%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP078^2 : 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 : n018.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 08:55:22 AM UTC 2026
% Result : Theorem 4.87s 2.14s
% Output : Refutation 4.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP078^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n018.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:25:16 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP078^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer Hilbert_Function problem prob_85__11621656_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : Hilbert_Function/prob_85__11621656_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 393 ( 144 unt; 63 typ; 0 def)
% 0.20/0.46 % Number of atoms : 827 ( 318 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.20/0.46 % Number of connectives : 3345 ( 68 ~; 3 |; 30 &;2920 @)
% 0.20/0.46 % ( 0 <=>; 324 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 19 ( 6 avg)
% 0.20/0.46 % Number of types : 5 ( 4 usr)
% 0.20/0.46 % Number of type conns : 217 ( 217 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 62 ( 59 usr; 4 con; 0-6 aty)
% 0.20/0.46 % Number of variables : 943 ( 35 ^; 826 !; 23 ?; 943 :)
% 0.20/0.46 % ( 59 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.46 % SPC : TH1_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:21:51.858
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (7)
% 0.20/0.46 thf(ty_t_Multiset_Omultiset,type,
% 0.20/0.46 multiset: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Real_Oreal,type,
% 0.20/0.46 real: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_List_Olist,type,
% 0.20/0.46 list: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Set_Oset,type,
% 0.20/0.46 set: $tType > $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Nat_Onat,type,
% 0.20/0.46 nat: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_t_Int_Oint,type,
% 0.20/0.46 int: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_tf_a,type,
% 0.20/0.46 a: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (56)
% 0.20/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.20/0.46 preorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Odioid,type,
% 0.20/0.46 dioid:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_HOL_Otype,type,
% 0.20/0.46 type:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Osize,type,
% 0.20/0.46 size:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.46 zero:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.20/0.46 field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Oring__1,type,
% 0.20/0.46 ring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.20/0.46 ord:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Oring__char__0,type,
% 0.20/0.46 ring_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.20/0.46 order:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.46 group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Osemiring__1,type,
% 0.20/0.46 semiring_1:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.20/0.46 linorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.20/0.46 semiring_char_0:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.20/0.46 wellorder:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.20/0.46 division_ring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.20/0.46 linordered_idom:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.20/0.46 linordered_field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.20/0.46 boolean_algebra:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.20/0.46 ordered_ab_group_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.20/0.46 linord219039673up_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.46 linord1659791738miring:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.20/0.46 archim1804426504_field:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.46 canoni770627133id_add:
% 0.20/0.46 !>[A: $tType] : $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.20/0.46 inverse_inverse:
% 0.20/0.46 !>[A: $tType] : ( A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Obij__betw,type,
% 0.20/0.46 bij_betw:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.20/0.46 comp:
% 0.20/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on,type,
% 0.20/0.46 inj_on:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.20/0.46 uminus_uminus:
% 0.20/0.46 !>[A: $tType] : ( A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.46 zero_zero:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If,type,
% 0.20/0.46 if:
% 0.20/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Int_Onat,type,
% 0.20/0.46 nat2: int > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.20/0.46 ring_1_of_int:
% 0.20/0.46 !>[A: $tType] : ( int > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Ogen__length,type,
% 0.20/0.46 gen_length:
% 0.20/0.46 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist_Omap,type,
% 0.20/0.46 map:
% 0.20/0.46 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Olist__ex,type,
% 0.20/0.46 list_ex:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Omap__tailrec,type,
% 0.20/0.46 map_tailrec:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Onth,type,
% 0.20/0.46 nth:
% 0.20/0.46 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_List_Oupt,type,
% 0.20/0.46 upt: nat > nat > ( list @ nat ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Multiset_Oimage__mset,type,
% 0.20/0.46 image_mset:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Multiset_Ois__empty,type,
% 0.20/0.46 is_empty:
% 0.20/0.46 !>[A: $tType] : ( ( multiset @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Multiset_Omset,type,
% 0.20/0.46 mset:
% 0.20/0.46 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Multiset_Omset__set,type,
% 0.20/0.46 mset_set:
% 0.20/0.46 !>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Multiset_Osize__multiset,type,
% 0.20/0.46 size_multiset:
% 0.20/0.46 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.20/0.46 semiring_1_of_nat:
% 0.20/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.20/0.46 size_size:
% 0.20/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.46 ord_less:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.20/0.46 ord_less_eq:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage,type,
% 0.20/0.46 image:
% 0.20/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 0.20/0.46 set_or1433837966ssThan:
% 0.20/0.46 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.20/0.46 set_ord_lessThan:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_f,type,
% 0.20/0.46 f: nat > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_xs,type,
% 0.20/0.46 xs: list @ a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_ys,type,
% 0.20/0.46 ys: list @ a ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (254)
% 0.20/0.46 thf(fact_0__C1_C,axiom,
% 0.20/0.46 inj_on @ nat @ nat @ f @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ).
% 0.20/0.46
% 0.20/0.46 % "1"
% 0.20/0.46 thf(fact_1_calculation,axiom,
% 0.20/0.46 ( ( mset @ a @ xs )
% 0.20/0.46 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % calculation
% 0.20/0.46 thf(fact_2_assms_I2_J,axiom,
% 0.20/0.46 ! [I: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ xs ) )
% 0.20/0.46 => ( ( nth @ a @ xs @ I )
% 0.20/0.46 = ( nth @ a @ ys @ ( f @ I ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms(2)
% 0.20/0.46 thf(fact_3__C2_C,axiom,
% 0.20/0.46 ( ( image @ nat @ nat @ f @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) )
% 0.20/0.46 = ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ ys ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % "2"
% 0.20/0.46 thf(fact_4_zero__fun__apply,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType] :
% 0.20/0.46 ( ( zero @ A )
% 0.20/0.46 => ( ( zero_zero @ ( B > A ) )
% 0.20/0.46 = ( ^ [X: B] : ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_fun_apply
% 0.20/0.46 thf(fact_5_assms_I1_J,axiom,
% 0.20/0.46 bij_betw @ nat @ nat @ f @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ ys ) ) ).
% 0.20/0.46
% 0.20/0.46 % assms(1)
% 0.20/0.46 thf(fact_6__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062,axiom,
% 0.20/0.46 ( ( mset @ a @ ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) )
% 0.20/0.46 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>mset (map ((!) ys \<circ> f) [0..<length xs]) = image_mset ((!) ys) (image_mset f (mset_set {0..<length xs}))\<close>
% 0.20/0.46 thf(fact_7__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062,axiom,
% 0.20/0.46 ( xs
% 0.20/0.46 = ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>xs = map ((!) ys \<circ> f) [0..<length xs]\<close>
% 0.20/0.46 thf(fact_8_Ex__list__of__length,axiom,
% 0.20/0.46 ! [A: $tType,N: nat] :
% 0.20/0.46 ? [Xs: list @ A] :
% 0.20/0.46 ( ( size_size @ ( list @ A ) @ Xs )
% 0.20/0.46 = N ) ).
% 0.20/0.46
% 0.20/0.46 % Ex_list_of_length
% 0.20/0.46 thf(fact_9_neq__if__length__neq,axiom,
% 0.20/0.46 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 0.20/0.46 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.20/0.46 != ( size_size @ ( list @ A ) @ Ys ) )
% 0.20/0.46 => ( Xs2 != Ys ) ) ).
% 0.20/0.46
% 0.20/0.46 % neq_if_length_neq
% 0.20/0.46 thf(fact_10_size__neq__size__imp__neq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( size @ A )
% 0.20/0.46 => ! [X2: A,Y: A] :
% 0.20/0.46 ( ( ( size_size @ A @ X2 )
% 0.20/0.46 != ( size_size @ A @ Y ) )
% 0.20/0.46 => ( X2 != Y ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % size_neq_size_imp_neq
% 0.20/0.46 thf(fact_11_not__gr__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [N: A] :
% 0.20/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.20/0.46 = ( N
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_gr_zero
% 0.20/0.46 thf(fact_12_bot__nat__0_Onot__eq__extremum,axiom,
% 0.20/0.46 ! [A2: nat] :
% 0.20/0.46 ( ( A2
% 0.20/0.46 != ( zero_zero @ nat ) )
% 0.20/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_nat_0.not_eq_extremum
% 0.20/0.46 thf(fact_13_less__nat__zero__code,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_nat_zero_code
% 0.20/0.46 thf(fact_14_neq0__conv,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( N
% 0.20/0.46 != ( zero_zero @ nat ) )
% 0.20/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % neq0_conv
% 0.20/0.46 thf(fact_15_map__map,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs2: list @ C] :
% 0.20/0.46 ( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs2 ) )
% 0.20/0.46 = ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % map_map
% 0.20/0.46 thf(fact_16_List_Omap_Ocomp,axiom,
% 0.20/0.46 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 0.20/0.46 ( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
% 0.20/0.46 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % List.map.comp
% 0.20/0.46 thf(fact_17_map__comp__map,axiom,
% 0.20/0.46 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
% 0.20/0.46 ( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
% 0.20/0.46 = ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % map_comp_map
% 0.20/0.46 thf(fact_18_list_Omap__comp,axiom,
% 0.20/0.46 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
% 0.20/0.46 ( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
% 0.20/0.46 = ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.20/0.46
% 0.20/0.46 % list.map_comp
% 0.20/0.46 thf(fact_19_List_Omap_Ocompositionality,axiom,
% 0.20/0.46 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
% 0.20/0.46 ( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
% 0.20/0.46 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% 0.20/0.46
% 0.20/0.46 % List.map.compositionality
% 0.20/0.46 thf(fact_20_length__map,axiom,
% 0.20/0.46 ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
% 0.20/0.46 ( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs2 ) )
% 0.20/0.46 = ( size_size @ ( list @ B ) @ Xs2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % length_map
% 0.20/0.46 thf(fact_21_nth__map,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType,N: nat,Xs2: list @ A,F: A > B] :
% 0.20/0.46 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.20/0.46 => ( ( nth @ B @ ( map @ A @ B @ F @ Xs2 ) @ N )
% 0.20/0.46 = ( F @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nth_map
% 0.20/0.46 thf(fact_22__092_060open_062mset_Axs_A_061_Amset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_092_060close_062,axiom,
% 0.20/0.46 ( ( mset @ a @ xs )
% 0.20/0.46 = ( mset @ a @ ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % \<open>mset xs = mset (map ((!) ys \<circ> f) [0..<length xs])\<close>
% 0.20/0.46 thf(fact_23_nat__neq__iff,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( M != N )
% 0.20/0.46 = ( ( ord_less @ nat @ M @ N )
% 0.20/0.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_neq_iff
% 0.20/0.46 thf(fact_24_less__not__refl,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl
% 0.20/0.46 thf(fact_25_less__not__refl2,axiom,
% 0.20/0.46 ! [N: nat,M: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ N @ M )
% 0.20/0.46 => ( M != N ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl2
% 0.20/0.46 thf(fact_26_less__not__refl3,axiom,
% 0.20/0.46 ! [S: nat,T: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ S @ T )
% 0.20/0.46 => ( S != T ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_not_refl3
% 0.20/0.46 thf(fact_27_measure__induct,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( wellorder @ B )
% 0.20/0.46 => ! [F: A > B,P: A > $o,A2: A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ! [Y2: A] :
% 0.20/0.46 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.20/0.46 => ( P @ Y2 ) )
% 0.20/0.46 => ( P @ X3 ) )
% 0.20/0.46 => ( P @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % measure_induct
% 0.20/0.46 thf(fact_28_less__irrefl__nat,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.20/0.46
% 0.20/0.46 % less_irrefl_nat
% 0.20/0.46 thf(fact_29_nat__less__induct,axiom,
% 0.20/0.46 ! [P: nat > $o,N: nat] :
% 0.20/0.46 ( ! [N2: nat] :
% 0.20/0.46 ( ! [M2: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.20/0.46 => ( P @ M2 ) )
% 0.20/0.46 => ( P @ N2 ) )
% 0.20/0.46 => ( P @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % nat_less_induct
% 0.20/0.46 thf(fact_30_infinite__descent,axiom,
% 0.20/0.46 ! [P: nat > $o,N: nat] :
% 0.20/0.46 ( ! [N2: nat] :
% 0.20/0.46 ( ~ ( P @ N2 )
% 0.20/0.46 => ? [M2: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.20/0.46 & ~ ( P @ M2 ) ) )
% 0.20/0.46 => ( P @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_descent
% 0.20/0.46 thf(fact_31_linorder__neqE__nat,axiom,
% 0.20/0.46 ! [X2: nat,Y: nat] :
% 0.20/0.46 ( ( X2 != Y )
% 0.20/0.46 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 0.20/0.46 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % linorder_neqE_nat
% 0.20/0.46 thf(fact_32_measure__induct__rule,axiom,
% 0.20/0.46 ! [B: $tType,A: $tType] :
% 0.20/0.46 ( ( wellorder @ B )
% 0.20/0.46 => ! [F: A > B,P: A > $o,A2: A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ! [Y2: A] :
% 0.20/0.46 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.20/0.46 => ( P @ Y2 ) )
% 0.20/0.46 => ( P @ X3 ) )
% 0.20/0.46 => ( P @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % measure_induct_rule
% 0.20/0.46 thf(fact_33_infinite__descent__measure,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,V2: A > nat,X2: A] :
% 0.20/0.46 ( ! [X3: A] :
% 0.20/0.46 ( ~ ( P @ X3 )
% 0.20/0.46 => ? [Y2: A] :
% 0.20/0.46 ( ( ord_less @ nat @ ( V2 @ Y2 ) @ ( V2 @ X3 ) )
% 0.20/0.47 & ~ ( P @ Y2 ) ) )
% 0.20/0.47 => ( P @ X2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_descent_measure
% 0.20/0.47 thf(fact_34_map__eq__imp__length__eq,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: B > A,Xs2: list @ B,G: C > A,Ys: list @ C] :
% 0.20/0.47 ( ( ( map @ B @ A @ F @ Xs2 )
% 0.20/0.47 = ( map @ C @ A @ G @ Ys ) )
% 0.20/0.47 => ( ( size_size @ ( list @ B ) @ Xs2 )
% 0.20/0.47 = ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % map_eq_imp_length_eq
% 0.20/0.47 thf(fact_35_zero__less__iff__neq__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.20/0.47 = ( N
% 0.20/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_less_iff_neq_zero
% 0.20/0.47 thf(fact_36_gr__implies__not__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [M: A,N: A] :
% 0.20/0.47 ( ( ord_less @ A @ M @ N )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_implies_not_zero
% 0.20/0.47 thf(fact_37_not__less__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_less_zero
% 0.20/0.47 thf(fact_38_gr__zeroI,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( N
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_zeroI
% 0.20/0.47 thf(fact_39_infinite__descent0__measure,axiom,
% 0.20/0.47 ! [A: $tType,V2: A > nat,P: A > $o,X2: A] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( ( V2 @ X3 )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 => ( P @ X3 ) )
% 0.20/0.47 => ( ! [X3: A] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
% 0.20/0.47 => ( ~ ( P @ X3 )
% 0.20/0.47 => ? [Y2: A] :
% 0.20/0.47 ( ( ord_less @ nat @ ( V2 @ Y2 ) @ ( V2 @ X3 ) )
% 0.20/0.47 & ~ ( P @ Y2 ) ) ) )
% 0.20/0.47 => ( P @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_descent0_measure
% 0.20/0.47 thf(fact_40_bot__nat__0_Oextremum__strict,axiom,
% 0.20/0.47 ! [A2: nat] :
% 0.20/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_nat_0.extremum_strict
% 0.20/0.47 thf(fact_41_infinite__descent0,axiom,
% 0.20/0.47 ! [P: nat > $o,N: nat] :
% 0.20/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.20/0.47 => ( ! [N2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.47 => ( ~ ( P @ N2 )
% 0.20/0.47 => ? [M2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.20/0.47 & ~ ( P @ M2 ) ) ) )
% 0.20/0.47 => ( P @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % infinite_descent0
% 0.20/0.47 thf(fact_42_gr__implies__not0,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ M @ N )
% 0.20/0.47 => ( N
% 0.20/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr_implies_not0
% 0.20/0.47 thf(fact_43_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.47 = ( P @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_44_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A] :
% 0.20/0.47 ( ( collect @ A
% 0.20/0.47 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.20/0.47 = A3 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_45_Collect__cong,axiom,
% 0.20/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( P @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collect @ A @ P )
% 0.20/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_46_ext,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( F @ X3 )
% 0.20/0.47 = ( G @ X3 ) )
% 0.20/0.47 => ( F = G ) ) ).
% 0.20/0.47
% 0.20/0.47 % ext
% 0.20/0.47 thf(fact_47_less__zeroE,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_zeroE
% 0.20/0.47 thf(fact_48_not__less0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_less0
% 0.20/0.47 thf(fact_49_not__gr0,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_gr0
% 0.20/0.47 thf(fact_50_gr0I,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( N
% 0.20/0.47 != ( zero_zero @ nat ) )
% 0.20/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.20/0.47
% 0.20/0.47 % gr0I
% 0.20/0.47 thf(fact_51_length__induct,axiom,
% 0.20/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
% 0.20/0.47 ( ! [Xs: list @ A] :
% 0.20/0.47 ( ! [Ys2: list @ A] :
% 0.20/0.47 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.20/0.47 => ( P @ Ys2 ) )
% 0.20/0.47 => ( P @ Xs ) )
% 0.20/0.47 => ( P @ Xs2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % length_induct
% 0.20/0.47 thf(fact_52_all__nat__less__eq,axiom,
% 0.20/0.47 ! [N: nat,P: nat > $o] :
% 0.20/0.47 ( ( ! [M3: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ M3 @ N )
% 0.20/0.47 => ( P @ M3 ) ) )
% 0.20/0.47 = ( ! [X: nat] :
% 0.20/0.47 ( ( member @ nat @ X @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.20/0.47 => ( P @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_nat_less_eq
% 0.20/0.47 thf(fact_53_ex__nat__less__eq,axiom,
% 0.20/0.47 ! [N: nat,P: nat > $o] :
% 0.20/0.47 ( ( ? [M3: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ M3 @ N )
% 0.20/0.47 & ( P @ M3 ) ) )
% 0.20/0.47 = ( ? [X: nat] :
% 0.20/0.47 ( ( member @ nat @ X @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.20/0.47 & ( P @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_nat_less_eq
% 0.20/0.47 thf(fact_54_list__eq__iff__nth__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) )
% 0.20/0.47 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 0.20/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.20/0.47 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 0.20/0.47 & ! [I2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.20/0.47 => ( ( nth @ A @ Xs3 @ I2 )
% 0.20/0.47 = ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_eq_iff_nth_eq
% 0.20/0.47 thf(fact_55_Skolem__list__nth,axiom,
% 0.20/0.47 ! [A: $tType,K: nat,P: nat > A > $o] :
% 0.20/0.47 ( ( ! [I2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.20/0.47 => ? [X4: A] : ( P @ I2 @ X4 ) ) )
% 0.20/0.47 = ( ? [Xs3: list @ A] :
% 0.20/0.47 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.20/0.47 = K )
% 0.20/0.47 & ! [I2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ I2 @ K )
% 0.20/0.47 => ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Skolem_list_nth
% 0.20/0.47 thf(fact_56_nth__equalityI,axiom,
% 0.20/0.47 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 0.20/0.47 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.20/0.47 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.20/0.47 => ( ! [I3: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.20/0.47 => ( ( nth @ A @ Xs2 @ I3 )
% 0.20/0.47 = ( nth @ A @ Ys @ I3 ) ) )
% 0.20/0.47 => ( Xs2 = Ys ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nth_equalityI
% 0.20/0.47 thf(fact_57_zero__fun__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( zero @ B )
% 0.20/0.47 => ( ( zero_zero @ ( A > B ) )
% 0.20/0.47 = ( ^ [X: A] : ( zero_zero @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_fun_def
% 0.20/0.47 thf(fact_58_zero__reorient,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( zero @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = X2 )
% 0.20/0.47 = ( X2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_reorient
% 0.20/0.47 thf(fact_59_mset__map,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
% 0.20/0.47 ( ( mset @ A @ ( map @ B @ A @ F @ Xs2 ) )
% 0.20/0.47 = ( image_mset @ B @ A @ F @ ( mset @ B @ Xs2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mset_map
% 0.20/0.47 thf(fact_60_size__mset,axiom,
% 0.20/0.47 ! [A: $tType,Xs2: list @ A] :
% 0.20/0.47 ( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs2 ) )
% 0.20/0.47 = ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_mset
% 0.20/0.47 thf(fact_61_lessThan__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [I: A,K: A] :
% 0.20/0.47 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 0.20/0.47 = ( ord_less @ A @ I @ K ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lessThan_iff
% 0.20/0.47 thf(fact_62_image__mset__mset__set,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( image_mset @ A @ B @ F @ ( mset_set @ A @ A3 ) )
% 0.20/0.47 = ( mset_set @ B @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_mset_mset_set
% 0.20/0.47 thf(fact_63_lessThan__atLeast0,axiom,
% 0.20/0.47 ( ( set_ord_lessThan @ nat )
% 0.20/0.47 = ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lessThan_atLeast0
% 0.20/0.47 thf(fact_64_bij__betw__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( bij_betw @ A @ B )
% 0.20/0.47 = ( ^ [F2: A > B,A4: set @ A,B2: set @ B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.20/0.47 & ( ( image @ A @ B @ F2 @ A4 )
% 0.20/0.47 = B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_def
% 0.20/0.47 thf(fact_65_bij__betw__imageI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( ( image @ A @ B @ F @ A3 )
% 0.20/0.47 = B3 )
% 0.20/0.47 => ( bij_betw @ A @ B @ F @ A3 @ B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_imageI
% 0.20/0.47 thf(fact_66_inj__on__imp__bij__betw,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( bij_betw @ A @ B @ F @ A3 @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imp_bij_betw
% 0.20/0.47 thf(fact_67_comp__inj__on,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: A > B,A3: set @ A,G: B > C] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_68_inj__on__imageI,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A3 )
% 0.20/0.47 => ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_69_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: A > B,A3: set @ A,F3: B > C] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( inj_on @ B @ C @ F3 @ ( image @ A @ B @ F @ A3 ) )
% 0.20/0.47 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_70_comp__apply,axiom,
% 0.20/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.20/0.47 ( ( comp @ B @ A @ C )
% 0.20/0.47 = ( ^ [F2: B > A,G2: C > B,X: C] : ( F2 @ ( G2 @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_apply
% 0.20/0.47 thf(fact_71_lessThan__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X2: A,Y: A] :
% 0.20/0.47 ( ( ( set_ord_lessThan @ A @ X2 )
% 0.20/0.47 = ( set_ord_lessThan @ A @ Y ) )
% 0.20/0.47 = ( X2 = Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lessThan_eq_iff
% 0.20/0.47 thf(fact_72_image__mset__empty,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.20/0.47 ( ( image_mset @ B @ A @ F @ ( zero_zero @ ( multiset @ B ) ) )
% 0.20/0.47 = ( zero_zero @ ( multiset @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_mset_empty
% 0.20/0.47 thf(fact_73_image__mset__is__empty__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,M4: multiset @ B] :
% 0.20/0.47 ( ( ( image_mset @ B @ A @ F @ M4 )
% 0.20/0.47 = ( zero_zero @ ( multiset @ A ) ) )
% 0.20/0.47 = ( M4
% 0.20/0.47 = ( zero_zero @ ( multiset @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_mset_is_empty_iff
% 0.20/0.47 thf(fact_74_size__eq__0__iff__empty,axiom,
% 0.20/0.47 ! [A: $tType,M4: multiset @ A] :
% 0.20/0.47 ( ( ( size_size @ ( multiset @ A ) @ M4 )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( M4
% 0.20/0.47 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_eq_0_iff_empty
% 0.20/0.47 thf(fact_75_size__empty,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
% 0.20/0.47 = ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_empty
% 0.20/0.47 thf(fact_76_size__image__mset,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,M4: multiset @ B] :
% 0.20/0.47 ( ( size_size @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M4 ) )
% 0.20/0.47 = ( size_size @ ( multiset @ B ) @ M4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_image_mset
% 0.20/0.47 thf(fact_77_mset__upt,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( mset @ nat @ ( upt @ M @ N ) )
% 0.20/0.47 = ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mset_upt
% 0.20/0.47 thf(fact_78_inj__on__strict__subset,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ B3 )
% 0.20/0.47 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.47 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_strict_subset
% 0.20/0.47 thf(fact_79_comp__eq__dest__lhs,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B4: A > C,C2: A > B,V: A] :
% 0.20/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 0.20/0.47 = C2 )
% 0.20/0.47 => ( ( A2 @ ( B4 @ V ) )
% 0.20/0.47 = ( C2 @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest_lhs
% 0.20/0.47 thf(fact_80_comp__eq__elim,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B4: A > C,C2: D > B,D2: A > D] :
% 0.20/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 0.20/0.47 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.20/0.47 => ! [V3: A] :
% 0.20/0.47 ( ( A2 @ ( B4 @ V3 ) )
% 0.20/0.47 = ( C2 @ ( D2 @ V3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_elim
% 0.20/0.47 thf(fact_81_comp__eq__dest,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B4: A > C,C2: D > B,D2: A > D,V: A] :
% 0.20/0.47 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 0.20/0.47 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.20/0.47 => ( ( A2 @ ( B4 @ V ) )
% 0.20/0.47 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest
% 0.20/0.47 thf(fact_82_comp__assoc,axiom,
% 0.20/0.47 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.20/0.47 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.20/0.47 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_assoc
% 0.20/0.47 thf(fact_83_comp__def,axiom,
% 0.20/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.20/0.47 ( ( comp @ B @ C @ A )
% 0.20/0.47 = ( ^ [F2: B > C,G2: A > B,X: A] : ( F2 @ ( G2 @ X ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_def
% 0.20/0.47 thf(fact_84_inj__on__inverseI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F: A > B] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A3 )
% 0.20/0.47 => ( ( G @ ( F @ X3 ) )
% 0.20/0.47 = X3 ) )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_inverseI
% 0.20/0.47 thf(fact_85_inj__on__contraD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( X2 != Y )
% 0.20/0.47 => ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( ( F @ X2 )
% 0.20/0.47 != ( F @ Y ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_contraD
% 0.20/0.47 thf(fact_86_inj__on__eq__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( ( ( F @ X2 )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 = ( X2 = Y ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_eq_iff
% 0.20/0.47 thf(fact_87_inj__on__cong,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,G: A > B] :
% 0.20/0.47 ( ! [A5: A] :
% 0.20/0.47 ( ( member @ A @ A5 @ A3 )
% 0.20/0.47 => ( ( F @ A5 )
% 0.20/0.47 = ( G @ A5 ) ) )
% 0.20/0.47 => ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_cong
% 0.20/0.47 thf(fact_88_inj__on__def,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( inj_on @ A @ B )
% 0.20/0.47 = ( ^ [F2: A > B,A4: set @ A] :
% 0.20/0.47 ! [X: A] :
% 0.20/0.47 ( ( member @ A @ X @ A4 )
% 0.20/0.47 => ! [Y4: A] :
% 0.20/0.47 ( ( member @ A @ Y4 @ A4 )
% 0.20/0.47 => ( ( ( F2 @ X )
% 0.20/0.47 = ( F2 @ Y4 ) )
% 0.20/0.47 => ( X = Y4 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_def
% 0.20/0.47 thf(fact_89_inj__onI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
% 0.20/0.47 ( ! [X3: A,Y5: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y5 @ A3 )
% 0.20/0.47 => ( ( ( F @ X3 )
% 0.20/0.47 = ( F @ Y5 ) )
% 0.20/0.47 => ( X3 = Y5 ) ) ) )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_onI
% 0.20/0.47 thf(fact_90_inj__onD,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ F @ A3 )
% 0.20/0.47 => ( ( ( F @ X2 )
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( ( member @ A @ Y @ A3 )
% 0.20/0.47 => ( X2 = Y ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_onD
% 0.20/0.47 thf(fact_91_bij__betw__iff__bijections,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( bij_betw @ A @ B )
% 0.20/0.47 = ( ^ [F2: A > B,A4: set @ A,B2: set @ B] :
% 0.20/0.47 ? [G2: B > A] :
% 0.20/0.47 ( ! [X: A] :
% 0.20/0.47 ( ( member @ A @ X @ A4 )
% 0.20/0.47 => ( ( member @ B @ ( F2 @ X ) @ B2 )
% 0.20/0.47 & ( ( G2 @ ( F2 @ X ) )
% 0.20/0.47 = X ) ) )
% 0.20/0.47 & ! [X: B] :
% 0.20/0.47 ( ( member @ B @ X @ B2 )
% 0.20/0.47 => ( ( member @ A @ ( G2 @ X ) @ A4 )
% 0.20/0.47 & ( ( F2 @ ( G2 @ X ) )
% 0.20/0.47 = X ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_iff_bijections
% 0.20/0.47 thf(fact_92_bij__betw__apply,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B,A2: A] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.20/0.47 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_apply
% 0.20/0.47 thf(fact_93_bij__betw__cong,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A3: set @ A,F: A > B,G: A > B,A6: set @ B] :
% 0.20/0.47 ( ! [A5: A] :
% 0.20/0.47 ( ( member @ A @ A5 @ A3 )
% 0.20/0.47 => ( ( F @ A5 )
% 0.20/0.47 = ( G @ A5 ) ) )
% 0.20/0.47 => ( ( bij_betw @ A @ B @ F @ A3 @ A6 )
% 0.20/0.47 = ( bij_betw @ A @ B @ G @ A3 @ A6 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_cong
% 0.20/0.47 thf(fact_94_bij__betw__inv,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ? [G3: B > A] : ( bij_betw @ B @ A @ G3 @ B3 @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_inv
% 0.20/0.47 thf(fact_95_bij__betwE,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ! [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ A3 )
% 0.20/0.47 => ( member @ B @ ( F @ X5 ) @ B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betwE
% 0.20/0.47 thf(fact_96_ex__mset,axiom,
% 0.20/0.47 ! [A: $tType,X6: multiset @ A] :
% 0.20/0.47 ? [Xs: list @ A] :
% 0.20/0.47 ( ( mset @ A @ Xs )
% 0.20/0.47 = X6 ) ).
