
% 2.70/1.53 => ( ! [N2: nat] :
% 2.70/1.53 ( ( N2
% 2.70/1.53 != ( zero_zero @ nat ) )
% 2.70/1.53 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
% 2.70/1.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 2.70/1.53 => ( P @ E2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_112_zero__le,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( canoni770627133id_add @ A )
% 2.70/1.53 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_113_le__numeral__extra_I3_J,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_114_ge__less__neq__conv,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linorder @ A )
% 2.70/1.53 => ( ( ord_less_eq @ A )
% 2.70/1.53 = ( ^ [A4: A,N3: A] :
% 2.70/1.53 ! [X3: A] :
% 2.70/1.53 ( ( ord_less @ A @ X3 @ A4 )
% 2.70/1.53 => ( N3 != X3 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_115_less__ge__neq__conv,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linorder @ A )
% 2.70/1.53 => ( ( ord_less @ A )
% 2.70/1.53 = ( ^ [N3: A,A4: A] :
% 2.70/1.53 ! [X3: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A4 @ X3 )
% 2.70/1.53 => ( N3 != X3 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_116_greater__le__neq__conv,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linorder @ A )
% 2.70/1.53 => ( ( ord_less @ A )
% 2.70/1.53 = ( ^ [A4: A,N3: A] :
% 2.70/1.53 ! [X3: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ X3 @ A4 )
% 2.70/1.53 => ( N3 != X3 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_117_le__greater__neq__conv,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linorder @ A )
% 2.70/1.53 => ( ( ord_less_eq @ A )
% 2.70/1.53 = ( ^ [N3: A,A4: A] :
% 2.70/1.53 ! [X3: A] :
% 2.70/1.53 ( ( ord_less @ A @ A4 @ X3 )
% 2.70/1.53 => ( N3 != X3 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_118_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.70/1.53 ! [A2: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.70/1.53 => ( A2
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_119_bot__nat__0_Oextremum__unique,axiom,
% 2.70/1.53 ! [A2: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( A2
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_120_le__0__eq,axiom,
% 2.70/1.53 ! [N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( N
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_121_less__eq__nat_Osimps_I1_J,axiom,
% 2.70/1.53 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_122_less__mono__imp__le__mono,axiom,
% 2.70/1.53 ! [F: nat > nat,I2: nat,J: nat] :
% 2.70/1.53 ( ! [I3: nat,J2: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ I3 @ J2 )
% 2.70/1.53 => ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 2.70/1.53 => ( ( ord_less_eq @ nat @ I2 @ J )
% 2.70/1.53 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_123_le__neq__implies__less,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ M @ N )
% 2.70/1.53 => ( ( M != N )
% 2.70/1.53 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_124_less__or__eq__imp__le,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 | ( M = N ) )
% 2.70/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_125_le__eq__less__or__eq,axiom,
% 2.70/1.53 ( ( ord_less_eq @ nat )
% 2.70/1.53 = ( ^ [M3: nat,N3: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M3 @ N3 )
% 2.70/1.53 | ( M3 = N3 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_126_less__imp__le__nat,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_127_nat__less__le,axiom,
% 2.70/1.53 ( ( ord_less @ nat )
% 2.70/1.53 = ( ^ [M3: nat,N3: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ M3 @ N3 )
% 2.70/1.53 & ( M3 != N3 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_128_zero__less__power,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_129_nat__power__less__imp__less,axiom,
% 2.70/1.53 ! [I2: nat,M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
% 2.70/1.53 => ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N ) )
% 2.70/1.53 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_130_power__strict__mono,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_semidom @ A )
% 2.70/1.53 => ! [A2: A,B2: A,N: nat] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_131_zero__power,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_1 @ A )
% 2.70/1.53 => ! [N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_132_of__nat__0__le__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linord1659791738miring @ A )
% 2.70/1.53 => ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_133_ex__least__nat__le,axiom,
% 2.70/1.53 ! [P: nat > $o,N: nat] :
% 2.70/1.53 ( ( P @ N )
% 2.70/1.53 => ( ~ ( P @ ( zero_zero @ nat ) )
% 2.70/1.53 => ? [K3: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ K3 @ N )
% 2.70/1.53 & ! [I4: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ I4 @ K3 )
% 2.70/1.53 => ~ ( P @ I4 ) )
% 2.70/1.53 & ( P @ K3 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_134_count__le__length,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_135_list__slice__length,axiom,
% 2.70/1.53 ! [A: $tType,Xs: list @ A,K: nat] :
% 2.70/1.53 ( ( size_size @ ( list @ ( list @ A ) ) @ ( listSl1903280966_slice @ A @ Xs @ K ) )
% 2.70/1.53 = ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_136_inverse__le__iff__le,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_137_inverse__le__iff__le__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_138_div__less,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 => ( ( divide_divide @ nat @ M @ N )
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_139_inverse__positive__iff__positive,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.70/1.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_140_inverse__negative__iff__negative,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.70/1.53 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_141_inverse__less__iff__less__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_142_inverse__eq__iff__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 = ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 = ( A2 = B2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_143_inverse__inverse__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 2.70/1.53 = A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_144_division__ring__divide__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_145_divide__cancel__right,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ! [A2: A,C: A,B2: A] :
% 2.70/1.53 ( ( ( divide_divide @ A @ A2 @ C )
% 2.70/1.53 = ( divide_divide @ A @ B2 @ C ) )
% 2.70/1.53 = ( ( C
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 | ( A2 = B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_146_divide__cancel__left,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ! [C: A,A2: A,B2: A] :
% 2.70/1.53 ( ( ( divide_divide @ A @ C @ A2 )
% 2.70/1.53 = ( divide_divide @ A @ C @ B2 ) )
% 2.70/1.53 = ( ( C
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 | ( A2 = B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_147_divide__eq__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ( divide_divide @ A @ A2 @ B2 )
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 = ( ( A2
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 | ( B2
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_148_inverse__nonzero__iff__nonzero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 = ( A2