% 0.20/0.47
% 0.20/0.47 % ex_mset
% 0.20/0.47 thf(fact_97_image__eq__imp__comp,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A3: set @ B,G: C > A,B3: set @ C,H: A > D] :
% 0.20/0.47 ( ( ( image @ B @ A @ F @ A3 )
% 0.20/0.47 = ( image @ C @ A @ G @ B3 ) )
% 0.20/0.47 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A3 )
% 0.20/0.47 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_eq_imp_comp
% 0.20/0.47 thf(fact_98_image__comp,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 0.20/0.47 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
% 0.20/0.47 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_comp
% 0.20/0.47 thf(fact_99_nonempty__has__size,axiom,
% 0.20/0.47 ! [A: $tType,S2: multiset @ A] :
% 0.20/0.47 ( ( S2
% 0.20/0.47 != ( zero_zero @ ( multiset @ A ) ) )
% 0.20/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonempty_has_size
% 0.20/0.47 thf(fact_100_atLeastLessThan__inj_I2_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [A2: A,B4: A,C2: A,D2: A] :
% 0.20/0.47 ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 0.20/0.47 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 0.20/0.47 => ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.20/0.47 => ( B4 = D2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % atLeastLessThan_inj(2)
% 0.20/0.47 thf(fact_101_atLeastLessThan__inj_I1_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [A2: A,B4: A,C2: A,D2: A] :
% 0.20/0.47 ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 0.20/0.47 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 0.20/0.47 => ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.20/0.47 => ( A2 = C2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % atLeastLessThan_inj(1)
% 0.20/0.47 thf(fact_102_atLeastLessThan__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [A2: A,B4: A,C2: A,D2: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ C2 @ D2 )
% 0.20/0.47 => ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 0.20/0.47 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 0.20/0.47 = ( ( A2 = C2 )
% 0.20/0.47 & ( B4 = D2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % atLeastLessThan_eq_iff
% 0.20/0.47 thf(fact_103_lessThan__strict__subset__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [M: A,N: A] :
% 0.20/0.47 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 0.20/0.47 = ( ord_less @ A @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lessThan_strict_subset_iff
% 0.20/0.47 thf(fact_104_inj__on__image__iff,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F: A > A] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ A3 )
% 0.20/0.47 => ! [Xa: A] :
% 0.20/0.47 ( ( member @ A @ Xa @ A3 )
% 0.20/0.47 => ( ( ( G @ ( F @ X3 ) )
% 0.20/0.47 = ( G @ ( F @ Xa ) ) )
% 0.20/0.47 = ( ( G @ X3 )
% 0.20/0.47 = ( G @ Xa ) ) ) ) )
% 0.20/0.47 => ( ( inj_on @ A @ A @ F @ A3 )
% 0.20/0.47 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A3 ) )
% 0.20/0.47 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_image_iff
% 0.20/0.47 thf(fact_105_inj__on__imageI2,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F: A > C,A3: set @ A] :
% 0.20/0.47 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F ) @ A3 )
% 0.20/0.47 => ( inj_on @ A @ C @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI2
% 0.20/0.47 thf(fact_106_bij__betw__imp__surj__on,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ( ( image @ A @ B @ F @ A3 )
% 0.20/0.47 = B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_imp_surj_on
% 0.20/0.47 thf(fact_107_bij__betw__comp__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: A > B,A3: set @ A,A6: set @ B,F3: B > C,A7: set @ C] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ A6 )
% 0.20/0.47 => ( ( bij_betw @ B @ C @ F3 @ A6 @ A7 )
% 0.20/0.47 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A3 @ A7 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_comp_iff
% 0.20/0.47 thf(fact_108_bij__betw__trans,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,C: $tType,F: A > B,A3: set @ A,B3: set @ B,G: B > C,C3: set @ C] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ( ( bij_betw @ B @ C @ G @ B3 @ C3 )
% 0.20/0.47 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A3 @ C3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_trans
% 0.20/0.47 thf(fact_109_multiset_Omap__comp,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: multiset @ A] :
% 0.20/0.47 ( ( image_mset @ B @ C @ G @ ( image_mset @ A @ B @ F @ V ) )
% 0.20/0.47 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % multiset.map_comp
% 0.20/0.47 thf(fact_110_image__mset_Ocompositionality,axiom,
% 0.20/0.47 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Multiset: multiset @ A] :
% 0.20/0.47 ( ( image_mset @ B @ C @ F @ ( image_mset @ A @ B @ G @ Multiset ) )
% 0.20/0.47 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Multiset ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_mset.compositionality
% 0.20/0.47 thf(fact_111_image__mset_Ocomp,axiom,
% 0.20/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 0.20/0.47 ( ( comp @ ( multiset @ B ) @ ( multiset @ C ) @ ( multiset @ A ) @ ( image_mset @ B @ C @ F ) @ ( image_mset @ A @ B @ G ) )
% 0.20/0.47 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_mset.comp
% 0.20/0.47 thf(fact_112_bij__betw__imp__inj__on,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 0.20/0.47 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 0.20/0.47 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_imp_inj_on
% 0.20/0.47 thf(fact_113_mset__eq__length,axiom,
% 0.20/0.47 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 0.20/0.47 ( ( ( mset @ A @ Xs2 )
% 0.20/0.47 = ( mset @ A @ Ys ) )
% 0.20/0.47 => ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.20/0.47 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mset_eq_length
% 0.20/0.47 thf(fact_114_image__eqI,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,B4: A,F: B > A,X2: B,A3: set @ B] :
% 0.20/0.47 ( ( B4
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 => ( ( member @ B @ X2 @ A3 )
% 0.20/0.47 => ( member @ A @ B4 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_eqI
% 0.20/0.47 thf(fact_115_mbs_Oless__not__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( size @ A )
% 0.20/0.47 => ! [X2: A,A3: set @ A,Y: A] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( ( ord_less @ nat @ ( size_size @ A @ X2 ) @ ( size_size @ A @ Y ) )
% 0.20/0.47 => ( X2 != Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mbs.less_not_eq
% 0.20/0.47 thf(fact_116_Sup_OSUP__image,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
% 0.20/0.47 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
% 0.20/0.47 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Sup.SUP_image
% 0.20/0.47 thf(fact_117_Inf_OINF__image,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
% 0.20/0.47 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
% 0.20/0.47 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Inf.INF_image
% 0.20/0.47 thf(fact_118_less__numeral__extra_I3_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_numeral_extra(3)
% 0.20/0.47 thf(fact_119_field__lbound__gt__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [D1: A,D22: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 0.20/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 0.20/0.47 => ? [E: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 0.20/0.47 & ( ord_less @ A @ E @ D1 )
% 0.20/0.47 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % field_lbound_gt_zero
% 0.20/0.47 thf(fact_120_length__code,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( size_size @ ( list @ A ) )
% 0.20/0.47 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % length_code
% 0.20/0.47 thf(fact_121_psubsetD,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.20/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.20/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % psubsetD
% 0.20/0.47 thf(fact_122_psubset__trans,axiom,
% 0.20/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.47 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.47 => ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
% 0.20/0.47 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % psubset_trans
% 0.20/0.47 thf(fact_123_imageI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F: A > B] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % imageI
% 0.20/0.47 thf(fact_124_image__iff,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,Z2: A,F: B > A,A3: set @ B] :
% 0.20/0.47 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 = ( ? [X: B] :
% 0.20/0.47 ( ( member @ B @ X @ A3 )
% 0.20/0.47 & ( Z2
% 0.20/0.47 = ( F @ X ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_iff
% 0.20/0.47 thf(fact_125_bex__imageD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.20/0.47 ( ? [X5: A] :
% 0.20/0.47 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 & ( P @ X5 ) )
% 0.20/0.47 => ? [X3: B] :
% 0.20/0.47 ( ( member @ B @ X3 @ A3 )
% 0.20/0.47 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bex_imageD
% 0.20/0.47 thf(fact_126_image__cong,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,M4: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 0.20/0.47 ( ( M4 = N3 )
% 0.20/0.47 => ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ N3 )
% 0.20/0.47 => ( ( F @ X3 )
% 0.20/0.47 = ( G @ X3 ) ) )
% 0.20/0.47 => ( ( image @ A @ B @ F @ M4 )
% 0.20/0.47 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_cong
% 0.20/0.47 thf(fact_127_ball__imageD,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 0.20/0.47 ( ! [X3: A] :
% 0.20/0.47 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 0.20/0.47 => ( P @ X3 ) )
% 0.20/0.47 => ! [X5: B] :
% 0.20/0.47 ( ( member @ B @ X5 @ A3 )
% 0.20/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ball_imageD
% 0.20/0.47 thf(fact_128_rev__image__eqI,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,X2: A,A3: set @ A,B4: B,F: A > B] :
% 0.20/0.47 ( ( member @ A @ X2 @ A3 )
% 0.20/0.47 => ( ( B4
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 => ( member @ B @ B4 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % rev_image_eqI
% 0.20/0.47 thf(fact_129_Inf_OINF__cong,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.20/0.47 ( ( A3 = B3 )
% 0.20/0.47 => ( ! [X3: B] :
% 0.20/0.47 ( ( member @ B @ X3 @ B3 )
% 0.20/0.47 => ( ( C3 @ X3 )
% 0.20/0.47 = ( D3 @ X3 ) ) )
% 0.20/0.47 => ( ( Inf @ ( image @ B @ A @ C3 @ A3 ) )
% 0.20/0.47 = ( Inf @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Inf.INF_cong
% 0.20/0.47 thf(fact_130_Sup_OSUP__cong,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.20/0.47 ( ( A3 = B3 )
% 0.20/0.47 => ( ! [X3: B] :
% 0.20/0.47 ( ( member @ B @ X3 @ B3 )
% 0.20/0.47 => ( ( C3 @ X3 )
% 0.20/0.47 = ( D3 @ X3 ) ) )
% 0.20/0.47 => ( ( Sup @ ( image @ B @ A @ C3 @ A3 ) )
% 0.20/0.47 = ( Sup @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Sup.SUP_cong
% 0.20/0.47 thf(fact_131_Multiset_Ois__empty__def,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( is_empty @ A )
% 0.20/0.47 = ( ^ [A4: multiset @ A] :
% 0.20/0.47 ( A4
% 0.20/0.47 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Multiset.is_empty_def
% 0.20/0.47 thf(fact_132_map__eq__map__tailrec,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( map @ A @ B )
% 0.20/0.47 = ( map_tailrec @ A @ B ) ) ).
% 0.20/0.47
% 0.20/0.47 % map_eq_map_tailrec
% 0.20/0.47 thf(fact_133_list__ex__length,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( list_ex @ A )
% 0.20/0.47 = ( ^ [P2: A > $o,Xs3: list @ A] :
% 0.20/0.47 ? [N4: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 0.20/0.47 & ( P2 @ ( nth @ A @ Xs3 @ N4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % list_ex_length
% 0.20/0.47 thf(fact_134_of__nat__0__less__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_0_less_iff
% 0.20/0.47 thf(fact_135_of__nat__fun__apply,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType] :
% 0.20/0.47 ( ( semiring_1 @ A )
% 0.20/0.47 => ( ( semiring_1_of_nat @ ( B > A ) )
% 0.20/0.47 = ( ^ [N4: nat,X: B] : ( semiring_1_of_nat @ A @ N4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_fun_apply
% 0.20/0.47 thf(fact_136_of__nat__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( M = N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_eq_iff
% 0.20/0.47 thf(fact_137_of__nat__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_1 @ A )
% 0.20/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_0
% 0.20/0.47 thf(fact_138_of__nat__0__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( ( zero_zero @ nat )
% 0.20/0.47 = N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_0_eq_iff
% 0.20/0.47 thf(fact_139_of__nat__eq__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [M: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( M
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_eq_0_iff
% 0.20/0.47 thf(fact_140_of__nat__less__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_less_iff
% 0.20/0.47 thf(fact_141_bij__betw__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N3: set @ nat,A3: set @ A] :
% 0.20/0.47 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 @ A3 )
% 0.20/0.47 = ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 )
% 0.20/0.47 = A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bij_betw_of_nat
% 0.20/0.47 thf(fact_142_image__int__atLeastLessThan,axiom,
% 0.20/0.47 ! [A2: nat,B4: nat] :
% 0.20/0.47 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1433837966ssThan @ nat @ A2 @ B4 ) )
% 0.20/0.47 = ( set_or1433837966ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_int_atLeastLessThan
% 0.20/0.47 thf(fact_143_inj__on__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A )
% 0.20/0.47 => ! [N3: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_of_nat
% 0.20/0.47 thf(fact_144_of__nat__fun,axiom,
% 0.20/0.47 ! [A: $tType,B: $tType] :
% 0.20/0.47 ( ( semiring_1 @ B )
% 0.20/0.47 => ( ( semiring_1_of_nat @ ( A > B ) )
% 0.20/0.47 = ( ^ [N4: nat,X: A] : ( semiring_1_of_nat @ B @ N4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_fun
% 0.20/0.47 thf(fact_145_of__nat__less__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat] :
% 0.20/0.47 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_less_0_iff
% 0.20/0.47 thf(fact_146_of__nat__less__imp__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_less_imp_less
% 0.20/0.47 thf(fact_147_less__imp__of__nat__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ M @ N )
% 0.20/0.47 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_of_nat_less
% 0.20/0.47 thf(fact_148_pos__int__cases,axiom,
% 0.20/0.47 ! [K: int] :
% 0.20/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.20/0.47 => ~ ! [N2: nat] :
% 0.20/0.47 ( ( K
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N2 ) )
% 0.20/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % pos_int_cases
% 0.20/0.47 thf(fact_149_zero__less__imp__eq__int,axiom,
% 0.20/0.47 ! [K: int] :
% 0.20/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.20/0.47 => ? [N2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.47 & ( K
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_less_imp_eq_int
% 0.20/0.47 thf(fact_150_reals__Archimedean2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( archim1804426504_field @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reals_Archimedean2
% 0.20/0.47 thf(fact_151_size__multiset__empty,axiom,
% 0.20/0.47 ! [A: $tType,F: A > nat] :
% 0.20/0.47 ( ( size_multiset @ A @ F @ ( zero_zero @ ( multiset @ A ) ) )
% 0.20/0.47 = ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_multiset_empty
% 0.20/0.47 thf(fact_152_size__multiset__eq__0__iff__empty,axiom,
% 0.20/0.47 ! [A: $tType,F: A > nat,M4: multiset @ A] :
% 0.20/0.47 ( ( ( size_multiset @ A @ F @ M4 )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( M4
% 0.20/0.47 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_multiset_eq_0_iff_empty
% 0.20/0.47 thf(fact_153_int__int__eq,axiom,
% 0.20/0.47 ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ( semiring_1_of_nat @ int @ M )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N ) )
% 0.20/0.47 = ( M = N ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_int_eq
% 0.20/0.47 thf(fact_154_less__int__code_I1_J,axiom,
% 0.20/0.47 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_int_code(1)
% 0.20/0.47 thf(fact_155_size__multiset__o__map,axiom,
% 0.20/0.47 ! [B: $tType,A: $tType,G: B > nat,F: A > B] :
% 0.20/0.47 ( ( comp @ ( multiset @ B ) @ nat @ ( multiset @ A ) @ ( size_multiset @ B @ G ) @ ( image_mset @ A @ B @ F ) )
% 0.20/0.47 = ( size_multiset @ A @ ( comp @ B @ nat @ A @ G @ F ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % size_multiset_o_map
% 0.20/0.47 thf(fact_156_nat__int__comparison_I2_J,axiom,
% 0.20/0.47 ( ( ord_less @ nat )
% 0.20/0.47 = ( ^ [A8: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_int_comparison(2)
% 0.20/0.47 thf(fact_157_int__ops_I1_J,axiom,
% 0.20/0.47 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 0.20/0.47 = ( zero_zero @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_ops(1)
% 0.20/0.47 thf(fact_158_neg__int__cases,axiom,
% 0.20/0.47 ! [K: int] :
% 0.20/0.47 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.20/0.47 => ~ ! [N2: nat] :
% 0.20/0.47 ( ( K
% 0.20/0.47 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
% 0.20/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_int_cases
% 0.20/0.47 thf(fact_159_add_Oinverse__inverse,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.inverse_inverse
% 0.20/0.47 thf(fact_160_neg__equal__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.20/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.20/0.47 = ( A2 = B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_iff_equal
% 0.20/0.47 thf(fact_161_add_Oinverse__neutral,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % add.inverse_neutral
% 0.20/0.47 thf(fact_162_neg__0__equal__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ( zero_zero @ A )
% 0.20/0.47 = A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_0_equal_iff_equal
% 0.20/0.47 thf(fact_163_neg__equal__0__iff__equal,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_0_iff_equal
% 0.20/0.47 thf(fact_164_equal__neg__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equal_neg_zero
% 0.20/0.47 thf(fact_165_neg__equal__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.20/0.47 = A2 )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_equal_zero
% 0.20/0.47 thf(fact_166_neg__less__iff__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [B4: A,A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less @ A @ A2 @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_less_iff_less
% 0.20/0.47 thf(fact_167_inj__uminus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A3: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_uminus
% 0.20/0.47 thf(fact_168_neg__less__0__iff__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_less_0_iff_less
% 0.20/0.47 thf(fact_169_neg__0__less__iff__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_0_less_iff_less
% 0.20/0.47 thf(fact_170_neg__less__pos,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.20/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_less_pos
% 0.20/0.47 thf(fact_171_less__neg__neg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_neg_neg
% 0.20/0.47 thf(fact_172_negative__eq__positive,axiom,
% 0.20/0.47 ! [N: nat,M: nat] :
% 0.20/0.47 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ M ) )
% 0.20/0.47 = ( ( N
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 & ( M
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % negative_eq_positive
% 0.20/0.47 thf(fact_173_int__cases2,axiom,
% 0.20/0.47 ! [Z2: int] :
% 0.20/0.47 ( ! [N2: nat] :
% 0.20/0.47 ( Z2
% 0.20/0.47 != ( semiring_1_of_nat @ int @ N2 ) )
% 0.20/0.47 => ~ ! [N2: nat] :
% 0.20/0.47 ( Z2
% 0.20/0.47 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_cases2
% 0.20/0.47 thf(fact_174_minus__less__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B4 )
% 0.20/0.47 = ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % minus_less_iff
% 0.20/0.47 thf(fact_175_less__minus__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B4 ) )
% 0.20/0.47 = ( ord_less @ A @ B4 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_minus_iff
% 0.20/0.47 thf(fact_176_uminus__int__code_I1_J,axiom,
% 0.20/0.47 ( ( uminus_uminus @ int @ ( zero_zero @ int ) )
% 0.20/0.47 = ( zero_zero @ int ) ) ).
% 0.20/0.47
% 0.20/0.47 % uminus_int_code(1)
% 0.20/0.47 thf(fact_177_verit__negate__coefficient_I2_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % verit_negate_coefficient(2)
% 0.20/0.47 thf(fact_178_equation__minus__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.20/0.47 = ( B4
% 0.20/0.47 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equation_minus_iff
% 0.20/0.47 thf(fact_179_minus__equation__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( group_add @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.20/0.47 = B4 )
% 0.20/0.47 = ( ( uminus_uminus @ A @ B4 )
% 0.20/0.47 = A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % minus_equation_iff
% 0.20/0.47 thf(fact_180_not__int__zless__negative,axiom,
% 0.20/0.47 ! [N: nat,M: nat] :
% 0.20/0.47 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % not_int_zless_negative
% 0.20/0.47 thf(fact_181_int__cases4,axiom,
% 0.20/0.47 ! [M: int] :
% 0.20/0.47 ( ! [N2: nat] :
% 0.20/0.47 ( M
% 0.20/0.47 != ( semiring_1_of_nat @ int @ N2 ) )
% 0.20/0.47 => ~ ! [N2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.47 => ( M
% 0.20/0.47 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_cases4
% 0.20/0.47 thf(fact_182_verit__comp__simplify1_I1_J,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( order @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % verit_comp_simplify1(1)
% 0.20/0.47 thf(fact_183_nat__int__comparison_I1_J,axiom,
% 0.20/0.47 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 0.20/0.47 = ( ^ [A8: nat,B5: nat] :
% 0.20/0.47 ( ( semiring_1_of_nat @ int @ A8 )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_int_comparison(1)
% 0.20/0.47 thf(fact_184_int__if,axiom,
% 0.20/0.47 ! [P: $o,A2: nat,B4: nat] :
% 0.20/0.47 ( ( P
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B4 ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 0.20/0.47 & ( ~ P
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B4 ) )
% 0.20/0.47 = ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_if
% 0.20/0.47 thf(fact_185_int__cases3,axiom,
% 0.20/0.47 ! [K: int] :
% 0.20/0.47 ( ( K
% 0.20/0.47 != ( zero_zero @ int ) )
% 0.20/0.47 => ( ! [N2: nat] :
% 0.20/0.47 ( ( K
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N2 ) )
% 0.20/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 0.20/0.47 => ~ ! [N2: nat] :
% 0.20/0.47 ( ( K
% 0.20/0.47 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
% 0.20/0.47 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_cases3
% 0.20/0.47 thf(fact_186_norm__frac_Oinduct,axiom,
% 0.20/0.47 ! [P: int > int > $o,A0: int,A1: int] :
% 0.20/0.47 ( ! [A5: int,B6: int] :
% 0.20/0.47 ( ( ( ord_less @ int @ B6 @ ( zero_zero @ int ) )
% 0.20/0.47 => ( P @ ( uminus_uminus @ int @ A5 ) @ ( uminus_uminus @ int @ B6 ) ) )
% 0.20/0.47 => ( P @ A5 @ B6 ) )
% 0.20/0.47 => ( P @ A0 @ A1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % norm_frac.induct
% 0.20/0.47 thf(fact_187_compl__less__swap1,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( boolean_algebra @ A )
% 0.20/0.47 => ! [Y: A,X2: A] :
% 0.20/0.47 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 0.20/0.47 => ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % compl_less_swap1
% 0.20/0.47 thf(fact_188_compl__less__swap2,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( boolean_algebra @ A )
% 0.20/0.47 => ! [Y: A,X2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 0.20/0.47 => ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % compl_less_swap2
% 0.20/0.47 thf(fact_189_compl__less__compl__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( boolean_algebra @ A )
% 0.20/0.47 => ! [X2: A,Y: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 0.20/0.47 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % compl_less_compl_iff
% 0.20/0.47 thf(fact_190_ex__inverse__of__nat__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( archim1804426504_field @ A )
% 0.20/0.47 => ! [X2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 0.20/0.47 => ? [N2: nat] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.47 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_inverse_of_nat_less
% 0.20/0.47 thf(fact_191_zero__less__nat__eq,axiom,
% 0.20/0.47 ! [Z2: int] :
% 0.20/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) )
% 0.20/0.47 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % zero_less_nat_eq
% 0.20/0.47 thf(fact_192_nat__int,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.20/0.47 = N ) ).
% 0.20/0.47
% 0.20/0.47 % nat_int
% 0.20/0.47 thf(fact_193_zless__nat__conj,axiom,
% 0.20/0.47 ! [W: int,Z2: int] :
% 0.20/0.47 ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.20/0.47 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
% 0.20/0.47 & ( ord_less @ int @ W @ Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % zless_nat_conj
% 0.20/0.47 thf(fact_194_nat__zminus__int,axiom,
% 0.20/0.47 ! [N: nat] :
% 0.20/0.47 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.20/0.47 = ( zero_zero @ nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_zminus_int
% 0.20/0.47 thf(fact_195_nat__zero__as__int,axiom,
% 0.20/0.47 ( ( zero_zero @ nat )
% 0.20/0.47 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_zero_as_int
% 0.20/0.47 thf(fact_196_forall__pos__mono,axiom,
% 0.20/0.47 ! [P: real > $o,E2: real] :
% 0.20/0.47 ( ! [D4: real,E: real] :
% 0.20/0.47 ( ( ord_less @ real @ D4 @ E )
% 0.20/0.47 => ( ( P @ D4 )
% 0.20/0.47 => ( P @ E ) ) )
% 0.20/0.47 => ( ! [N2: nat] :
% 0.20/0.47 ( ( N2
% 0.20/0.47 != ( zero_zero @ nat ) )
% 0.20/0.47 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
% 0.20/0.47 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 0.20/0.47 => ( P @ E2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % forall_pos_mono
% 0.20/0.47 thf(fact_197_real__arch__inverse,axiom,
% 0.20/0.47 ! [E2: real] :
% 0.20/0.47 ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 0.20/0.47 = ( ? [N4: nat] :
% 0.20/0.47 ( ( N4
% 0.20/0.47 != ( zero_zero @ nat ) )
% 0.20/0.47 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) )
% 0.20/0.47 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) @ E2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % real_arch_inverse
% 0.20/0.47 thf(fact_198_nat__mono__iff,axiom,
% 0.20/0.47 ! [Z2: int,W: int] :
% 0.20/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
% 0.20/0.47 => ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 0.20/0.47 = ( ord_less @ int @ W @ Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_mono_iff
% 0.20/0.47 thf(fact_199_zless__nat__eq__int__zless,axiom,
% 0.20/0.47 ! [M: nat,Z2: int] :
% 0.20/0.47 ( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) )
% 0.20/0.47 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % zless_nat_eq_int_zless
% 0.20/0.47 thf(fact_200_split__nat,axiom,
% 0.20/0.47 ! [P: nat > $o,I: int] :
% 0.20/0.47 ( ( P @ ( nat2 @ I ) )
% 0.20/0.47 = ( ! [N4: nat] :
% 0.20/0.47 ( ( I
% 0.20/0.47 = ( semiring_1_of_nat @ int @ N4 ) )
% 0.20/0.47 => ( P @ N4 ) )
% 0.20/0.47 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
% 0.20/0.47 => ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % split_nat
% 0.20/0.47 thf(fact_201_inverse__positive__iff__positive,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.20/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_positive_iff_positive
% 0.20/0.47 thf(fact_202_inverse__negative__iff__negative,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_negative_iff_negative
% 0.20/0.47 thf(fact_203_inverse__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_zero
% 0.20/0.47 thf(fact_204_inverse__nonzero__iff__nonzero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( A2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_nonzero_iff_nonzero
% 0.20/0.47 thf(fact_205_inverse__less__iff__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 = ( ord_less @ A @ B4 @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_less_iff_less
% 0.20/0.47 thf(fact_206_inverse__less__iff__less__neg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 = ( ord_less @ A @ B4 @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_less_iff_less_neg
% 0.20/0.47 thf(fact_207_linordered__field__no__lb,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [X5: A] :
% 0.20/0.47 ? [Y5: A] : ( ord_less @ A @ Y5 @ X5 ) ) ).
% 0.20/0.47
% 0.20/0.47 % linordered_field_no_lb
% 0.20/0.47 thf(fact_208_linordered__field__no__ub,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [X5: A] :
% 0.20/0.47 ? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % linordered_field_no_ub
% 0.20/0.47 thf(fact_209_field__class_Ofield__inverse__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( field @ A )
% 0.20/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % field_class.field_inverse_zero
% 0.20/0.47 thf(fact_210_inverse__zero__imp__zero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 => ( A2
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_zero_imp_zero
% 0.20/0.47 thf(fact_211_nonzero__inverse__eq__imp__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.20/0.47 = ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 => ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ( B4
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( A2 = B4 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonzero_inverse_eq_imp_eq
% 0.20/0.47 thf(fact_212_nonzero__inverse__inverse__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 0.20/0.47 = A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonzero_inverse_inverse_eq
% 0.20/0.47 thf(fact_213_nonzero__imp__inverse__nonzero,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ( inverse_inverse @ A @ A2 )
% 0.20/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonzero_imp_inverse_nonzero
% 0.20/0.47 thf(fact_214_inverse__less__imp__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.47 => ( ord_less @ A @ B4 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_less_imp_less
% 0.20/0.47 thf(fact_215_less__imp__inverse__less,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B4 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_inverse_less
% 0.20/0.47 thf(fact_216_inverse__less__imp__less__neg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ B4 @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_less_imp_less_neg
% 0.20/0.47 thf(fact_217_less__imp__inverse__less__neg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ B4 )
% 0.20/0.47 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B4 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_imp_inverse_less_neg
% 0.20/0.47 thf(fact_218_inverse__negative__imp__negative,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_negative_imp_negative
% 0.20/0.47 thf(fact_219_inverse__positive__imp__positive,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.20/0.47 => ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_positive_imp_positive
% 0.20/0.47 thf(fact_220_negative__imp__inverse__negative,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % negative_imp_inverse_negative
% 0.20/0.47 thf(fact_221_positive__imp__inverse__positive,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % positive_imp_inverse_positive
% 0.20/0.47 thf(fact_222_nonzero__inverse__minus__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( division_ring @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( A2
% 0.20/0.47 != ( zero_zero @ A ) )
% 0.20/0.47 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonzero_inverse_minus_eq
% 0.20/0.47 thf(fact_223_image__atLeastZeroLessThan__int,axiom,
% 0.20/0.47 ! [U: int] :
% 0.20/0.47 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 0.20/0.47 => ( ( set_or1433837966ssThan @ int @ ( zero_zero @ int ) @ U )
% 0.20/0.47 = ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % image_atLeastZeroLessThan_int
% 0.20/0.47 thf(fact_224_of__int__of__nat,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_1 @ A )
% 0.20/0.47 => ( ( ring_1_of_int @ A )
% 0.20/0.47 = ( ^ [K2: int] : ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K2 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_of_nat
% 0.20/0.47 thf(fact_225_of__int__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_char_0 @ A )
% 0.20/0.47 => ! [W: int,Z2: int] :
% 0.20/0.47 ( ( ( ring_1_of_int @ A @ W )
% 0.20/0.47 = ( ring_1_of_int @ A @ Z2 ) )
% 0.20/0.47 = ( W = Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_eq_iff
% 0.20/0.47 thf(fact_226_le__zero__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( canoni770627133id_add @ A )
% 0.20/0.47 => ! [N: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.20/0.47 = ( N
% 0.20/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % le_zero_eq
% 0.20/0.47 thf(fact_227_neg__le__iff__le,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [B4: A,A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A2 @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_le_iff_le
% 0.20/0.47 thf(fact_228_of__nat__le__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat,N: nat] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.47 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_le_iff
% 0.20/0.47 thf(fact_229_ivl__subset,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [I: A,J: A,M: A,N: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ I @ J ) @ ( set_or1433837966ssThan @ A @ M @ N ) )
% 0.20/0.47 = ( ( ord_less_eq @ A @ J @ I )
% 0.20/0.47 | ( ( ord_less_eq @ A @ M @ I )
% 0.20/0.47 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % ivl_subset
% 0.20/0.47 thf(fact_230_lessThan__subset__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linorder @ A )
% 0.20/0.47 => ! [X2: A,Y: A] :
% 0.20/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y ) )
% 0.20/0.47 = ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % lessThan_subset_iff
% 0.20/0.47 thf(fact_231_neg__0__le__iff__le,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_0_le_iff_le
% 0.20/0.47 thf(fact_232_neg__le__0__iff__le,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_le_0_iff_le
% 0.20/0.47 thf(fact_233_less__eq__neg__nonpos,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % less_eq_neg_nonpos
% 0.20/0.47 thf(fact_234_neg__less__eq__nonneg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord219039673up_add @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % neg_less_eq_nonneg
% 0.20/0.47 thf(fact_235_inverse__nonpositive__iff__nonpositive,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_nonpositive_iff_nonpositive
% 0.20/0.47 thf(fact_236_inverse__nonnegative__iff__nonnegative,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_nonnegative_iff_nonnegative
% 0.20/0.47 thf(fact_237_atLeastLessThan__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ord @ A )
% 0.20/0.47 => ! [I: A,L: A,U: A] :
% 0.20/0.47 ( ( member @ A @ I @ ( set_or1433837966ssThan @ A @ L @ U ) )
% 0.20/0.47 = ( ( ord_less_eq @ A @ L @ I )
% 0.20/0.47 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % atLeastLessThan_iff
% 0.20/0.47 thf(fact_238_of__int__eq__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_char_0 @ A )
% 0.20/0.47 => ! [Z2: int] :
% 0.20/0.47 ( ( ( ring_1_of_int @ A @ Z2 )
% 0.20/0.47 = ( zero_zero @ A ) )
% 0.20/0.47 = ( Z2
% 0.20/0.47 = ( zero_zero @ int ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_eq_0_iff
% 0.20/0.47 thf(fact_239_of__int__0__eq__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_char_0 @ A )
% 0.20/0.47 => ! [Z2: int] :
% 0.20/0.47 ( ( ( zero_zero @ A )
% 0.20/0.47 = ( ring_1_of_int @ A @ Z2 ) )
% 0.20/0.47 = ( Z2
% 0.20/0.47 = ( zero_zero @ int ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_0_eq_iff
% 0.20/0.47 thf(fact_240_of__int__0,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_1 @ A )
% 0.20/0.47 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 0.20/0.47 = ( zero_zero @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_0
% 0.20/0.47 thf(fact_241_of__int__le__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_idom @ A )
% 0.20/0.47 => ! [W: int,Z2: int] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
% 0.20/0.47 = ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_le_iff
% 0.20/0.47 thf(fact_242_of__int__less__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_idom @ A )
% 0.20/0.47 => ! [W: int,Z2: int] :
% 0.20/0.47 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
% 0.20/0.47 = ( ord_less @ int @ W @ Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_less_iff
% 0.20/0.47 thf(fact_243_of__int__minus,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_1 @ A )
% 0.20/0.47 => ! [Z2: int] :
% 0.20/0.47 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
% 0.20/0.47 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_minus
% 0.20/0.47 thf(fact_244_of__int__of__nat__eq,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( ring_1 @ A )
% 0.20/0.47 => ! [N: nat] :
% 0.20/0.47 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 0.20/0.47 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_of_nat_eq
% 0.20/0.47 thf(fact_245_negative__zle,axiom,
% 0.20/0.47 ! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 0.20/0.47
% 0.20/0.47 % negative_zle
% 0.20/0.47 thf(fact_246_of__nat__le__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linord1659791738miring @ A )
% 0.20/0.47 => ! [M: nat] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( M
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_nat_le_0_iff
% 0.20/0.47 thf(fact_247_inverse__le__iff__le,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 0.20/0.47 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ B4 @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_le_iff_le
% 0.20/0.47 thf(fact_248_inverse__le__iff__le__neg,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_field @ A )
% 0.20/0.47 => ! [A2: A,B4: A] :
% 0.20/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 0.20/0.47 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 0.20/0.47 = ( ord_less_eq @ A @ B4 @ A2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inverse_le_iff_le_neg
% 0.20/0.47 thf(fact_249_nat__le__0,axiom,
% 0.20/0.47 ! [Z2: int] :
% 0.20/0.47 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
% 0.20/0.47 => ( ( nat2 @ Z2 )
% 0.20/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_le_0
% 0.20/0.47 thf(fact_250_nat__0__iff,axiom,
% 0.20/0.47 ! [I: int] :
% 0.20/0.47 ( ( ( nat2 @ I )
% 0.20/0.47 = ( zero_zero @ nat ) )
% 0.20/0.47 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nat_0_iff
% 0.20/0.47 thf(fact_251_int__nat__eq,axiom,
% 0.20/0.47 ! [Z2: int] :
% 0.20/0.47 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
% 0.20/0.47 = Z2 ) )
% 0.20/0.47 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.20/0.47 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
% 0.20/0.47 = ( zero_zero @ int ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % int_nat_eq
% 0.20/0.47 thf(fact_252_of__int__le__0__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_idom @ A )
% 0.20/0.47 => ! [Z2: int] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
% 0.20/0.47 = ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_le_0_iff
% 0.20/0.47 thf(fact_253_of__int__0__le__iff,axiom,
% 0.20/0.47 ! [A: $tType] :
% 0.20/0.47 ( ( linordered_idom @ A )
% 0.20/0.47 => ! [Z2: int] :
% 0.20/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 0.20/0.47 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % of_int_0_le_iff
% 0.20/0.47
% 0.20/0.47 % Type constructors (72)
% 0.20/0.47 thf(tcon_Multiset_Omultiset___Orderings_Opreorder,axiom,
% 0.20/0.47 ! [A9: $tType] :
% 0.20/0.47 ( ( preorder @ A9 )
% 0.20/0.47 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Orderings_Opreorder_1,axiom,
% 0.20/0.47 preorder @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
% 0.20/0.47 preorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 0.20/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Odioid,axiom,
% 0.20/0.47 dioid @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 0.20/0.47 preorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom,
% 0.20/0.47 preorder @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Rings_Odioid_6,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( dioid @ A10 )
% 0.20/0.47 => ( dioid @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Opreorder_7,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( preorder @ A10 )
% 0.20/0.47 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( dioid @ A10 )
% 0.20/0.47 => ( canoni770627133id_add @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Groups_Oordered__ab__group__add,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( ordered_ab_group_add @ A10 )
% 0.20/0.47 => ( ordered_ab_group_add @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( boolean_algebra @ A10 )
% 0.20/0.47 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Nat_Osemiring__char__0,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( semiring_char_0 @ A10 )
% 0.20/0.47 => ( semiring_char_0 @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Rings_Osemiring__1,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( semiring_1 @ A10 )
% 0.20/0.47 => ( semiring_1 @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Groups_Ogroup__add,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( group_add @ A10 )
% 0.20/0.47 => ( group_add @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( order @ A10 )
% 0.20/0.47 => ( order @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Nat_Oring__char__0,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( ring_char_0 @ A10 )
% 0.20/0.47 => ( ring_char_0 @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( ord @ A10 )
% 0.20/0.47 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Rings_Oring__1,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( ring_1 @ A10 )
% 0.20/0.47 => ( ring_1 @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_fun___Groups_Ozero,axiom,
% 0.20/0.47 ! [A9: $tType,A10: $tType] :
% 0.20/0.47 ( ( zero @ A10 )
% 0.20/0.47 => ( zero @ ( A9 > A10 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.20/0.47 linord1659791738miring @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 0.20/0.47 linord219039673up_add @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add_8,axiom,
% 0.20/0.47 ordered_ab_group_add @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.20/0.47 linordered_idom @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Nat_Osemiring__char__0_9,axiom,
% 0.20/0.47 semiring_char_0 @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 0.20/0.47 linorder @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Rings_Osemiring__1_10,axiom,
% 0.20/0.47 semiring_1 @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Groups_Ogroup__add_11,axiom,
% 0.20/0.47 group_add @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Orderings_Oorder_12,axiom,
% 0.20/0.47 order @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Nat_Oring__char__0_13,axiom,
% 0.20/0.47 ring_char_0 @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Orderings_Oord_14,axiom,
% 0.20/0.47 ord @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Rings_Oring__1_15,axiom,
% 0.20/0.47 ring_1 @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Int_Oint___Groups_Ozero_16,axiom,
% 0.20/0.47 zero @ int ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_17,axiom,
% 0.20/0.47 canoni770627133id_add @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_18,axiom,
% 0.20/0.47 linord1659791738miring @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.20/0.47 wellorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_19,axiom,
% 0.20/0.47 semiring_char_0 @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder_20,axiom,
% 0.20/0.47 linorder @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1_21,axiom,
% 0.20/0.47 semiring_1 @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_22,axiom,
% 0.20/0.47 order @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Orderings_Oord_23,axiom,
% 0.20/0.47 ord @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Groups_Ozero_24,axiom,
% 0.20/0.47 zero @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.20/0.47 size @ nat ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_25,axiom,
% 0.20/0.47 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oorder_26,axiom,
% 0.20/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Set_Oset___Orderings_Oord_27,axiom,
% 0.20/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_28,axiom,
% 0.20/0.47 boolean_algebra @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_29,axiom,
% 0.20/0.47 linorder @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_30,axiom,
% 0.20/0.47 order @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_HOL_Obool___Orderings_Oord_31,axiom,
% 0.20/0.47 ord @ $o ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_List_Olist___Nat_Osize_32,axiom,
% 0.20/0.47 ! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 0.20/0.47 archim1804426504_field @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_33,axiom,
% 0.20/0.47 linord1659791738miring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_34,axiom,
% 0.20/0.47 linord219039673up_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_35,axiom,
% 0.20/0.47 ordered_ab_group_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 0.20/0.47 linordered_field @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Olinordered__idom_36,axiom,
% 0.20/0.47 linordered_idom @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 0.20/0.47 division_ring @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_37,axiom,
% 0.20/0.47 semiring_char_0 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Orderings_Olinorder_38,axiom,
% 0.20/0.47 linorder @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Osemiring__1_39,axiom,
% 0.20/0.47 semiring_1 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ogroup__add_40,axiom,
% 0.20/0.47 group_add @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Orderings_Oorder_41,axiom,
% 0.20/0.47 order @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Nat_Oring__char__0_42,axiom,
% 0.20/0.47 ring_char_0 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Orderings_Oord_43,axiom,
% 0.20/0.47 ord @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Rings_Oring__1_44,axiom,
% 0.20/0.47 ring_1 @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 0.20/0.47 field @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Real_Oreal___Groups_Ozero_45,axiom,
% 0.20/0.47 zero @ real ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Multiset_Omultiset___Orderings_Oorder_46,axiom,
% 0.20/0.47 ! [A9: $tType] :
% 0.20/0.47 ( ( preorder @ A9 )
% 0.20/0.47 => ( order @ ( multiset @ A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Multiset_Omultiset___Orderings_Oord_47,axiom,
% 0.20/0.47 ! [A9: $tType] :
% 0.20/0.47 ( ( preorder @ A9 )
% 0.20/0.47 => ( ord @ ( multiset @ A9 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Multiset_Omultiset___Groups_Ozero_48,axiom,
% 0.20/0.47 ! [A9: $tType] : ( zero @ ( multiset @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(tcon_Multiset_Omultiset___Nat_Osize_49,axiom,
% 0.20/0.47 ! [A9: $tType] : ( size @ ( multiset @ A9 ) ) ).
% 0.20/0.47
% 0.20/0.47 % Helper facts (3)
% 0.20/0.47 thf(help_If_3_1_T,axiom,
% 0.20/0.47 ! [P: $o] :
% 0.20/0.47 ( ( P = $true )
% 0.20/0.47 | ( P = $false ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_2_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X2: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $false @ X2 @ Y )
% 0.20/0.47 = Y ) ).
% 0.20/0.47
% 0.20/0.47 thf(help_If_1_1_T,axiom,
% 0.20/0.47 ! [A: $tType,X2: A,Y: A] :
% 0.20/0.47 ( ( if @ A @ $true @ X2 @ Y )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 % Conjectures (1)
% 0.20/0.47 thf(conj_0,conjecture,
% 0.20/0.47 ( ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) )
% 0.20/0.47 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ ys ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 3.12/1.82 ---- Embedded in THF ---
% 3.12/1.82 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.12/1.82 %%% Generated on Tue Feb 24 08:25:17 EST 2026
% 3.12/1.82 %%% using '$$dhol' embedding, version 1.3.0.
% 3.12/1.82 %%% Logic specification used:
% 3.12/1.82 %%% thf(spec, logic, $$dhol).
% 3.12/1.82
% 3.12/1.82 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2DTF409.p
% 3.12/1.82 thf(ty_t_Multiset_Omultiset, type, multiset: ($tType > $tType)).
% 3.12/1.82 thf(ty_t_Real_Oreal, type, real: $tType).
% 3.12/1.82 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 3.12/1.82 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.12/1.82 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.12/1.82 thf(ty_t_Int_Oint, type, int: $tType).
% 3.12/1.82 thf(ty_tf_a, type, a: $tType).
% 3.12/1.82 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Rings_Odioid, type, dioid: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 3.12/1.82 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 3.12/1.82 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 3.12/1.82 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 3.12/1.82 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 3.12/1.82 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 3.12/1.82 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 3.12/1.82 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.12/1.82 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 3.12/1.82 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 3.12/1.82 thf(sy_c_List_Ogen__length, type, gen_length: (!> [A:$tType]: ((nat > ((list @ A) > nat))))).
% 3.12/1.82 thf(sy_c_List_Olist_Omap, type, map: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((list @ A) > (list @ Aa)))))).
% 3.12/1.82 thf(sy_c_List_Olist__ex, type, list_ex: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 3.12/1.82 thf(sy_c_List_Omap__tailrec, type, map_tailrec: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > (list @ B)))))).
% 3.12/1.82 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 3.12/1.82 thf(sy_c_List_Oupt, type, upt: (nat > (nat > (list @ nat)))).
% 3.12/1.82 thf(sy_c_Multiset_Oimage__mset, type, image_mset: (!> [A:$tType,B:$tType]: (((A > B) > ((multiset @ A) > (multiset @ B)))))).
% 3.12/1.82 thf(sy_c_Multiset_Ois__empty, type, is_empty: (!> [A:$tType]: (((multiset @ A) > $o)))).
% 3.12/1.82 thf(sy_c_Multiset_Omset, type, mset: (!> [A:$tType]: (((list @ A) > (multiset @ A))))).
% 3.12/1.82 thf(sy_c_Multiset_Omset__set, type, mset_set: (!> [B:$tType]: (((set @ B) > (multiset @ B))))).
% 3.12/1.82 thf(sy_c_Multiset_Osize__multiset, type, size_multiset: (!> [A:$tType]: (((A > nat) > ((multiset @ A) > nat))))).
% 3.12/1.82 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 3.12/1.82 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 3.12/1.82 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.12/1.82 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.12/1.82 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.12/1.82 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.12/1.82 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan, type, set_or1433837966ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 3.12/1.82 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 3.12/1.82 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.12/1.82 thf(sy_v_f, type, f: (nat > nat)).
% 3.12/1.82 thf(sy_v_xs, type, xs: (list @ a)).
% 3.12/1.82 thf(sy_v_ys, type, ys: (list @ a)).
% 3.12/1.82 thf(fact_0__C1_C, axiom, ((((inj_on @ nat) @ nat) @ f) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs)))).
% 3.12/1.82 thf(fact_1_calculation, axiom, (((mset @ a) @ xs) = ((((image_mset @ nat) @ a) @ ((nth @ a) @ ys)) @ ((((image_mset @ nat) @ nat) @ f) @ ((mset_set @ nat) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs))))))).
% 3.12/1.82 thf(fact_2_assms_I2_J, axiom, (! [I:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ a)) @ xs)) => ((((nth @ a) @ xs) @ I) = (((nth @ a) @ ys) @ (f @ I))))))).
% 3.12/1.82 thf(fact_3__C2_C, axiom, (((((image @ nat) @ nat) @ f) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs))) = (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ ys)))).
% 3.12/1.82 thf(fact_4_zero__fun__apply, axiom, (! [A:$tType,B:$tType]: (((zero @ A) => ((zero_zero @ (B > A)) = (^ [X:B]: ((zero_zero @ A)))))))).
% 3.12/1.82 thf(fact_5_assms_I1_J, axiom, (((((bij_betw @ nat) @ nat) @ f) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ a)) @ xs))) @ ((set_ord_lessThan @ nat) @ ((size_size @ (list @ a)) @ ys)))).
% 3.12/1.82 thf(fact_6__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062, axiom, (((mset @ a) @ ((((map @ nat) @ a) @ (((((comp @ nat) @ a) @ nat) @ ((nth @ a) @ ys)) @ f)) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs)))) = ((((image_mset @ nat) @ a) @ ((nth @ a) @ ys)) @ ((((image_mset @ nat) @ nat) @ f) @ ((mset_set @ nat) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs))))))).
% 3.12/1.82 thf(fact_7__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062, axiom, (xs = ((((map @ nat) @ a) @ (((((comp @ nat) @ a) @ nat) @ ((nth @ a) @ ys)) @ f)) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs))))).
% 3.12/1.82 thf(fact_8_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs:(list @ A)]: ((((size_size @ (list @ A)) @ Xs) = N)))))).
% 3.12/1.82 thf(fact_9_neq__if__length__neq, axiom, (! [A:$tType,Xs2:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) != ((size_size @ (list @ A)) @ Ys)) => (Xs2 != Ys))))).
% 3.12/1.82 thf(fact_10_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X2:A,Y:A]: (((((size_size @ A) @ X2) != ((size_size @ A) @ Y)) => (X2 != Y)))))))).
% 3.12/1.82 thf(fact_11_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_12_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 3.12/1.82 thf(fact_13_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 3.12/1.82 thf(fact_14_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 3.12/1.82 thf(fact_15_map__map, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),Xs2:(list @ C)]: ((((((map @ B) @ A) @ F) @ ((((map @ C) @ B) @ G) @ Xs2)) = ((((map @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ Xs2))))).
% 3.12/1.82 thf(fact_16_List_Omap_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C),G:(A > B)]: (((((((comp @ (list @ B)) @ (list @ C)) @ (list @ A)) @ (((map @ B) @ C) @ F)) @ (((map @ A) @ B) @ G)) = (((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)))))).
% 3.12/1.82 thf(fact_17_map__comp__map, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),G:(A > C)]: (((((((comp @ (list @ C)) @ (list @ B)) @ (list @ A)) @ (((map @ C) @ B) @ F)) @ (((map @ A) @ C) @ G)) = (((map @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F) @ G)))))).
% 3.12/1.82 thf(fact_18_list_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(list @ A)]: ((((((map @ B) @ C) @ G) @ ((((map @ A) @ B) @ F) @ V)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 3.12/1.82 thf(fact_19_List_Omap_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C),G:(A > B),List:(list @ A)]: ((((((map @ B) @ C) @ F) @ ((((map @ A) @ B) @ G) @ List)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)) @ List))))).
% 3.12/1.82 thf(fact_20_length__map, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs2:(list @ B)]: ((((size_size @ (list @ A)) @ ((((map @ B) @ A) @ F) @ Xs2)) = ((size_size @ (list @ B)) @ Xs2))))).
% 3.12/1.82 thf(fact_21_nth__map, axiom, (! [B:$tType,A:$tType,N:nat,Xs2:(list @ A),F:(A > B)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2)) => ((((nth @ B) @ ((((map @ A) @ B) @ F) @ Xs2)) @ N) = (F @ (((nth @ A) @ Xs2) @ N))))))).
% 3.12/1.82 thf(fact_22__092_060open_062mset_Axs_A_061_Amset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_092_060close_062, axiom, (((mset @ a) @ xs) = ((mset @ a) @ ((((map @ nat) @ a) @ (((((comp @ nat) @ a) @ nat) @ ((nth @ a) @ ys)) @ f)) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs)))))).
% 3.12/1.82 thf(fact_23_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 3.12/1.82 thf(fact_24_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 3.12/1.82 thf(fact_25_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 3.12/1.82 thf(fact_26_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 3.12/1.82 thf(fact_27_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2)))))))).
% 3.12/1.82 thf(fact_28_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 3.12/1.82 thf(fact_29_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 3.12/1.82 thf(fact_30_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 3.12/1.82 thf(fact_31_linorder__neqE__nat, axiom, (! [X2:nat,Y:nat]: (((X2 != Y) => ((~ ((((ord_less @ nat) @ X2) @ Y))) => (((ord_less @ nat) @ Y) @ X2)))))).
% 3.12/1.82 thf(fact_32_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2)))))))).
% 3.12/1.82 thf(fact_33_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V2:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V2 @ Y2)) @ (V2 @ X3)) & (~ ((P @ Y2))))))))) => (P @ X2))))).
% 3.12/1.82 thf(fact_34_map__eq__imp__length__eq, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),Xs2:(list @ B),G:(C > A),Ys:(list @ C)]: (((((((map @ B) @ A) @ F) @ Xs2) = ((((map @ C) @ A) @ G) @ Ys)) => (((size_size @ (list @ B)) @ Xs2) = ((size_size @ (list @ C)) @ Ys)))))).
% 3.12/1.82 thf(fact_35_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_36_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_37_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 3.12/1.82 thf(fact_38_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 3.12/1.82 thf(fact_39_infinite__descent0__measure, axiom, (! [A:$tType,V2:(A > nat),P:(A > $o),X2:A]: (((! [X3:A]: ((((V2 @ X3) = (zero_zero @ nat)) => (P @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V2 @ X3)) => ((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V2 @ Y2)) @ (V2 @ X3)) & (~ ((P @ Y2)))))))))) => (P @ X2)))))).
% 3.12/1.82 thf(fact_40_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 3.12/1.82 thf(fact_41_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 3.12/1.82 thf(fact_42_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 3.12/1.82 thf(fact_43_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.12/1.82 thf(fact_44_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 3.12/1.82 thf(fact_45_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.12/1.82 thf(fact_46_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 3.12/1.82 thf(fact_47_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 3.12/1.82 thf(fact_48_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 3.12/1.82 thf(fact_49_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 3.12/1.82 thf(fact_50_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 3.12/1.82 thf(fact_51_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs2:(list @ A)]: (((! [Xs:(list @ A)]: (((! [Ys2:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs)) => (P @ Ys2)))) => (P @ Xs)))) => (P @ Xs2))))).
% 3.12/1.82 thf(fact_52_all__nat__less__eq, axiom, (! [N:nat,P:(nat > $o)]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N) => (P @ M3)))) = (! [X:nat]: (((((member @ nat) @ X) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ N)) => (P @ X)))))))).
% 3.12/1.82 thf(fact_53_ex__nat__less__eq, axiom, (! [N:nat,P:(nat > $o)]: (((? [M3:nat]: (((((ord_less @ nat) @ M3) @ N) & (P @ M3)))) = (? [X:nat]: (((((member @ nat) @ X) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ N)) & (P @ X)))))))).
% 3.12/1.82 thf(fact_54_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y3:(list @ A),Z:(list @ A)]: ((Y3 = Z))) = (^ [Xs3:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys3)) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs3)) => ((((nth @ A) @ Xs3) @ I2) = (((nth @ A) @ Ys3) @ I2)))))))))))).
% 3.12/1.82 thf(fact_55_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => (? [X4:A]: (((P @ I2) @ X4)))))) = (? [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (! [I2:nat]: (((((ord_less @ nat) @ I2) @ K) => ((P @ I2) @ (((nth @ A) @ Xs3) @ I2)))))))))))).
% 3.12/1.82 thf(fact_56_nth__equalityI, axiom, (! [A:$tType,Xs2:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys)) => ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs2)) => ((((nth @ A) @ Xs2) @ I3) = (((nth @ A) @ Ys) @ I3))))) => (Xs2 = Ys)))))).
% 3.12/1.82 thf(fact_57_zero__fun__def, axiom, (! [B:$tType,A:$tType]: (((zero @ B) => ((zero_zero @ (A > B)) = (^ [X:A]: ((zero_zero @ B)))))))).
% 3.12/1.82 thf(fact_58_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_59_mset__map, axiom, (! [A:$tType,B:$tType,F:(B > A),Xs2:(list @ B)]: ((((mset @ A) @ ((((map @ B) @ A) @ F) @ Xs2)) = ((((image_mset @ B) @ A) @ F) @ ((mset @ B) @ Xs2)))))).
% 3.12/1.82 thf(fact_60_size__mset, axiom, (! [A:$tType,Xs2:(list @ A)]: ((((size_size @ (multiset @ A)) @ ((mset @ A) @ Xs2)) = ((size_size @ (list @ A)) @ Xs2))))).
% 3.12/1.82 thf(fact_61_lessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))).
% 3.12/1.82 thf(fact_62_image__mset__mset__set, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((((image_mset @ A) @ B) @ F) @ ((mset_set @ A) @ A3)) = ((mset_set @ B) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 3.12/1.82 thf(fact_63_lessThan__atLeast0, axiom, ((set_ord_lessThan @ nat) = ((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)))).
% 3.12/1.82 thf(fact_64_bij__betw__def, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F2:(A > B),A4:(set @ A),B2:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A4) & (((((image @ A) @ B) @ F2) @ A4) = B2)))))))).
% 3.12/1.82 thf(fact_65_bij__betw__imageI, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ A),B3:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((((image @ A) @ B) @ F) @ A3) = B3) => (((((bij_betw @ A) @ B) @ F) @ A3) @ B3)))))).
% 3.12/1.82 thf(fact_66_inj__on__imp__bij__betw, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((((bij_betw @ A) @ B) @ F) @ A3) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 3.12/1.82 thf(fact_67_comp__inj__on, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A3:(set @ A),G:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((((inj_on @ B) @ C) @ G) @ ((((image @ A) @ B) @ F) @ A3)) => ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ A3)))))).
% 3.12/1.82 thf(fact_68_inj__on__imageI, axiom, (! [B:$tType,C:$tType,A:$tType,G:(C > B),F:(A > C),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ G) @ F)) @ A3) => ((((inj_on @ C) @ B) @ G) @ ((((image @ A) @ C) @ F) @ A3)))))).
% 3.12/1.82 thf(fact_69_comp__inj__on__iff, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A3:(set @ A),F3:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((((inj_on @ B) @ C) @ F3) @ ((((image @ A) @ B) @ F) @ A3)) = ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F)) @ A3)))))).
% 3.12/1.82 thf(fact_70_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G2:(C > B),X:C]: ((F2 @ (G2 @ X)))))))).
% 3.12/1.82 thf(fact_71_lessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((set_ord_lessThan @ A) @ X2) = ((set_ord_lessThan @ A) @ Y)) = (X2 = Y)))))))).
% 3.12/1.82 thf(fact_72_image__mset__empty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((image_mset @ B) @ A) @ F) @ (zero_zero @ (multiset @ B))) = (zero_zero @ (multiset @ A)))))).
% 3.12/1.82 thf(fact_73_image__mset__is__empty__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),M4:(multiset @ B)]: (((((((image_mset @ B) @ A) @ F) @ M4) = (zero_zero @ (multiset @ A))) = (M4 = (zero_zero @ (multiset @ B))))))).