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_149_inverse__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_150_inverse__divide,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
% 2.70/1.53 = ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_151_inverse__nonnegative__iff__nonnegative,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.70/1.53 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_152_inverse__nonpositive__iff__nonpositive,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.70/1.53 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_153_inverse__less__iff__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_154_div__neg__neg__trivial,axiom,
% 2.70/1.53 ! [K: int,L: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less @ int @ L @ K )
% 2.70/1.53 => ( ( divide_divide @ int @ K @ L )
% 2.70/1.53 = ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_155_div__pos__pos__trivial,axiom,
% 2.70/1.53 ! [K: int,L: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ( ( ord_less @ int @ K @ L )
% 2.70/1.53 => ( ( divide_divide @ int @ K @ L )
% 2.70/1.53 = ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_156_less__eq__real__def,axiom,
% 2.70/1.53 ( ( ord_less_eq @ real )
% 2.70/1.53 = ( ^ [X3: real,Y3: real] :
% 2.70/1.53 ( ( ord_less @ real @ X3 @ Y3 )
% 2.70/1.53 | ( X3 = Y3 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_157_le__refl,axiom,
% 2.70/1.53 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_158_le__trans,axiom,
% 2.70/1.53 ! [I2: nat,J: nat,K: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ I2 @ J )
% 2.70/1.53 => ( ( ord_less_eq @ nat @ J @ K )
% 2.70/1.53 => ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_159_eq__imp__le,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( M = N )
% 2.70/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_160_le__antisym,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ M @ N )
% 2.70/1.53 => ( ( ord_less_eq @ nat @ N @ M )
% 2.70/1.53 => ( M = N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_161_nat__le__linear,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ M @ N )
% 2.70/1.53 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_162_Nat_Oex__has__greatest__nat,axiom,
% 2.70/1.53 ! [P: nat > $o,K: nat,B2: nat] :
% 2.70/1.53 ( ( P @ K )
% 2.70/1.53 => ( ! [Y4: nat] :
% 2.70/1.53 ( ( P @ Y4 )
% 2.70/1.53 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 2.70/1.53 => ? [X2: nat] :
% 2.70/1.53 ( ( P @ X2 )
% 2.70/1.53 & ! [Y2: nat] :
% 2.70/1.53 ( ( P @ Y2 )
% 2.70/1.53 => ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_163_real__of__nat__div4,axiom,
% 2.70/1.53 ! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_164_linordered__field__no__lb,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X4: A] :
% 2.70/1.53 ? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_165_linordered__field__no__ub,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X4: A] :
% 2.70/1.53 ? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_166_inverse__eq__imp__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 = ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( A2 = B2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_167_div__le__mono,axiom,
% 2.70/1.53 ! [M: nat,N: nat,K: nat] :
% 2.70/1.53 ( ( ord_less_eq @ nat @ M @ N )
% 2.70/1.53 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_168_div__le__dividend,axiom,
% 2.70/1.53 ! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_169_nonzero__imp__inverse__nonzero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 != ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_170_nonzero__inverse__inverse__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 2.70/1.53 = A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_171_nonzero__inverse__eq__imp__eq,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 = ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ( B2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( A2 = B2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_172_inverse__zero__imp__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( division_ring @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ( inverse_inverse @ A @ A2 )
% 2.70/1.53 = ( zero_zero @ A ) )
% 2.70/1.53 => ( A2
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_173_field__class_Ofield__inverse__zero,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( field @ A )
% 2.70/1.53 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_174_divide__le__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 2.70/1.53 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
% 2.70/1.53 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_175_divide__right__mono,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A,C: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_176_zero__le__divide__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
% 2.70/1.53 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
% 2.70/1.53 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_177_divide__nonneg__nonneg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_178_divide__nonneg__nonpos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_179_divide__nonpos__nonneg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_180_divide__nonpos__nonpos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_181_divide__right__mono__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A,C: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C ) @ ( divide_divide @ A @ A2 @ C ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_182_divide__neg__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_183_divide__neg__pos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_184_divide__pos__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_185_divide__pos__pos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_186_divide__less__0__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 2.70/1.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 2.70/1.53 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_187_divide__less__cancel,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,C: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) )
% 2.70/1.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.70/1.53 => ( ord_less @ A @ A2 @ B2 ) )
% 2.70/1.53 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ B2 @ A2 ) )
% 2.70/1.53 & ( C
% 2.70/1.53 != ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_188_zero__less__divide__iff,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
% 2.70/1.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 2.70/1.53 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_189_divide__strict__right__mono,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A,C: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_190_divide__strict__right__mono__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [B2: A,A2: A,C: A] :
% 2.70/1.53 ( ( ord_less @ A @ B2 @ A2 )
% 2.70/1.53 => ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_191_inverse__less__imp__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_192_less__imp__inverse__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_193_inverse__less__imp__less__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_194_less__imp__inverse__less__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_195_inverse__negative__imp__negative,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_196_inverse__positive__imp__positive,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.70/1.53 => ( ( A2