% 3.12/1.82 thf(fact_74_size__eq__0__iff__empty, axiom, (! [A:$tType,M4:(multiset @ A)]: (((((size_size @ (multiset @ A)) @ M4) = (zero_zero @ nat)) = (M4 = (zero_zero @ (multiset @ A))))))).
% 3.12/1.82 thf(fact_75_size__empty, axiom, (! [A:$tType]: ((((size_size @ (multiset @ A)) @ (zero_zero @ (multiset @ A))) = (zero_zero @ nat))))).
% 3.12/1.82 thf(fact_76_size__image__mset, axiom, (! [A:$tType,B:$tType,F:(B > A),M4:(multiset @ B)]: ((((size_size @ (multiset @ A)) @ ((((image_mset @ B) @ A) @ F) @ M4)) = ((size_size @ (multiset @ B)) @ M4))))).
% 3.12/1.82 thf(fact_77_mset__upt, axiom, (! [M:nat,N:nat]: ((((mset @ nat) @ ((upt @ M) @ N)) = ((mset_set @ nat) @ (((set_or1433837966ssThan @ nat) @ M) @ N)))))).
% 3.12/1.82 thf(fact_78_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:(set @ A),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B3) => ((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A3)) @ ((((image @ A) @ B) @ F) @ B3))))))).
% 3.12/1.82 thf(fact_79_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B4:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B4) = C2) => ((A2 @ (B4 @ V)) = (C2 @ V)))))).
% 3.12/1.82 thf(fact_80_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B4:(A > C),C2:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B4) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => (! [V3:A]: (((A2 @ (B4 @ V3)) = (C2 @ (D2 @ V3))))))))).
% 3.12/1.82 thf(fact_81_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B4:(A > C),C2:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B4) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => ((A2 @ (B4 @ V)) = (C2 @ (D2 @ V))))))).
% 3.12/1.82 thf(fact_82_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H)))))).
% 3.12/1.82 thf(fact_83_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(A > B),X:A]: ((F2 @ (G2 @ X)))))))).
% 3.12/1.82 thf(fact_84_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(B > A),F:(A > B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => ((G @ (F @ X3)) = X3)))) => ((((inj_on @ A) @ B) @ F) @ A3))))).
% 3.12/1.82 thf(fact_85_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((X2 != Y) => ((((member @ A) @ X2) @ A3) => ((((member @ A) @ Y) @ A3) => ((F @ X2) != (F @ Y))))))))).
% 3.12/1.82 thf(fact_86_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => ((((member @ A) @ X2) @ A3) => ((((member @ A) @ Y) @ A3) => (((F @ X2) = (F @ Y)) = (X2 = Y)))))))).
% 3.12/1.82 thf(fact_87_inj__on__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F:(A > B),G:(A > B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((F @ A5) = (G @ A5))))) => (((((inj_on @ A) @ B) @ F) @ A3) = ((((inj_on @ A) @ B) @ G) @ A3)))))).
% 3.12/1.82 thf(fact_88_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F2:(A > B),A4:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A4) => (! [Y4:A]: (((((member @ A) @ Y4) @ A4) => (((F2 @ X) = (F2 @ Y4)) => (X = Y4)))))))))))))).
% 3.12/1.82 thf(fact_89_inj__onI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F:(A > B)]: (((! [X3:A,Y5:A]: (((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y5) @ A3) => (((F @ X3) = (F @ Y5)) => (X3 = Y5)))))) => ((((inj_on @ A) @ B) @ F) @ A3))))).
% 3.12/1.82 thf(fact_90_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),X2:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A3) => (((F @ X2) = (F @ Y)) => ((((member @ A) @ X2) @ A3) => ((((member @ A) @ Y) @ A3) => (X2 = Y)))))))).
% 3.12/1.82 thf(fact_91_bij__betw__iff__bijections, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F2:(A > B),A4:(set @ A),B2:(set @ B)]: ((? [G2:(B > A)]: (((! [X:A]: (((((member @ A) @ X) @ A4) => ((((member @ B) @ (F2 @ X)) @ B2) & ((G2 @ (F2 @ X)) = X))))) & (! [X:B]: (((((member @ B) @ X) @ B2) => ((((member @ A) @ (G2 @ X)) @ A4) & ((F2 @ (G2 @ X)) = X)))))))))))))).
% 3.12/1.82 thf(fact_92_bij__betw__apply, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ A),B3:(set @ B),A2:A]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => ((((member @ A) @ A2) @ A3) => (((member @ B) @ (F @ A2)) @ B3)))))).
% 3.12/1.82 thf(fact_93_bij__betw__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F:(A > B),G:(A > B),A6:(set @ B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((F @ A5) = (G @ A5))))) => ((((((bij_betw @ A) @ B) @ F) @ A3) @ A6) = (((((bij_betw @ A) @ B) @ G) @ A3) @ A6)))))).
% 3.12/1.82 thf(fact_94_bij__betw__inv, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => (? [G3:(B > A)]: ((((((bij_betw @ B) @ A) @ G3) @ B3) @ A3))))))).
% 3.12/1.82 thf(fact_95_bij__betwE, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => (! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ B) @ (F @ X5)) @ B3)))))))).
% 3.12/1.82 thf(fact_96_ex__mset, axiom, (! [A:$tType,X6:(multiset @ A)]: ((? [Xs:(list @ A)]: ((((mset @ A) @ Xs) = X6)))))).
% 3.12/1.82 thf(fact_97_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(B > A),A3:(set @ B),G:(C > A),B3:(set @ C),H:(A > D)]: (((((((image @ B) @ A) @ F) @ A3) = ((((image @ C) @ A) @ G) @ B3)) => (((((image @ B) @ D) @ (((((comp @ A) @ D) @ B) @ H) @ F)) @ A3) = ((((image @ C) @ D) @ (((((comp @ A) @ D) @ C) @ H) @ G)) @ B3)))))).
% 3.12/1.82 thf(fact_98_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),R:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G) @ R)) = ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ R))))).
% 3.12/1.82 thf(fact_99_nonempty__has__size, axiom, (! [A:$tType,S2:(multiset @ A)]: (((S2 != (zero_zero @ (multiset @ A))) = (((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (multiset @ A)) @ S2)))))).
% 3.12/1.82 thf(fact_100_atLeastLessThan__inj_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B4:A,C2:A,D2:A]: ((((((set_or1433837966ssThan @ A) @ A2) @ B4) = (((set_or1433837966ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A2) @ B4) => ((((ord_less @ A) @ C2) @ D2) => (B4 = D2)))))))))).
% 3.12/1.82 thf(fact_101_atLeastLessThan__inj_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B4:A,C2:A,D2:A]: ((((((set_or1433837966ssThan @ A) @ A2) @ B4) = (((set_or1433837966ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A2) @ B4) => ((((ord_less @ A) @ C2) @ D2) => (A2 = C2)))))))))).
% 3.12/1.82 thf(fact_102_atLeastLessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B4:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B4) => ((((ord_less @ A) @ C2) @ D2) => (((((set_or1433837966ssThan @ A) @ A2) @ B4) = (((set_or1433837966ssThan @ A) @ C2) @ D2)) = ((A2 = C2) & (B4 = D2))))))))))).
% 3.12/1.82 thf(fact_103_lessThan__strict__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [M:A,N:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M)) @ ((set_ord_lessThan @ A) @ N)) = (((ord_less @ A) @ M) @ N)))))))).
% 3.12/1.82 thf(fact_104_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A3:(set @ A),G:(A > B),F:(A > A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => (((G @ (F @ X3)) = (G @ (F @ Xa))) = ((G @ X3) = (G @ Xa))))))))) => (((((inj_on @ A) @ A) @ F) @ A3) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F) @ A3)) = ((((inj_on @ A) @ B) @ G) @ A3))))))).
% 3.12/1.82 thf(fact_105_inj__on__imageI2, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),F:(A > C),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F3) @ F)) @ A3) => ((((inj_on @ A) @ C) @ F) @ A3))))).
% 3.12/1.82 thf(fact_106_bij__betw__imp__surj__on, axiom, (! [A:$tType,B:$tType,F:(A > B),A3:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => (((((image @ A) @ B) @ F) @ A3) = B3))))).
% 3.12/1.82 thf(fact_107_bij__betw__comp__iff, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),A3:(set @ A),A6:(set @ B),F3:(B > C),A7:(set @ C)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ A6) => ((((((bij_betw @ B) @ C) @ F3) @ A6) @ A7) = (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F)) @ A3) @ A7)))))).
% 3.12/1.82 thf(fact_108_bij__betw__trans, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),A3:(set @ A),B3:(set @ B),G:(B > C),C3:(set @ C)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => ((((((bij_betw @ B) @ C) @ G) @ B3) @ C3) => (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ A3) @ C3)))))).
% 3.12/1.82 thf(fact_109_multiset_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(multiset @ A)]: ((((((image_mset @ B) @ C) @ G) @ ((((image_mset @ A) @ B) @ F) @ V)) = ((((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 3.12/1.82 thf(fact_110_image__mset_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C),G:(A > B),Multiset:(multiset @ A)]: ((((((image_mset @ B) @ C) @ F) @ ((((image_mset @ A) @ B) @ G) @ Multiset)) = ((((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)) @ Multiset))))).
% 3.12/1.82 thf(fact_111_image__mset_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C),G:(A > B)]: (((((((comp @ (multiset @ B)) @ (multiset @ C)) @ (multiset @ A)) @ (((image_mset @ B) @ C) @ F)) @ (((image_mset @ A) @ B) @ G)) = (((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)))))).
% 3.12/1.82 thf(fact_112_bij__betw__imp__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A3) @ B3) => ((((inj_on @ A) @ B) @ F) @ A3))))).
% 3.12/1.82 thf(fact_113_mset__eq__length, axiom, (! [A:$tType,Xs2:(list @ A),Ys:(list @ A)]: (((((mset @ A) @ Xs2) = ((mset @ A) @ Ys)) => (((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys)))))).
% 3.12/1.82 thf(fact_114_image__eqI, axiom, (! [A:$tType,B:$tType,B4:A,F:(B > A),X2:B,A3:(set @ B)]: (((B4 = (F @ X2)) => ((((member @ B) @ X2) @ A3) => (((member @ A) @ B4) @ ((((image @ B) @ A) @ F) @ A3))))))).
% 3.12/1.82 thf(fact_115_mbs_Oless__not__eq, axiom, (! [A:$tType]: (((size @ A) => (! [X2:A,A3:(set @ A),Y:A]: (((((member @ A) @ X2) @ A3) => ((((ord_less @ nat) @ ((size_size @ A) @ X2)) @ ((size_size @ A) @ Y)) => (X2 != Y))))))))).
% 3.12/1.82 thf(fact_116_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G:(B > A),F:(C > B),A3:(set @ C)]: (((Sup @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A3))) = (Sup @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A3)))))).
% 3.12/1.82 thf(fact_117_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G:(B > A),F:(C > B),A3:(set @ C)]: (((Inf @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A3))) = (Inf @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A3)))))).
% 3.12/1.82 thf(fact_118_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 3.12/1.82 thf(fact_119_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D22:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D22) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D22))))))))))))).
% 3.12/1.82 thf(fact_120_length__code, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = ((gen_length @ A) @ (zero_zero @ nat)))))).
% 3.12/1.82 thf(fact_121_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 3.12/1.82 thf(fact_122_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 3.12/1.82 thf(fact_123_imageI, axiom, (! [B:$tType,A:$tType,X2:A,A3:(set @ A),F:(A > B)]: (((((member @ A) @ X2) @ A3) => (((member @ B) @ (F @ X2)) @ ((((image @ A) @ B) @ F) @ A3)))))).
% 3.12/1.82 thf(fact_124_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F:(B > A),A3:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F) @ A3)) = (? [X:B]: (((((member @ B) @ X) @ A3) & (Z2 = (F @ X))))))))).
% 3.12/1.82 thf(fact_125_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A3)) & (P @ X5)))) => (? [X3:B]: (((((member @ B) @ X3) @ A3) & (P @ (F @ X3))))))))).
% 3.12/1.82 thf(fact_126_image__cong, axiom, (! [B:$tType,A:$tType,M4:(set @ A),N3:(set @ A),F:(A > B),G:(A > B)]: (((M4 = N3) => ((! [X3:A]: (((((member @ A) @ X3) @ N3) => ((F @ X3) = (G @ X3))))) => (((((image @ A) @ B) @ F) @ M4) = ((((image @ A) @ B) @ G) @ N3))))))).
% 3.12/1.82 thf(fact_127_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F) @ A3)) => (P @ X3)))) => (! [X5:B]: (((((member @ B) @ X5) @ A3) => (P @ (F @ X5))))))))).
% 3.12/1.82 thf(fact_128_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X2:A,A3:(set @ A),B4:B,F:(A > B)]: (((((member @ A) @ X2) @ A3) => ((B4 = (F @ X2)) => (((member @ B) @ B4) @ ((((image @ A) @ B) @ F) @ A3))))))).
% 3.12/1.82 thf(fact_129_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B3:(set @ B),C3:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((C3 @ X3) = (D3 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C3) @ A3)) = (Inf @ ((((image @ B) @ A) @ D3) @ B3)))))))).
% 3.12/1.82 thf(fact_130_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B3:(set @ B),C3:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A3 = B3) => ((! [X3:B]: (((((member @ B) @ X3) @ B3) => ((C3 @ X3) = (D3 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C3) @ A3)) = (Sup @ ((((image @ B) @ A) @ D3) @ B3)))))))).
% 3.12/1.82 thf(fact_131_Multiset_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A4:(multiset @ A)]: ((A4 = (zero_zero @ (multiset @ A))))))))).
% 3.12/1.82 thf(fact_132_map__eq__map__tailrec, axiom, (! [B:$tType,A:$tType]: ((((map @ A) @ B) = ((map_tailrec @ A) @ B))))).
% 3.12/1.82 thf(fact_133_list__ex__length, axiom, (! [A:$tType]: (((list_ex @ A) = (^ [P2:(A > $o),Xs3:(list @ A)]: ((? [N4:nat]: (((((ord_less @ nat) @ N4) @ ((size_size @ (list @ A)) @ Xs3)) & (P2 @ (((nth @ A) @ Xs3) @ N4))))))))))).
% 3.12/1.82 thf(fact_134_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 3.12/1.82 thf(fact_135_of__nat__fun__apply, axiom, (! [B:$tType,A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ (B > A)) = (^ [N4:nat,X:B]: (((semiring_1_of_nat @ A) @ N4)))))))).
% 3.12/1.82 thf(fact_136_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 3.12/1.82 thf(fact_137_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 3.12/1.82 thf(fact_138_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 3.12/1.82 thf(fact_139_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 3.12/1.82 thf(fact_140_of__nat__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 3.12/1.82 thf(fact_141_bij__betw__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N3:(set @ nat),A3:(set @ A)]: (((((((bij_betw @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N3) @ A3) = (((((image @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N3) = A3)))))))).
% 3.12/1.82 thf(fact_142_image__int__atLeastLessThan, axiom, (! [A2:nat,B4:nat]: ((((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1433837966ssThan @ nat) @ A2) @ B4)) = (((set_or1433837966ssThan @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B4)))))).
% 3.12/1.82 thf(fact_143_inj__on__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N3:(set @ nat)]: (((((inj_on @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N3))))))).
% 3.12/1.82 thf(fact_144_of__nat__fun, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ B) => ((semiring_1_of_nat @ (A > B)) = (^ [N4:nat,X:A]: (((semiring_1_of_nat @ B) @ N4)))))))).
% 3.12/1.82 thf(fact_145_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 3.12/1.82 thf(fact_146_of__nat__less__imp__less, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) => (((ord_less @ nat) @ M) @ N)))))))).
% 3.12/1.82 thf(fact_147_less__imp__of__nat__less, axiom, (! [A:$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))))))))).
% 3.12/1.82 thf(fact_148_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)))))))))))).
% 3.12/1.82 thf(fact_149_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))))))))).
% 3.12/1.82 thf(fact_150_reals__Archimedean2, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X2:A]: ((? [N2:nat]: ((((ord_less @ A) @ X2) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 3.12/1.82 thf(fact_151_size__multiset__empty, axiom, (! [A:$tType,F:(A > nat)]: (((((size_multiset @ A) @ F) @ (zero_zero @ (multiset @ A))) = (zero_zero @ nat))))).
% 3.12/1.82 thf(fact_152_size__multiset__eq__0__iff__empty, axiom, (! [A:$tType,F:(A > nat),M4:(multiset @ A)]: ((((((size_multiset @ A) @ F) @ M4) = (zero_zero @ nat)) = (M4 = (zero_zero @ (multiset @ A))))))).
% 3.12/1.82 thf(fact_153_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 3.12/1.82 thf(fact_154_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 3.12/1.82 thf(fact_155_size__multiset__o__map, axiom, (! [B:$tType,A:$tType,G:(B > nat),F:(A > B)]: (((((((comp @ (multiset @ B)) @ nat) @ (multiset @ A)) @ ((size_multiset @ B) @ G)) @ (((image_mset @ A) @ B) @ F)) = ((size_multiset @ A) @ (((((comp @ B) @ nat) @ A) @ G) @ F)))))).
% 3.12/1.82 thf(fact_156_nat__int__comparison_I2_J, axiom, ((ord_less @ nat) = (^ [A8:nat,B5:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A8)) @ ((semiring_1_of_nat @ int) @ B5)))))).
% 3.12/1.82 thf(fact_157_int__ops_I1_J, axiom, (((semiring_1_of_nat @ int) @ (zero_zero @ nat)) = (zero_zero @ int))).
% 3.12/1.82 thf(fact_158_neg__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (~ ((! [N2:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))))))))).
% 3.12/1.82 thf(fact_159_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 3.12/1.82 thf(fact_160_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B4:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B4)) = (A2 = B4)))))))).
% 3.12/1.82 thf(fact_161_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 3.12/1.82 thf(fact_162_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))).
% 3.12/1.82 thf(fact_163_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_164_equal__neg__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_165_neg__equal__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_166_neg__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B4:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ B4)))))))).
% 3.12/1.82 thf(fact_167_inj__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:(set @ A)]: (((((inj_on @ A) @ A) @ (uminus_uminus @ A)) @ A3))))))).
% 3.12/1.82 thf(fact_168_neg__less__0__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.12/1.82 thf(fact_169_neg__0__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_170_neg__less__pos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.12/1.82 thf(fact_171_less__neg__neg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_172_negative__eq__positive, axiom, (! [N:nat,M:nat]: (((((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ int) @ M)) = ((N = (zero_zero @ nat)) & (M = (zero_zero @ nat))))))).
% 3.12/1.82 thf(fact_173_int__cases2, axiom, (! [Z2:int]: (((! [N2:nat]: ((Z2 != ((semiring_1_of_nat @ int) @ N2)))) => (~ ((! [N2:nat]: ((Z2 != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))))))))))).
% 3.12/1.82 thf(fact_174_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B4) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B4)) @ A2)))))))).
% 3.12/1.82 thf(fact_175_less__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ B4)) = (((ord_less @ A) @ B4) @ ((uminus_uminus @ A) @ A2))))))))).
% 3.12/1.82 thf(fact_176_uminus__int__code_I1_J, axiom, (((uminus_uminus @ int) @ (zero_zero @ int)) = (zero_zero @ int))).
% 3.12/1.82 thf(fact_177_verit__negate__coefficient_I2_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ B4) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A2))))))))).
% 3.12/1.82 thf(fact_178_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B4:A]: (((A2 = ((uminus_uminus @ A) @ B4)) = (B4 = ((uminus_uminus @ A) @ A2))))))))).
% 3.12/1.82 thf(fact_179_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B4:A]: (((((uminus_uminus @ A) @ A2) = B4) = (((uminus_uminus @ A) @ B4) = A2)))))))).
% 3.12/1.82 thf(fact_180_not__int__zless__negative, axiom, (! [N:nat,M:nat]: ((~ ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M)))))))).
% 3.12/1.82 thf(fact_181_int__cases4, axiom, (! [M:int]: (((! [N2:nat]: ((M != ((semiring_1_of_nat @ int) @ N2)))) => (~ ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (M != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2)))))))))))).
% 3.12/1.82 thf(fact_182_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 3.12/1.82 thf(fact_183_nat__int__comparison_I1_J, axiom, ((^ [Y3:nat,Z:nat]: ((Y3 = Z))) = (^ [A8:nat,B5:nat]: ((((semiring_1_of_nat @ int) @ A8) = ((semiring_1_of_nat @ int) @ B5)))))).
% 3.12/1.82 thf(fact_184_int__if, axiom, (! [P:$o,A2:nat,B4:nat]: (((P => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B4)) = ((semiring_1_of_nat @ int) @ A2))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ A2) @ B4)) = ((semiring_1_of_nat @ int) @ B4))))))).
% 3.12/1.82 thf(fact_185_int__cases3, axiom, (! [K:int]: (((K != (zero_zero @ int)) => ((! [N2:nat]: (((K = ((semiring_1_of_nat @ int) @ N2)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))) => (~ ((! [N2:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))))))).
% 3.12/1.82 thf(fact_186_norm__frac_Oinduct, axiom, (! [P:(int > (int > $o)),A0:int,A1:int]: (((! [A5:int,B6:int]: ((((((ord_less @ int) @ B6) @ (zero_zero @ int)) => ((P @ ((uminus_uminus @ int) @ A5)) @ ((uminus_uminus @ int) @ B6))) => ((P @ A5) @ B6)))) => ((P @ A0) @ A1))))).
% 3.12/1.82 thf(fact_187_compl__less__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less @ A) @ Y) @ ((uminus_uminus @ A) @ X2)) => (((ord_less @ A) @ X2) @ ((uminus_uminus @ A) @ Y))))))))).
% 3.12/1.82 thf(fact_188_compl__less__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y:A,X2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ Y)) @ X2) => (((ord_less @ A) @ ((uminus_uminus @ A) @ X2)) @ Y)))))))).
% 3.12/1.82 thf(fact_189_compl__less__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X2:A,Y:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ X2)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less @ A) @ Y) @ X2)))))))).
% 3.12/1.82 thf(fact_190_ex__inverse__of__nat__less, axiom, (! [A:$tType]: (((archim1804426504_field @ A) => (! [X2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X2) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ X2))))))))))).
% 3.12/1.82 thf(fact_191_zero__less__nat__eq, axiom, (! [Z2:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (nat2 @ Z2)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z2))))).
% 3.12/1.82 thf(fact_192_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 3.12/1.82 thf(fact_193_zless__nat__conj, axiom, (! [W:int,Z2:int]: (((((ord_less @ nat) @ (nat2 @ W)) @ (nat2 @ Z2)) = ((((ord_less @ int) @ (zero_zero @ int)) @ Z2) & (((ord_less @ int) @ W) @ Z2)))))).
% 3.12/1.82 thf(fact_194_nat__zminus__int, axiom, (! [N:nat]: (((nat2 @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) = (zero_zero @ nat))))).
% 3.12/1.82 thf(fact_195_nat__zero__as__int, axiom, ((zero_zero @ nat) = (nat2 @ (zero_zero @ int)))).
% 3.12/1.82 thf(fact_196_forall__pos__mono, axiom, (! [P:(real > $o),E2:real]: (((! [D4:real,E:real]: (((((ord_less @ real) @ D4) @ E) => ((P @ D4) => (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))))))).
% 3.12/1.82 thf(fact_197_real__arch__inverse, axiom, (! [E2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E2) = (? [N4:nat]: (((N4 != (zero_zero @ nat)) & ((((ord_less @ real) @ (zero_zero @ real)) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N4))) & (((ord_less @ real) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N4))) @ E2))))))))).
% 3.12/1.82 thf(fact_198_nat__mono__iff, axiom, (! [Z2:int,W:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z2) => ((((ord_less @ nat) @ (nat2 @ W)) @ (nat2 @ Z2)) = (((ord_less @ int) @ W) @ Z2)))))).
% 3.12/1.82 thf(fact_199_zless__nat__eq__int__zless, axiom, (! [M:nat,Z2:int]: (((((ord_less @ nat) @ M) @ (nat2 @ Z2)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ M)) @ Z2))))).
% 3.12/1.82 thf(fact_200_split__nat, axiom, (! [P:(nat > $o),I:int]: (((P @ (nat2 @ I)) = ((! [N4:nat]: (((I = ((semiring_1_of_nat @ int) @ N4)) => (P @ N4)))) & ((((ord_less @ int) @ I) @ (zero_zero @ int)) => (P @ (zero_zero @ nat)))))))).
% 3.12/1.82 thf(fact_201_inverse__positive__iff__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.12/1.82 thf(fact_202_inverse__negative__iff__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_203_inverse__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 3.12/1.82 thf(fact_204_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_205_inverse__less__iff__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B4) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) = (((ord_less @ A) @ B4) @ A2)))))))))).
% 3.12/1.82 thf(fact_206_inverse__less__iff__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B4) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) = (((ord_less @ A) @ B4) @ A2)))))))))).
% 3.12/1.82 thf(fact_207_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X5:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X5))))))))).
% 3.12/1.82 thf(fact_208_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X5:A]: ((? [X_1:A]: ((((ord_less @ A) @ X5) @ X_1))))))))).
% 3.12/1.82 thf(fact_209_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: (((field @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 3.12/1.82 thf(fact_210_inverse__zero__imp__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_211_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B4:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B4)) => ((A2 != (zero_zero @ A)) => ((B4 != (zero_zero @ A)) => (A2 = B4)))))))))).
% 3.12/1.82 thf(fact_212_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2)))))))).
% 3.12/1.82 thf(fact_213_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A2) != (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_214_inverse__less__imp__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ B4) @ A2))))))))).
% 3.12/1.82 thf(fact_215_less__imp__inverse__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ B4) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B4)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 3.12/1.82 thf(fact_216_inverse__less__imp__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) => ((((ord_less @ A) @ B4) @ (zero_zero @ A)) => (((ord_less @ A) @ B4) @ A2))))))))).
% 3.12/1.82 thf(fact_217_less__imp__inverse__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ B4) => ((((ord_less @ A) @ B4) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B4)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 3.12/1.82 thf(fact_218_inverse__negative__imp__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))).
% 3.12/1.82 thf(fact_219_inverse__positive__imp__positive, axiom, (! [A:$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))))))))).
% 3.12/1.82 thf(fact_220_negative__imp__inverse__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_221_positive__imp__inverse__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2))))))))).
% 3.12/1.82 thf(fact_222_nonzero__inverse__minus__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((uminus_uminus @ A) @ A2)) = ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ A2)))))))))).
% 3.12/1.82 thf(fact_223_image__atLeastZeroLessThan__int, axiom, (! [U:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ U) => ((((set_or1433837966ssThan @ int) @ (zero_zero @ int)) @ U) = ((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ ((set_ord_lessThan @ nat) @ (nat2 @ U)))))))).
% 3.12/1.82 thf(fact_224_of__int__of__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [K2:int]: (((((if @ A) @ (((ord_less @ int) @ K2) @ (zero_zero @ int))) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((uminus_uminus @ int) @ K2))))) @ ((semiring_1_of_nat @ A) @ (nat2 @ K2)))))))))).
% 3.12/1.82 thf(fact_225_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W:int,Z2:int]: (((((ring_1_of_int @ A) @ W) = ((ring_1_of_int @ A) @ Z2)) = (W = Z2)))))))).
% 3.12/1.82 thf(fact_226_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_227_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B4:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B4)))))))).
% 3.12/1.82 thf(fact_228_of__nat__le__iff, axiom, (! [A:$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)))))))).
% 3.12/1.82 thf(fact_229_ivl__subset, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,J:A,M:A,N:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1433837966ssThan @ A) @ I) @ J)) @ (((set_or1433837966ssThan @ A) @ M) @ N)) = ((((ord_less_eq @ A) @ J) @ I) | ((((ord_less_eq @ A) @ M) @ I) & (((ord_less_eq @ A) @ J) @ N)))))))))).
% 3.12/1.82 thf(fact_230_lessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X2)) @ ((set_ord_lessThan @ A) @ Y)) = (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 3.12/1.82 thf(fact_231_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_232_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.12/1.82 thf(fact_233_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_234_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 3.12/1.82 thf(fact_235_inverse__nonpositive__iff__nonpositive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 3.12/1.82 thf(fact_236_inverse__nonnegative__iff__nonnegative, axiom, (! [A:$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)))))))).
% 3.12/1.82 thf(fact_237_atLeastLessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or1433837966ssThan @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))).
% 3.12/1.82 thf(fact_238_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: (((((ring_1_of_int @ A) @ Z2) = (zero_zero @ A)) = (Z2 = (zero_zero @ int))))))))).
% 3.12/1.82 thf(fact_239_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z2)) = (Z2 = (zero_zero @ int))))))))).
% 3.12/1.82 thf(fact_240_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))).
% 3.12/1.82 thf(fact_241_of__int__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ W) @ Z2)))))))).
% 3.12/1.82 thf(fact_242_of__int__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W:int,Z2:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ W) @ Z2)))))))).
% 3.12/1.82 thf(fact_243_of__int__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:int]: ((((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ Z2)) = ((uminus_uminus @ A) @ ((ring_1_of_int @ A) @ Z2))))))))).
% 3.12/1.82 thf(fact_244_of__int__of__nat__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: ((((ring_1_of_int @ A) @ ((semiring_1_of_nat @ int) @ N)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 3.12/1.82 thf(fact_245_negative__zle, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N))) @ ((semiring_1_of_nat @ int) @ M))))).
% 3.12/1.82 thf(fact_246_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 3.12/1.82 thf(fact_247_inverse__le__iff__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B4) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) = (((ord_less_eq @ A) @ B4) @ A2)))))))))).
% 3.12/1.82 thf(fact_248_inverse__le__iff__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B4:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B4) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B4)) = (((ord_less_eq @ A) @ B4) @ A2)))))))))).
% 3.12/1.82 thf(fact_249_nat__le__0, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ Z2) @ (zero_zero @ int)) => ((nat2 @ Z2) = (zero_zero @ nat)))))).
% 3.12/1.82 thf(fact_250_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))).
% 3.12/1.82 thf(fact_251_int__nat__eq, axiom, (! [Z2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((semiring_1_of_nat @ int) @ (nat2 @ Z2)) = Z2)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z2)) = (zero_zero @ int))))))).
% 3.12/1.82 thf(fact_252_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z2) @ (zero_zero @ int))))))))).
% 3.12/1.82 thf(fact_253_of__int__0__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2)))))))).
% 3.12/1.82 thf(tcon_Multiset_Omultiset___Orderings_Opreorder, axiom, (! [A9:$tType]: (((preorder @ A9) => (preorder @ (multiset @ A9)))))).
% 3.12/1.82 thf(tcon_Real_Oreal___Orderings_Opreorder_1, axiom, (preorder @ real)).
% 3.12/1.82 thf(tcon_HOL_Obool___Orderings_Opreorder_2, axiom, (preorder @ $o)).
% 3.12/1.82 thf(tcon_Set_Oset___Orderings_Opreorder_3, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 3.12/1.82 thf(tcon_Nat_Onat___Rings_Odioid, axiom, (dioid @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Orderings_Opreorder_4, axiom, (preorder @ nat)).
% 3.12/1.82 thf(tcon_Int_Oint___Orderings_Opreorder_5, axiom, (preorder @ int)).
% 3.12/1.82 thf(tcon_fun___Rings_Odioid_6, axiom, (! [A9:$tType,A10:$tType]: (((dioid @ A10) => (dioid @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Orderings_Opreorder_7, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add, axiom, (! [A9:$tType,A10:$tType]: (((dioid @ A10) => (canoni770627133id_add @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Groups_Oordered__ab__group__add, axiom, (! [A9:$tType,A10:$tType]: (((ordered_ab_group_add @ A10) => (ordered_ab_group_add @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A9:$tType,A10:$tType]: (((boolean_algebra @ A10) => (boolean_algebra @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Nat_Osemiring__char__0, axiom, (! [A9:$tType,A10:$tType]: (((semiring_char_0 @ A10) => (semiring_char_0 @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Rings_Osemiring__1, axiom, (! [A9:$tType,A10:$tType]: (((semiring_1 @ A10) => (semiring_1 @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Groups_Ogroup__add, axiom, (! [A9:$tType,A10:$tType]: (((group_add @ A10) => (group_add @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Nat_Oring__char__0, axiom, (! [A9:$tType,A10:$tType]: (((ring_char_0 @ A10) => (ring_char_0 @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Rings_Oring__1, axiom, (! [A9:$tType,A10:$tType]: (((ring_1 @ A10) => (ring_1 @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_fun___Groups_Ozero, axiom, (! [A9:$tType,A10:$tType]: (((zero @ A10) => (zero @ (A9 > A10)))))).
% 3.12/1.82 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord219039673up_add @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add_8, axiom, (ordered_ab_group_add @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Nat_Osemiring__char__0_9, axiom, (semiring_char_0 @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Rings_Osemiring__1_10, axiom, (semiring_1 @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Groups_Ogroup__add_11, axiom, (group_add @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Orderings_Oorder_12, axiom, (order @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Nat_Oring__char__0_13, axiom, (ring_char_0 @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Orderings_Oord_14, axiom, (ord @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Rings_Oring__1_15, axiom, (ring_1 @ int)).
% 3.12/1.82 thf(tcon_Int_Oint___Groups_Ozero_16, axiom, (zero @ int)).
% 3.12/1.82 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_17, axiom, (canoni770627133id_add @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_18, axiom, (linord1659791738miring @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_19, axiom, (semiring_char_0 @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Orderings_Olinorder_20, axiom, (linorder @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Rings_Osemiring__1_21, axiom, (semiring_1 @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Orderings_Oorder_22, axiom, (order @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Orderings_Oord_23, axiom, (ord @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Groups_Ozero_24, axiom, (zero @ nat)).
% 3.12/1.82 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 3.12/1.82 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_25, axiom, (! [A9:$tType]: ((boolean_algebra @ (set @ A9))))).
% 3.12/1.82 thf(tcon_Set_Oset___Orderings_Oorder_26, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 3.12/1.82 thf(tcon_Set_Oset___Orderings_Oord_27, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 3.12/1.82 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_28, axiom, (boolean_algebra @ $o)).
% 3.12/1.82 thf(tcon_HOL_Obool___Orderings_Olinorder_29, axiom, (linorder @ $o)).
% 3.12/1.82 thf(tcon_HOL_Obool___Orderings_Oorder_30, axiom, (order @ $o)).
% 3.12/1.82 thf(tcon_HOL_Obool___Orderings_Oord_31, axiom, (ord @ $o)).
% 3.12/1.82 thf(tcon_List_Olist___Nat_Osize_32, axiom, (! [A9:$tType]: ((size @ (list @ A9))))).
% 3.12/1.82 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field, axiom, (archim1804426504_field @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_33, axiom, (linord1659791738miring @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_34, axiom, (linord219039673up_add @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_35, axiom, (ordered_ab_group_add @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Fields_Olinordered__field, axiom, (linordered_field @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Rings_Olinordered__idom_36, axiom, (linordered_idom @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Fields_Odivision__ring, axiom, (division_ring @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_37, axiom, (semiring_char_0 @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Orderings_Olinorder_38, axiom, (linorder @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Rings_Osemiring__1_39, axiom, (semiring_1 @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Groups_Ogroup__add_40, axiom, (group_add @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Orderings_Oorder_41, axiom, (order @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Nat_Oring__char__0_42, axiom, (ring_char_0 @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Orderings_Oord_43, axiom, (ord @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Rings_Oring__1_44, axiom, (ring_1 @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Fields_Ofield, axiom, (field @ real)).
% 3.12/1.82 thf(tcon_Real_Oreal___Groups_Ozero_45, axiom, (zero @ real)).
% 3.12/1.82 thf(tcon_Multiset_Omultiset___Orderings_Oorder_46, axiom, (! [A9:$tType]: (((preorder @ A9) => (order @ (multiset @ A9)))))).
% 3.12/1.82 thf(tcon_Multiset_Omultiset___Orderings_Oord_47, axiom, (! [A9:$tType]: (((preorder @ A9) => (ord @ (multiset @ A9)))))).
% 3.12/1.82 thf(tcon_Multiset_Omultiset___Groups_Ozero_48, axiom, (! [A9:$tType]: ((zero @ (multiset @ A9))))).
% 3.12/1.82 thf(tcon_Multiset_Omultiset___Nat_Osize_49, axiom, (! [A9:$tType]: ((size @ (multiset @ A9))))).
% 3.12/1.82 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.12/1.82 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $false) @ X2) @ Y) = Y)))).
% 3.12/1.82 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $true) @ X2) @ Y) = X2)))).
% 3.12/1.82 thf(conj_0, conjecture, (((((image_mset @ nat) @ a) @ ((nth @ a) @ ys)) @ ((((image_mset @ nat) @ nat) @ f) @ ((mset_set @ nat) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ xs))))) = ((((image_mset @ nat) @ a) @ ((nth @ a) @ ys)) @ ((mset_set @ nat) @ (((set_or1433837966ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ a)) @ ys)))))).
% 3.12/1.82 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2DTF409.p
% 3.12/1.82 ------------------------
% 3.12/1.84 ---- Cleaned THF ---
% 3.12/1.84 thf(ty_t_Multiset_Omultiset,type,
% 3.12/1.84 multiset: $tType > $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_t_Real_Oreal,type,
% 3.12/1.84 real: $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_t_List_Olist,type,
% 3.12/1.84 list: $tType > $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_t_Set_Oset,type,
% 3.12/1.84 set: $tType > $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_t_Nat_Onat,type,
% 3.12/1.84 nat: $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_t_Int_Oint,type,
% 3.12/1.84 int: $tType ).
% 3.12/1.84
% 3.12/1.84 thf(ty_tf_a,type,
% 3.12/1.84 a: $tType ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Orderings_Opreorder,type,
% 3.12/1.84 preorder:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Rings_Odioid,type,
% 3.12/1.84 dioid:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_HOL_Otype,type,
% 3.12/1.84 type:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Nat_Osize,type,
% 3.12/1.84 size:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Groups_Ozero,type,
% 3.12/1.84 zero:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Fields_Ofield,type,
% 3.12/1.84 field:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Rings_Oring__1,type,
% 3.12/1.84 ring_1:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Orderings_Oord,type,
% 3.12/1.84 ord:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Nat_Oring__char__0,type,
% 3.12/1.84 ring_char_0:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Orderings_Oorder,type,
% 3.12/1.84 order:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Groups_Ogroup__add,type,
% 3.12/1.84 group_add:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Rings_Osemiring__1,type,
% 3.12/1.84 semiring_1:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Orderings_Olinorder,type,
% 3.12/1.84 linorder:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Nat_Osemiring__char__0,type,
% 3.12/1.84 semiring_char_0:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Orderings_Owellorder,type,
% 3.12/1.84 wellorder:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Fields_Odivision__ring,type,
% 3.12/1.84 division_ring:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Rings_Olinordered__idom,type,
% 3.12/1.84 linordered_idom:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Fields_Olinordered__field,type,
% 3.12/1.84 linordered_field:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 3.12/1.84 boolean_algebra:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.12/1.84 ordered_ab_group_add:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 3.12/1.84 linord219039673up_add:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 3.12/1.84 linord1659791738miring:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 3.12/1.84 archim1804426504_field:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 3.12/1.84 canoni770627133id_add:
% 3.12/1.84 !>[A: $tType] : $o ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 3.12/1.84 inverse_inverse:
% 3.12/1.84 !>[A: $tType] : ( A > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Fun_Obij__betw,type,
% 3.12/1.84 bij_betw:
% 3.12/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Fun_Ocomp,type,
% 3.12/1.84 comp:
% 3.12/1.84 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Fun_Oinj__on,type,
% 3.12/1.84 inj_on:
% 3.12/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 3.12/1.84 uminus_uminus:
% 3.12/1.84 !>[A: $tType] : ( A > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 3.12/1.84 zero_zero:
% 3.12/1.84 !>[A: $tType] : A ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_If,type,
% 3.12/1.84 if:
% 3.12/1.84 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Int_Onat,type,
% 3.12/1.84 nat2: int > nat ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 3.12/1.84 ring_1_of_int:
% 3.12/1.84 !>[A: $tType] : ( int > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Ogen__length,type,
% 3.12/1.84 gen_length:
% 3.12/1.84 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Olist_Omap,type,
% 3.12/1.84 map:
% 3.12/1.84 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Olist__ex,type,
% 3.12/1.84 list_ex:
% 3.12/1.84 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Omap__tailrec,type,
% 3.12/1.84 map_tailrec:
% 3.12/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Onth,type,
% 3.12/1.84 nth:
% 3.12/1.84 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_List_Oupt,type,
% 3.12/1.84 upt: nat > nat > ( list @ nat ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Multiset_Oimage__mset,type,
% 3.12/1.84 image_mset:
% 3.12/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Multiset_Ois__empty,type,
% 3.12/1.84 is_empty:
% 3.12/1.84 !>[A: $tType] : ( ( multiset @ A ) > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Multiset_Omset,type,
% 3.12/1.84 mset:
% 3.12/1.84 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Multiset_Omset__set,type,
% 3.12/1.84 mset_set:
% 3.12/1.84 !>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Multiset_Osize__multiset,type,
% 3.12/1.84 size_multiset:
% 3.12/1.84 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 3.12/1.84 semiring_1_of_nat:
% 3.12/1.84 !>[A: $tType] : ( nat > A ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Nat_Osize__class_Osize,type,
% 3.12/1.84 size_size:
% 3.12/1.84 !>[A: $tType] : ( A > nat ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.12/1.84 ord_less:
% 3.12/1.84 !>[A: $tType] : ( A > A > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.12/1.84 ord_less_eq:
% 3.12/1.84 !>[A: $tType] : ( A > A > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Set_OCollect,type,
% 3.12/1.84 collect:
% 3.12/1.84 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Set_Oimage,type,
% 3.12/1.84 image:
% 3.12/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 3.12/1.84 set_or1433837966ssThan:
% 3.12/1.84 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 3.12/1.84 set_ord_lessThan:
% 3.12/1.84 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_c_member,type,
% 3.12/1.84 member:
% 3.12/1.84 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.12/1.84
% 3.12/1.84 thf(sy_v_f,type,
% 3.12/1.84 f: nat > nat ).
% 3.12/1.84
% 3.12/1.84 thf(sy_v_xs,type,
% 3.12/1.84 xs: list @ a ).
% 3.12/1.84
% 3.12/1.84 thf(sy_v_ys,type,
% 3.12/1.84 ys: list @ a ).
% 3.12/1.84
% 3.12/1.84 thf(fact_0__C1_C,axiom,
% 3.12/1.84 inj_on @ nat @ nat @ f @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_1_calculation,axiom,
% 3.12/1.84 ( ( mset @ a @ xs )
% 3.12/1.84 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_2_assms_I2_J,axiom,
% 3.12/1.84 ! [I: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ xs ) )
% 3.12/1.84 => ( ( nth @ a @ xs @ I )
% 3.12/1.84 = ( nth @ a @ ys @ ( f @ I ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_3__C2_C,axiom,
% 3.12/1.84 ( ( image @ nat @ nat @ f @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) )
% 3.12/1.84 = ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ ys ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_4_zero__fun__apply,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType] :
% 3.12/1.84 ( ( zero @ A )
% 3.12/1.84 => ( ( zero_zero @ ( B > A ) )
% 3.12/1.84 = ( ^ [X: B] : ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_5_assms_I1_J,axiom,
% 3.12/1.84 bij_betw @ nat @ nat @ f @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ ys ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_6__092_060open_062mset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_A_061_Aimage__mset_A_I_I_B_J_Ays_J_A_Iimage__mset_Af_A_Imset__set_A_1230_O_O_060length_Axs_125_J_J_092_060close_062,axiom,
% 3.12/1.84 ( ( mset @ a @ ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) )
% 3.12/1.84 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_7__092_060open_062xs_A_061_Amap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_092_060close_062,axiom,
% 3.12/1.84 ( xs
% 3.12/1.84 = ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_8_Ex__list__of__length,axiom,
% 3.12/1.84 ! [A: $tType,N: nat] :
% 3.12/1.84 ? [Xs: list @ A] :
% 3.12/1.84 ( ( size_size @ ( list @ A ) @ Xs )
% 3.12/1.84 = N ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_9_neq__if__length__neq,axiom,
% 3.12/1.84 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 3.12/1.84 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 3.12/1.84 != ( size_size @ ( list @ A ) @ Ys ) )
% 3.12/1.84 => ( Xs2 != Ys ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_10_size__neq__size__imp__neq,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( size @ A )
% 3.12/1.84 => ! [X2: A,Y: A] :
% 3.12/1.84 ( ( ( size_size @ A @ X2 )
% 3.12/1.84 != ( size_size @ A @ Y ) )
% 3.12/1.84 => ( X2 != Y ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_11_not__gr__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( canoni770627133id_add @ A )
% 3.12/1.84 => ! [N: A] :
% 3.12/1.84 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 3.12/1.84 = ( N
% 3.12/1.84 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_12_bot__nat__0_Onot__eq__extremum,axiom,
% 3.12/1.84 ! [A2: nat] :
% 3.12/1.84 ( ( A2
% 3.12/1.84 != ( zero_zero @ nat ) )
% 3.12/1.84 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_13_less__nat__zero__code,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_14_neq0__conv,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ( ( N
% 3.12/1.84 != ( zero_zero @ nat ) )
% 3.12/1.84 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_15_map__map,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs2: list @ C] :
% 3.12/1.84 ( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs2 ) )
% 3.12/1.84 = ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_16_List_Omap_Ocomp,axiom,
% 3.12/1.84 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 3.12/1.84 ( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
% 3.12/1.84 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_17_map__comp__map,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
% 3.12/1.84 ( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
% 3.12/1.84 = ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_18_list_Omap__comp,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
% 3.12/1.84 ( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
% 3.12/1.84 = ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_19_List_Omap_Ocompositionality,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
% 3.12/1.84 ( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
% 3.12/1.84 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_20_length__map,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
% 3.12/1.84 ( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs2 ) )
% 3.12/1.84 = ( size_size @ ( list @ B ) @ Xs2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_21_nth__map,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,N: nat,Xs2: list @ A,F: A > B] :
% 3.12/1.84 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 3.12/1.84 => ( ( nth @ B @ ( map @ A @ B @ F @ Xs2 ) @ N )
% 3.12/1.84 = ( F @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_22__092_060open_062mset_Axs_A_061_Amset_A_Imap_A_I_I_B_J_Ays_A_092_060circ_062_Af_J_A_0910_O_O_060length_Axs_093_J_092_060close_062,axiom,
% 3.12/1.84 ( ( mset @ a @ xs )
% 3.12/1.84 = ( mset @ a @ ( map @ nat @ a @ ( comp @ nat @ a @ nat @ ( nth @ a @ ys ) @ f ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_23_nat__neq__iff,axiom,
% 3.12/1.84 ! [M: nat,N: nat] :
% 3.12/1.84 ( ( M != N )
% 3.12/1.84 = ( ( ord_less @ nat @ M @ N )
% 3.12/1.84 | ( ord_less @ nat @ N @ M ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_24_less__not__refl,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ N @ N ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_25_less__not__refl2,axiom,
% 3.12/1.84 ! [N: nat,M: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ N @ M )
% 3.12/1.84 => ( M != N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_26_less__not__refl3,axiom,
% 3.12/1.84 ! [S: nat,T: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ S @ T )
% 3.12/1.84 => ( S != T ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_27_measure__induct,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( wellorder @ B )
% 3.12/1.84 => ! [F: A > B,P: A > $o,A2: A] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ! [Y2: A] :
% 3.12/1.84 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 3.12/1.84 => ( P @ Y2 ) )
% 3.12/1.84 => ( P @ X3 ) )
% 3.12/1.84 => ( P @ A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_28_less__irrefl__nat,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ N @ N ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_29_nat__less__induct,axiom,
% 3.12/1.84 ! [P: nat > $o,N: nat] :
% 3.12/1.84 ( ! [N2: nat] :
% 3.12/1.84 ( ! [M2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M2 @ N2 )
% 3.12/1.84 => ( P @ M2 ) )
% 3.12/1.84 => ( P @ N2 ) )
% 3.12/1.84 => ( P @ N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_30_infinite__descent,axiom,
% 3.12/1.84 ! [P: nat > $o,N: nat] :
% 3.12/1.84 ( ! [N2: nat] :
% 3.12/1.84 ( ~ ( P @ N2 )
% 3.12/1.84 => ? [M2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M2 @ N2 )
% 3.12/1.84 & ~ ( P @ M2 ) ) )
% 3.12/1.84 => ( P @ N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_31_linorder__neqE__nat,axiom,
% 3.12/1.84 ! [X2: nat,Y: nat] :
% 3.12/1.84 ( ( X2 != Y )
% 3.12/1.84 => ( ~ ( ord_less @ nat @ X2 @ Y )
% 3.12/1.84 => ( ord_less @ nat @ Y @ X2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_32_measure__induct__rule,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( wellorder @ B )
% 3.12/1.84 => ! [F: A > B,P: A > $o,A2: A] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ! [Y2: A] :
% 3.12/1.84 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
% 3.12/1.84 => ( P @ Y2 ) )
% 3.12/1.84 => ( P @ X3 ) )
% 3.12/1.84 => ( P @ A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_33_infinite__descent__measure,axiom,
% 3.12/1.84 ! [A: $tType,P: A > $o,V2: A > nat,X2: A] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ~ ( P @ X3 )
% 3.12/1.84 => ? [Y2: A] :
% 3.12/1.84 ( ( ord_less @ nat @ ( V2 @ Y2 ) @ ( V2 @ X3 ) )
% 3.12/1.84 & ~ ( P @ Y2 ) ) )
% 3.12/1.84 => ( P @ X2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_34_map__eq__imp__length__eq,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,C: $tType,F: B > A,Xs2: list @ B,G: C > A,Ys: list @ C] :
% 3.12/1.84 ( ( ( map @ B @ A @ F @ Xs2 )
% 3.12/1.84 = ( map @ C @ A @ G @ Ys ) )
% 3.12/1.84 => ( ( size_size @ ( list @ B ) @ Xs2 )
% 3.12/1.84 = ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_35_zero__less__iff__neq__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( canoni770627133id_add @ A )
% 3.12/1.84 => ! [N: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 3.12/1.84 = ( N
% 3.12/1.84 != ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_36_gr__implies__not__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( canoni770627133id_add @ A )
% 3.12/1.84 => ! [M: A,N: A] :
% 3.12/1.84 ( ( ord_less @ A @ M @ N )
% 3.12/1.84 => ( N
% 3.12/1.84 != ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_37_not__less__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( canoni770627133id_add @ A )
% 3.12/1.84 => ! [N: A] :
% 3.12/1.84 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_38_gr__zeroI,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( canoni770627133id_add @ A )
% 3.12/1.84 => ! [N: A] :
% 3.12/1.84 ( ( N
% 3.12/1.84 != ( zero_zero @ A ) )
% 3.12/1.84 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_39_infinite__descent0__measure,axiom,
% 3.12/1.84 ! [A: $tType,V2: A > nat,P: A > $o,X2: A] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( ( V2 @ X3 )
% 3.12/1.84 = ( zero_zero @ nat ) )
% 3.12/1.84 => ( P @ X3 ) )
% 3.12/1.84 => ( ! [X3: A] :
% 3.12/1.84 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
% 3.12/1.84 => ( ~ ( P @ X3 )
% 3.12/1.84 => ? [Y2: A] :
% 3.12/1.84 ( ( ord_less @ nat @ ( V2 @ Y2 ) @ ( V2 @ X3 ) )
% 3.12/1.84 & ~ ( P @ Y2 ) ) ) )
% 3.12/1.84 => ( P @ X2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_40_bot__nat__0_Oextremum__strict,axiom,
% 3.12/1.84 ! [A2: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_41_infinite__descent0,axiom,
% 3.12/1.84 ! [P: nat > $o,N: nat] :
% 3.12/1.84 ( ( P @ ( zero_zero @ nat ) )
% 3.12/1.84 => ( ! [N2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.12/1.84 => ( ~ ( P @ N2 )
% 3.12/1.84 => ? [M2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M2 @ N2 )
% 3.12/1.84 & ~ ( P @ M2 ) ) ) )
% 3.12/1.84 => ( P @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_42_gr__implies__not0,axiom,
% 3.12/1.84 ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M @ N )
% 3.12/1.84 => ( N
% 3.12/1.84 != ( zero_zero @ nat ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_43_mem__Collect__eq,axiom,
% 3.12/1.84 ! [A: $tType,A2: A,P: A > $o] :
% 3.12/1.84 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.12/1.84 = ( P @ A2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_44_Collect__mem__eq,axiom,
% 3.12/1.84 ! [A: $tType,A3: set @ A] :
% 3.12/1.84 ( ( collect @ A
% 3.12/1.84 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 3.12/1.84 = A3 ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_45_Collect__cong,axiom,
% 3.12/1.84 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( P @ X3 )
% 3.12/1.84 = ( Q @ X3 ) )
% 3.12/1.84 => ( ( collect @ A @ P )
% 3.12/1.84 = ( collect @ A @ Q ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_46_ext,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( F @ X3 )
% 3.12/1.84 = ( G @ X3 ) )
% 3.12/1.84 => ( F = G ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_47_less__zeroE,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_48_not__less0,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_49_not__gr0,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 3.12/1.84 = ( N
% 3.12/1.84 = ( zero_zero @ nat ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_50_gr0I,axiom,
% 3.12/1.84 ! [N: nat] :
% 3.12/1.84 ( ( N
% 3.12/1.84 != ( zero_zero @ nat ) )
% 3.12/1.84 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_51_length__induct,axiom,
% 3.12/1.84 ! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
% 3.12/1.84 ( ! [Xs: list @ A] :
% 3.12/1.84 ( ! [Ys2: list @ A] :
% 3.12/1.84 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 3.12/1.84 => ( P @ Ys2 ) )
% 3.12/1.84 => ( P @ Xs ) )
% 3.12/1.84 => ( P @ Xs2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_52_all__nat__less__eq,axiom,
% 3.12/1.84 ! [N: nat,P: nat > $o] :
% 3.12/1.84 ( ( ! [M3: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M3 @ N )
% 3.12/1.84 => ( P @ M3 ) ) )
% 3.12/1.84 = ( ! [X: nat] :
% 3.12/1.84 ( ( member @ nat @ X @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 3.12/1.84 => ( P @ X ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_53_ex__nat__less__eq,axiom,
% 3.12/1.84 ! [N: nat,P: nat > $o] :
% 3.12/1.84 ( ( ? [M3: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M3 @ N )
% 3.12/1.84 & ( P @ M3 ) ) )
% 3.12/1.84 = ( ? [X: nat] :
% 3.12/1.84 ( ( member @ nat @ X @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 3.12/1.84 & ( P @ X ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_54_list__eq__iff__nth__eq,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) )
% 3.12/1.84 = ( ^ [Xs3: list @ A,Ys3: list @ A] :
% 3.12/1.84 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 3.12/1.84 = ( size_size @ ( list @ A ) @ Ys3 ) )
% 3.12/1.84 & ! [I2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 3.12/1.84 => ( ( nth @ A @ Xs3 @ I2 )
% 3.12/1.84 = ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_55_Skolem__list__nth,axiom,
% 3.12/1.84 ! [A: $tType,K: nat,P: nat > A > $o] :
% 3.12/1.84 ( ( ! [I2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ I2 @ K )
% 3.12/1.84 => ? [X4: A] : ( P @ I2 @ X4 ) ) )
% 3.12/1.84 = ( ? [Xs3: list @ A] :
% 3.12/1.84 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 3.12/1.84 = K )
% 3.12/1.84 & ! [I2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ I2 @ K )
% 3.12/1.84 => ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_56_nth__equalityI,axiom,
% 3.12/1.84 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 3.12/1.84 ( ( ( size_size @ ( list @ A ) @ Xs2 )
% 3.12/1.84 = ( size_size @ ( list @ A ) @ Ys ) )
% 3.12/1.84 => ( ! [I3: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 3.12/1.84 => ( ( nth @ A @ Xs2 @ I3 )
% 3.12/1.84 = ( nth @ A @ Ys @ I3 ) ) )
% 3.12/1.84 => ( Xs2 = Ys ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_57_zero__fun__def,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( zero @ B )
% 3.12/1.84 => ( ( zero_zero @ ( A > B ) )
% 3.12/1.84 = ( ^ [X: A] : ( zero_zero @ B ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_58_zero__reorient,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( zero @ A )
% 3.12/1.84 => ! [X2: A] :
% 3.12/1.84 ( ( ( zero_zero @ A )
% 3.12/1.84 = X2 )
% 3.12/1.84 = ( X2
% 3.12/1.84 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_59_mset__map,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
% 3.12/1.84 ( ( mset @ A @ ( map @ B @ A @ F @ Xs2 ) )
% 3.12/1.84 = ( image_mset @ B @ A @ F @ ( mset @ B @ Xs2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_60_size__mset,axiom,
% 3.12/1.84 ! [A: $tType,Xs2: list @ A] :
% 3.12/1.84 ( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs2 ) )
% 3.12/1.84 = ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_61_lessThan__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ord @ A )
% 3.12/1.84 => ! [I: A,K: A] :
% 3.12/1.84 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 3.12/1.84 = ( ord_less @ A @ I @ K ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_62_image__mset__mset__set,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( image_mset @ A @ B @ F @ ( mset_set @ A @ A3 ) )
% 3.12/1.84 = ( mset_set @ B @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_63_lessThan__atLeast0,axiom,
% 3.12/1.84 ( ( set_ord_lessThan @ nat )
% 3.12/1.84 = ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_64_bij__betw__def,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( bij_betw @ A @ B )
% 3.12/1.84 = ( ^ [F2: A > B,A4: set @ A,B2: set @ B] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F2 @ A4 )
% 3.12/1.84 & ( ( image @ A @ B @ F2 @ A4 )
% 3.12/1.84 = B2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_65_bij__betw__imageI,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( ( image @ A @ B @ F @ A3 )
% 3.12/1.84 = B3 )
% 3.12/1.84 => ( bij_betw @ A @ B @ F @ A3 @ B3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_66_inj__on__imp__bij__betw,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( bij_betw @ A @ B @ F @ A3 @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_67_comp__inj__on,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F: A > B,A3: set @ A,G: B > C] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A3 ) )
% 3.12/1.84 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_68_inj__on__imageI,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A3: set @ A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A3 )
% 3.12/1.84 => ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_69_comp__inj__on__iff,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F: A > B,A3: set @ A,F3: B > C] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( inj_on @ B @ C @ F3 @ ( image @ A @ B @ F @ A3 ) )