% 2.70/1.53 != ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_197_negative__imp__inverse__negative,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_198_positive__imp__inverse__positive,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_199_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ( divide_divide @ nat @ M @ N )
% 2.70/1.53 = ( zero_zero @ nat ) )
% 2.70/1.53 = ( ( ord_less @ nat @ M @ N )
% 2.70/1.53 | ( N
% 2.70/1.53 = ( zero_zero @ nat ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_200_div__greater__zero__iff,axiom,
% 2.70/1.53 ! [M: nat,N: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
% 2.70/1.53 = ( ( ord_less_eq @ nat @ N @ M )
% 2.70/1.53 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_201_div__le__mono2,axiom,
% 2.70/1.53 ! [M: nat,N: nat,K: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.70/1.53 => ( ( ord_less_eq @ nat @ M @ N )
% 2.70/1.53 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_202_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( euclid1852923125th_nat @ A )
% 2.70/1.53 => ! [M: nat,N: nat] :
% 2.70/1.53 ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
% 2.70/1.53 = ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_203_divide__nonpos__pos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_204_divide__nonpos__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_205_divide__nonneg__pos,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_206_divide__nonneg__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_207_divide__le__cancel,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,C: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C ) @ ( divide_divide @ A @ B2 @ C ) )
% 2.70/1.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
% 2.70/1.53 => ( ord_less_eq @ A @ A2 @ B2 ) )
% 2.70/1.53 & ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_208_frac__less2,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A,W: A,Z: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.70/1.53 => ( ( ord_less @ A @ W @ Z )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_209_frac__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [X: A,Y: A,W: A,Z: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 2.70/1.53 => ( ( ord_less @ A @ X @ Y )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.70/1.53 => ( ( ord_less_eq @ A @ W @ Z )
% 2.70/1.53 => ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_210_frac__le,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [Y: A,X: A,W: A,Z: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 2.70/1.53 => ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
% 2.70/1.53 => ( ( ord_less_eq @ A @ W @ Z )
% 2.70/1.53 => ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W ) ) ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_211_le__imp__inverse__le__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_212_inverse__le__imp__le__neg,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.70/1.53 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_213_le__imp__inverse__le,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_214_inverse__le__imp__le,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_field @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.70/1.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_215_bits__div__by__0,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_bits @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_216_bits__div__0,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semiring_bits @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_217_div__by__0,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semidom_divide @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_218_div__0,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( semidom_divide @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_219_complete__real,axiom,
% 2.70/1.53 ! [S2: set @ real] :
% 2.70/1.53 ( ? [X4: real] : ( member2 @ real @ X4 @ S2 )
% 2.70/1.53 => ( ? [Z2: real] :
% 2.70/1.53 ! [X2: real] :
% 2.70/1.53 ( ( member2 @ real @ X2 @ S2 )
% 2.70/1.53 => ( ord_less_eq @ real @ X2 @ Z2 ) )
% 2.70/1.53 => ? [Y4: real] :
% 2.70/1.53 ( ! [X4: real] :
% 2.70/1.53 ( ( member2 @ real @ X4 @ S2 )
% 2.70/1.53 => ( ord_less_eq @ real @ X4 @ Y4 ) )
% 2.70/1.53 & ! [Z2: real] :
% 2.70/1.53 ( ! [X2: real] :
% 2.70/1.53 ( ( member2 @ real @ X2 @ S2 )
% 2.70/1.53 => ( ord_less_eq @ real @ X2 @ Z2 ) )
% 2.70/1.53 => ( ord_less_eq @ real @ Y4 @ Z2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_220_linorder__neqE__linordered__idom,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linordered_idom @ A )
% 2.70/1.53 => ! [X: A,Y: A] :
% 2.70/1.53 ( ( X != Y )
% 2.70/1.53 => ( ~ ( ord_less @ A @ X @ Y )
% 2.70/1.53 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_221_nonneg1__imp__zdiv__pos__iff,axiom,
% 2.70/1.53 ! [A2: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 2.70/1.53 = ( ( ord_less_eq @ int @ B2 @ A2 )
% 2.70/1.53 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_222_pos__imp__zdiv__nonneg__iff,axiom,
% 2.70/1.53 ! [B2: int,A2: int] :
% 2.70/1.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 2.70/1.53 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_223_div__neg__pos__less0,axiom,
% 2.70/1.53 ! [A2: int,B2: int] :
% 2.70/1.53 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 2.70/1.53 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_224_neg__imp__zdiv__neg__iff,axiom,
% 2.70/1.53 ! [B2: int,A2: int] :
% 2.70/1.53 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
% 2.70/1.53 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_225_pos__imp__zdiv__neg__iff,axiom,
% 2.70/1.53 ! [B2: int,A2: int] :
% 2.70/1.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 2.70/1.53 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
% 2.70/1.53 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_226_zdiv__int,axiom,
% 2.70/1.53 ! [A2: nat,B2: nat] :
% 2.70/1.53 ( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
% 2.70/1.53 = ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_227_div__positive,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( unique1598680935umeral @ A )
% 2.70/1.53 => ! [B2: A,A2: A] :
% 2.70/1.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.70/1.53 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.70/1.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_228_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( unique1598680935umeral @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ A @ A2 @ B2 )
% 2.70/1.53 => ( ( divide_divide @ A @ A2 @ B2 )
% 2.70/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_229_zdiv__mono1,axiom,
% 2.70/1.53 ! [A2: int,A5: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ A2 @ A5 )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A5 @ B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_230_zdiv__mono2,axiom,
% 2.70/1.53 ! [A2: int,B3: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
% 2.70/1.53 => ( ( ord_less_eq @ int @ B3 @ B2 )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_231_zdiv__eq__0__iff,axiom,
% 2.70/1.53 ! [I2: int,K: int] :
% 2.70/1.53 ( ( ( divide_divide @ int @ I2 @ K )
% 2.70/1.53 = ( zero_zero @ int ) )
% 2.70/1.53 = ( ( K
% 2.70/1.53 = ( zero_zero @ int ) )
% 2.70/1.53 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
% 2.70/1.53 & ( ord_less @ int @ I2 @ K ) )
% 2.70/1.53 | ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
% 2.70/1.53 & ( ord_less @ int @ K @ I2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_232_zdiv__mono1__neg,axiom,
% 2.70/1.53 ! [A2: int,A5: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ A2 @ A5 )
% 2.70/1.53 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A5 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_233_zdiv__mono2__neg,axiom,
% 2.70/1.53 ! [A2: int,B3: int,B2: int] :
% 2.70/1.53 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
% 2.70/1.53 => ( ( ord_less_eq @ int @ B3 @ B2 )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B3 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_234_div__int__pos__iff,axiom,
% 2.70/1.53 ! [K: int,L: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
% 2.70/1.53 = ( ( K
% 2.70/1.53 = ( zero_zero @ int ) )
% 2.70/1.53 | ( L
% 2.70/1.53 = ( zero_zero @ int ) )
% 2.70/1.53 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
% 2.70/1.53 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 2.70/1.53 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_235_div__positive__int,axiom,
% 2.70/1.53 ! [L: int,K: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ L @ K )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 2.70/1.53 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_236_div__nonneg__neg__le0,axiom,
% 2.70/1.53 ! [A2: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
% 2.70/1.53 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_237_div__nonpos__pos__le0,axiom,
% 2.70/1.53 ! [A2: int,B2: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
% 2.70/1.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_238_pos__imp__zdiv__pos__iff,axiom,
% 2.70/1.53 ! [K: int,I2: int] :
% 2.70/1.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.70/1.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K ) )
% 2.70/1.53 = ( ord_less_eq @ int @ K @ I2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_239_neg__imp__zdiv__nonneg__iff,axiom,
% 2.70/1.53 ! [B2: int,A2: int] :
% 2.70/1.53 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
% 2.70/1.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
% 2.70/1.53 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_240_realpow__pos__nth__unique,axiom,
% 2.70/1.53 ! [N: nat,A2: real] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.70/1.53 => ? [X2: real] :
% 2.70/1.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
% 2.70/1.53 & ( ( power_power @ real @ X2 @ N )
% 2.70/1.53 = A2 )
% 2.70/1.53 & ! [Y2: real] :
% 2.70/1.53 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
% 2.70/1.53 & ( ( power_power @ real @ Y2 @ N )
% 2.70/1.53 = A2 ) )
% 2.70/1.53 => ( Y2 = X2 ) ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_241_realpow__pos__nth,axiom,
% 2.70/1.53 ! [N: nat,A2: real] :
% 2.70/1.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.70/1.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
% 2.70/1.53 => ? [R: real] :
% 2.70/1.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
% 2.70/1.53 & ( ( power_power @ real @ R @ N )
% 2.70/1.53 = A2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_242_nat__int__comparison_I3_J,axiom,
% 2.70/1.53 ( ( ord_less_eq @ nat )
% 2.70/1.53 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_243_nat__int__comparison_I2_J,axiom,
% 2.70/1.53 ( ( ord_less @ nat )
% 2.70/1.53 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_244_verit__la__disequality,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( linorder @ A )
% 2.70/1.53 => ! [A2: A,B2: A] :
% 2.70/1.53 ( ( A2 = B2 )
% 2.70/1.53 | ~ ( ord_less_eq @ A @ A2 @ B2 )
% 2.70/1.53 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_245_verit__comp__simplify1_I1_J,axiom,
% 2.70/1.53 ! [A: $tType] :
% 2.70/1.53 ( ( order @ A )
% 2.70/1.53 => ! [A2: A] :
% 2.70/1.53 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_246_verit__la__generic,axiom,
% 2.70/1.53 ! [A2: int,X: int] :
% 2.70/1.53 ( ( ord_less_eq @ int @ A2 @ X )
% 2.70/1.53 | ( A2 = X )
% 2.70/1.53 | ( ord_less_eq @ int @ X @ A2 ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_247_int__if,axiom,
% 2.70/1.53 ! [P: $o,A2: nat,B2: nat] :
% 2.70/1.53 ( ( P
% 2.70/1.53 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
% 2.70/1.53 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 2.70/1.53 & ( ~ P
% 2.70/1.53 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
% 2.70/1.53 = ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_248_nat__int__comparison_I1_J,axiom,
% 2.70/1.53 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 2.70/1.53 = ( ^ [A4: nat,B4: nat] :
% 2.70/1.53 ( ( semiring_1_of_nat @ int @ A4 )
% 2.70/1.53 = ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_249_verit__comp__simplify1_I3_J,axiom,
% 2.70/1.53 ! [B: $tType] :
% 2.70/1.53 ( ( linorder @ B )
% 2.70/1.53 => ! [B3: B,A5: B] :
% 2.70/1.53 ( ( ~ ( ord_less_eq @ B @ B3 @ A5 ) )
% 2.70/1.53 = ( ord_less @ B @ A5 @ B3 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_250_int__ops_I1_J,axiom,
% 2.70/1.53 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 2.70/1.53 = ( zero_zero @ int ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_251_div__gr__imp__gr__divisor,axiom,
% 2.70/1.53 ! [X: nat,N: nat,M: nat] :
% 2.70/1.53 ( ( ord_less @ nat @ X @ ( divide_divide @ nat @ N @ M ) )
% 2.70/1.53 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(fact_252_div__eq__0__conv,axiom,
% 2.70/1.53 ! [N: nat,M: nat] :
% 2.70/1.53 ( ( ( divide_divide @ nat @ N @ M )
% 2.70/1.53 = ( zero_zero @ nat ) )
% 2.70/1.53 = ( ( M
% 2.70/1.53 = ( zero_zero @ nat ) )
% 2.70/1.53 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.70/1.53 ! [A6: $tType,A7: $tType] :
% 2.70/1.53 ( ( order @ A7 )
% 2.70/1.53 => ( order @ ( A6 > A7 ) ) ) ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 2.70/1.53 euclid1852923125th_nat @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 2.70/1.53 unique1598680935umeral @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
% 2.70/1.53 semiri134348788visors @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.70/1.53 linord1659791738miring @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.70/1.53 linordered_semidom @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.70/1.53 linordered_idom @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 2.70/1.53 semidom_divide @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Parity_Osemiring__bits,axiom,
% 2.70/1.53 semiring_bits @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.70/1.53 semiring_char_0 @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 2.70/1.53 linorder @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.70/1.53 semiring_1 @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Orderings_Oorder_1,axiom,
% 2.70/1.53 order @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.70/1.53 zero @ int ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_2,axiom,
% 2.70/1.53 euclid1852923125th_nat @ nat ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_3,axiom,
% 2.70/1.53 unique1598680935umeral @ nat ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.70/1.53 canoni770627133id_add @ nat ).