% 3.12/1.84 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_70_comp__apply,axiom,
% 3.12/1.84 ! [C: $tType,A: $tType,B: $tType] :
% 3.12/1.84 ( ( comp @ B @ A @ C )
% 3.12/1.84 = ( ^ [F2: B > A,G2: C > B,X: C] : ( F2 @ ( G2 @ X ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_71_lessThan__eq__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linorder @ A )
% 3.12/1.84 => ! [X2: A,Y: A] :
% 3.12/1.84 ( ( ( set_ord_lessThan @ A @ X2 )
% 3.12/1.84 = ( set_ord_lessThan @ A @ Y ) )
% 3.12/1.84 = ( X2 = Y ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_72_image__mset__empty,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: B > A] :
% 3.12/1.84 ( ( image_mset @ B @ A @ F @ ( zero_zero @ ( multiset @ B ) ) )
% 3.12/1.84 = ( zero_zero @ ( multiset @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_73_image__mset__is__empty__iff,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,M4: multiset @ B] :
% 3.12/1.84 ( ( ( image_mset @ B @ A @ F @ M4 )
% 3.12/1.84 = ( zero_zero @ ( multiset @ A ) ) )
% 3.12/1.84 = ( M4
% 3.12/1.84 = ( zero_zero @ ( multiset @ B ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_74_size__eq__0__iff__empty,axiom,
% 3.12/1.84 ! [A: $tType,M4: multiset @ A] :
% 3.12/1.84 ( ( ( size_size @ ( multiset @ A ) @ M4 )
% 3.12/1.84 = ( zero_zero @ nat ) )
% 3.12/1.84 = ( M4
% 3.12/1.84 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_75_size__empty,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
% 3.12/1.84 = ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_76_size__image__mset,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,M4: multiset @ B] :
% 3.12/1.84 ( ( size_size @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M4 ) )
% 3.12/1.84 = ( size_size @ ( multiset @ B ) @ M4 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_77_mset__upt,axiom,
% 3.12/1.84 ! [M: nat,N: nat] :
% 3.12/1.84 ( ( mset @ nat @ ( upt @ M @ N ) )
% 3.12/1.84 = ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ M @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_78_inj__on__strict__subset,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,B3: set @ A,A3: set @ A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ B3 )
% 3.12/1.84 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 3.12/1.84 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_79_comp__eq__dest__lhs,axiom,
% 3.12/1.84 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B4: A > C,C2: A > B,V: A] :
% 3.12/1.84 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 3.12/1.84 = C2 )
% 3.12/1.84 => ( ( A2 @ ( B4 @ V ) )
% 3.12/1.84 = ( C2 @ V ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_80_comp__eq__elim,axiom,
% 3.12/1.84 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B4: A > C,C2: D > B,D2: A > D] :
% 3.12/1.84 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 3.12/1.84 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.12/1.84 => ! [V3: A] :
% 3.12/1.84 ( ( A2 @ ( B4 @ V3 ) )
% 3.12/1.84 = ( C2 @ ( D2 @ V3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_81_comp__eq__dest,axiom,
% 3.12/1.84 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B4: A > C,C2: D > B,D2: A > D,V: A] :
% 3.12/1.84 ( ( ( comp @ C @ B @ A @ A2 @ B4 )
% 3.12/1.84 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.12/1.84 => ( ( A2 @ ( B4 @ V ) )
% 3.12/1.84 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_82_comp__assoc,axiom,
% 3.12/1.84 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 3.12/1.84 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 3.12/1.84 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_83_comp__def,axiom,
% 3.12/1.84 ! [A: $tType,C: $tType,B: $tType] :
% 3.12/1.84 ( ( comp @ B @ C @ A )
% 3.12/1.84 = ( ^ [F2: B > C,G2: A > B,X: A] : ( F2 @ ( G2 @ X ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_84_inj__on__inverseI,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,A3: set @ A,G: B > A,F: A > B] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( member @ A @ X3 @ A3 )
% 3.12/1.84 => ( ( G @ ( F @ X3 ) )
% 3.12/1.84 = X3 ) )
% 3.12/1.84 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_85_inj__on__contraD,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( X2 != Y )
% 3.12/1.84 => ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( ( member @ A @ Y @ A3 )
% 3.12/1.84 => ( ( F @ X2 )
% 3.12/1.84 != ( F @ Y ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_86_inj__on__eq__iff,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( ( member @ A @ Y @ A3 )
% 3.12/1.84 => ( ( ( F @ X2 )
% 3.12/1.84 = ( F @ Y ) )
% 3.12/1.84 = ( X2 = Y ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_87_inj__on__cong,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,A3: set @ A,F: A > B,G: A > B] :
% 3.12/1.84 ( ! [A5: A] :
% 3.12/1.84 ( ( member @ A @ A5 @ A3 )
% 3.12/1.84 => ( ( F @ A5 )
% 3.12/1.84 = ( G @ A5 ) ) )
% 3.12/1.84 => ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 = ( inj_on @ A @ B @ G @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_88_inj__on__def,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( inj_on @ A @ B )
% 3.12/1.84 = ( ^ [F2: A > B,A4: set @ A] :
% 3.12/1.84 ! [X: A] :
% 3.12/1.84 ( ( member @ A @ X @ A4 )
% 3.12/1.84 => ! [Y4: A] :
% 3.12/1.84 ( ( member @ A @ Y4 @ A4 )
% 3.12/1.84 => ( ( ( F2 @ X )
% 3.12/1.84 = ( F2 @ Y4 ) )
% 3.12/1.84 => ( X = Y4 ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_89_inj__onI,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
% 3.12/1.84 ( ! [X3: A,Y5: A] :
% 3.12/1.84 ( ( member @ A @ X3 @ A3 )
% 3.12/1.84 => ( ( member @ A @ Y5 @ A3 )
% 3.12/1.84 => ( ( ( F @ X3 )
% 3.12/1.84 = ( F @ Y5 ) )
% 3.12/1.84 => ( X3 = Y5 ) ) ) )
% 3.12/1.84 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_90_inj__onD,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A,Y: A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ F @ A3 )
% 3.12/1.84 => ( ( ( F @ X2 )
% 3.12/1.84 = ( F @ Y ) )
% 3.12/1.84 => ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( ( member @ A @ Y @ A3 )
% 3.12/1.84 => ( X2 = Y ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_91_bij__betw__iff__bijections,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( bij_betw @ A @ B )
% 3.12/1.84 = ( ^ [F2: A > B,A4: set @ A,B2: set @ B] :
% 3.12/1.84 ? [G2: B > A] :
% 3.12/1.84 ( ! [X: A] :
% 3.12/1.84 ( ( member @ A @ X @ A4 )
% 3.12/1.84 => ( ( member @ B @ ( F2 @ X ) @ B2 )
% 3.12/1.84 & ( ( G2 @ ( F2 @ X ) )
% 3.12/1.84 = X ) ) )
% 3.12/1.84 & ! [X: B] :
% 3.12/1.84 ( ( member @ B @ X @ B2 )
% 3.12/1.84 => ( ( member @ A @ ( G2 @ X ) @ A4 )
% 3.12/1.84 & ( ( F2 @ ( G2 @ X ) )
% 3.12/1.84 = X ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_92_bij__betw__apply,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B,A2: A] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ( ( member @ A @ A2 @ A3 )
% 3.12/1.84 => ( member @ B @ ( F @ A2 ) @ B3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_93_bij__betw__cong,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,A3: set @ A,F: A > B,G: A > B,A6: set @ B] :
% 3.12/1.84 ( ! [A5: A] :
% 3.12/1.84 ( ( member @ A @ A5 @ A3 )
% 3.12/1.84 => ( ( F @ A5 )
% 3.12/1.84 = ( G @ A5 ) ) )
% 3.12/1.84 => ( ( bij_betw @ A @ B @ F @ A3 @ A6 )
% 3.12/1.84 = ( bij_betw @ A @ B @ G @ A3 @ A6 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_94_bij__betw__inv,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ? [G3: B > A] : ( bij_betw @ B @ A @ G3 @ B3 @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_95_bij__betwE,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ! [X5: A] :
% 3.12/1.84 ( ( member @ A @ X5 @ A3 )
% 3.12/1.84 => ( member @ B @ ( F @ X5 ) @ B3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_96_ex__mset,axiom,
% 3.12/1.84 ! [A: $tType,X6: multiset @ A] :
% 3.12/1.84 ? [Xs: list @ A] :
% 3.12/1.84 ( ( mset @ A @ Xs )
% 3.12/1.84 = X6 ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_97_image__eq__imp__comp,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A3: set @ B,G: C > A,B3: set @ C,H: A > D] :
% 3.12/1.84 ( ( ( image @ B @ A @ F @ A3 )
% 3.12/1.84 = ( image @ C @ A @ G @ B3 ) )
% 3.12/1.84 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A3 )
% 3.12/1.84 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_98_image__comp,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 3.12/1.84 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
% 3.12/1.84 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_99_nonempty__has__size,axiom,
% 3.12/1.84 ! [A: $tType,S2: multiset @ A] :
% 3.12/1.84 ( ( S2
% 3.12/1.84 != ( zero_zero @ ( multiset @ A ) ) )
% 3.12/1.84 = ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_100_atLeastLessThan__inj_I2_J,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linorder @ A )
% 3.12/1.84 => ! [A2: A,B4: A,C2: A,D2: A] :
% 3.12/1.84 ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 3.12/1.84 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 3.12/1.84 => ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.84 => ( ( ord_less @ A @ C2 @ D2 )
% 3.12/1.84 => ( B4 = D2 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_101_atLeastLessThan__inj_I1_J,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linorder @ A )
% 3.12/1.84 => ! [A2: A,B4: A,C2: A,D2: A] :
% 3.12/1.84 ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 3.12/1.84 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 3.12/1.84 => ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.84 => ( ( ord_less @ A @ C2 @ D2 )
% 3.12/1.84 => ( A2 = C2 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_102_atLeastLessThan__eq__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linorder @ A )
% 3.12/1.84 => ! [A2: A,B4: A,C2: A,D2: A] :
% 3.12/1.84 ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.84 => ( ( ord_less @ A @ C2 @ D2 )
% 3.12/1.84 => ( ( ( set_or1433837966ssThan @ A @ A2 @ B4 )
% 3.12/1.84 = ( set_or1433837966ssThan @ A @ C2 @ D2 ) )
% 3.12/1.84 = ( ( A2 = C2 )
% 3.12/1.84 & ( B4 = D2 ) ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_103_lessThan__strict__subset__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linorder @ A )
% 3.12/1.84 => ! [M: A,N: A] :
% 3.12/1.84 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
% 3.12/1.84 = ( ord_less @ A @ M @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_104_inj__on__image__iff,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,A3: set @ A,G: A > B,F: A > A] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( member @ A @ X3 @ A3 )
% 3.12/1.84 => ! [Xa: A] :
% 3.12/1.84 ( ( member @ A @ Xa @ A3 )
% 3.12/1.84 => ( ( ( G @ ( F @ X3 ) )
% 3.12/1.84 = ( G @ ( F @ Xa ) ) )
% 3.12/1.84 = ( ( G @ X3 )
% 3.12/1.84 = ( G @ Xa ) ) ) ) )
% 3.12/1.84 => ( ( inj_on @ A @ A @ F @ A3 )
% 3.12/1.84 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A3 ) )
% 3.12/1.84 = ( inj_on @ A @ B @ G @ A3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_105_inj__on__imageI2,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F: A > C,A3: set @ A] :
% 3.12/1.84 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F ) @ A3 )
% 3.12/1.84 => ( inj_on @ A @ C @ F @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_106_bij__betw__imp__surj__on,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ( ( image @ A @ B @ F @ A3 )
% 3.12/1.84 = B3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_107_bij__betw__comp__iff,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,C: $tType,F: A > B,A3: set @ A,A6: set @ B,F3: B > C,A7: set @ C] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ A6 )
% 3.12/1.84 => ( ( bij_betw @ B @ C @ F3 @ A6 @ A7 )
% 3.12/1.84 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A3 @ A7 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_108_bij__betw__trans,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,C: $tType,F: A > B,A3: set @ A,B3: set @ B,G: B > C,C3: set @ C] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ( ( bij_betw @ B @ C @ G @ B3 @ C3 )
% 3.12/1.84 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A3 @ C3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_109_multiset_Omap__comp,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: multiset @ A] :
% 3.12/1.84 ( ( image_mset @ B @ C @ G @ ( image_mset @ A @ B @ F @ V ) )
% 3.12/1.84 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_110_image__mset_Ocompositionality,axiom,
% 3.12/1.84 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Multiset: multiset @ A] :
% 3.12/1.84 ( ( image_mset @ B @ C @ F @ ( image_mset @ A @ B @ G @ Multiset ) )
% 3.12/1.84 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Multiset ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_111_image__mset_Ocomp,axiom,
% 3.12/1.84 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 3.12/1.84 ( ( comp @ ( multiset @ B ) @ ( multiset @ C ) @ ( multiset @ A ) @ ( image_mset @ B @ C @ F ) @ ( image_mset @ A @ B @ G ) )
% 3.12/1.84 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_112_bij__betw__imp__inj__on,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B3: set @ B] :
% 3.12/1.84 ( ( bij_betw @ A @ B @ F @ A3 @ B3 )
% 3.12/1.84 => ( inj_on @ A @ B @ F @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_113_mset__eq__length,axiom,
% 3.12/1.84 ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
% 3.12/1.84 ( ( ( mset @ A @ Xs2 )
% 3.12/1.84 = ( mset @ A @ Ys ) )
% 3.12/1.84 => ( ( size_size @ ( list @ A ) @ Xs2 )
% 3.12/1.84 = ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_114_image__eqI,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,B4: A,F: B > A,X2: B,A3: set @ B] :
% 3.12/1.84 ( ( B4
% 3.12/1.84 = ( F @ X2 ) )
% 3.12/1.84 => ( ( member @ B @ X2 @ A3 )
% 3.12/1.84 => ( member @ A @ B4 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_115_mbs_Oless__not__eq,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( size @ A )
% 3.12/1.84 => ! [X2: A,A3: set @ A,Y: A] :
% 3.12/1.84 ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( ( ord_less @ nat @ ( size_size @ A @ X2 ) @ ( size_size @ A @ Y ) )
% 3.12/1.84 => ( X2 != Y ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_116_Sup_OSUP__image,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
% 3.12/1.84 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
% 3.12/1.84 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_117_Inf_OINF__image,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
% 3.12/1.84 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
% 3.12/1.84 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_118_less__numeral__extra_I3_J,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_119_field__lbound__gt__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linordered_field @ A )
% 3.12/1.84 => ! [D1: A,D22: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
% 3.12/1.84 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
% 3.12/1.84 => ? [E: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( zero_zero @ A ) @ E )
% 3.12/1.84 & ( ord_less @ A @ E @ D1 )
% 3.12/1.84 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_120_length__code,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( size_size @ ( list @ A ) )
% 3.12/1.84 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_121_psubsetD,axiom,
% 3.12/1.84 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 3.12/1.84 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 3.12/1.84 => ( ( member @ A @ C2 @ A3 )
% 3.12/1.84 => ( member @ A @ C2 @ B3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_122_psubset__trans,axiom,
% 3.12/1.84 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.12/1.84 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 3.12/1.84 => ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
% 3.12/1.84 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_123_imageI,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,X2: A,A3: set @ A,F: A > B] :
% 3.12/1.84 ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_124_image__iff,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,Z2: A,F: B > A,A3: set @ B] :
% 3.12/1.84 ( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A3 ) )
% 3.12/1.84 = ( ? [X: B] :
% 3.12/1.84 ( ( member @ B @ X @ A3 )
% 3.12/1.84 & ( Z2
% 3.12/1.84 = ( F @ X ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_125_bex__imageD,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 3.12/1.84 ( ? [X5: A] :
% 3.12/1.84 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A3 ) )
% 3.12/1.84 & ( P @ X5 ) )
% 3.12/1.84 => ? [X3: B] :
% 3.12/1.84 ( ( member @ B @ X3 @ A3 )
% 3.12/1.84 & ( P @ ( F @ X3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_126_image__cong,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,M4: set @ A,N3: set @ A,F: A > B,G: A > B] :
% 3.12/1.84 ( ( M4 = N3 )
% 3.12/1.84 => ( ! [X3: A] :
% 3.12/1.84 ( ( member @ A @ X3 @ N3 )
% 3.12/1.84 => ( ( F @ X3 )
% 3.12/1.84 = ( G @ X3 ) ) )
% 3.12/1.84 => ( ( image @ A @ B @ F @ M4 )
% 3.12/1.84 = ( image @ A @ B @ G @ N3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_127_ball__imageD,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
% 3.12/1.84 ( ! [X3: A] :
% 3.12/1.84 ( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
% 3.12/1.84 => ( P @ X3 ) )
% 3.12/1.84 => ! [X5: B] :
% 3.12/1.84 ( ( member @ B @ X5 @ A3 )
% 3.12/1.84 => ( P @ ( F @ X5 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_128_rev__image__eqI,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,X2: A,A3: set @ A,B4: B,F: A > B] :
% 3.12/1.84 ( ( member @ A @ X2 @ A3 )
% 3.12/1.84 => ( ( B4
% 3.12/1.84 = ( F @ X2 ) )
% 3.12/1.84 => ( member @ B @ B4 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_129_Inf_OINF__cong,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 3.12/1.84 ( ( A3 = B3 )
% 3.12/1.84 => ( ! [X3: B] :
% 3.12/1.84 ( ( member @ B @ X3 @ B3 )
% 3.12/1.84 => ( ( C3 @ X3 )
% 3.12/1.84 = ( D3 @ X3 ) ) )
% 3.12/1.84 => ( ( Inf @ ( image @ B @ A @ C3 @ A3 ) )
% 3.12/1.84 = ( Inf @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_130_Sup_OSUP__cong,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType,A3: set @ B,B3: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 3.12/1.84 ( ( A3 = B3 )
% 3.12/1.84 => ( ! [X3: B] :
% 3.12/1.84 ( ( member @ B @ X3 @ B3 )
% 3.12/1.84 => ( ( C3 @ X3 )
% 3.12/1.84 = ( D3 @ X3 ) ) )
% 3.12/1.84 => ( ( Sup @ ( image @ B @ A @ C3 @ A3 ) )
% 3.12/1.84 = ( Sup @ ( image @ B @ A @ D3 @ B3 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_131_Multiset_Ois__empty__def,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( is_empty @ A )
% 3.12/1.84 = ( ^ [A4: multiset @ A] :
% 3.12/1.84 ( A4
% 3.12/1.84 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_132_map__eq__map__tailrec,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( map @ A @ B )
% 3.12/1.84 = ( map_tailrec @ A @ B ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_133_list__ex__length,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( list_ex @ A )
% 3.12/1.84 = ( ^ [P2: A > $o,Xs3: list @ A] :
% 3.12/1.84 ? [N4: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
% 3.12/1.84 & ( P2 @ ( nth @ A @ Xs3 @ N4 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_134_of__nat__0__less__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ! [N: nat] :
% 3.12/1.84 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.84 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_135_of__nat__fun__apply,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType] :
% 3.12/1.84 ( ( semiring_1 @ A )
% 3.12/1.84 => ( ( semiring_1_of_nat @ ( B > A ) )
% 3.12/1.84 = ( ^ [N4: nat,X: B] : ( semiring_1_of_nat @ A @ N4 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_136_of__nat__eq__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_char_0 @ A )
% 3.12/1.84 => ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ( semiring_1_of_nat @ A @ M )
% 3.12/1.84 = ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.84 = ( M = N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_137_of__nat__0,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_1 @ A )
% 3.12/1.84 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 3.12/1.84 = ( zero_zero @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_138_of__nat__0__eq__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_char_0 @ A )
% 3.12/1.84 => ! [N: nat] :
% 3.12/1.84 ( ( ( zero_zero @ A )
% 3.12/1.84 = ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.84 = ( ( zero_zero @ nat )
% 3.12/1.84 = N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_139_of__nat__eq__0__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_char_0 @ A )
% 3.12/1.84 => ! [M: nat] :
% 3.12/1.84 ( ( ( semiring_1_of_nat @ A @ M )
% 3.12/1.84 = ( zero_zero @ A ) )
% 3.12/1.84 = ( M
% 3.12/1.84 = ( zero_zero @ nat ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_140_of__nat__less__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.84 = ( ord_less @ nat @ M @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_141_bij__betw__of__nat,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_char_0 @ A )
% 3.12/1.84 => ! [N3: set @ nat,A3: set @ A] :
% 3.12/1.84 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 @ A3 )
% 3.12/1.84 = ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 )
% 3.12/1.84 = A3 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_142_image__int__atLeastLessThan,axiom,
% 3.12/1.84 ! [A2: nat,B4: nat] :
% 3.12/1.84 ( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1433837966ssThan @ nat @ A2 @ B4 ) )
% 3.12/1.84 = ( set_or1433837966ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_143_inj__on__of__nat,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( semiring_char_0 @ A )
% 3.12/1.84 => ! [N3: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_144_of__nat__fun,axiom,
% 3.12/1.84 ! [A: $tType,B: $tType] :
% 3.12/1.84 ( ( semiring_1 @ B )
% 3.12/1.84 => ( ( semiring_1_of_nat @ ( A > B ) )
% 3.12/1.84 = ( ^ [N4: nat,X: A] : ( semiring_1_of_nat @ B @ N4 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_145_of__nat__less__0__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ! [M: nat] :
% 3.12/1.84 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_146_of__nat__less__imp__less,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.84 => ( ord_less @ nat @ M @ N ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_147_less__imp__of__nat__less,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord1659791738miring @ A )
% 3.12/1.84 => ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ M @ N )
% 3.12/1.84 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_148_pos__int__cases,axiom,
% 3.12/1.84 ! [K: int] :
% 3.12/1.84 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 3.12/1.84 => ~ ! [N2: nat] :
% 3.12/1.84 ( ( K
% 3.12/1.84 = ( semiring_1_of_nat @ int @ N2 ) )
% 3.12/1.84 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_149_zero__less__imp__eq__int,axiom,
% 3.12/1.84 ! [K: int] :
% 3.12/1.84 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 3.12/1.84 => ? [N2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.12/1.84 & ( K
% 3.12/1.84 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_150_reals__Archimedean2,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( archim1804426504_field @ A )
% 3.12/1.84 => ! [X2: A] :
% 3.12/1.84 ? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_151_size__multiset__empty,axiom,
% 3.12/1.84 ! [A: $tType,F: A > nat] :
% 3.12/1.84 ( ( size_multiset @ A @ F @ ( zero_zero @ ( multiset @ A ) ) )
% 3.12/1.84 = ( zero_zero @ nat ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_152_size__multiset__eq__0__iff__empty,axiom,
% 3.12/1.84 ! [A: $tType,F: A > nat,M4: multiset @ A] :
% 3.12/1.84 ( ( ( size_multiset @ A @ F @ M4 )
% 3.12/1.84 = ( zero_zero @ nat ) )
% 3.12/1.84 = ( M4
% 3.12/1.84 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_153_int__int__eq,axiom,
% 3.12/1.84 ! [M: nat,N: nat] :
% 3.12/1.84 ( ( ( semiring_1_of_nat @ int @ M )
% 3.12/1.84 = ( semiring_1_of_nat @ int @ N ) )
% 3.12/1.84 = ( M = N ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_154_less__int__code_I1_J,axiom,
% 3.12/1.84 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_155_size__multiset__o__map,axiom,
% 3.12/1.84 ! [B: $tType,A: $tType,G: B > nat,F: A > B] :
% 3.12/1.84 ( ( comp @ ( multiset @ B ) @ nat @ ( multiset @ A ) @ ( size_multiset @ B @ G ) @ ( image_mset @ A @ B @ F ) )
% 3.12/1.84 = ( size_multiset @ A @ ( comp @ B @ nat @ A @ G @ F ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_156_nat__int__comparison_I2_J,axiom,
% 3.12/1.84 ( ( ord_less @ nat )
% 3.12/1.84 = ( ^ [A8: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A8 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_157_int__ops_I1_J,axiom,
% 3.12/1.84 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
% 3.12/1.84 = ( zero_zero @ int ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_158_neg__int__cases,axiom,
% 3.12/1.84 ! [K: int] :
% 3.12/1.84 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 3.12/1.84 => ~ ! [N2: nat] :
% 3.12/1.84 ( ( K
% 3.12/1.84 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
% 3.12/1.84 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_159_add_Oinverse__inverse,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = A2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_160_neg__equal__iff__equal,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( ( uminus_uminus @ A @ A2 )
% 3.12/1.84 = ( uminus_uminus @ A @ B4 ) )
% 3.12/1.84 = ( A2 = B4 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_161_add_Oinverse__neutral,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 3.12/1.84 = ( zero_zero @ A ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_162_neg__0__equal__iff__equal,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ( zero_zero @ A )
% 3.12/1.84 = ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = ( ( zero_zero @ A )
% 3.12/1.84 = A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_163_neg__equal__0__iff__equal,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ( uminus_uminus @ A @ A2 )
% 3.12/1.84 = ( zero_zero @ A ) )
% 3.12/1.84 = ( A2
% 3.12/1.84 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_164_equal__neg__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord219039673up_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( A2
% 3.12/1.84 = ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = ( A2
% 3.12/1.84 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_165_neg__equal__zero,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord219039673up_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ( uminus_uminus @ A @ A2 )
% 3.12/1.84 = A2 )
% 3.12/1.84 = ( A2
% 3.12/1.84 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_166_neg__less__iff__less,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [B4: A,A2: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = ( ord_less @ A @ A2 @ B4 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_167_inj__uminus,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A3: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A3 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_168_neg__less__0__iff__less,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.12/1.84 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_169_neg__0__less__iff__less,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_170_neg__less__pos,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord219039673up_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.12/1.84 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_171_less__neg__neg,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( linord219039673up_add @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.84 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_172_negative__eq__positive,axiom,
% 3.12/1.84 ! [N: nat,M: nat] :
% 3.12/1.84 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
% 3.12/1.84 = ( semiring_1_of_nat @ int @ M ) )
% 3.12/1.84 = ( ( N
% 3.12/1.84 = ( zero_zero @ nat ) )
% 3.12/1.84 & ( M
% 3.12/1.84 = ( zero_zero @ nat ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_173_int__cases2,axiom,
% 3.12/1.84 ! [Z2: int] :
% 3.12/1.84 ( ! [N2: nat] :
% 3.12/1.84 ( Z2
% 3.12/1.84 != ( semiring_1_of_nat @ int @ N2 ) )
% 3.12/1.84 => ~ ! [N2: nat] :
% 3.12/1.84 ( Z2
% 3.12/1.84 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_174_minus__less__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B4 )
% 3.12/1.84 = ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_175_less__minus__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B4 ) )
% 3.12/1.84 = ( ord_less @ A @ B4 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_176_uminus__int__code_I1_J,axiom,
% 3.12/1.84 ( ( uminus_uminus @ int @ ( zero_zero @ int ) )
% 3.12/1.84 = ( zero_zero @ int ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_177_verit__negate__coefficient_I2_J,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( ordered_ab_group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.84 => ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_178_equation__minus__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( A2
% 3.12/1.84 = ( uminus_uminus @ A @ B4 ) )
% 3.12/1.84 = ( B4
% 3.12/1.84 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_179_minus__equation__iff,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( group_add @ A )
% 3.12/1.84 => ! [A2: A,B4: A] :
% 3.12/1.84 ( ( ( uminus_uminus @ A @ A2 )
% 3.12/1.84 = B4 )
% 3.12/1.84 = ( ( uminus_uminus @ A @ B4 )
% 3.12/1.84 = A2 ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_180_not__int__zless__negative,axiom,
% 3.12/1.84 ! [N: nat,M: nat] :
% 3.12/1.84 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_181_int__cases4,axiom,
% 3.12/1.84 ! [M: int] :
% 3.12/1.84 ( ! [N2: nat] :
% 3.12/1.84 ( M
% 3.12/1.84 != ( semiring_1_of_nat @ int @ N2 ) )
% 3.12/1.84 => ~ ! [N2: nat] :
% 3.12/1.84 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.12/1.84 => ( M
% 3.12/1.84 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_182_verit__comp__simplify1_I1_J,axiom,
% 3.12/1.84 ! [A: $tType] :
% 3.12/1.84 ( ( order @ A )
% 3.12/1.84 => ! [A2: A] :
% 3.12/1.84 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 3.12/1.84
% 3.12/1.84 thf(fact_183_nat__int__comparison_I1_J,axiom,
% 3.12/1.84 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
% 3.12/1.84 = ( ^ [A8: nat,B5: nat] :
% 3.12/1.85 ( ( semiring_1_of_nat @ int @ A8 )
% 3.12/1.85 = ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_184_int__if,axiom,
% 3.12/1.85 ! [P: $o,A2: nat,B4: nat] :
% 3.12/1.85 ( ( P
% 3.12/1.85 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B4 ) )
% 3.12/1.85 = ( semiring_1_of_nat @ int @ A2 ) ) )
% 3.12/1.85 & ( ~ P
% 3.12/1.85 => ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B4 ) )
% 3.12/1.85 = ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_185_int__cases3,axiom,
% 3.12/1.85 ! [K: int] :
% 3.12/1.85 ( ( K
% 3.12/1.85 != ( zero_zero @ int ) )
% 3.12/1.85 => ( ! [N2: nat] :
% 3.12/1.85 ( ( K
% 3.12/1.85 = ( semiring_1_of_nat @ int @ N2 ) )
% 3.12/1.85 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
% 3.12/1.85 => ~ ! [N2: nat] :
% 3.12/1.85 ( ( K
% 3.12/1.85 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
% 3.12/1.85 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_186_norm__frac_Oinduct,axiom,
% 3.12/1.85 ! [P: int > int > $o,A0: int,A1: int] :
% 3.12/1.85 ( ! [A5: int,B6: int] :
% 3.12/1.85 ( ( ( ord_less @ int @ B6 @ ( zero_zero @ int ) )
% 3.12/1.85 => ( P @ ( uminus_uminus @ int @ A5 ) @ ( uminus_uminus @ int @ B6 ) ) )
% 3.12/1.85 => ( P @ A5 @ B6 ) )
% 3.12/1.85 => ( P @ A0 @ A1 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_187_compl__less__swap1,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( boolean_algebra @ A )
% 3.12/1.85 => ! [Y: A,X2: A] :
% 3.12/1.85 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
% 3.12/1.85 => ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_188_compl__less__swap2,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( boolean_algebra @ A )
% 3.12/1.85 => ! [Y: A,X2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
% 3.12/1.85 => ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_189_compl__less__compl__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( boolean_algebra @ A )
% 3.12/1.85 => ! [X2: A,Y: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