% 2.70/1.53
% 2.70/1.53 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_4,axiom,
% 2.70/1.53 semiri134348788visors @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_5,axiom,
% 2.70/1.54 linord1659791738miring @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
% 2.70/1.54 linordered_semidom @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Rings_Osemidom__divide_7,axiom,
% 2.70/1.54 semidom_divide @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Parity_Osemiring__bits_8,axiom,
% 2.70/1.54 semiring_bits @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.70/1.54 wellorder @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9,axiom,
% 2.70/1.54 semiring_char_0 @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
% 2.70/1.54 linorder @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
% 2.70/1.54 semiring_1 @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
% 2.70/1.54 order @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Groups_Ozero_13,axiom,
% 2.70/1.54 zero @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.70/1.54 size @ nat ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
% 2.70/1.54 ! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
% 2.70/1.54 linorder @ $o ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
% 2.70/1.54 order @ $o ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_List_Olist___Nat_Osize_17,axiom,
% 2.70/1.54 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 2.70/1.54 archim1804426504_field @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_18,axiom,
% 2.70/1.54 semiri134348788visors @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_19,axiom,
% 2.70/1.54 linord1659791738miring @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_20,axiom,
% 2.70/1.54 linordered_semidom @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 2.70/1.54 linordered_field @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Olinordered__idom_21,axiom,
% 2.70/1.54 linordered_idom @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Osemidom__divide_22,axiom,
% 2.70/1.54 semidom_divide @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 2.70/1.54 division_ring @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_23,axiom,
% 2.70/1.54 semiring_char_0 @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Orderings_Olinorder_24,axiom,
% 2.70/1.54 linorder @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Rings_Osemiring__1_25,axiom,
% 2.70/1.54 semiring_1 @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Orderings_Oorder_26,axiom,
% 2.70/1.54 order @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 2.70/1.54 field @ real ).
% 2.70/1.54
% 2.70/1.54 thf(tcon_Real_Oreal___Groups_Ozero_27,axiom,
% 2.70/1.54 zero @ real ).
% 2.70/1.54
% 2.70/1.54 thf(help_If_3_1_T,axiom,
% 2.70/1.54 ! [P: $o] :
% 2.70/1.54 ( ( P = $true )
% 2.70/1.54 | ( P = $false ) ) ).
% 2.70/1.54
% 2.70/1.54 thf(help_If_2_1_T,axiom,
% 2.70/1.54 ! [A: $tType,X: A,Y: A] :
% 2.70/1.54 ( ( if @ A @ $false @ X @ Y )
% 2.70/1.54 = Y ) ).
% 2.70/1.54
% 2.70/1.54 thf(help_If_1_1_T,axiom,
% 2.70/1.54 ! [A: $tType,X: A,Y: A] :
% 2.70/1.54 ( ( if @ A @ $true @ X @ Y )
% 2.70/1.54 = X ) ).
% 2.70/1.54
% 2.70/1.54 thf(conj_0,conjecture,
% 2.70/1.54 ( ( listSl1903280966_slice @ a @ ( nil @ a ) @ k )
% 2.70/1.54 = ( nil @ ( list @ a ) ) ) ).
% 2.70/1.54 ------------------------
% 2.70/1.55 % (7468)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/2Mi)
% 2.70/1.55 % (7469)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_7234 for (2999ds/2Mi)
% 2.70/1.55 % (7469)Instruction limit reached!
% 2.70/1.55 % (7469)------------------------------
% 2.70/1.55 % (7469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (7469)Termination reason: Unknown
% 2.70/1.55 % (7469)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (7469)Memory used [KB]: 1279
% 2.70/1.55 % (7469)Time elapsed: 0.002 s
% 2.70/1.55 % (7469)Instructions burned: 2 (million)
% 2.70/1.55 % (7469)------------------------------
% 2.70/1.55 % (7469)------------------------------
% 2.70/1.55 % (7468)Instruction limit reached!
% 2.70/1.55 % (7468)------------------------------
% 2.70/1.55 % (7468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (7468)Termination reason: Unknown
% 2.70/1.55 % (7468)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (7468)Memory used [KB]: 1407
% 2.70/1.55 % (7468)Time elapsed: 0.003 s
% 2.70/1.55 % (7468)Instructions burned: 4 (million)
% 2.70/1.55 % (7468)------------------------------
% 2.70/1.55 % (7468)------------------------------
% 2.70/1.56 % (7465)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/183Mi)
% 2.70/1.56 % (7466)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_7234 for (2999ds/4Mi)
% 2.70/1.56 % (7467)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_7234 for (2999ds/27Mi)
% 2.70/1.56 % (7471)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/18Mi)
% 2.70/1.56 % (7470)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_7234 for (2999ds/275Mi)
% 2.70/1.56 % (7477)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/3Mi)
% 2.70/1.56 % (7466)Instruction limit reached!
% 2.70/1.56 % (7466)------------------------------
% 2.70/1.56 % (7466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (7466)Termination reason: Unknown
% 2.70/1.56 % (7466)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (7466)Memory used [KB]: 1407
% 2.70/1.56 % (7466)Time elapsed: 0.005 s
% 2.70/1.56 % (7466)Instructions burned: 5 (million)
% 2.70/1.56 % (7466)------------------------------
% 2.70/1.56 % (7466)------------------------------
% 2.70/1.56 % (7479)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_7234 for (2999ds/37Mi)
% 2.70/1.56 % (7477)Instruction limit reached!
% 2.70/1.56 % (7477)------------------------------
% 2.70/1.56 % (7477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (7477)Termination reason: Unknown
% 2.70/1.56 % (7477)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (7477)Memory used [KB]: 1407
% 2.70/1.56 % (7477)Time elapsed: 0.003 s
% 2.70/1.56 % (7477)Instructions burned: 5 (million)
% 2.70/1.56 % (7477)------------------------------
% 2.70/1.56 % (7477)------------------------------
% 2.70/1.57 % (7471)Instruction limit reached!
% 2.70/1.57 % (7471)------------------------------
% 2.70/1.57 % (7471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (7471)Termination reason: Unknown
% 2.70/1.57 % (7471)Termination phase: Property scanning
% 2.70/1.57
% 2.70/1.57 % (7471)Memory used [KB]: 1663
% 2.70/1.57 % (7471)Time elapsed: 0.014 s
% 2.70/1.57 % (7471)Instructions burned: 19 (million)
% 2.70/1.57 % (7471)------------------------------
% 2.70/1.57 % (7471)------------------------------
% 2.70/1.57 % (7486)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/3Mi)
% 2.70/1.57 % (7479)Instruction limit reached!