% 3.12/1.85 = ( ord_less @ A @ Y @ X2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_190_ex__inverse__of__nat__less,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( archim1804426504_field @ A )
% 3.12/1.85 => ! [X2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 3.12/1.85 => ? [N2: nat] :
% 3.12/1.85 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 3.12/1.85 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_191_zero__less__nat__eq,axiom,
% 3.12/1.85 ! [Z2: int] :
% 3.12/1.85 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) )
% 3.12/1.85 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_192_nat__int,axiom,
% 3.12/1.85 ! [N: nat] :
% 3.12/1.85 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 3.12/1.85 = N ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_193_zless__nat__conj,axiom,
% 3.12/1.85 ! [W: int,Z2: int] :
% 3.12/1.85 ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 3.12/1.85 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
% 3.12/1.85 & ( ord_less @ int @ W @ Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_194_nat__zminus__int,axiom,
% 3.12/1.85 ! [N: nat] :
% 3.12/1.85 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
% 3.12/1.85 = ( zero_zero @ nat ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_195_nat__zero__as__int,axiom,
% 3.12/1.85 ( ( zero_zero @ nat )
% 3.12/1.85 = ( nat2 @ ( zero_zero @ int ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_196_forall__pos__mono,axiom,
% 3.12/1.85 ! [P: real > $o,E2: real] :
% 3.12/1.85 ( ! [D4: real,E: real] :
% 3.12/1.85 ( ( ord_less @ real @ D4 @ E )
% 3.12/1.85 => ( ( P @ D4 )
% 3.12/1.85 => ( P @ E ) ) )
% 3.12/1.85 => ( ! [N2: nat] :
% 3.12/1.85 ( ( N2
% 3.12/1.85 != ( zero_zero @ nat ) )
% 3.12/1.85 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
% 3.12/1.85 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 3.12/1.85 => ( P @ E2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_197_real__arch__inverse,axiom,
% 3.12/1.85 ! [E2: real] :
% 3.12/1.85 ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
% 3.12/1.85 = ( ? [N4: nat] :
% 3.12/1.85 ( ( N4
% 3.12/1.85 != ( zero_zero @ nat ) )
% 3.12/1.85 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) )
% 3.12/1.85 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N4 ) ) @ E2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_198_nat__mono__iff,axiom,
% 3.12/1.85 ! [Z2: int,W: int] :
% 3.12/1.85 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
% 3.12/1.85 => ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
% 3.12/1.85 = ( ord_less @ int @ W @ Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_199_zless__nat__eq__int__zless,axiom,
% 3.12/1.85 ! [M: nat,Z2: int] :
% 3.12/1.85 ( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) )
% 3.12/1.85 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_200_split__nat,axiom,
% 3.12/1.85 ! [P: nat > $o,I: int] :
% 3.12/1.85 ( ( P @ ( nat2 @ I ) )
% 3.12/1.85 = ( ! [N4: nat] :
% 3.12/1.85 ( ( I
% 3.12/1.85 = ( semiring_1_of_nat @ int @ N4 ) )
% 3.12/1.85 => ( P @ N4 ) )
% 3.12/1.85 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
% 3.12/1.85 => ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_201_inverse__positive__iff__positive,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 3.12/1.85 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_202_inverse__negative__iff__negative,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.12/1.85 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_203_inverse__zero,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 3.12/1.85 = ( zero_zero @ A ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_204_inverse__nonzero__iff__nonzero,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ( inverse_inverse @ A @ A2 )
% 3.12/1.85 = ( zero_zero @ A ) )
% 3.12/1.85 = ( A2
% 3.12/1.85 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_205_inverse__less__iff__less,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 3.12/1.85 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 3.12/1.85 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 = ( ord_less @ A @ B4 @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_206_inverse__less__iff__less__neg,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 = ( ord_less @ A @ B4 @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_207_linordered__field__no__lb,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [X5: A] :
% 3.12/1.85 ? [Y5: A] : ( ord_less @ A @ Y5 @ X5 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_208_linordered__field__no__ub,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [X5: A] :
% 3.12/1.85 ? [X_1: A] : ( ord_less @ A @ X5 @ X_1 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_209_field__class_Ofield__inverse__zero,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( field @ A )
% 3.12/1.85 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 3.12/1.85 = ( zero_zero @ A ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_210_inverse__zero__imp__zero,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ( inverse_inverse @ A @ A2 )
% 3.12/1.85 = ( zero_zero @ A ) )
% 3.12/1.85 => ( A2
% 3.12/1.85 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_211_nonzero__inverse__eq__imp__eq,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ( inverse_inverse @ A @ A2 )
% 3.12/1.85 = ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 => ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ( B4
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( A2 = B4 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_212_nonzero__inverse__inverse__eq,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
% 3.12/1.85 = A2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_213_nonzero__imp__inverse__nonzero,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ( inverse_inverse @ A @ A2 )
% 3.12/1.85 != ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_214_inverse__less__imp__less,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 3.12/1.85 => ( ord_less @ A @ B4 @ A2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_215_less__imp__inverse__less,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.85 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 3.12/1.85 => ( ord_less @ A @ ( inverse_inverse @ A @ B4 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_216_inverse__less__imp__less__neg,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ord_less @ A @ B4 @ A2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_217_less__imp__inverse__less__neg,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ A2 @ B4 )
% 3.12/1.85 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ord_less @ A @ ( inverse_inverse @ A @ B4 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_218_inverse__negative__imp__negative,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_219_inverse__positive__imp__positive,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 3.12/1.85 => ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_220_negative__imp__inverse__negative,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_221_positive__imp__inverse__positive,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 3.12/1.85 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_222_nonzero__inverse__minus__eq,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( division_ring @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( A2
% 3.12/1.85 != ( zero_zero @ A ) )
% 3.12/1.85 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.85 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_223_image__atLeastZeroLessThan__int,axiom,
% 3.12/1.85 ! [U: int] :
% 3.12/1.85 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 3.12/1.85 => ( ( set_or1433837966ssThan @ int @ ( zero_zero @ int ) @ U )
% 3.12/1.85 = ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_224_of__int__of__nat,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_1 @ A )
% 3.12/1.85 => ( ( ring_1_of_int @ A )
% 3.12/1.85 = ( ^ [K2: int] : ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K2 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K2 ) ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_225_of__int__eq__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_char_0 @ A )
% 3.12/1.85 => ! [W: int,Z2: int] :
% 3.12/1.85 ( ( ( ring_1_of_int @ A @ W )
% 3.12/1.85 = ( ring_1_of_int @ A @ Z2 ) )
% 3.12/1.85 = ( W = Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_226_le__zero__eq,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( canoni770627133id_add @ A )
% 3.12/1.85 => ! [N: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 3.12/1.85 = ( N
% 3.12/1.85 = ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_227_neg__le__iff__le,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ordered_ab_group_add @ A )
% 3.12/1.85 => ! [B4: A,A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ A2 @ B4 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_228_of__nat__le__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linord1659791738miring @ A )
% 3.12/1.85 => ! [M: nat,N: nat] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 3.12/1.85 = ( ord_less_eq @ nat @ M @ N ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_229_ivl__subset,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linorder @ A )
% 3.12/1.85 => ! [I: A,J: A,M: A,N: A] :
% 3.12/1.85 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ I @ J ) @ ( set_or1433837966ssThan @ A @ M @ N ) )
% 3.12/1.85 = ( ( ord_less_eq @ A @ J @ I )
% 3.12/1.85 | ( ( ord_less_eq @ A @ M @ I )
% 3.12/1.85 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_230_lessThan__subset__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linorder @ A )
% 3.12/1.85 => ! [X2: A,Y: A] :
% 3.12/1.85 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y ) )
% 3.12/1.85 = ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_231_neg__0__le__iff__le,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ordered_ab_group_add @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_232_neg__le__0__iff__le,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ordered_ab_group_add @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.12/1.85 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_233_less__eq__neg__nonpos,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linord219039673up_add @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_234_neg__less__eq__nonneg,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linord219039673up_add @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 3.12/1.85 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_235_inverse__nonpositive__iff__nonpositive,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 3.12/1.85 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_236_inverse__nonnegative__iff__nonnegative,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_237_atLeastLessThan__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ord @ A )
% 3.12/1.85 => ! [I: A,L: A,U: A] :
% 3.12/1.85 ( ( member @ A @ I @ ( set_or1433837966ssThan @ A @ L @ U ) )
% 3.12/1.85 = ( ( ord_less_eq @ A @ L @ I )
% 3.12/1.85 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_238_of__int__eq__0__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_char_0 @ A )
% 3.12/1.85 => ! [Z2: int] :
% 3.12/1.85 ( ( ( ring_1_of_int @ A @ Z2 )
% 3.12/1.85 = ( zero_zero @ A ) )
% 3.12/1.85 = ( Z2
% 3.12/1.85 = ( zero_zero @ int ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_239_of__int__0__eq__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_char_0 @ A )
% 3.12/1.85 => ! [Z2: int] :
% 3.12/1.85 ( ( ( zero_zero @ A )
% 3.12/1.85 = ( ring_1_of_int @ A @ Z2 ) )
% 3.12/1.85 = ( Z2
% 3.12/1.85 = ( zero_zero @ int ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_240_of__int__0,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_1 @ A )
% 3.12/1.85 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
% 3.12/1.85 = ( zero_zero @ A ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_241_of__int__le__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_idom @ A )
% 3.12/1.85 => ! [W: int,Z2: int] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
% 3.12/1.85 = ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_242_of__int__less__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_idom @ A )
% 3.12/1.85 => ! [W: int,Z2: int] :
% 3.12/1.85 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
% 3.12/1.85 = ( ord_less @ int @ W @ Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_243_of__int__minus,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_1 @ A )
% 3.12/1.85 => ! [Z2: int] :
% 3.12/1.85 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
% 3.12/1.85 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_244_of__int__of__nat__eq,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( ring_1 @ A )
% 3.12/1.85 => ! [N: nat] :
% 3.12/1.85 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
% 3.12/1.85 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_245_negative__zle,axiom,
% 3.12/1.85 ! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_246_of__nat__le__0__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linord1659791738miring @ A )
% 3.12/1.85 => ! [M: nat] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
% 3.12/1.85 = ( M
% 3.12/1.85 = ( zero_zero @ nat ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_247_inverse__le__iff__le,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 3.12/1.85 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
% 3.12/1.85 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ B4 @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_248_inverse__le__iff__le__neg,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_field @ A )
% 3.12/1.85 => ! [A2: A,B4: A] :
% 3.12/1.85 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
% 3.12/1.85 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B4 ) )
% 3.12/1.85 = ( ord_less_eq @ A @ B4 @ A2 ) ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_249_nat__le__0,axiom,
% 3.12/1.85 ! [Z2: int] :
% 3.12/1.85 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
% 3.12/1.85 => ( ( nat2 @ Z2 )
% 3.12/1.85 = ( zero_zero @ nat ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_250_nat__0__iff,axiom,
% 3.12/1.85 ! [I: int] :
% 3.12/1.85 ( ( ( nat2 @ I )
% 3.12/1.85 = ( zero_zero @ nat ) )
% 3.12/1.85 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_251_int__nat__eq,axiom,
% 3.12/1.85 ! [Z2: int] :
% 3.12/1.85 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 3.12/1.85 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
% 3.12/1.85 = Z2 ) )
% 3.12/1.85 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 3.12/1.85 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
% 3.12/1.85 = ( zero_zero @ int ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_252_of__int__le__0__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_idom @ A )
% 3.12/1.85 => ! [Z2: int] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
% 3.12/1.85 = ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(fact_253_of__int__0__le__iff,axiom,
% 3.12/1.85 ! [A: $tType] :
% 3.12/1.85 ( ( linordered_idom @ A )
% 3.12/1.85 => ! [Z2: int] :
% 3.12/1.85 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 3.12/1.85 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Multiset_Omultiset___Orderings_Opreorder,axiom,
% 3.12/1.85 ! [A9: $tType] :
% 3.12/1.85 ( ( preorder @ A9 )
% 3.12/1.85 => ( preorder @ ( multiset @ A9 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Orderings_Opreorder_1,axiom,
% 3.12/1.85 preorder @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
% 3.12/1.85 preorder @ $o ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
% 3.12/1.85 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Rings_Odioid,axiom,
% 3.12/1.85 dioid @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
% 3.12/1.85 preorder @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom,
% 3.12/1.85 preorder @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Rings_Odioid_6,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( dioid @ A10 )
% 3.12/1.85 => ( dioid @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Orderings_Opreorder_7,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( preorder @ A10 )
% 3.12/1.85 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( dioid @ A10 )
% 3.12/1.85 => ( canoni770627133id_add @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Groups_Oordered__ab__group__add,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( ordered_ab_group_add @ A10 )
% 3.12/1.85 => ( ordered_ab_group_add @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( boolean_algebra @ A10 )
% 3.12/1.85 => ( boolean_algebra @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Nat_Osemiring__char__0,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( semiring_char_0 @ A10 )
% 3.12/1.85 => ( semiring_char_0 @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Rings_Osemiring__1,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( semiring_1 @ A10 )
% 3.12/1.85 => ( semiring_1 @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Groups_Ogroup__add,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( group_add @ A10 )
% 3.12/1.85 => ( group_add @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( order @ A10 )
% 3.12/1.85 => ( order @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Nat_Oring__char__0,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( ring_char_0 @ A10 )
% 3.12/1.85 => ( ring_char_0 @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Orderings_Oord,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( ord @ A10 )
% 3.12/1.85 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Rings_Oring__1,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( ring_1 @ A10 )
% 3.12/1.85 => ( ring_1 @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_fun___Groups_Ozero,axiom,
% 3.12/1.85 ! [A9: $tType,A10: $tType] :
% 3.12/1.85 ( ( zero @ A10 )
% 3.12/1.85 => ( zero @ ( A9 > A10 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 3.12/1.85 linord1659791738miring @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 3.12/1.85 linord219039673up_add @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add_8,axiom,
% 3.12/1.85 ordered_ab_group_add @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 3.12/1.85 linordered_idom @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Nat_Osemiring__char__0_9,axiom,
% 3.12/1.85 semiring_char_0 @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 3.12/1.85 linorder @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Rings_Osemiring__1_10,axiom,
% 3.12/1.85 semiring_1 @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Groups_Ogroup__add_11,axiom,
% 3.12/1.85 group_add @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Orderings_Oorder_12,axiom,
% 3.12/1.85 order @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Nat_Oring__char__0_13,axiom,
% 3.12/1.85 ring_char_0 @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Orderings_Oord_14,axiom,
% 3.12/1.85 ord @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Rings_Oring__1_15,axiom,
% 3.12/1.85 ring_1 @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Int_Oint___Groups_Ozero_16,axiom,
% 3.12/1.85 zero @ int ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_17,axiom,
% 3.12/1.85 canoni770627133id_add @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_18,axiom,
% 3.12/1.85 linord1659791738miring @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.12/1.85 wellorder @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_19,axiom,
% 3.12/1.85 semiring_char_0 @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Orderings_Olinorder_20,axiom,
% 3.12/1.85 linorder @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Rings_Osemiring__1_21,axiom,
% 3.12/1.85 semiring_1 @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Orderings_Oorder_22,axiom,
% 3.12/1.85 order @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Orderings_Oord_23,axiom,
% 3.12/1.85 ord @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Groups_Ozero_24,axiom,
% 3.12/1.85 zero @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 3.12/1.85 size @ nat ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_25,axiom,
% 3.12/1.85 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Set_Oset___Orderings_Oorder_26,axiom,
% 3.12/1.85 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Set_Oset___Orderings_Oord_27,axiom,
% 3.12/1.85 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_28,axiom,
% 3.12/1.85 boolean_algebra @ $o ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_HOL_Obool___Orderings_Olinorder_29,axiom,
% 3.12/1.85 linorder @ $o ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_HOL_Obool___Orderings_Oorder_30,axiom,
% 3.12/1.85 order @ $o ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_HOL_Obool___Orderings_Oord_31,axiom,
% 3.12/1.85 ord @ $o ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_List_Olist___Nat_Osize_32,axiom,
% 3.12/1.85 ! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
% 3.12/1.85 archim1804426504_field @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_33,axiom,
% 3.12/1.85 linord1659791738miring @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_34,axiom,
% 3.12/1.85 linord219039673up_add @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_35,axiom,
% 3.12/1.85 ordered_ab_group_add @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
% 3.12/1.85 linordered_field @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Rings_Olinordered__idom_36,axiom,
% 3.12/1.85 linordered_idom @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
% 3.12/1.85 division_ring @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_37,axiom,
% 3.12/1.85 semiring_char_0 @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Orderings_Olinorder_38,axiom,
% 3.12/1.85 linorder @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Rings_Osemiring__1_39,axiom,
% 3.12/1.85 semiring_1 @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Groups_Ogroup__add_40,axiom,
% 3.12/1.85 group_add @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Orderings_Oorder_41,axiom,
% 3.12/1.85 order @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Nat_Oring__char__0_42,axiom,
% 3.12/1.85 ring_char_0 @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Orderings_Oord_43,axiom,
% 3.12/1.85 ord @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Rings_Oring__1_44,axiom,
% 3.12/1.85 ring_1 @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Fields_Ofield,axiom,
% 3.12/1.85 field @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Real_Oreal___Groups_Ozero_45,axiom,
% 3.12/1.85 zero @ real ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Multiset_Omultiset___Orderings_Oorder_46,axiom,
% 3.12/1.85 ! [A9: $tType] :
% 3.12/1.85 ( ( preorder @ A9 )
% 3.12/1.85 => ( order @ ( multiset @ A9 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Multiset_Omultiset___Orderings_Oord_47,axiom,
% 3.12/1.85 ! [A9: $tType] :
% 3.12/1.85 ( ( preorder @ A9 )
% 3.12/1.85 => ( ord @ ( multiset @ A9 ) ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Multiset_Omultiset___Groups_Ozero_48,axiom,
% 3.12/1.85 ! [A9: $tType] : ( zero @ ( multiset @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(tcon_Multiset_Omultiset___Nat_Osize_49,axiom,
% 3.12/1.85 ! [A9: $tType] : ( size @ ( multiset @ A9 ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(help_If_3_1_T,axiom,
% 3.12/1.85 ! [P: $o] :
% 3.12/1.85 ( ( P = $true )
% 3.12/1.85 | ( P = $false ) ) ).
% 3.12/1.85
% 3.12/1.85 thf(help_If_2_1_T,axiom,
% 3.12/1.85 ! [A: $tType,X2: A,Y: A] :
% 3.12/1.85 ( ( if @ A @ $false @ X2 @ Y )
% 3.12/1.85 = Y ) ).
% 3.12/1.85
% 3.12/1.85 thf(help_If_1_1_T,axiom,
% 3.12/1.85 ! [A: $tType,X2: A,Y: A] :
% 3.12/1.85 ( ( if @ A @ $true @ X2 @ Y )
% 3.12/1.85 = X2 ) ).
% 3.12/1.85
% 3.12/1.85 thf(conj_0,conjecture,
% 3.12/1.85 ( ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( image_mset @ nat @ nat @ f @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ xs ) ) ) ) )
% 3.12/1.85 = ( image_mset @ nat @ a @ ( nth @ a @ ys ) @ ( mset_set @ nat @ ( set_or1433837966ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ a ) @ ys ) ) ) ) ) ).
% 3.12/1.85 ------------------------
% 3.12/1.87 % (544)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_409 for (2999ds/27Mi)
% 3.12/1.87 % (547)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_409 for (2999ds/275Mi)
% 3.12/1.87 % (546)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_409 for (2999ds/2Mi)
% 3.12/1.87 % (546)Instruction limit reached!
% 3.12/1.87 % (546)------------------------------
% 3.12/1.87 % (546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.87 % (546)Termination reason: Unknown
% 3.12/1.87 % (546)Termination phase: shuffling
% 3.12/1.87
% 3.12/1.87 % (546)Memory used [KB]: 1407
% 3.12/1.87 % (546)Time elapsed: 0.002 s
% 3.12/1.87 % (546)Instructions burned: 3 (million)
% 3.12/1.87 % (546)------------------------------
% 3.12/1.87 % (546)------------------------------
% 3.12/1.87 % (548)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/18Mi)
% 3.12/1.87 % (544)Instruction limit reached!
% 3.12/1.87 % (544)------------------------------
% 3.12/1.87 % (544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.87 % (544)Termination reason: Unknown
% 3.12/1.87 % (544)Termination phase: Property scanning
% 3.12/1.87
% 3.12/1.87 % (544)Memory used [KB]: 1791
% 3.12/1.87 % (544)Time elapsed: 0.010 s
% 3.12/1.87 % (544)Instructions burned: 28 (million)
% 3.12/1.87 % (544)------------------------------
% 3.12/1.87 % (544)------------------------------
% 3.12/1.88 % (548)Instruction limit reached!
% 3.12/1.88 % (548)------------------------------
% 3.12/1.88 % (548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.88 % (548)Termination reason: Unknown
% 3.12/1.88 % (548)Termination phase: Property scanning
% 3.12/1.88
% 3.12/1.88 % (548)Memory used [KB]: 1791
% 3.12/1.88 % (548)Time elapsed: 0.008 s
% 3.12/1.88 % (548)Instructions burned: 20 (million)
% 3.12/1.88 % (548)------------------------------
% 3.12/1.88 % (548)------------------------------
% 3.12/1.88 % (542)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/183Mi)
% 3.12/1.88 % (545)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/2Mi)
% 3.12/1.88 % (545)Instruction limit reached!
% 3.12/1.88 % (545)------------------------------
% 3.12/1.88 % (545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.88 % (545)Termination reason: Unknown
% 3.12/1.88 % (545)Termination phase: shuffling
% 3.12/1.88
% 3.12/1.88 % (545)Memory used [KB]: 1407
% 3.12/1.88 % (545)Time elapsed: 0.002 s
% 3.12/1.88 % (545)Instructions burned: 3 (million)
% 3.12/1.88 % (545)------------------------------
% 3.12/1.88 % (545)------------------------------
% 3.12/1.88 % (549)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/3Mi)
% 3.12/1.88 % (543)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_409 for (2999ds/4Mi)
% 3.12/1.88 % (549)Instruction limit reached!
% 3.12/1.88 % (549)------------------------------
% 3.12/1.88 % (549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.88 % (549)Termination reason: Unknown
% 3.12/1.88 % (549)Termination phase: shuffling
% 3.12/1.88
% 3.12/1.88 % (549)Memory used [KB]: 1407
% 3.12/1.88 % (549)Time elapsed: 0.002 s
% 3.12/1.88 % (549)Instructions burned: 3 (million)
% 3.12/1.88 % (549)------------------------------
% 3.12/1.88 % (549)------------------------------
% 3.12/1.88 % (543)Instruction limit reached!
% 3.12/1.88 % (543)------------------------------
% 3.12/1.88 % (543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.88 % (543)Termination reason: Unknown
% 3.12/1.88 % (543)Termination phase: shuffling
% 3.12/1.88
% 3.12/1.88 % (543)Memory used [KB]: 1407
% 3.12/1.88 % (543)Time elapsed: 0.004 s
% 3.12/1.88 % (543)Instructions burned: 5 (million)
% 3.12/1.88 % (543)------------------------------
% 3.12/1.88 % (543)------------------------------
% 3.12/1.88 % (550)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_409 for (2999ds/37Mi)
% 3.12/1.88 % (551)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_409 for (2999ds/15Mi)
% 3.12/1.89 % (552)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_409 for (2999ds/3Mi)
% 3.12/1.89 % (552)Instruction limit reached!
% 3.12/1.89 % (552)------------------------------
% 3.12/1.89 % (552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.89 % (552)Termination reason: Unknown
% 3.12/1.89 % (552)Termination phase: shuffling
% 3.12/1.89
% 3.12/1.89 % (552)Memory used [KB]: 1407
% 3.12/1.89 % (552)Time elapsed: 0.002 s
% 3.12/1.89 % (552)Instructions burned: 3 (million)
% 3.12/1.89 % (552)------------------------------
% 3.12/1.89 % (552)------------------------------
% 3.12/1.89 % (553)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_409 for (2999ds/1041Mi)
% 3.12/1.89 % (551)Instruction limit reached!
% 3.12/1.89 % (551)------------------------------
% 3.12/1.89 % (551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.89 % (551)Termination reason: Unknown
% 3.12/1.89 % (551)Termination phase: shuffling
% 3.12/1.89
% 3.12/1.89 % (551)Memory used [KB]: 1663
% 3.12/1.89 % (551)Time elapsed: 0.006 s
% 3.12/1.89 % (551)Instructions burned: 15 (million)
% 3.12/1.89 % (551)------------------------------
% 3.12/1.89 % (551)------------------------------
% 3.12/1.90 % (550)Instruction limit reached!
% 3.12/1.90 % (550)------------------------------
% 3.12/1.90 % (550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.12/1.90 % (550)Termination reason: Unknown
% 3.12/1.90 % (550)Termination phase: Property scanning
% 3.12/1.90
% 3.12/1.90 % (550)Memory used [KB]: 2174
% 3.12/1.90 % (550)Time elapsed: 0.012 s
% 3.12/1.90 % (550)Instructions burned: 40 (million)
% 3.12/1.90 % (550)------------------------------
% 3.12/1.90 % (550)------------------------------
% 3.17/1.90 % (558)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/16Mi)
% 3.17/1.90 % (559)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_409 for (2999ds/3Mi)
% 3.17/1.90 % (554)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/7Mi)
% 3.17/1.90 % (559)Instruction limit reached!
% 3.17/1.90 % (559)------------------------------
% 3.17/1.90 % (559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.90 % (559)Termination reason: Unknown
% 3.17/1.90 % (559)Termination phase: shuffling
% 3.17/1.90
% 3.17/1.90 % (559)Memory used [KB]: 1407
% 3.17/1.90 % (559)Time elapsed: 0.002 s
% 3.17/1.90 % (559)Instructions burned: 3 (million)
% 3.17/1.90 % (559)------------------------------
% 3.17/1.90 % (559)------------------------------
% 3.17/1.90 % (558)Instruction limit reached!
% 3.17/1.90 % (558)------------------------------
% 3.17/1.90 % (558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.90 % (558)Termination reason: Unknown
% 3.17/1.90 % (558)Termination phase: Property scanning
% 3.17/1.90
% 3.17/1.90 % (558)Memory used [KB]: 1791
% 3.17/1.90 % (558)Time elapsed: 0.007 s
% 3.17/1.90 % (558)Instructions burned: 18 (million)
% 3.17/1.90 % (558)------------------------------
% 3.17/1.90 % (558)------------------------------
% 3.17/1.90 % (554)Instruction limit reached!
% 3.17/1.90 % (554)------------------------------
% 3.17/1.90 % (554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.90 % (554)Termination reason: Unknown
% 3.17/1.90 % (554)Termination phase: shuffling
% 3.17/1.90
% 3.17/1.90 % (554)Memory used [KB]: 1535
% 3.17/1.90 % (554)Time elapsed: 0.006 s
% 3.17/1.90 % (554)Instructions burned: 8 (million)
% 3.17/1.90 % (554)------------------------------
% 3.17/1.90 % (554)------------------------------
% 3.17/1.90 % (560)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/3Mi)
% 3.17/1.91 % (560)Instruction limit reached!
% 3.17/1.91 % (560)------------------------------
% 3.17/1.91 % (560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.91 % (560)Termination reason: Unknown
% 3.17/1.91 % (560)Termination phase: shuffling
% 3.17/1.91
% 3.17/1.91 % (560)Memory used [KB]: 1407
% 3.17/1.91 % (560)Time elapsed: 0.003 s
% 3.17/1.91 % (560)Instructions burned: 5 (million)
% 3.17/1.91 % (560)------------------------------
% 3.17/1.91 % (560)------------------------------
% 3.17/1.91 % (561)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/7Mi)
% 3.17/1.91 % (561)Instruction limit reached!