% 2.70/1.57 % (7479)------------------------------
% 2.70/1.57 % (7479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (7479)Termination reason: Unknown
% 2.70/1.57 % (7479)Termination phase: Property scanning
% 2.70/1.57
% 2.70/1.57 % (7479)Memory used [KB]: 2046
% 2.70/1.57 % (7479)Time elapsed: 0.012 s
% 2.70/1.57 % (7479)Instructions burned: 39 (million)
% 2.70/1.57 % (7479)------------------------------
% 2.70/1.57 % (7479)------------------------------
% 2.70/1.57 % (7486)Instruction limit reached!
% 2.70/1.57 % (7486)------------------------------
% 2.70/1.57 % (7486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (7486)Termination reason: Unknown
% 2.70/1.57 % (7486)Termination phase: shuffling
% 2.70/1.57
% 2.70/1.57 % (7486)Memory used [KB]: 1279
% 2.70/1.57 % (7486)Time elapsed: 0.002 s
% 2.70/1.57 % (7486)Instructions burned: 3 (million)
% 2.70/1.57 % (7486)------------------------------
% 2.70/1.57 % (7486)------------------------------
% 2.70/1.58 % (7467)Instruction limit reached!
% 2.70/1.58 % (7467)------------------------------
% 2.70/1.58 % (7467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (7467)Termination reason: Unknown
% 2.70/1.58 % (7467)Termination phase: Property scanning
% 2.70/1.58
% 2.70/1.58 % (7467)Memory used [KB]: 1663
% 2.70/1.58 % (7467)Time elapsed: 0.018 s
% 2.70/1.58 % (7467)Instructions burned: 28 (million)
% 2.70/1.58 % (7467)------------------------------
% 2.70/1.58 % (7467)------------------------------
% 2.70/1.58 % (7484)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_7234 for (2999ds/15Mi)
% 2.70/1.58 % (7491)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/7Mi)
% 2.70/1.58 % (7493)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/16Mi)
% 2.70/1.59 % (7491)Instruction limit reached!
% 2.70/1.59 % (7491)------------------------------
% 2.70/1.59 % (7491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (7491)Termination reason: Unknown
% 2.70/1.59 % (7491)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (7491)Memory used [KB]: 1535
% 2.70/1.59 % (7491)Time elapsed: 0.004 s
% 2.70/1.59 % (7491)Instructions burned: 8 (million)
% 2.70/1.59 % (7491)------------------------------
% 2.70/1.59 % (7491)------------------------------
% 2.70/1.59 % (7494)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/3Mi)
% 2.70/1.59 % (7490)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7234 for (2999ds/1041Mi)
% 2.70/1.59 % (7493)Instruction limit reached!
% 2.70/1.59 % (7493)------------------------------
% 2.70/1.59 % (7493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (7493)Termination reason: Unknown
% 2.70/1.59 % (7493)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (7493)Memory used [KB]: 1663
% 2.70/1.59 % (7493)Time elapsed: 0.007 s
% 2.70/1.59 % (7493)Instructions burned: 17 (million)
% 2.70/1.59 % (7493)------------------------------
% 2.70/1.59 % (7493)------------------------------
% 2.70/1.59 % (7484)Instruction limit reached!
% 2.70/1.59 % (7484)------------------------------
% 2.70/1.59 % (7484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (7484)Termination reason: Unknown
% 2.70/1.59 % (7484)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (7484)Memory used [KB]: 1663
% 2.70/1.59 % (7484)Time elapsed: 0.012 s
% 2.70/1.59 % (7484)Instructions burned: 16 (million)
% 2.70/1.59 % (7484)------------------------------
% 2.70/1.59 % (7484)------------------------------
% 2.70/1.59 % (7494)Instruction limit reached!
% 2.70/1.59 % (7494)------------------------------
% 2.70/1.59 % (7494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (7494)Termination reason: Unknown
% 2.70/1.59 % (7494)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (7494)Memory used [KB]: 1407
% 2.70/1.59 % (7494)Time elapsed: 0.003 s
% 2.70/1.59 % (7494)Instructions burned: 4 (million)
% 2.70/1.59 % (7494)------------------------------
% 2.70/1.59 % (7494)------------------------------
% 2.70/1.59 % (7470)First to succeed.
% 2.70/1.59 % (7470)Refutation found. Thanks to Tanya!
% 2.70/1.59 % SZS status Theorem for DTF2THF_7234
% 2.70/1.59 % SZS output start Proof for DTF2THF_7234
% 2.70/1.59 thf(type_def_6, type, list: ($tType) > $tType).
% 2.70/1.59 thf(type_def_7, type, nat: $tType).
% 2.70/1.59 thf(type_def_8, type, option: ($tType) > $tType).
% 2.70/1.59 thf(type_def_9, type, set: ($tType) > $tType).
% 2.70/1.59 thf(type_def_10, type, int: $tType).
% 2.70/1.59 thf(type_def_11, type, real: $tType).
% 2.70/1.59 thf(type_def_12, type, a: $tType).
% 2.70/1.59 thf(func_def_0, type, option: ($tType) > $tType).
% 2.70/1.59 thf(func_def_1, type, real: $tType).
% 2.70/1.59 thf(func_def_2, type, list: ($tType) > $tType).
% 2.70/1.59 thf(func_def_3, type, set: ($tType) > $tType).
% 2.70/1.59 thf(func_def_4, type, nat: $tType).
% 2.70/1.59 thf(func_def_5, type, int: $tType).
% 2.70/1.59 thf(func_def_6, type, a: $tType).
% 2.70/1.59 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_8, type, size: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_9, type, zero: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_10, type, field: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_11, type, order: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_12, type, semiring_1: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_14, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_15, type, wellorder: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_16, type, division_ring: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_17, type, semiring_bits: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_18, type, semidom_divide: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_20, type, linordered_field: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_21, type, linordered_semidom: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_22, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_23, type, semiri134348788visors: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_24, type, archim1804426504_field: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_25, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_26, type, unique1598680935umeral: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_27, type, euclid1852923125th_nat: !>[X0: $tType]:($o)).
% 2.70/1.59 thf(func_def_28, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 2.70/1.59 thf(func_def_29, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.70/1.59 thf(func_def_30, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.70/1.59 thf(func_def_31, type, listSl1903280966_slice: !>[X0: $tType]:(list @ X0 > nat > list @ list @ X0)).
% 2.70/1.59 thf(func_def_32, type, listSl1123830174ce_aux: !>[X0: $tType]:(list @ X0 > nat > nat > list @ list @ X0)).
% 2.70/1.59 thf(func_def_33, type, bind: !>[X0: $tType, X1: $tType]:(list @ X0 > (X0 > list @ X1) > list @ X1)).
% 2.70/1.59 thf(func_def_34, type, concat: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.70/1.59 thf(func_def_35, type, count_list: !>[X0: $tType]:(list @ X0 > X0 > nat)).
% 2.70/1.59 thf(func_def_36, type, gen_length: !>[X0: $tType]:(nat > list @ X0 > nat)).
% 2.70/1.59 thf(func_def_37, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.70/1.59 thf(func_def_38, type, list_ex1: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.70/1.59 thf(func_def_39, type, map_filter: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > list @ X0 > list @ X1)).
% 2.70/1.59 thf(func_def_40, type, maps: !>[X0: $tType, X1: $tType]:((X0 > list @ X1) > list @ X0 > list @ X1)).
% 2.70/1.59 thf(func_def_41, type, member: !>[X0: $tType]:(list @ X0 > X0 > $o)).
% 2.70/1.59 thf(func_def_42, type, null: !>[X0: $tType]:(list @ X0 > $o)).
% 2.70/1.59 thf(func_def_43, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.70/1.59 thf(func_def_44, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.70/1.59 thf(func_def_45, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.70/1.59 thf(func_def_46, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.70/1.59 thf(func_def_47, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 2.70/1.59 thf(func_def_48, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.70/1.59 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.70/1.59 thf(func_def_50, type, member2: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.70/1.59 thf(func_def_51, type, k: nat).
% 2.70/1.59 thf(func_def_65, type, sK0: !>[X0: $tType]:(nat > list @ X0)).
% 2.70/1.59 thf(func_def_66, type, sK1: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 2.70/1.59 thf(func_def_67, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.70/1.59 thf(func_def_68, type, sK3: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.70/1.59 thf(func_def_69, type, sK4: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.70/1.59 thf(func_def_70, type, sK5: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_71, type, sK6: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_72, type, sK7: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_73, type, sK8: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.70/1.59 thf(func_def_75, type, ph10: !>[X0: $tType]:(X0)).
% 2.70/1.59 thf(f1179,plain,(
% 2.70/1.59 $false),
% 2.70/1.59 inference(trivial_inequality_removal,[status(thm)],[f1178])).
% 2.70/1.59 thf(f1178,plain,(
% 2.70/1.59 (nil @ list @ a != nil @ list @ a)),
% 2.70/1.59 inference(superposition,[status(thm)],[f951,f1058])).
% 2.70/1.59 thf(f1058,plain,(
% 2.70/1.59 ( ! [X0 : $tType,X1 : list @ X0] : ((nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ k))) )),
% 2.70/1.59 inference(superposition,[status(thm)],[f1007,f1041])).
% 2.70/1.59 thf(f1041,plain,(
% 2.70/1.59 (zero_zero @ nat = k)),
% 2.70/1.59 inference(trivial_inequality_removal,[status(thm)],[f1032])).
% 2.70/1.59 thf(f1032,plain,(
% 2.70/1.59 (zero_zero @ nat = k) | ($true != $true)),
% 2.70/1.59 inference(superposition,[status(thm)],[f1027,f989])).
% 2.70/1.59 thf(f989,plain,(
% 2.70/1.59 ( ! [X0 : nat] : (($true = (ord_less @ nat @ zero_zero @ nat @ X0)) | (zero_zero @ nat = X0)) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f932])).
% 2.70/1.59 thf(f932,plain,(
% 2.70/1.59 ! [X0 : nat] : (((zero_zero @ nat != X0) | ($true != (ord_less @ nat @ zero_zero @ nat @ X0))) & (($true = (ord_less @ nat @ zero_zero @ nat @ X0)) | (zero_zero @ nat = X0)))),
% 2.70/1.59 inference(nnf_transformation,[status(thm)],[f440])).
% 2.70/1.59 thf(f440,plain,(
% 2.70/1.59 ! [X0 : nat] : ((zero_zero @ nat != X0) <=> ($true = (ord_less @ nat @ zero_zero @ nat @ X0)))),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f439])).
% 2.70/1.59 thf(f439,plain,(
% 2.70/1.59 ! [X0 : nat] : ((zero_zero @ nat != X0) = (ord_less @ nat @ zero_zero @ nat @ X0))),
% 2.70/1.59 inference(rectify,[status(thm)],[f19])).
% 2.70/1.59 thf(f19,axiom,(
% 2.70/1.59 ! [X7 : nat] : ((zero_zero @ nat != X7) = (ord_less @ nat @ zero_zero @ nat @ X7))),
% 2.70/1.59 file('/export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2THF_7234.p',fact_18_neq0__conv)).
% 2.70/1.59 thf(f1027,plain,(
% 2.70/1.59 ((ord_less @ nat @ zero_zero @ nat @ k) != $true)),
% 2.70/1.59 inference(forward_demodulation,[status(thm)],[f1026,f1015])).
% 2.70/1.59 thf(f1015,plain,(
% 2.70/1.59 ( ! [X0 : $tType] : ((zero_zero @ nat = (size_size @ list @ X0 @ nil @ X0))) )),
% 2.70/1.59 inference(equality_resolution,[status(thm)],[f978])).