% 3.17/1.91 % (561)------------------------------
% 3.17/1.91 % (561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.91 % (561)Termination reason: Unknown
% 3.17/1.91 % (561)Termination phase: shuffling
% 3.17/1.91
% 3.17/1.91 % (561)Memory used [KB]: 1535
% 3.17/1.91 % (561)Time elapsed: 0.004 s
% 3.17/1.91 % (561)Instructions burned: 9 (million)
% 3.17/1.91 % (561)------------------------------
% 3.17/1.91 % (561)------------------------------
% 3.17/1.91 % (562)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/3Mi)
% 3.17/1.91 % (562)Instruction limit reached!
% 3.17/1.91 % (562)------------------------------
% 3.17/1.91 % (562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.91 % (562)Termination reason: Unknown
% 3.17/1.91 % (562)Termination phase: shuffling
% 3.17/1.91
% 3.17/1.91 % (562)Memory used [KB]: 1407
% 3.17/1.91 % (562)Time elapsed: 0.002 s
% 3.17/1.91 % (562)Instructions burned: 4 (million)
% 3.17/1.91 % (562)------------------------------
% 3.17/1.91 % (562)------------------------------
% 3.17/1.92 % (564)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/18Mi)
% 3.17/1.92 % (563)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/4Mi)
% 3.17/1.92 % (565)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/710Mi)
% 3.17/1.92 % (564)Instruction limit reached!
% 3.17/1.92 % (564)------------------------------
% 3.17/1.92 % (564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.92 % (564)Termination reason: Unknown
% 3.17/1.92 % (564)Termination phase: Property scanning
% 3.17/1.92
% 3.17/1.92 % (564)Memory used [KB]: 1791
% 3.17/1.92 % (564)Time elapsed: 0.007 s
% 3.17/1.92 % (564)Instructions burned: 18 (million)
% 3.17/1.92 % (564)------------------------------
% 3.17/1.92 % (564)------------------------------
% 3.17/1.92 % (566)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/6Mi)
% 3.17/1.92 % (563)Instruction limit reached!
% 3.17/1.92 % (563)------------------------------
% 3.17/1.92 % (563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.92 % (563)Termination reason: Unknown
% 3.17/1.92 % (563)Termination phase: shuffling
% 3.17/1.92
% 3.17/1.92 % (563)Memory used [KB]: 1407
% 3.17/1.92 % (563)Time elapsed: 0.004 s
% 3.17/1.92 % (563)Instructions burned: 5 (million)
% 3.17/1.92 % (563)------------------------------
% 3.17/1.92 % (563)------------------------------
% 3.17/1.92 % (566)Instruction limit reached!
% 3.17/1.92 % (566)------------------------------
% 3.17/1.92 % (566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.92 % (566)Termination reason: Unknown
% 3.17/1.92 % (566)Termination phase: shuffling
% 3.17/1.92
% 3.17/1.92 % (566)Memory used [KB]: 1535
% 3.17/1.92 % (566)Time elapsed: 0.003 s
% 3.17/1.92 % (566)Instructions burned: 6 (million)
% 3.17/1.92 % (566)------------------------------
% 3.17/1.92 % (566)------------------------------
% 3.17/1.93 % (567)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/902Mi)
% 3.17/1.94 % (569)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/5Mi)
% 3.17/1.94 % (569)Instruction limit reached!
% 3.17/1.94 % (569)------------------------------
% 3.17/1.94 % (569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.94 % (569)Termination reason: Unknown
% 3.17/1.94 % (569)Termination phase: shuffling
% 3.17/1.94
% 3.17/1.94 % (569)Memory used [KB]: 1407
% 3.17/1.94 % (569)Time elapsed: 0.005 s
% 3.17/1.94 % (569)Instructions burned: 6 (million)
% 3.17/1.94 % (569)------------------------------
% 3.17/1.94 % (569)------------------------------
% 3.17/1.94 % (568)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/21Mi)
% 3.17/1.95 % (570)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/6Mi)
% 3.17/1.95 % (570)Instruction limit reached!
% 3.17/1.95 % (570)------------------------------
% 3.17/1.95 % (570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.95 % (570)Termination reason: Unknown
% 3.17/1.95 % (570)Termination phase: shuffling
% 3.17/1.95
% 3.17/1.95 % (570)Memory used [KB]: 1407
% 3.17/1.95 % (570)Time elapsed: 0.003 s
% 3.17/1.95 % (570)Instructions burned: 6 (million)
% 3.17/1.95 % (570)------------------------------
% 3.17/1.95 % (570)------------------------------
% 3.17/1.95 % (568)Instruction limit reached!
% 3.17/1.95 % (568)------------------------------
% 3.17/1.95 % (568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.95 % (568)Termination reason: Unknown
% 3.17/1.95 % (568)Termination phase: shuffling
% 3.17/1.95
% 3.17/1.95 % (568)Memory used [KB]: 1918
% 3.17/1.95 % (568)Time elapsed: 0.014 s
% 3.17/1.95 % (568)Instructions burned: 22 (million)
% 3.17/1.95 % (568)------------------------------
% 3.17/1.95 % (568)------------------------------
% 3.17/1.96 % (571)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/377Mi)
% 3.17/1.96 % (567)Refutation not found, incomplete strategy
% 3.17/1.96 % (567)------------------------------
% 3.17/1.96 % (567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.96 % (567)Termination reason: Refutation not found, incomplete strategy
% 3.17/1.96
% 3.17/1.96
% 3.17/1.96 % (567)Memory used [KB]: 7036
% 3.17/1.96 % (567)Time elapsed: 0.032 s
% 3.17/1.96 % (567)Instructions burned: 97 (million)
% 3.17/1.96 % (567)------------------------------
% 3.17/1.96 % (567)------------------------------
% 3.17/1.97 % (572)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_409 for (2999ds/779Mi)
% 3.17/1.97 % (573)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/19Mi)
% 3.17/1.98 % (542)Instruction limit reached!
% 3.17/1.98 % (542)------------------------------
% 3.17/1.98 % (542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.98 % (542)Termination reason: Unknown
% 3.17/1.98 % (542)Termination phase: Saturation
% 3.17/1.98
% 3.17/1.98 % (542)Memory used [KB]: 7164
% 3.17/1.98 % (542)Time elapsed: 0.102 s
% 3.17/1.98 % (542)Instructions burned: 183 (million)
% 3.17/1.98 % (542)------------------------------
% 3.17/1.98 % (542)------------------------------
% 3.17/1.98 % (573)Instruction limit reached!
% 3.17/1.98 % (573)------------------------------
% 3.17/1.98 % (573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.98 % (573)Termination reason: Unknown
% 3.17/1.98 % (573)Termination phase: shuffling
% 3.17/1.98
% 3.17/1.98 % (573)Memory used [KB]: 1791
% 3.17/1.98 % (573)Time elapsed: 0.007 s
% 3.17/1.98 % (573)Instructions burned: 20 (million)
% 3.17/1.98 % (573)------------------------------
% 3.17/1.98 % (573)------------------------------
% 3.17/1.99 % (575)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/17Mi)
% 3.17/1.99 % (575)Instruction limit reached!
% 3.17/1.99 % (575)------------------------------
% 3.17/1.99 % (575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.99 % (575)Termination reason: Unknown
% 3.17/1.99 % (575)Termination phase: shuffling
% 3.17/1.99
% 3.17/1.99 % (575)Memory used [KB]: 1791
% 3.17/1.99 % (575)Time elapsed: 0.007 s
% 3.17/1.99 % (575)Instructions burned: 17 (million)
% 3.17/1.99 % (575)------------------------------
% 3.17/1.99 % (575)------------------------------
% 3.17/2.00 % (574)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/879Mi)
% 4.12/2.02 % (576)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/3Mi)
% 4.12/2.02 % (576)Instruction limit reached!
% 4.12/2.02 % (576)------------------------------
% 4.12/2.02 % (576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.02 % (576)Termination reason: Unknown
% 4.12/2.02 % (576)Termination phase: shuffling
% 4.12/2.02
% 4.12/2.02 % (576)Memory used [KB]: 1407
% 4.12/2.02 % (576)Time elapsed: 0.003 s
% 4.12/2.02 % (576)Instructions burned: 4 (million)
% 4.12/2.02 % (576)------------------------------
% 4.12/2.02 % (576)------------------------------
% 4.12/2.03 % (547)Instruction limit reached!
% 4.12/2.03 % (547)------------------------------
% 4.12/2.03 % (547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.03 % (547)Termination reason: Unknown
% 4.12/2.03 % (547)Termination phase: Saturation
% 4.12/2.03
% 4.12/2.03 % (547)Memory used [KB]: 7291
% 4.12/2.03 % (547)Time elapsed: 0.155 s
% 4.12/2.03 % (547)Instructions burned: 275 (million)
% 4.12/2.03 % (547)------------------------------
% 4.12/2.03 % (547)------------------------------
% 4.12/2.03 % (577)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/30Mi)
% 4.12/2.04 % (578)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/127Mi)
% 4.12/2.04 % (577)Instruction limit reached!
% 4.12/2.04 % (577)------------------------------
% 4.12/2.04 % (577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.04 % (577)Termination reason: Unknown
% 4.12/2.04 % (577)Termination phase: Property scanning
% 4.12/2.04
% 4.12/2.04 % (577)Memory used [KB]: 1918
% 4.12/2.04 % (577)Time elapsed: 0.011 s
% 4.12/2.04 % (577)Instructions burned: 30 (million)
% 4.12/2.04 % (577)------------------------------
% 4.12/2.04 % (577)------------------------------
% 4.12/2.05 % (579)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_409 for (2998ds/100Mi)
% 4.12/2.07 % (571)Instruction limit reached!
% 4.12/2.07 % (571)------------------------------
% 4.12/2.07 % (571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.07 % (571)Termination reason: Unknown
% 4.12/2.07 % (571)Termination phase: Saturation
% 4.12/2.07
% 4.12/2.07 % (571)Memory used [KB]: 7931
% 4.12/2.07 % (571)Time elapsed: 0.106 s
% 4.12/2.07 % (571)Instructions burned: 377 (million)
% 4.12/2.07 % (571)------------------------------
% 4.12/2.07 % (571)------------------------------
% 4.12/2.07 % (578)Instruction limit reached!
% 4.12/2.07 % (578)------------------------------
% 4.12/2.07 % (578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.07 % (578)Termination reason: Unknown
% 4.12/2.07 % (578)Termination phase: Saturation
% 4.12/2.07
% 4.12/2.07 % (578)Memory used [KB]: 6908
% 4.12/2.07 % (578)Time elapsed: 0.039 s
% 4.12/2.07 % (578)Instructions burned: 130 (million)
% 4.12/2.07 % (578)------------------------------
% 4.12/2.07 % (578)------------------------------
% 4.12/2.08 % (580)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/3Mi)
% 4.12/2.08 % (580)Instruction limit reached!
% 4.12/2.08 % (580)------------------------------
% 4.12/2.08 % (580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.08 % (580)Termination reason: Unknown
% 4.12/2.08 % (580)Termination phase: shuffling
% 4.12/2.08
% 4.12/2.08 % (580)Memory used [KB]: 1407
% 4.12/2.08 % (580)Time elapsed: 0.003 s
% 4.12/2.08 % (580)Instructions burned: 5 (million)
% 4.12/2.08 % (580)------------------------------
% 4.12/2.08 % (580)------------------------------
% 4.12/2.08 % (579)Instruction limit reached!
% 4.12/2.08 % (579)------------------------------
% 4.12/2.08 % (579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.08 % (579)Termination reason: Unknown
% 4.12/2.08 % (579)Termination phase: Saturation
% 4.12/2.08
% 4.12/2.08 % (579)Memory used [KB]: 6780
% 4.12/2.08 % (579)Time elapsed: 0.029 s
% 4.12/2.08 % (579)Instructions burned: 101 (million)
% 4.12/2.08 % (579)------------------------------
% 4.12/2.08 % (579)------------------------------
% 4.12/2.08 % (581)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/20Mi)
% 4.12/2.09 % (582)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/86Mi)
% 4.12/2.09 % (583)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/107Mi)
% 4.12/2.09 % (581)Instruction limit reached!
% 4.12/2.09 % (581)------------------------------
% 4.12/2.09 % (581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.09 % (581)Termination reason: Unknown
% 4.12/2.09 % (581)Termination phase: Property scanning
% 4.12/2.09
% 4.12/2.09 % (581)Memory used [KB]: 1791
% 4.12/2.09 % (581)Time elapsed: 0.008 s
% 4.12/2.09 % (581)Instructions burned: 21 (million)
% 4.12/2.09 % (581)------------------------------
% 4.12/2.09 % (581)------------------------------
% 4.12/2.10 % (584)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/39Mi)
% 4.12/2.11 % (584)Instruction limit reached!
% 4.12/2.11 % (584)------------------------------
% 4.12/2.11 % (584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.11 % (584)Termination reason: Unknown
% 4.12/2.11 % (584)Termination phase: Property scanning
% 4.12/2.11
% 4.12/2.11 % (584)Memory used [KB]: 2174
% 4.12/2.11 % (584)Time elapsed: 0.013 s
% 4.12/2.11 % (584)Instructions burned: 42 (million)
% 4.12/2.11 % (584)------------------------------
% 4.12/2.11 % (584)------------------------------
% 4.12/2.11 % (582)Instruction limit reached!
% 4.12/2.11 % (582)------------------------------
% 4.12/2.11 % (582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.12/2.11 % (582)Termination reason: Unknown
% 4.12/2.11 % (582)Termination phase: Saturation
% 4.12/2.11
% 4.12/2.11 % (582)Memory used [KB]: 6652
% 4.12/2.11 % (582)Time elapsed: 0.027 s
% 4.12/2.11 % (582)Instructions burned: 88 (million)
% 4.12/2.11 % (582)------------------------------
% 4.12/2.11 % (582)------------------------------
% 4.87/2.12 % (583)Instruction limit reached!
% 4.87/2.12 % (583)------------------------------
% 4.87/2.12 % (583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.87/2.12 % (583)Termination reason: Unknown
% 4.87/2.12 % (583)Termination phase: Saturation
% 4.87/2.12
% 4.87/2.12 % (583)Memory used [KB]: 6908
% 4.87/2.12 % (583)Time elapsed: 0.033 s
% 4.87/2.12 % (583)Instructions burned: 109 (million)
% 4.87/2.12 % (583)------------------------------
% 4.87/2.12 % (583)------------------------------
% 4.87/2.12 % (585)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/448Mi)
% 4.87/2.12 % (586)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/46Mi)
% 4.87/2.13 % (587)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_409 for (2997ds/98Mi)
% 4.87/2.14 % (586)Instruction limit reached!
% 4.87/2.14 % (586)------------------------------
% 4.87/2.14 % (586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.87/2.14 % (586)Termination reason: Unknown
% 4.87/2.14 % (586)Termination phase: Property scanning
% 4.87/2.14
% 4.87/2.14 % (586)Memory used [KB]: 2174
% 4.87/2.14 % (586)Time elapsed: 0.016 s
% 4.87/2.14 % (586)Instructions burned: 48 (million)
% 4.87/2.14 % (586)------------------------------
% 4.87/2.14 % (586)------------------------------
% 4.87/2.14 % (565)First to succeed.
% 4.87/2.14 % (565)Refutation found. Thanks to Tanya!
% 4.87/2.14 % SZS status Theorem for DTF2THF_409
% 4.87/2.14 % SZS output start Proof for DTF2THF_409
% 4.87/2.14 thf(type_def_6, type, set: ($tType) > $tType).
% 4.87/2.14 thf(type_def_7, type, int: $tType).
% 4.87/2.14 thf(type_def_8, type, nat: $tType).
% 4.87/2.14 thf(type_def_9, type, list: ($tType) > $tType).
% 4.87/2.14 thf(type_def_10, type, multiset: ($tType) > $tType).
% 4.87/2.14 thf(type_def_11, type, a: $tType).
% 4.87/2.14 thf(type_def_12, type, real: $tType).
% 4.87/2.14 thf(func_def_0, type, multiset: ($tType) > $tType).
% 4.87/2.14 thf(func_def_1, type, real: $tType).
% 4.87/2.14 thf(func_def_2, type, list: ($tType) > $tType).
% 4.87/2.14 thf(func_def_3, type, set: ($tType) > $tType).
% 4.87/2.14 thf(func_def_4, type, nat: $tType).
% 4.87/2.14 thf(func_def_5, type, int: $tType).
% 4.87/2.14 thf(func_def_6, type, a: $tType).
% 4.87/2.14 thf(func_def_7, type, preorder: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_8, type, dioid: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_9, type, type: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_10, type, size: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_11, type, zero: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_12, type, field: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_13, type, ring_1: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_14, type, ord: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_15, type, ring_char_0: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_16, type, order: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_17, type, group_add: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_18, type, semiring_1: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_19, type, linorder: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_20, type, semiring_char_0: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_21, type, wellorder: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_22, type, division_ring: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_23, type, linordered_idom: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_24, type, linordered_field: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_25, type, boolean_algebra: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_26, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_27, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_28, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_29, type, archim1804426504_field: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_30, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 4.87/2.14 thf(func_def_31, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 4.87/2.14 thf(func_def_32, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 4.87/2.14 thf(func_def_33, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 4.87/2.14 thf(func_def_34, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 4.87/2.14 thf(func_def_35, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 4.87/2.14 thf(func_def_36, type, zero_zero: !>[X0: $tType]:(X0)).
% 4.87/2.14 thf(func_def_37, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 4.87/2.14 thf(func_def_38, type, nat2: int > nat).
% 4.87/2.14 thf(func_def_39, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 4.87/2.14 thf(func_def_40, type, gen_length: !>[X0: $tType]:(nat > list @ X0 > nat)).
% 4.87/2.14 thf(func_def_41, type, map: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1)).
% 4.87/2.14 thf(func_def_42, type, list_ex: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 4.87/2.14 thf(func_def_43, type, map_tailrec: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1)).
% 4.87/2.14 thf(func_def_44, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 4.87/2.14 thf(func_def_45, type, upt: nat > nat > list @ nat).
% 4.87/2.14 thf(func_def_46, type, image_mset: !>[X0: $tType, X1: $tType]:((X0 > X1) > multiset @ X0 > multiset @ X1)).
% 4.87/2.14 thf(func_def_47, type, is_empty: !>[X0: $tType]:(multiset @ X0 > $o)).
% 4.87/2.14 thf(func_def_48, type, mset: !>[X0: $tType]:(list @ X0 > multiset @ X0)).
% 4.87/2.14 thf(func_def_49, type, mset_set: !>[X0: $tType]:(set @ X0 > multiset @ X0)).
% 4.87/2.14 thf(func_def_50, type, size_multiset: !>[X0: $tType]:((X0 > nat) > multiset @ X0 > nat)).
% 4.87/2.14 thf(func_def_51, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 4.87/2.14 thf(func_def_52, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 4.87/2.14 thf(func_def_53, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 4.87/2.14 thf(func_def_54, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 4.87/2.14 thf(func_def_55, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 4.87/2.14 thf(func_def_56, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 4.87/2.14 thf(func_def_57, type, set_or1433837966ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 4.87/2.14 thf(func_def_58, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 4.87/2.14 thf(func_def_59, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 4.87/2.14 thf(func_def_60, type, f: nat > nat).
% 4.87/2.14 thf(func_def_61, type, xs: list @ a).
% 4.87/2.14 thf(func_def_62, type, ys: list @ a).
% 4.87/2.14 thf(func_def_77, type, sK0: (nat > $o) > nat > nat).
% 4.87/2.14 thf(func_def_78, type, sK1: (nat > $o) > nat > nat).
% 4.87/2.14 thf(func_def_79, type, sK2: !>[X0: $tType]:(nat > (nat > X0 > $o) > X0)).
% 4.87/2.14 thf(func_def_80, type, sK3: !>[X0: $tType]:(list @ X0 > (nat > X0 > $o) > nat > nat)).
% 4.87/2.14 thf(func_def_81, type, sK4: !>[X0: $tType]:((nat > X0 > $o) > nat > list @ X0)).
% 4.87/2.14 thf(func_def_82, type, sK5: !>[X0: $tType]:((nat > X0 > $o) > nat > nat)).
% 4.87/2.14 thf(func_def_83, type, sK6: !>[X0: $tType]:((list @ X0 > $o) > list @ X0)).
% 4.87/2.14 thf(func_def_84, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 4.87/2.14 thf(func_def_85, type, sK8: !>[X0: $tType]:(list @ X0 > list @ X0 > nat)).
% 4.87/2.14 thf(func_def_86, type, sK9: !>[X0: $tType]:(nat > list @ X0)).
% 4.87/2.14 thf(func_def_87, type, sK10: (nat > $o) > nat > nat).
% 4.87/2.14 thf(func_def_88, type, sK11: (nat > $o) > nat > nat).
% 4.87/2.14 thf(func_def_91, type, ph13: !>[X0: $tType]:(X0)).
% 4.87/2.14 thf(f1999,plain,(
% 4.87/2.14 $false),
% 4.87/2.14 inference(subsumption_resolution,[status(thm)],[f1998,f997])).
% 4.87/2.14 thf(f997,plain,(
% 4.87/2.14 ((mset @ a @ xs) != (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(backward_demodulation,[status(thm)],[f995,f988])).
% 4.87/2.14 thf(f988,plain,(
% 4.87/2.14 ( ! [X0 : nat,X1 : nat] : (((mset @ nat @ (upt @ X0 @ X1)) = (mset_set @ nat @ (set_or1433837966ssThan @ nat @ X0 @ X1)))) )),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f897])).
% 4.87/2.14 thf(f897,plain,(
% 4.87/2.14 ! [X0 : nat,X1 : nat] : ((mset @ nat @ (upt @ X0 @ X1)) = (mset_set @ nat @ (set_or1433837966ssThan @ nat @ X0 @ X1)))),
% 4.87/2.14 inference(rectify,[status(thm)],[f78])).
% 4.87/2.14 thf(f78,axiom,(
% 4.87/2.14 ! [X17 : nat,X6 : nat] : ((mset @ nat @ (upt @ X17 @ X6)) = (mset_set @ nat @ (set_or1433837966ssThan @ nat @ X17 @ X6)))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',fact_77_mset__upt)).
% 4.87/2.14 thf(f995,plain,(
% 4.87/2.14 ((mset @ a @ xs) != (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(forward_demodulation,[status(thm)],[f979,f950])).
% 4.87/2.14 thf(f950,plain,(
% 4.87/2.14 ((mset @ a @ xs) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))))),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f2])).
% 4.87/2.14 thf(f2,axiom,(
% 4.87/2.14 ((mset @ a @ xs) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',fact_1_calculation)).
% 4.87/2.14 thf(f979,plain,(
% 4.87/2.14 ((image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))) != (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f891])).
% 4.87/2.14 thf(f891,plain,(
% 4.87/2.14 ((image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))) != (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(flattening,[status(thm)],[f331])).
% 4.87/2.14 thf(f331,negated_conjecture,(
% 4.87/2.14 ~((image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(negated_conjecture,[status(cth)],[f330])).
% 4.87/2.14 thf(f330,conjecture,(
% 4.87/2.14 ((image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))))) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',conj_0)).
% 4.87/2.14 thf(f1998,plain,(
% 4.87/2.14 ((mset @ a @ xs) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ ys)))))),
% 4.87/2.14 inference(backward_demodulation,[status(thm)],[f996,f1997])).
% 4.87/2.14 thf(f1997,plain,(
% 4.87/2.14 ((mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ ys))) = (image_mset @ nat @ nat @ f @ (mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ xs)))))),
% 4.87/2.14 inference(forward_demodulation,[status(thm)],[f1996,f988])).
% 4.87/2.14 thf(f1996,plain,(
% 4.87/2.14 ((image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))) = (mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ ys))))),
% 4.87/2.14 inference(forward_demodulation,[status(thm)],[f1995,f988])).
% 4.87/2.14 thf(f1995,plain,(
% 4.87/2.14 ((image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))) = (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys))))),
% 4.87/2.14 inference(forward_demodulation,[status(thm)],[f1988,f986])).
% 4.87/2.14 thf(f986,plain,(
% 4.87/2.14 ((image @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))) = (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f4])).
% 4.87/2.14 thf(f4,axiom,(
% 4.87/2.14 ((image @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))) = (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ ys)))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',fact_3__C2_C)).
% 4.87/2.14 thf(f1988,plain,(
% 4.87/2.14 ((image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))) = (mset_set @ nat @ (image @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))))),
% 4.87/2.14 inference(trivial_inequality_removal,[status(thm)],[f1981])).
% 4.87/2.14 thf(f1981,plain,(
% 4.87/2.14 ($true != $true) | ((image_mset @ nat @ nat @ f @ (mset_set @ nat @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))) = (mset_set @ nat @ (image @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))))),
% 4.87/2.14 inference(superposition,[status(thm)],[f942,f969])).
% 4.87/2.14 thf(f969,plain,(
% 4.87/2.14 ((inj_on @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))) = $true)),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f708])).
% 4.87/2.14 thf(f708,plain,(
% 4.87/2.14 ((inj_on @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs))) = $true)),
% 4.87/2.14 inference(fool_elimination,[status(thm)],[f707])).
% 4.87/2.14 thf(f707,plain,(
% 4.87/2.14 (inj_on @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))),
% 4.87/2.14 inference(rectify,[status(thm)],[f1])).
% 4.87/2.14 thf(f1,axiom,(
% 4.87/2.14 (inj_on @ nat @ nat @ f @ (set_or1433837966ssThan @ nat @ zero_zero @ nat @ (size_size @ list @ a @ xs)))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',fact_0__C1_C)).
% 4.87/2.14 thf(f942,plain,(
% 4.87/2.14 ( ! [X1 : $tType,X0 : $tType,X2 : X1 > X0,X3 : set @ X1] : (((inj_on @ X1 @ X0 @ X2 @ X3) != $true) | ((image_mset @ X1 @ X0 @ X2 @ (mset_set @ X1 @ X3)) = (mset_set @ X0 @ (image @ X1 @ X0 @ X2 @ X3)))) )),
% 4.87/2.14 inference(cnf_transformation,[status(thm)],[f908])).
% 4.87/2.14 thf(f908,plain,(
% 4.87/2.14 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0,X3 : set @ X1] : (((inj_on @ X1 @ X0 @ X2 @ X3) != $true) | ((image_mset @ X1 @ X0 @ X2 @ (mset_set @ X1 @ X3)) = (mset_set @ X0 @ (image @ X1 @ X0 @ X2 @ X3))))),
% 4.87/2.14 inference(ennf_transformation,[status(thm)],[f628])).
% 4.87/2.14 thf(f628,plain,(
% 4.87/2.14 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0,X3 : set @ X1] : (((inj_on @ X1 @ X0 @ X2 @ X3) = $true) => ((image_mset @ X1 @ X0 @ X2 @ (mset_set @ X1 @ X3)) = (mset_set @ X0 @ (image @ X1 @ X0 @ X2 @ X3))))),
% 4.87/2.14 inference(fool_elimination,[status(thm)],[f627])).
% 4.87/2.14 thf(f627,plain,(
% 4.87/2.14 ! [X0 : $tType,X1 : $tType,X2 : X1 > X0,X3 : set @ X1] : ((inj_on @ X1 @ X0 @ X2 @ X3) => ((image_mset @ X1 @ X0 @ X2 @ (mset_set @ X1 @ X3)) = (mset_set @ X0 @ (image @ X1 @ X0 @ X2 @ X3))))),
% 4.87/2.14 inference(rectify,[status(thm)],[f63])).
% 4.87/2.14 thf(f63,axiom,(
% 4.87/2.14 ! [X1 : $tType,X0 : $tType,X13 : X0 > X1,X26 : set @ X0] : ((inj_on @ X0 @ X1 @ X13 @ X26) => ((image_mset @ X0 @ X1 @ X13 @ (mset_set @ X0 @ X26)) = (mset_set @ X1 @ (image @ X0 @ X1 @ X13 @ X26))))),
% 4.87/2.14 file('/export/starexec/sandbox/tmp/tmp.5v6FZq027F/DTF2THF_409.p',fact_62_image__mset__mset__set)).
% 4.87/2.14 thf(f996,plain,(
% 4.87/2.14 ((mset @ a @ xs) = (image_mset @ nat @ a @ (nth @ a @ ys) @ (image_mset @ nat @ nat @ f @ (mset @ nat @ (upt @ zero_zero @ nat @ (size_size @ list @ a @ xs))))))),
% 4.87/2.14 inference(backward_demodulation,[status(thm)],[f950,f988])).
% 4.87/2.14 % SZS output end Proof for DTF2THF_409
% 4.87/2.14 % (565)------------------------------
% 4.87/2.14 % (565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.87/2.14 % (565)Termination reason: Refutation
% 4.87/2.14
% 4.87/2.14 % (565)Memory used [KB]: 7931
% 4.87/2.14 % (565)Time elapsed: 0.219 s
% 4.87/2.14 % (565)Instructions burned: 434 (million)
% 4.87/2.14 % (565)------------------------------
% 4.87/2.14 % (565)------------------------------
% 4.87/2.14 % (541)Success in time 0.288 s
%------------------------------------------------------------------------------