% 2.70/1.59 thf(f978,plain,(
% 2.70/1.59 ( ! [X0 : $tType,X1 : list @ X0] : ((zero_zero @ nat = (size_size @ list @ X0 @ X1)) | (nil @ X0 != X1)) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f928])).
% 2.70/1.59 thf(f928,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0] : (((zero_zero @ nat = (size_size @ list @ X0 @ X1)) | (nil @ X0 != X1)) & ((nil @ X0 = X1) | (zero_zero @ nat != (size_size @ list @ X0 @ X1))))),
% 2.70/1.59 inference(nnf_transformation,[status(thm)],[f542])).
% 2.70/1.59 thf(f542,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0] : ((zero_zero @ nat = (size_size @ list @ X0 @ X1)) <=> (nil @ X0 = X1))),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f541])).
% 2.70/1.59 thf(f541,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0] : ((zero_zero @ nat = (size_size @ list @ X0 @ X1)) = (nil @ X0 = X1))),
% 2.70/1.59 inference(rectify,[status(thm)],[f12])).
% 2.70/1.59 thf(f12,axiom,(
% 2.70/1.59 ! [X0 : $tType,X2 : list @ X0] : ((zero_zero @ nat = (size_size @ list @ X0 @ X2)) = (nil @ X0 = X2))),
% 2.70/1.59 file('/export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2THF_7234.p',fact_11_length__0__conv)).
% 2.70/1.59 thf(f1026,plain,(
% 2.70/1.59 ((ord_less @ nat @ (size_size @ list @ a @ nil @ a) @ k) != $true)),
% 2.70/1.59 inference(trivial_inequality_removal,[status(thm)],[f1025])).
% 2.70/1.59 thf(f1025,plain,(
% 2.70/1.59 (nil @ list @ a != nil @ list @ a) | ((ord_less @ nat @ (size_size @ list @ a @ nil @ a) @ k) != $true)),
% 2.70/1.59 inference(superposition,[status(thm)],[f951,f955])).
% 2.70/1.59 thf(f955,plain,(
% 2.70/1.59 ( ! [X0 : $tType,X2 : nat,X1 : list @ X0] : ((nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ X2)) | ((ord_less @ nat @ (size_size @ list @ X0 @ X1) @ X2) != $true)) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f890])).
% 2.70/1.59 thf(f890,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0,X2 : nat] : (((ord_less @ nat @ (size_size @ list @ X0 @ X1) @ X2) != $true) | (nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ X2)))),
% 2.70/1.59 inference(ennf_transformation,[status(thm)],[f390])).
% 2.70/1.59 thf(f390,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0,X2 : nat] : (((ord_less @ nat @ (size_size @ list @ X0 @ X1) @ X2) = $true) => (nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ X2)))),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f389])).
% 2.70/1.59 thf(f389,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0,X2 : nat] : ((ord_less @ nat @ (size_size @ list @ X0 @ X1) @ X2) => (nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ X2)))),
% 2.70/1.59 inference(rectify,[status(thm)],[f4])).
% 2.70/1.59 thf(f4,axiom,(
% 2.70/1.59 ! [X0 : $tType,X2 : list @ X0,X4 : nat] : ((ord_less @ nat @ (size_size @ list @ X0 @ X2) @ X4) => (nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X2 @ X4)))),
% 2.70/1.59 file('/export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2THF_7234.p',fact_3_list__slice__less)).
% 2.70/1.59 thf(f1007,plain,(
% 2.70/1.59 ( ! [X0 : $tType,X1 : list @ X0] : ((nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ zero_zero @ nat))) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f883])).
% 2.70/1.59 thf(f883,plain,(
% 2.70/1.59 ! [X0 : $tType,X1 : list @ X0] : (nil @ list @ X0 = (listSl1903280966_slice @ X0 @ X1 @ zero_zero @ nat))),
% 2.70/1.59 inference(rectify,[status(thm)],[f1])).
% 2.70/1.59 thf(f1,axiom,(
% 2.70/1.59 ! [X0 : $tType,X2 : list @ X0] : ((listSl1903280966_slice @ X0 @ X2 @ zero_zero @ nat) = nil @ list @ X0)),
% 2.70/1.59 file('/export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2THF_7234.p',fact_0_list__slice__0)).
% 2.70/1.59 thf(f951,plain,(
% 2.70/1.59 ((listSl1903280966_slice @ a @ nil @ a @ k) != nil @ list @ a)),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f869])).
% 2.70/1.59 thf(f869,plain,(
% 2.70/1.59 ((listSl1903280966_slice @ a @ nil @ a @ k) != nil @ list @ a)),
% 2.70/1.59 inference(flattening,[status(thm)],[f305])).
% 2.70/1.59 thf(f305,negated_conjecture,(
% 2.70/1.59 ~((listSl1903280966_slice @ a @ nil @ a @ k) = nil @ list @ a)),
% 2.70/1.59 inference(negated_conjecture,[status(cth)],[f304])).
% 2.70/1.59 thf(f304,conjecture,(
% 2.70/1.59 ((listSl1903280966_slice @ a @ nil @ a @ k) = nil @ list @ a)),
% 2.70/1.59 file('/export/starexec/sandbox/tmp/tmp.cWSF0Mhyxp/DTF2THF_7234.p',conj_0)).
% 2.70/1.59 % SZS output end Proof for DTF2THF_7234
% 2.70/1.59 % (7470)------------------------------
% 2.70/1.59 % (7470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (7470)Termination reason: Refutation
% 2.70/1.59
% 2.70/1.59 % (7470)Memory used [KB]: 6396
% 2.70/1.59 % (7470)Time elapsed: 0.035 s
% 2.70/1.59 % (7470)Instructions burned: 59 (million)
% 2.70/1.59 % (7470)------------------------------
% 2.70/1.59 % (7470)------------------------------
% 2.70/1.59 % (7462)Success in time 0.055 s
%------------------------------------------------------------------------------